year stringdate 1938-01-01 00:00:00 2023-01-01 00:00:00 | id stringlengths 6 10 | problem stringlengths 47 947 | solution stringlengths 140 4.53k | answer_type stringclasses 2
values | source stringclasses 2
values | type stringclasses 100
values | original_problem stringlengths 47 1.65k | original_solution stringlengths 0 4.54k | variation int64 0 0 |
|---|---|---|---|---|---|---|---|---|---|
1938 | 1938_1 | A solid is bounded by two bases in the horizontal planes $z = h/2$ and $z = -h/2$, and by such a surface that the area of every section in a horizontal plane is given by a formula of the sort \[ \text{Area} = a_0 z^3 + a_1 z^2 + a_2 z + a_3 \] (where as special cases some of the coefficients may be $0$). Find the volum... | The volume in question is given by \[ V = \int_{-h/2}^{h/2} (a_0 z^3 + a_1 z^2 + a_2 z + a_3) dz \] \[ = \frac{a_1 h^3}{12} + a_3 h. \] On the other hand, the base areas and $M$ are given by \[ B_1 = \frac{a_0 h^3}{8} + \frac{a_1 h^2}{4} + \frac{a_2 h}{2} + a_3, \] \[ B_2 = -\frac{a_0 h^3}{8} + \frac{a_1 h^2}{4} - \fra... | algebraic | putnam (modified boxing) | Geometry Calculus | A solid is bounded by two bases in the horizontal planes $z = h/2$ and $z = -h/2$, and by such a surface that the area of every section in a horizontal plane is given by a formula of the sort \[ \text{Area} = a_0 z^3 + a_1 z^2 + a_2 z + a_3 \] (where as special cases some of the coefficients may be $0$). Show that the ... | The volume in question is given by \[ V = \int_{-h/2}^{h/2} (a_0 z^3 + a_1 z^2 + a_2 z + a_3) dz \] \[ = \frac{a_1 h^3}{12} + a_3 h. \] On the other hand, the base areas and $M$ are given by \[ B_1 = \frac{a_0 h^3}{8} + \frac{a_1 h^2}{4} + \frac{a_2 h}{2} + a_3, \] \[ B_2 = -\frac{a_0 h^3}{8} + \frac{a_1 h^2}{4} - \fra... | 0 |
1938 | 1938_2 | A can buoy is to be made of three pieces, namely, a cylinder and two equal cones, the altitude of each cone being equal to the altitude of the cylinder. For a given area of surface, find the ratio of $h_0/r_0$ where $h_0$ is the hight at which the maximum volume is achieved and $r_0$ is the corresponding radius at maxi... | Let $r$ be the radius of the cylinder, and $h$ its altitude. The given condition is \[ S = 2\pi rh + 2(\pi r\sqrt{h^2 + r^2}) = \text{constant}, \] and the volume of the buoy is \[ V = \pi r^2 h + \frac{2\pi r^2 h}{3} = \frac{5\pi r^2 h}{3}. \] The required problem is to find the maximum value of $V$ subject to conditi... | numerical | putnam (modified boxing) | Geometry Calculus | A can buoy is to be made of three pieces, namely, a cylinder and two equal cones, the altitude of each cone being equal to the altitude of the cylinder. For a given area of surface, what shape will have the greatest volume? | Let $r$ be the radius of the cylinder, and $h$ its altitude. The given condition is \[ S = 2\pi rh + 2(\pi r\sqrt{h^2 + r^2}) = \text{constant}, \] and the volume of the buoy is \[ V = \pi r^2 h + \frac{2\pi r^2 h}{3} = \frac{5\pi r^2 h}{3}. \] The required problem is to find the maximum value of $V$ subject to conditi... | 0 |
1938 | 1938_3 | If a particle moves in the plane, we may express its coordinates $x$ and $y$ as functions of the time $t$. If $x = t^3 - t$ and $y = t^4 + t$, show that the curve has a point of inflection at $t = 0$ and find the maximum speed/absolute value of velocity it achieves durings its entire movement.$. | If the velocity vector at time $t$ is of length $v$ and has direction $\theta$, then $\dot{x} = v \cos \theta$, $\dot{y} = v \sin \theta$, and $\ddot{x} = \dot{v} \cos \theta - v \dot{\theta} \sin \theta$, $\ddot{y} = \dot{v} \sin \theta + v \dot{\theta} \cos \theta$. Thus $\dot{x} \ddot{y} - \ddot{x} \dot{y} = v^2 \do... | numerical | putnam | Calculus Analysis | If a particle moves in the plane, we may express its coordinates $x$ and $y$ as functions of the time $t$. If $x = t^3 - t$ and $y = t^4 + t$, show that the curve has a point of inflection at $t = 0$ and that the velocity of the moving particle has a maximum at $t = 0$. | First Solution. If the velocity vector at time $t$ is of length $v$ and has direction $\theta$, then $\dot{x} = v \cos \theta$, $\dot{y} = v \sin \theta$, and $\ddot{x} = \dot{v} \cos \theta - v \dot{\theta} \sin \theta$, $\ddot{y} = \dot{v} \sin \theta + v \dot{\theta} \cos \theta$. Thus $\dot{x} \ddot{y} - \ddot{x} \... | 0 |
1938 | 1938_5 | Evaluate the following limits:\n\n(i) $\lim_{n \to \infty} \frac{n^2}{e^n}$.\n\n(ii) $\lim_{x \to 0} \frac{1}{x} \int_0^x (1 + \sin 2t)^{1/t} \, dt and return the sume as the final answer.$ | \textbf{(i)} It follows from L'Hospital's rule that\n\[\lim_{x \to \infty} \frac{x^2}{e^x} = \lim_{x \to \infty} \frac{2x}{e^x} = \lim_{x \to \infty} \frac{2}{e^x} = 0,\] whence the desired limit is zero.\n\nAlternatively, one could use the fact that for $x > 0$,\n\[ e^x = \sum_{n=0}^\infty \frac{x^n}{n!} > \frac{x^3}{... | numerical | putnam (modified boxing) | Calculus | Evaluate the following limits:\n\n(i) $\lim_{n \to \infty} \frac{n^2}{e^n}$.\n\n(ii) $\lim_{x \to 0} \frac{1}{x} \int_0^x (1 + \sin 2t)^{1/t} \, dt.$ | \textbf{Solution.}\n\n\textbf{(i)} It follows from L'Hospital's rule that\n\[\lim_{x \to \infty} \frac{x^2}{e^x} = \lim_{x \to \infty} \frac{2x}{e^x} = \lim_{x \to \infty} \frac{2}{e^x} = 0,\] whence the desired limit is zero.\n\nAlternatively, one could use the fact that for $x > 0$,\n\[ e^x = \sum_{n=0}^\infty \frac{... | 0 |
1938 | 1938_8_i | Let $A_{ik}$ be the cofactor of $a_{ik}$ in the determinant\n\[ d = \begin{vmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{vmatrix}. \] Let $D$ be the corresponding determinant with $a_{ik}$ replaced by $A_{ik... | Let $\alpha$ be the matrix of the given determinant with elements $a_{ik}$ and let $\beta$ be the matrix of the cofactors $A_{ik}$, and let $\gamma$ be the transpose of $\beta$. Then the product matrix $\alpha \gamma$ is a diagonal matrix with all entries on the main diagonal equal to $d$.\n\nThus $\det(\alpha \gamma) ... | algebraic | putnam (modified boxing) | Linear Algebra | Let $A_{ik}$ be the cofactor of $a_{ik}$ in the determinant\n\[ d = \begin{vmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{vmatrix}. \] Let $D$ be the corresponding determinant with $a_{ik}$ replaced by $A_{ik... | Let $\alpha$ be the matrix of the given determinant with elements $a_{ik}$ and let $\beta$ be the matrix of the cofactors $A_{ik}$, and let $\gamma$ be the transpose of $\beta$. Then the product matrix $\alpha \gamma$ is a diagonal matrix with all entries on the main diagonal equal to $d$.\n\nThus $\det(\alpha \gamma) ... | 0 |
1938 | 1938_8_ii | Let $P(y) = Ay^2 + By + C$ be a quadratic polynomial in $y$. If the roots of the quadratic equation $P(y) - y = 0$ are $a$ and $b$ ($a \neq b$), show that $a$ and $b$ are roots of the biquadratic equation $P(P(y)) - y = 0$. Hence write down a quadratic equation which will give the other two roots, $c$ and $d$, of the b... | Since $a$ is a root of $P(y) - y = 0$, we have $P(a) = a$. Then $P(P(a)) = P(a) = a$, so $a$ is a root of $P(P(y)) - y = 0$. Similarly, $b$ is a root of this biquadratic.\n\nLet $Q(y) = P(P(y)) - y$. To find the other zeros of $Q$, note that $P(y) - y = Ay^2 + (B - 1)y + C = A(y - a)(y - b)$, whence $A(a + b) = 1 - B$.... | numerical | putnam (modified boxing) | Algebra | Let $P(y) = Ay^2 + By + C$ be a quadratic polynomial in $y$. If the roots of the quadratic equation $P(y) - y = 0$ are $a$ and $b$ ($a \neq b$), show that $a$ and $b$ are roots of the biquadratic equation $P(P(y)) - y = 0$. Hence write down a quadratic equation which will give the other two roots, $c$ and $d$, of the b... | Since $a$ is a root of $P(y) - y = 0$, we have $P(a) = a$. Then $P(P(a)) = P(a) = a$, so $a$ is a root of $P(P(y)) - y = 0$. Similarly, $b$ is a root of this biquadratic.\n\nLet $Q(y) = P(P(y)) - y$. To find the other zeros of $Q$, note that $P(y) - y = Ay^2 + (B - 1)y + C = A(y - a)(y - b)$, whence $A(a + b) = 1 - B$.... | 0 |
1938 | 1938_9 | Find all the ranges of the domain of the equation \[ y y'' - 2(y')^2 = 0 \] which pass through the point $x = 1, y = 1$ and return them in list format, i.e., inside [] being comma separated. | $1 / y^3$ is an integrating factor since\n\[ \frac{d}{dx} \left( \frac{y'}{y^2} \right) = \frac{y y'' - 2(y')^2}{y^3} = 0. \]\nTherefore $y' / y^2 = C$ and $-1 / y = Cx + D$ for appropriate constants $C$ and $D$. In order that the solution pass through $(1, 1)$, we require that $C + D = -1$. Hence\n\[ y = \frac{1}{1 + ... | algebraic | putnam (modified boxing) | Differential Equations | Find all the solutions of the equation \[ y y'' - 2(y')^2 = 0 \] which pass through the point $x = 1, y = 1$. | First Solution. $1 / y^3$ is an integrating factor since\n\[ \frac{d}{dx} \left( \frac{y'}{y^2} \right) = \frac{y y'' - 2(y')^2}{y^3} = 0. \]\nTherefore $y' / y^2 = C$ and $-1 / y = Cx + D$ for appropriate constants $C$ and $D$. In order that the solution pass through $(1, 1)$, we require that $C + D = -1$. Hence\n\[ y... | 0 |
1938 | 1938_10 | A horizontal disc of diameter 3 inches is rotating at 4 revolutions per minute. A light is shining at a distant point in the plane of the disc. An insect is placed at the edge of the disc furthest from the light, facing the light. It at once starts crawling, and crawls so as always to face the light, at 1 inch per seco... | Choose both rectangular and polar coordinate systems so that the origin is at the center of the disc, the insect is initially at $(3/2, 0)$, the distant light at $(-\infty, 0)$, and the disc rotates counterclockwise. Suppose that at time $t$ the insect’s position is $(x, y)$ in cartesian coordinates, and $(r, \theta)$ ... | numerical | putnam | Calculus Geometry Differential Equations | A horizontal disc of diameter 3 inches is rotating at 4 revolutions per minute. A light is shining at a distant point in the plane of the disc. An insect is placed at the edge of the disc furthest from the light, facing the light. It at once starts crawling, and crawls so as always to face the light, at 1 inch per seco... | Choose both rectangular and polar coordinate systems so that the origin is at the center of the disc, the insect is initially at $(3/2, 0)$, the distant light at $(-\infty, 0)$, and the disc rotates counterclockwise. Suppose that at time $t$ the insect’s position is $(x, y)$ in cartesian coordinates, and $(r, \theta)$ ... | 0 |
1938 | 1938_11 | Given the parabola $y^2 = 2mx$, what is the length of the shortest chord that is normal to the curve at one end? | Any point on the parabola has coordinates of the form $(2mt^2, 2mt)$. Let $AB$ be a chord normal to the parabola at $A$. Say $A = (2mt^2, 2mt)$ and $B = (2ms^2, 2ms)$. The slope of $AB$ is $1/(s + t)$, and the slope of the tangent at $A$ is $1/(2t)$. Hence $s + t = -1/(2t)$.\n\nThe length $L$ of $AB$ is given by\n\[ L^... | algebraic | putnam | Geometry Calculus | Given the parabola $y^2 = 2mx$, what is the length of the shortest chord that is normal to the curve at one end? | Any point on the parabola has coordinates of the form $(2mt^2, 2mt)$. Let $AB$ be a chord normal to the parabola at $A$. Say $A = (2mt^2, 2mt)$ and $B = (2ms^2, 2ms)$. The slope of $AB$ is $1/(s + t)$, and the slope of the tangent at $A$ is $1/(2t)$. Hence $s + t = -1/(2t)$.\n\nThe length $L$ of $AB$ is given by\n\[ L^... | 0 |
1938 | 1938_12 | 12. From the center of a rectangular hyperbola a perpendicular is dropped upon a variable tangent. Find the locus of the foot of the perpendicular. Obtain the equation of the locus in polar coordinates, and sketch the curve. | Let the axes be the asymptotes, so that $xy = a^2$ is the equation of the given hyperbola. Let the point $(h, k)$ be on the hyperbola. Then $hk = a^2$ and the equation of the tangent line at $(h, k)$ is $hy + kx - 2hk = 0$.\n\nThe $x$ and $y$ intercepts of this tangent line are $2h$ and $2k$ respectively. Let $(r, \the... | algebraic | putnam | Geometry Algebra | 12. From the center of a rectangular hyperbola a perpendicular is dropped upon a variable tangent. Find the locus of the foot of the perpendicular. Obtain the equation of the locus in polar coordinates, and sketch the curve. | Let the axes be the asymptotes, so that $xy = a^2$ is the equation of the given hyperbola. Let the point $(h, k)$ be on the hyperbola. Then $hk = a^2$ and the equation of the tangent line at $(h, k)$ is $hy + kx - 2hk = 0$.\n\nThe $x$ and $y$ intercepts of this tangent line are $2h$ and $2k$ respectively. Let $(r, \the... | 0 |
1938 | 1938_13 | Find the shortest distance between the plane $Ax + By + Cz + 1 = 0$ and the ellipsoid $x^2/a^2 + y^2/b^2 + z^2/c^2 = 1$. (For brevity, let\n\[ h = \frac{1}{\sqrt{A^2 + B^2 + C^2}} \quad \text{and} \quad m = \sqrt{a^2A^2 + b^2B^2 + c^2C^2}. \]\nState algebraically the condition that the plane shall lie outside the ellip... | If the given plane intersects the ellipsoid, then the minimum distance is zero. If the plane fails to intersect the ellipsoid, then the shortest distance is the distance between the given plane and the nearer of the two tangent planes to the ellipsoid that are parallel to the given plane.\n\nThe tangent plane to the el... | algebraic | putnam | Geometry Analysis | Find the shortest distance between the plane $Ax + By + Cz + 1 = 0$ and the ellipsoid $x^2/a^2 + y^2/b^2 + z^2/c^2 = 1$. (For brevity, let\n\[ h = \frac{1}{\sqrt{A^2 + B^2 + C^2}} \quad \text{and} \quad m = \sqrt{a^2A^2 + b^2B^2 + c^2C^2}. \]\nState algebraically the condition that the plane shall lie outside the ellip... | If the given plane intersects the ellipsoid, then the minimum distance is zero. If the plane fails to intersect the ellipsoid, then the shortest distance is the distance between the given plane and the nearer of the two tangent planes to the ellipsoid that are parallel to the given plane.\n\nThe tangent plane to the el... | 0 |
1939 | 1939_1 | Find the length of the curve $y^2 = x^3$ from the origin to the point where the tangent makes an angle of $45^\circ$ with the x-axis. | The arc in the first quadrant is represented by the equation $y = x^{3/2}$, and its slope is $\frac{3}{2}x^{1/2}$. The point $P(x_0, y_0)$ where the tangent makes an angle of $45^\circ$ is determined from the relation $\frac{3}{2}x_0^{1/2} = 1$, whence $x_0 = \frac{4}{9}$. The desired length is therefore\n\[ \int_0^{4/... | numerical | putnam | Calculus Geometry | Find the length of the curve $y^2 = x^3$ from the origin to the point where the tangent makes an angle of $45^\circ$ with the x-axis. | The arc in the first quadrant is represented by the equation $y = x^{3/2}$, and its slope is $\frac{3}{2}x^{1/2}$. The point $P(x_0, y_0)$ where the tangent makes an angle of $45^\circ$ is determined from the relation $\frac{3}{2}x_0^{1/2} = 1$, whence $x_0 = \frac{4}{9}$. The desired length is therefore\n\[ \int_0^{4/... | 0 |
1939 | 1939_2 | A point $P$ is taken on the curve $y = x^3$. The tangent at $P$ meets the curve again at $Q$. Given the slope of the curve at $Q$ is $n$ times the slope at $P$ find the value of $n$. | Let $P$ have coordinates $(x_0, y_0)$; then the slope at $P$ is $3x_0^2$. The equation of the tangent at $P$ is $y = 3x_0^2(x - x_0) + x_0^3$. The points of intersection of the tangent and the original curve are determined by the relation\n\[ x^3 = 3x_0^2(x - x_0) + x_0^3, \]\nwhich is equivalent to\n\[ (x - x_0)^2(x +... | numerical | putnam (modified boxing) | Geometry Calculus | A point $P$ is taken on the curve $y = x^3$. The tangent at $P$ meets the curve again at $Q$. Prove that the slope of the curve at $Q$ is four times the slope at $P$. | Let $P$ have coordinates $(x_0, y_0)$; then the slope at $P$ is $3x_0^2$. The equation of the tangent at $P$ is $y = 3x_0^2(x - x_0) + x_0^3$. The points of intersection of the tangent and the original curve are determined by the relation\n\[ x^3 = 3x_0^2(x - x_0) + x_0^3, \]\nwhich is equivalent to\n\[ (x - x_0)^2(x +... | 0 |
1939 | 1939_3 | Find the cubic equation whose roots are the cubes of the roots of
\[ x^3 + ax^2 + bx + c = 0. \] | First Solution. Let the roots of the given cubic equation be $x_1, x_2, x_3$. Then the roots of the desired equation are $x_1^3, x_2^3, x_3^3$. From
\[ x^3 + ax^2 + bx + c = (x - x_1)(x - x_2)(x - x_3), \]
it follows that
\[ x_1 + x_2 + x_3 = -a, \quad x_1x_2 + x_2x_3 + x_3x_1 = b, \quad x_1x_2x_3 = -c. \]
Let the desi... | algebraic | putnam | Algebra | Find the cubic equation whose roots are the cubes of the roots of
\[ x^3 + ax^2 + bx + c = 0. \] | First Solution. Let the roots of the given cubic equation be $x_1, x_2, x_3$. Then the roots of the desired equation are $x_1^3, x_2^3, x_3^3$. From
\[ x^3 + ax^2 + bx + c = (x - x_1)(x - x_2)(x - x_3), \]
it follows that
\[ x_1 + x_2 + x_3 = -a, \quad x_1x_2 + x_2x_3 + x_3x_1 = b, \quad x_1x_2x_3 = -c. \]
Let the desi... | 0 |
1939 | 1939_4 | Find the equations of the two straight lines each of which cuts all the four straight lines
\[
x = 1, y = 1, z = 0; \quad z = 1, x = 0; \quad y = 1, z = 0; \quad x = y = -6z and return the sum of the equations of the 2 lines.
\] | Suppose the required line $L$ meets the given lines in the points $A$, $B$, $C$, and $D$ respectively. Then
\[ A = (1, 0, a), \quad B = (b, 1, 0), \quad C = (0, c, 1), \quad D = (6d, 6d, -d) \]
for some numbers $a$, $b$, $c$, and $d$. Treat $A$, $B$, $C$, and $D$ as vectors. The condition that they be collinear is that... | algebraic | putnam (modified boxing) | Geometry | Find the equations of the two straight lines each of which cuts all the four straight lines
\[
x = 1, y = 1, z = 0; \quad z = 1, x = 0; \quad y = 1, z = 0; \quad x = y = -6z.
\] | Suppose the required line $L$ meets the given lines in the points $A$, $B$, $C$, and $D$ respectively. Then
\[ A = (1, 0, a), \quad B = (b, 1, 0), \quad C = (0, c, 1), \quad D = (6d, 6d, -d) \]
for some numbers $a$, $b$, $c$, and $d$. Treat $A$, $B$, $C$, and $D$ as vectors. The condition that they be collinear is that... | 0 |
1939 | 1939_5 | A heavy particle is attached to the end $A$ of a light rod $AB$ of length $a$. The rod is hinged at $B$ so that it can turn freely in a vertical plane. The rod is balanced in the vertical position above the hinge and then slightly disturbed. Find an expression for the time taken to pass from the horizontal position to ... | Let $m$ be the mass of the particle, and let $\theta$ be the angular position of the rod, measured from the vertical, at time $t$. The force of gravity $mg$ can be resolved into two components, $mg \cos \theta$ acting along the rod, and $mg \sin \theta$ acting perpendicular to the rod. The former is counterbalanced by ... | algebraic | putnam (modified boxing) | Calculus Differential Equations | A heavy particle is attached to the end $A$ of a light rod $AB$ of length $a$. The rod is hinged at $B$ so that it can turn freely in a vertical plane. The rod is balanced in the vertical position above the hinge and then slightly disturbed. Prove that the time taken to pass from the horizontal position to the lowest p... | Let $m$ be the mass of the particle, and let $\theta$ be the angular position of the rod, measured from the vertical, at time $t$. The force of gravity $mg$ can be resolved into two components, $mg \cos \theta$ acting along the rod, and $mg \sin \theta$ acting perpendicular to the rod. The former is counterbalanced by ... | 0 |
1939 | 1939_6_i | A circle of radius $a$ rolls on the inner side of the circumference of a circle of radius $3a$. Find the area contained within the closed curve generated by a point on the circumference of the rolling circle. | Take rectangular coordinates with the origin at the center of the large circle so that the generating point $P$ is in contact with the large circle at $A = (3a, 0)$. It can be seen from the diagram that when the small circle has rolled until the line of centers makes an angle $\theta$ with $OA$, the coordinates of $P$ ... | algebraic | putnam | Geometry Calculus | A circle of radius $a$ rolls on the inner side of the circumference of a circle of radius $3a$. Find the area contained within the closed curve generated by a point on the circumference of the rolling circle. | Take rectangular coordinates with the origin at the center of the large circle so that the generating point $P$ is in contact with the large circle at $A = (3a, 0)$. It can be seen from the diagram that when the small circle has rolled until the line of centers makes an angle $\theta$ with $OA$, the coordinates of $P$ ... | 0 |
1939 | 1939_6_ii | A shell strikes an airplane flying at a height $h$ above the ground. It is known that the shell was fired from a gun on the ground with a muzzle velocity of magnitude $V$, but the position of the gun and its angle of elevation are both unknown. Deduce that the gun is situated within a circle whose center lies directly ... | Choose rectangular coordinates with the $y$-axis vertical, the origin at the position of the gun, and the airplane over a point of the positive $x$-axis. Then the coordinates of the airplane are $(u, h)$ where $u \geq 0$.\nIf the gun is fired at time $t = 0$ with muzzle velocity $V$ and elevation angle $\alpha$, then (... | algebraic | putnam (modified boxing) | Calculus Geometry Trigonometry | A shell strikes an airplane flying at a height $h$ above the ground. It is known that the shell was fired from a gun on the ground with a muzzle velocity of magnitude $V$, but the position of the gun and its angle of elevation are both unknown. Deduce that the gun is situated within a circle whose center lies directly ... | Choose rectangular coordinates with the $y$-axis vertical, the origin at the position of the gun, and the airplane over a point of the positive $x$-axis. Then the coordinates of the airplane are $(u, h)$ where $u \geq 0$.\nIf the gun is fired at time $t = 0$ with muzzle velocity $V$ and elevation angle $\alpha$, then (... | 0 |
1939 | 1939_7_i | Find the curve touched by all the curves of the family \[ (y - k^2)^2 = x^2(k^2 - x^2). \] | We may use the graphs of \[ y = x^2(k^2 - x^2) \] and \[ y^2 = x^2(k^2 - x^2) \] as aids in sketching the family of curves:
The function $x^2(k^2 - x^2)$ assumes its maximum when $x^2 = k^2 - x^2$; i.e., when $x = \pm k/\sqrt{2}$. Hence the graph of the curve \[ f(x, y, k) = (y - k^2)^2 - x^2(k^2 - x^2) = 0 \] has low... | algebraic | putnam | Geometry | Find the curve touched by all the curves of the family \[ (y - k^2)^2 = x^2(k^2 - x^2). \] Make a rough sketch showing this curve and two curves of the family. | We may use the graphs of \[ y = x^2(k^2 - x^2) \] and \[ y^2 = x^2(k^2 - x^2) \] as aids in sketching the family of curves:
The function $x^2(k^2 - x^2)$ assumes its maximum when $x^2 = k^2 - x^2$; i.e., when $x = \pm k/\sqrt{2}$. Hence the graph of the curve \[ f(x, y, k) = (y - k^2)^2 - x^2(k^2 - x^2) = 0 \] has low... | 0 |
1939 | 1939_9 | Evaluate the definite integrals \[ \text{(i)} \int_1^3 \frac{dx}{\sqrt{(x-1)(3-x)}}, \quad \text{(ii)} \int_1^\infty \frac{dx}{e^{x+1} + e^{3-x}} and return their sum. \] | Part (i). Since the integrand is not defined at either bound of integration, one should write \[ \int_1^3 \frac{dx}{\sqrt{(x-1)(3-x)}} = \lim_{\epsilon \to 0^+} \int_{1+\epsilon}^{3-\delta} \frac{dx}{\sqrt{(x-1)(3-x)}} \] \[ = \lim_{\epsilon \to 0^+, \delta \to 0^+} \int_{1+\epsilon}^{3-\delta} \frac{dx}{\sqrt{1 - (x-2... | numerical | putnam (modified boxing) | Calculus | Evaluate the definite integrals \[ \text{(i)} \int_1^3 \frac{dx}{\sqrt{(x-1)(3-x)}}, \quad \text{(ii)} \int_1^\infty \frac{dx}{e^{x+1} + e^{3-x}}. \] | Part (i). Since the integrand is not defined at either bound of integration, one should write \[ \int_1^3 \frac{dx}{\sqrt{(x-1)(3-x)}} = \lim_{\epsilon \to 0^+} \int_{1+\epsilon}^{3-\delta} \frac{dx}{\sqrt{(x-1)(3-x)}} \] \[ = \lim_{\epsilon \to 0^+, \delta \to 0^+} \int_{1+\epsilon}^{3-\delta} \frac{dx}{\sqrt{1 - (x-2... | 0 |
1939 | 1939_10 | Given the power-series \[ a_0 + a_1 x + a_2 x^2 + \cdots \] in which \[ a_n = (n^2 + 1)3^n, \] show that there is a relation of the form \[ a_n + p a_{n+1} + q a_{n+2} + r a_{n+3} = 0, \] in which $p, q, r$ are constants independent of $n$. Find these constants and the sum of the power-series and given the final answer... | The desired relation is \[ (n^2 + 1)3^n + p((n+1)^2 + 1)3^{n+1} + q((n+2)^2 + 1)3^{n+2} + r((n+3)^2 + 1)3^{n+3} = 0, \] which is equivalent to \[ n^2(1 + 3p + 9q + 27r) + n(6p + 36q + 162r) + (1 + 6p + 45q + 270r) = 0. \] Equation (1) holds for all $n$ if and only if \[ 1 + 3p + 9q + 27r = 0, \] \[ p + 6q + 27r = 0, \]... | algebraic | putnam (modified boxing) | Algebra Analysis | Given the power-series \[ a_0 + a_1 x + a_2 x^2 + \cdots \] in which \[ a_n = (n^2 + 1)3^n, \] show that there is a relation of the form \[ a_n + p a_{n+1} + q a_{n+2} + r a_{n+3} = 0, \] in which $p, q, r$ are constants independent of $n$. Find these constants and the sum of the power-series. | The desired relation is \[ (n^2 + 1)3^n + p((n+1)^2 + 1)3^{n+1} + q((n+2)^2 + 1)3^{n+2} + r((n+3)^2 + 1)3^{n+3} = 0, \] which is equivalent to \[ n^2(1 + 3p + 9q + 27r) + n(6p + 36q + 162r) + (1 + 6p + 45q + 270r) = 0. \] Equation (1) holds for all $n$ if and only if \[ 1 + 3p + 9q + 27r = 0, \] \[ p + 6q + 27r = 0, \]... | 0 |
1939 | 1939_11 | Find the equation of the parabola which touches the $x$-axis at the point $(1, 0)$ and the $y$-axis at the point $(0, 2)$. Find the equation of the axis of the parabola. | Clearly the required parabola does not pass through the origin, and any conic not passing through the origin has an equation of the form \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + 1 = 0. \]
In order that this conic be tangent to the $x$-axis at $(1, 0)$, the equation obtained by setting $y = 0$ must have a double root at $x = 1... | algebraic | putnam | Algebra Geometry | Find the equation of the parabola which touches the $x$-axis at the point $(1, 0)$ and the $y$-axis at the point $(0, 2)$. Find the equation of the axis of the parabola and the coordinates of its vertex. | Clearly the required parabola does not pass through the origin, and any conic not passing through the origin has an equation of the form \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + 1 = 0. \]
In order that this conic be tangent to the $x$-axis at $(1, 0)$, the equation obtained by setting $y = 0$ must have a double root at $x = 1... | 0 |
1939 | 1939_12_i | Prove that \[ \int_1^a [x]f'(x) \,dx = [a]f(a) - \{f(1) + \dots + f([a])\}, \] where $a$ is greater than 1 and where $[x]$ denotes the greatest of the integers not exceeding $x$. Obtain a corresponding expression for \[ \int_1^a [x^2]f'(x) \,dx. \] | We have \[ \int_1^a [x]f'(x) \,dx = \int_1^2 1 \cdot f'(x) \,dx + \int_2^3 2 \cdot f'(x) \,dx + \dots + \int_{[a]}^a [a] \cdot f'(x) \,dx \]
\[ = f(2) - f(1) + 2(f(3) - f(2)) + \dots + [a](f(a) - f([a])) \]
\[ = [a]f(a) - \{f(1) + f(2) + \dots + f([a])\}. \]
For the second part, we have \[ \int_1^a [x^2]f'(x) \,dx = \... | algebraic | putnam | Calculus | (i) Prove that \[ \int_1^a [x]f'(x) \,dx = [a]f(a) - \{f(1) + \dots + f([a])\}, \] where $a$ is greater than 1 and where $[x]$ denotes the greatest of the integers not exceeding $x$. Obtain a corresponding expression for \[ \int_1^a [x^2]f'(x) \,dx. \] | Solution. We have \[ \int_1^a [x]f'(x) \,dx = \int_1^2 1 \cdot f'(x) \,dx + \int_2^3 2 \cdot f'(x) \,dx + \dots + \int_{[a]}^a [a] \cdot f'(x) \,dx \]
\[ = f(2) - f(1) + 2(f(3) - f(2)) + \dots + [a](f(a) - f([a])) \]
\[ = [a]f(a) - \{f(1) + f(2) + \dots + f([a])\}. \]
For the second part, we have \[ \int_1^a [x^2]f'(x... | 0 |
1939 | 1939_12_ii | A particle moves on a straight line, the only force acting on it being a resistance proportional to the velocity. If it started with a velocity of 1,000 ft. per sec. and had a velocity of 900 ft. per sec. when it had travelled 1,200 ft., calculate to the nearest hundredth of a second the time it took to travel this dis... | The differential equation governing the motion is \[ m \frac{d^2x}{dt^2} = -k \frac{dx}{dt}, \] and the boundary conditions are \[ x = 0, \quad \frac{dx}{dt} = 1000, \quad \text{when } t = 0 \] \[ x = 1200, \quad \frac{dx}{dt} = 900, \quad \text{when } t = T, \] where $T$ is the time required.
Let $b = k/m$. Then $d^2... | numerical | putnam | Differential Equations Calculus | (ii) A particle moves on a straight line, the only force acting on it being a resistance proportional to the velocity. If it started with a velocity of 1,000 ft. per sec. and had a velocity of 900 ft. per sec. when it had travelled 1,200 ft., calculate to the nearest hundredth of a second the time it took to travel thi... | Solution. The differential equation governing the motion is \[ m \frac{d^2x}{dt^2} = -k \frac{dx}{dt}, \] and the boundary conditions are \[ x = 0, \quad \frac{dx}{dt} = 1000, \quad \text{when } t = 0 \] \[ x = 1200, \quad \frac{dx}{dt} = 900, \quad \text{when } t = T, \] where $T$ is the time required.
Let $b = k/m$.... | 0 |
1939 | 1939_13_ii | Calculate the mutual gravitational attraction of two uniform rods, each of mass $m$ and length $2a$, placed parallel to one another and perpendicular to the line joining their centers at a distance $b$ apart. | We first find the vertical component of the force of attraction between a particle $P$ of mass $\mu$ situated at the point $(h, b)$ and a uniform rod of mass $m$ lying along the $x$-axis from $(0, 0)$ to $(2a, 0)$.
Consider a short segment $S$ of the rod of length $\Delta x$ and center at $Q = (x, 0)$. Let $\alpha, \b... | algebraic | putnam | Calculus | Calculate the mutual gravitational attraction of two uniform rods, each of mass $m$ and length $2a$, placed parallel to one another and perpendicular to the line joining their centers at a distance $b$ apart. In your answer let $a$ approach zero, and comment on the form of the result. | First Solution. We first find the vertical component of the force of attraction between a particle $P$ of mass $\mu$ situated at the point $(h, b)$ and a uniform rod of mass $m$ lying along the $x$-axis from $(0, 0)$ to $(2a, 0)$.
Consider a short segment $S$ of the rod of length $\Delta x$ and center at $Q = (x, 0)$.... | 0 |
1939 | 1939_14_i | If \[ u = 1 + \frac{x^3}{3!} + \frac{x^6}{6!} + \cdots, \]
\[ v = \frac{x}{1!} + \frac{x^4}{4!} + \frac{x^7}{7!} + \cdots, \]
\[ w = \frac{x^2}{2!} + \frac{x^5}{5!} + \frac{x^8}{8!} + \cdots, \]
find the value of
\[ u^3 + v^3 + w^3 - 3uvw. \] | The power series for $u$, $v$, and $w$ converge for all $x$, and
\[ \frac{du}{dx} = w, \quad \frac{dv}{dx} = u, \quad \frac{dw}{dx} = v, \]
as we see by differentiating them. Letting
\[ f = u^3 + v^3 + w^3 - 3uvw, \]
we have
\[ f' = 3u^2u' + 3v^2v' + 3w^2w' - 3uvw' - 3uv'w - 3u'vw \]
\[ = 3u^2w + 3v^2u + 3w^2v - 3uvw... | numerical | putnam (modified boxing) | Algebra Analysis | If \[ u = 1 + \frac{x^3}{3!} + \frac{x^6}{6!} + \cdots, \]
\[ v = \frac{x}{1!} + \frac{x^4}{4!} + \frac{x^7}{7!} + \cdots, \]
\[ w = \frac{x^2}{2!} + \frac{x^5}{5!} + \frac{x^8}{8!} + \cdots, \]
prove that
\[ u^3 + v^3 + w^3 - 3uvw = 1. \] | First Solution. The power series for $u$, $v$, and $w$ converge for all $x$, and
\[ \frac{du}{dx} = w, \quad \frac{dv}{dx} = u, \quad \frac{dw}{dx} = v, \]
as we see by differentiating them. Letting
\[ f = u^3 + v^3 + w^3 - 3uvw, \]
we have
\[ f' = 3u^2u' + 3v^2v' + 3w^2w' - 3uvw' - 3uv'w - 3u'vw \]
\[ = 3u^2w + 3v^2... | 0 |
1940 | 1940_1 | Given that $f(x)$ is a polynomial with integral coefficients, and there exists an integer $k$ such that none of the integers $f(1), f(2), \ldots, f(k)$ is divisible by $k$, then find the number of integral roots of $f(x)$. | Suppose $f$ has an integral root $r$. Then $f(x) = (x - r)g(x)$ where $g(x)$ is also a polynomial with integral coefficients. Then there are integers $p$ and $q$ such that $r = p + kq$ and $1 \leq p \leq k$. But
\[ f(p) = (p - r)g(p) = -kqg(p), \]
and hence $f(p)$ is divisible by $k$ contrary to the hypothesis. This c... | numerical | putnam (modified boxing) | Algebra Number Theory | Prove that if $f(x)$ is a polynomial with integral coefficients, and there exists an integer $k$ such that none of the integers $f(1), f(2), \ldots, f(k)$ is divisible by $k$, then $f(x)$ has no integral root. | Suppose $f$ has an integral root $r$. Then $f(x) = (x - r)g(x)$ where $g(x)$ is also a polynomial with integral coefficients. Then there are integers $p$ and $q$ such that $r = p + kq$ and $1 \leq p \leq k$. But
\[ f(p) = (p - r)g(p) = -kqg(p), \]
and hence $f(p)$ is divisible by $k$ contrary to the hypothesis. This c... | 0 |
1940 | 1940_3 | Find $f(x)$ such that
\[ \int [f(x)]^n dx = \left(\int f(x) dx\right)^n, \]
when constants of integration ($c$) are suitably chosen. | We assume that only real-valued continuous functions $f$ defined on an interval are to be considered. If we put $g(x) = \int f(x)^n dx$ and $h(x) = \int f(x) dx$, we are asked to find all pairs of $C^1$-functions $g$ and $h$ defined on an interval such that
\[ g(x) = h(x)^n \tag{1} \]
and
\[ g'(x) = h'(x)^n. \tag{2} \... | algebraic | putnam | Analysis Algebra | Find $f(x)$ such that
\[ \int [f(x)]^n dx = \left(\int f(x) dx\right)^n, \]
when constants of integration are suitably chosen. | We assume that only real-valued continuous functions $f$ defined on an interval are to be considered. If we put $g(x) = \int f(x)^n dx$ and $h(x) = \int f(x) dx$, we are asked to find all pairs of $C^1$-functions $g$ and $h$ defined on an interval such that
\[ g(x) = h(x)^n \tag{1} \]
and
\[ g'(x) = h'(x)^n. \tag{2} \... | 0 |
1940 | 1940_4 | The parabola $y^2 = -4px$ rolls without slipping around the parabola $y^2 = 4px$. Find the equation of the locus of the vertex of the rolling parabola. | If the rolling parabola and the fixed parabola are tangent at the point $Q$, it is obvious from symmetry that the vertex $V$ of the rolling parabola is the reflection of the origin (the vertex of the fixed parabola) in the tangent line at $Q$.
In the sketch, we have tacitly assumed $p > 0$. Suppose that $Q$ is the po... | algebraic | putnam | Geometry Analysis | The parabola $y^2 = -4px$ rolls without slipping around the parabola $y^2 = 4px$. Find the equation of the locus of the vertex of the rolling parabola. | If the rolling parabola and the fixed parabola are tangent at the point $Q$, it is obvious from symmetry that the vertex $V$ of the rolling parabola is the reflection of the origin (the vertex of the fixed parabola) in the tangent line at $Q$.
In the sketch, we have tacitly assumed $p > 0$. Suppose that $Q$ is the po... | 0 |
1940 | 1940_5 | The simultaneous equations $x^4 - x^2 = y^4 - y^2 = z^4 - z^2$ are satisfied by the points of $n$ straight lines and $m$ ellipses, and by no other points. Find $m+n$. | Let $L$ denote the locus of the given equations. Then a point is on $L$ if and only if its coordinates $(x, y, z)$ satisfy
\[ (x^2 + y^2 - 1)(x^2 - y^2) = 0 \tag{1} \]
\[ (y^2 + z^2 - 1)(y^2 - z^2) = 0 \tag{2} \]
\[ (z^2 + x^2 - 1)(z^2 - x^2) = 0 \tag{3} \]
Consider the loci $A$, $B$, $C$, $D$ defined as follows:
\[ ... | numerical | putnam (modified boxing) | Geometry Analysis | Prove that the simultaneous equations $x^4 - x^2 = y^4 - y^2 = z^4 - z^2$ are satisfied by the points of four straight lines and six ellipses, and by no other points. | Let $L$ denote the locus of the given equations. Then a point is on $L$ if and only if its coordinates $(x, y, z)$ satisfy
\[ (x^2 + y^2 - 1)(x^2 - y^2) = 0 \tag{1} \]
\[ (y^2 + z^2 - 1)(y^2 - z^2) = 0 \tag{2} \]
\[ (z^2 + x^2 - 1)(z^2 - x^2) = 0 \tag{3} \]
Consider the loci $A$, $B$, $C$, $D$ defined as follows:
\[ ... | 0 |
1940 | 1940_9 | A projectile, thrown with initial velocity $v_0$ in a direction making angle $\alpha$ with the horizontal, is acted on by no force except gravity. Find an expression for when the flight is the longest. \] | The differential equations of the motion (using $x$ for the horizontal coordinate and $y$ for the vertical coordinate and taking the origin at the initial point) are
\[ \frac{d^2x}{dt^2} = 0, \quad \frac{d^2y}{dt^2} = -g, \]
where $g$ is the acceleration due to gravity. Using the given initial conditions these can be ... | algebraic | putnam (modified boxing) | Algebra Analysis Trigonometry | A projectile, thrown with initial velocity $v_0$ in a direction making angle $\alpha$ with the horizontal, is acted on by no force except gravity. Find the length of its path until it strikes a horizontal plane through the starting point. Show that the flight is longest when
\[ \sin \alpha \log(\sec \alpha + \tan \alp... | The differential equations of the motion (using $x$ for the horizontal coordinate and $y$ for the vertical coordinate and taking the origin at the initial point) are
\[ \frac{d^2x}{dt^2} = 0, \quad \frac{d^2y}{dt^2} = -g, \]
where $g$ is the acceleration due to gravity. Using the given initial conditions these can be ... | 0 |
1940 | 1940_10_i | A cylindrical hole of radius $r$ is bored through a cylinder of radius $R$ ($r \leq R$) so that the axes intersect at right angles. Find an expression for the area of the larger cylinder which is inside the smaller where $m = \frac{r}{R}$, $x^2 + z^2 = R^2$, $x^2 + y^2 = r^2$ and $v = x/r$. Give the final answer in the... | Let the two cylindrical surfaces be $x^2+z^2=R^2$ and $x^2+y^2=r^2$, where $r \leq R$. The shaded area shown in the diagram is the part of the required area that lies in one octant. The equation of this surface is \[ z = \sqrt{R^2 - x^2}. \] The required area is \[ S = 8 \int \int \sqrt{1+\left(\frac{\partial z}{\parti... | algebraic | putnam (modified boxing) | Geometry Calculus | A cylindrical hole of radius $r$ is bored through a cylinder of radius $R$ ($r \leq R$) so that the axes intersect at right angles. (i) Show that the area of the larger cylinder which is inside the smaller can be expressed in the form \[ S = 8r^2 \int_0^1 \frac{1-v^2}{\sqrt{(1-v^2)(1-m^2v^2)}} dv \] where $m = \frac{r}... | Let the two cylindrical surfaces be $x^2+z^2=R^2$ and $x^2+y^2=r^2$, where $r \leq R$. The shaded area shown in the diagram is the part of the required area that lies in one octant. The equation of this surface is \[ z = \sqrt{R^2 - x^2}. \] The required area is \[ S = 8 \int \int \sqrt{1+\left(\frac{\partial z}{\parti... | 0 |
1940 | 1940_10_ii | A cylindrical hole of radius $r$ is bored through a cylinder of radius $R$ ($r \leq R$) so that the axes intersect at right angles. We are given that the area of the larger cylinder which is inside the smaller can be expressed in the form
\[ S = 8r^2 \int_0^1 \frac{1 - v^2}{\sqrt{(1 - v^2)(1 - m^2v^2)}} dv \quad \text{... | Let the two cylindrical surfaces be $x^2 + z^2 = R^2$ and $x^2 + y^2 = r^2$, where $r \leq R$. The shaded area shown in the diagram is the part of the required area that lies in one octant. The equation of this surface is
\[ z = \sqrt{R^2 - x^2}. \]
The required area is
\[ S = 8 \int \int \sqrt{1 + \left( \frac{\parti... | algebraic | putnam (modified boxing) | Geometry Calculus | A cylindrical hole of radius $r$ is bored through a cylinder of radius $R$ ($r \leq R$) so that the axes intersect at right angles.
\begin{enumerate}
\item[(i)] Show that the area of the larger cylinder which is inside the smaller can be expressed in the form
\[ S = 8r^2 \int_0^1 \frac{1 - v^2}{\sqrt{(1 - v^2)(1 - m^2v... | Let the two cylindrical surfaces be $x^2 + z^2 = R^2$ and $x^2 + y^2 = r^2$, where $r \leq R$. The shaded area shown in the diagram is the part of the required area that lies in one octant. The equation of this surface is
\[ z = \sqrt{R^2 - x^2}. \]
The required area is
\[ S = 8 \int \int \sqrt{1 + \left( \frac{\parti... | 0 |
1940 | 1940_12 | Find that the locus of the point of intersection of three mutually perpendicular planes tangent to the surface
\begin{equation}
ax^2 + by^2 + cz^2 = 1 \quad (abc \neq 0)
\end{equation} in terms of $x,y,z,a,b,c$. | We first find the conditions on the coefficients in order that the plane
\begin{equation}
\alpha x + \beta y + \gamma z = \delta
\end{equation}
be tangent to the quadric surface $Q$ given by (1).\nThe tangent plane to $Q$ at the point $(x_1, y_1, z_1)$ has the equation
\begin{equation}
ax_1 x + by_1 y + cz_1 z ... | algebraic | putnam (modified boxing) | Geometry Linear Algebra | Prove that the locus of the point of intersection of three mutually perpendicular planes tangent to the surface
\begin{equation}
ax^2 + by^2 + cz^2 = 1 \quad (abc \neq 0)
\end{equation}
is the sphere
\begin{equation}
x^2 + y^2 + z^2 = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}.
\end{equation} | We first find the conditions on the coefficients in order that the plane
\begin{equation}
\alpha x + \beta y + \gamma z = \delta
\end{equation}
be tangent to the quadric surface $Q$ given by (1).\nThe tangent plane to $Q$ at the point $(x_1, y_1, z_1)$ has the equation
\begin{equation}
ax_1 x + by_1 y + cz_1 z ... | 0 |
1940 | 1940_13 | Determine all rational values for which $a, b, c$ are the roots of \[ x^3 + ax^2 + bx + c = 0. \] Find the sum of all corresponding $a, b, c$ values and return it as a final ordered triple with the sums. | The conditions on the roots are equivalent to\n\n\begin{align} \tag{1} a + b + c &= -a, \\ \tag{2} ab + bc + ca &= b, \\ \tag{3} abc &= -c. \end{align}\n\nIf $c = 0$, then $ab = b$ and $2a + b = 0$, so either $b = 0$, $a = 0$, or $a = 1$, $b = -2$.\n\nIf $c \neq 0$, then $ab = -1$. If $a + b = 0$, then (2) becomes $ab ... | numerical | putnam (modified boxing) | Algebra | Determine all rational values for which $a, b, c$ are the roots of \[ x^3 + ax^2 + bx + c = 0. \] | The conditions on the roots are equivalent to\n\n\begin{align} \tag{1} a + b + c &= -a, \\ \tag{2} ab + bc + ca &= b, \\ \tag{3} abc &= -c. \end{align}\n\nIf $c = 0$, then $ab = b$ and $2a + b = 0$, so either $b = 0$, $a = 0$, or $a = 1$, $b = -2$.\n\nIf $c \neq 0$, then $ab = -1$. If $a + b = 0$, then (2) becomes $ab ... | 0 |
1940 | 1940_14 | Prove that \[ \begin{vmatrix} a_1^2 + k & a_1a_2 & a_1a_3 & \cdots & a_1a_n \\ a_2a_1 & a_2^2 + k & a_2a_3 & \cdots & a_2a_n \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_na_1 & a_na_2 & a_na_3 & \cdots & a_n^2 + k \end{vmatrix} \] is divisible by $k^{n-1}$ and find its other factor. | Let $B$ be the matrix\n\[ \begin{pmatrix} a_1^2 & a_1a_2 & a_1a_3 & \cdots & a_1a_n \\ a_2a_1 & a_2^2 & a_2a_3 & \cdots & a_2a_n \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_na_1 & a_na_2 & a_na_3 & \cdots & a_n^2 \end{pmatrix}. \]\n\n$B$ has rank at most one, since any two rows (or columns) are clearly dependent... | algebraic | putnam | Algebra Linear Algebra | Prove that \[ \begin{vmatrix} a_1^2 + k & a_1a_2 & a_1a_3 & \cdots & a_1a_n \\ a_2a_1 & a_2^2 + k & a_2a_3 & \cdots & a_2a_n \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_na_1 & a_na_2 & a_na_3 & \cdots & a_n^2 + k \end{vmatrix} \] is divisible by $k^{n-1}$ and find its other factor. | Let $B$ be the matrix\n\[ \begin{pmatrix} a_1^2 & a_1a_2 & a_1a_3 & \cdots & a_1a_n \\ a_2a_1 & a_2^2 & a_2a_3 & \cdots & a_2a_n \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_na_1 & a_na_2 & a_na_3 & \cdots & a_n^2 \end{pmatrix}. \]\n\n$B$ has rank at most one, since any two rows (or columns) are clearly dependent... | 0 |
1941 | 1941_2 | Find the $n$th derivative with respect to $x$ of \[ \int_0^x \left[1 + \frac{(x - t)}{1!} + \frac{(x - t)^2}{2!} + \cdots + \frac{(x - t)^{n-1}}{(n-1)!} \right] e^{nt} dt. \] If there are ranges and in turn multiple expressions return their sum. | Let \[ \phi_k(x) = \int_0^x \frac{(x - t)^k}{k!} e^{nt} dt. \] Then, for $k > 0$, \[ \phi_k'(x) = \phi_{k-1}(x). \] Also \[ \phi_0(x) = \int_0^x e^{nt} dt = \frac{e^{nx} - 1}{n}. \] Therefore, \[ \left(\frac{d}{dx}\right)^n \phi_k(x) = \left(\frac{d}{dx}\right)^{n-k} \phi_0(x) = n^{n-k-1} e^{nx} \text{ for } n > k. \] ... | algebraic | putnam (modified boxing) | Calculus Analysis | Find the $n$th derivative with respect to $x$ of \[ \int_0^x \left[1 + \frac{(x - t)}{1!} + \frac{(x - t)^2}{2!} + \cdots + \frac{(x - t)^{n-1}}{(n-1)!} \right] e^{nt} dt. \] | Let \[ \phi_k(x) = \int_0^x \frac{(x - t)^k}{k!} e^{nt} dt. \] Then, for $k > 0$, \[ \phi_k'(x) = \phi_{k-1}(x). \] Also \[ \phi_0(x) = \int_0^x e^{nt} dt = \frac{e^{nx} - 1}{n}. \] Therefore, \[ \left(\frac{d}{dx}\right)^n \phi_k(x) = \left(\frac{d}{dx}\right)^{n-k} \phi_0(x) = n^{n-k-1} e^{nx} \text{ for } n > k. \] ... | 0 |
1941 | 1941_4 | Let the roots $a, b, c$ of \[ f(x) \equiv x^3 + px^2 + qx + r = 0 \] be real, and let $a \leq b \leq c$. Prove that, if the interval $(b, c)$ is divided into six equal parts, a root of $f'(x) = 0$ will lie in the fourth part counting from the end $b$. What will be the form of $f(x)$ if the root in question of $f'(x) = ... | The proposition is valid for $f(x)$ if and only if it is valid for $f(x + b)$ so we can translate all the roots by $-b$ and thus arrange that the middle root is zero. It is no loss of generality, therefore, to assume that $b = 0$ to begin with. Hence we consider \[ f(x) = (x - a)(x - c) = x^3 - (a + c)x^2 + acx. \] The... | algebraic | putnam (modified boxing) | Algebra Analysis | Let the roots $a, b, c$ of \[ f(x) \equiv x^3 + px^2 + qx + r = 0 \] be real, and let $a \leq b \leq c$. Prove that, if the interval $(b, c)$ is divided into six equal parts, a root of $f'(x) = 0$ will lie in the fourth part counting from the end $b$. What will be the form of $f(x)$ if the root in question of $f'(x) = ... | The proposition is valid for $f(x)$ if and only if it is valid for $f(x + b)$ so we can translate all the roots by $-b$ and thus arrange that the middle root is zero. It is no loss of generality, therefore, to assume that $b = 0$ to begin with. Hence we consider \[ f(x) = (x - a)(x - c) = x^3 - (a + c)x^2 + acx. \] The... | 0 |
1941 | 1941_7_ii | A semi-ellipsoid of revolution is formed by revolving about the $x$-axis the area lying within the first quadrant of the ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1. \] Find the condition for which this semi-ellipsoid will balance in stable equilibrium, with its vertex resting on a horizontal plane in terms of $a$... | First Solution. Let $C$ be the center of gravity of the solid semi-ellipsoid $S$, and let $V$ be its vertex. Consider the sphere with center $C$ and radius $CV$. Suppose that near $V$ the sphere lies strictly inside $S$ (except for the point $V$, of course). Then if $S$ rests on a horizontal plane with point of contact... | algebraic | putnam (modified boxing) | Geometry Calculus | A semi-ellipsoid of revolution is formed by revolving about the $x$-axis the area lying within the first quadrant of the ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1. \] Show that this semi-ellipsoid will balance in stable equilibrium, with its vertex resting on a horizontal plane, when and only when \[ b\sqrt{8} \... | First Solution. Let $C$ be the center of gravity of the solid semi-ellipsoid $S$, and let $V$ be its vertex. Consider the sphere with center $C$ and radius $CV$. Suppose that near $V$ the sphere lies strictly inside $S$ (except for the point $V$, of course). Then if $S$ rests on a horizontal plane with point of contact... | 0 |
1941 | 1941_9 | Evaluate the following limits:\[\lim_{n \to \infty} \left(\frac{1}{\sqrt{n^2 + 1}} + \frac{1}{\sqrt{n^2 + 2^2}} + \cdots + \frac{1}{\sqrt{n^2 + n^2}}\right);\] \[\lim_{n \to \infty} \left(\frac{1}{\sqrt{n^2 + 1}} + \frac{1}{\sqrt{n^2 + 2}} + \cdots + \frac{1}{\sqrt{n^2 + n}}\right);\] \[\lim_{n \to \infty} \left(\frac{... | (i) For the first sum,\[ \frac{1}{\sqrt{n^2 + 1}} + \frac{1}{\sqrt{n^2 + 2^2}} + \cdots + \frac{1}{\sqrt{n^2 + n^2}} = \frac{1}{n}\left[ \frac{1}{\sqrt{1 + (\frac{1}{n})^2}} + \frac{1}{\sqrt{1 + (\frac{2}{n})^2}} + \cdots + \frac{1}{\sqrt{1 + (\frac{n}{n})^2}} \right]. \] This latter form is the lower Riemann sum for \... | numerical | putnam (modified boxing) | Calculus Analysis | Evaluate the following limits:\[\lim_{n \to \infty} \left(\frac{1}{\sqrt{n^2 + 1}} + \frac{1}{\sqrt{n^2 + 2^2}} + \cdots + \frac{1}{\sqrt{n^2 + n^2}}\right);\] \[\lim_{n \to \infty} \left(\frac{1}{\sqrt{n^2 + 1}} + \frac{1}{\sqrt{n^2 + 2}} + \cdots + \frac{1}{\sqrt{n^2 + n}}\right);\] \[\lim_{n \to \infty} \left(\frac{... | (i) For the first sum,\[ \frac{1}{\sqrt{n^2 + 1}} + \frac{1}{\sqrt{n^2 + 2^2}} + \cdots + \frac{1}{\sqrt{n^2 + n^2}} = \frac{1}{n}\left[ \frac{1}{\sqrt{1 + (\frac{1}{n})^2}} + \frac{1}{\sqrt{1 + (\frac{2}{n})^2}} + \cdots + \frac{1}{\sqrt{1 + (\frac{n}{n})^2}} \right]. \] This latter form is the lower Riemann sum for \... | 0 |
1941 | 1941_10 | Find the differential equation satisfied by the product $z$ of any two linearly independent integrals of the equation\[ y'' + y'P(x) + yQ(x) = 0. \] | Suppose $y_1$ and $y_2$ are two linearly independent solutions of the given differential equation. Then any two solutions have the form $u = ay_1 + by_2$ and $v = cy_1 + dy_2$. Since $uv$ falls in the linear space spanned by $y_1^2, y_1y_2, y_2^2$, we expect to find that it satisfies a linear differential equation of t... | algebraic | putnam | Differential Equations Analysis | Find the differential equation satisfied by the product $z$ of any two linearly independent integrals of the equation\[ y'' + y'P(x) + yQ(x) = 0. \] | Suppose $y_1$ and $y_2$ are two linearly independent solutions of the given differential equation. Then any two solutions have the form $u = ay_1 + by_2$ and $v = cy_1 + dy_2$. Since $uv$ falls in the linear space spanned by $y_1^2, y_1y_2, y_2^2$, we expect to find that it satisfies a linear differential equation of t... | 0 |
1941 | 1941_12 | A car is being driven so that its wheels, all of radius $a$ feet, have an angular velocity of $\omega$ radians per second. A particle is thrown off from the tire of one of these wheels, where it is supposed that $a\omega^2 > g$. Neglecting the resistance of the air, find the expression for the maximum height above the ... | If a particle is thrown into motion in a gravitational field starting at height $h$ and with upward component of velocity $v$, it will rise to the height $h + \frac{v^2}{2g}$. [The horizontal components of the motion do not influence the maximum height].
As long as the particle remains attached to the tire, it follows... | algebraic | putnam (modified boxing) | Calculus | A car is being driven so that its wheels, all of radius $a$ feet, have an angular velocity of $\omega$ radians per second. A particle is thrown off from the tire of one of these wheels, where it is supposed that $a\omega^2 > g$. Neglecting the resistance of the air, show that the maximum height above the roadway which ... | If a particle is thrown into motion in a gravitational field starting at height $h$ and with upward component of velocity $v$, it will rise to the height $h + \frac{v^2}{2g}$. [The horizontal components of the motion do not influence the maximum height].
As long as the particle remains attached to the tire, it follows... | 0 |
1942 | 1942_2 | If a polynomial $f(x)$ is divided by $(x - a)^2(x - b)$, where $a \neq b$, derive a formula for the remainder. | Since $f(x)$ is divided by a cubic polynomial, the remainder $R(x)$ will be of degree at most two in $x$, say \[ R(x) = Ax^2 + Bx + C. \]
Then\[ f(x) = (x - a)^2(x - b)Q(x) + Ax^2 + Bx + C \] and \[ f'(x) = 2(x - a)(x - b)Q(x) + (x - a)^2Q(x) + (x - a)^2(x - b)Q'(x) + 2Ax + B. \]
From these relations one gets \[ f(a)... | algebraic | putnam | Algebra | If a polynomial $f(x)$ is divided by $(x - a)^2(x - b)$, where $a \neq b$, derive a formula for the remainder. | Since $f(x)$ is divided by a cubic polynomial, the remainder $R(x)$ will be of degree at most two in $x$, say \[ R(x) = Ax^2 + Bx + C. \]
Then\[ f(x) = (x - a)^2(x - b)Q(x) + Ax^2 + Bx + C \] and \[ f'(x) = 2(x - a)(x - b)Q(x) + (x - a)^2Q(x) + (x - a)^2(x - b)Q'(x) + 2Ax + B. \]
From these relations one gets \[ f(a)... | 0 |
1942 | 1942_4 | Find the orthogonal trajectories of the family of conics $(x + 2y)^2 = a(x + y)$. At what angle do the curves of one family cut the curves of the other family at the origin? | The given family is a family of parabolas all tangent to the line $x + y = 0$ at the origin. For $a = 0$ the parabola degenerates to the double line $(x + 2y)^2 = 0$ which should be viewed as two degenerate parabolas, the ray in the fourth quadrant being the limiting case as $a$ goes to zero through positive values and... | numerical | putnam | Algebra Calculus Geometry | Find the orthogonal trajectories of the family of conics $(x + 2y)^2 = a(x + y)$. At what angle do the curves of one family cut the curves of the other family at the origin? | The given family is a family of parabolas all tangent to the line $x + y = 0$ at the origin. For $a = 0$ the parabola degenerates to the double line $(x + 2y)^2 = 0$ which should be viewed as two degenerate parabolas, the ray in the fourth quadrant being the limiting case as $a$ goes to zero through positive values and... | 0 |
1942 | 1942_5 | A circle of radius $a$ is revolved through $180^\circ$ about a line in its plane, distant $b$ from the center of the circle, where $b > a$. For what value of the ratio $b/a$ does the center of gravity of the solid thus generated lie on the surface of the solid? | We choose axes so that the generating circle starts in the $x$-$z$ plane and is revolved about the $z$-axis. The generated solid is half of a toroid (i.e., a solid bounded by a torus).
It is clear from symmetry that the centroid lies at a point $(0, \bar{y}, 0)$ on the $y$-axis, and the requirement of the problem is t... | algebraic | putnam | Geometry Calculus | A circle of radius $a$ is revolved through $180^\circ$ about a line in its plane, distant $b$ from the center of the circle, where $b > a$. For what value of the ratio $b/a$ does the center of gravity of the solid thus generated lie on the surface of the solid? | We choose axes so that the generating circle starts in the $x$-$z$ plane and is revolved about the $z$-axis. The generated solid is half of a toroid (i.e., a solid bounded by a torus).
It is clear from symmetry that the centroid lies at a point $(0, \bar{y}, 0)$ on the $y$-axis, and the requirement of the problem is t... | 0 |
1942 | 1942_10 | A particle moves under a central force inversely proportional to the $k$th power of the distance. If the particle describes a circle (the central force proceeding from a point on the circumference of the circle), find $k$. | Choose polar coordinates with pole at the center of force and initial ray a diameter of the circular orbit of the particle. The equation of the orbit is then
\[
r = A \cos \theta,
\]
where $A$ is the diameter of the circle.
The equations of motion are
\[
\frac{d^2r}{dt^2} - r\left(\frac{d\theta}{dt}\right)^2 = -\frac{... | numerical | putnam | Calculus | A particle moves under a central force inversely proportional to the $k$th power of the distance. If the particle describes a circle (the central force proceeding from a point on the circumference of the circle), find $k$. | Choose polar coordinates with pole at the center of force and initial ray a diameter of the circular orbit of the particle. The equation of the orbit is then
\[
r = A \cos \theta,
\]
where $A$ is the diameter of the circle.
The equations of motion are
\[
\frac{d^2r}{dt^2} - r\left(\frac{d\theta}{dt}\right)^2 = -\frac{... | 0 |
1946 | 1946_1 | Suppose that the function $f(x) = ax^2 + bx + c$, where $a$, $b$, $c$ are real constants, satisfies the condition $|f(x)| \leq 1$ for $|x| \leq 1$. Find the upper bound of $|f'(x)|$ for $|x| \leq 1$. | If $a \neq 0$, the graph of $y = ax^2 + bx + c$ is a parabola that opens upward, i.e., $a > 0$. Without loss of generality, assume $b \geq 0$. The vertex falls in the left half-plane, and the maximum value of $|f'(x)|$ for $|x| \leq 1$ occurs at $x = 1$. Therefore, $|f'(1)| = 2a + b$.
Now evaluate $f(1)$ and $f(0)$:
\... | numerical | putnam (modified boxing) | Algebra Analysis | Suppose that the function $f(x) = ax^2 + bx + c$, where $a$, $b$, $c$ are real constants, satisfies the condition $|f(x)| \leq 1$ for $|x| \leq 1$. Prove that $|f'(x)| \leq 4$ for $|x| \leq 1$. | If $a \neq 0$, the graph of $y = ax^2 + bx + c$ is a parabola that can be assumed to open upward, i.e., $a > 0$. Without loss of generality, assume $b \geq 0$. By symmetry, the vertex falls in the left half-plane, and the maximum value of $|f'(x)|$ for $|x| \leq 1$ occurs at $x = 1$. Therefore, $|f'(1)| = 2a + b$.
Now... | 0 |
1946 | 1946_5 | Find the smallest volume bounded by the coordinate planes and by a tangent plane to the ellipsoid
\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1.
\] | The tangent plane to the ellipsoid at the point $(x_1, y_1, z_1)$ has the equation
\[
\frac{xx_1}{a^2} + \frac{yy_1}{b^2} + \frac{zz_1}{c^2} = 1.
\]
Its intercepts on the $x$, $y$, and $z$-axes are respectively $\frac{a^2}{x_1}$, $\frac{b^2}{y_1}$, and $\frac{c^2}{z_1}$. The volume of the solid cut off by the tangent p... | algebraic | putnam | Geometry Analysis | Find the smallest volume bounded by the coordinate planes and by a tangent plane to the ellipsoid
\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1.
\] | The tangent plane to the ellipsoid at the point $(x_1, y_1, z_1)$ has the equation
\[
\frac{xx_1}{a^2} + \frac{yy_1}{b^2} + \frac{zz_1}{c^2} = 1.
\]
Its intercepts on the $x$, $y$, and $z$-axes are respectively $\frac{a^2}{x_1}$, $\frac{b^2}{y_1}$, and $\frac{c^2}{z_1}$. The volume of the solid cut off by the tangent p... | 0 |
1946 | 1946_7 | Let $K$ denote the circumference of a circular disc of radius one, and let $k$ denote a circular arc that joins two points $a$, $b$ on $K$ and lies otherwise in the given circular disc. Suppose that $k$ divides the circular disc into two parts of equal area. Find the lower bound of the length of $k$. | If $a$ and $b$ were diametrically opposite on $K$, there would exist no circular arc from $a$ to $b$ that bisects $K$. Hence we may choose coordinates such that $K$ is the unit circle $x^2 + y^2 = 1$ and $a$ and $b$ have coordinates $(c, d)$ and $(c, -d)$, respectively, where $c < 0$.
Now the arc $k$ divides the circu... | numerical | putnam (modified boxing) | Geometry | Let $K$ denote the circumference of a circular disc of radius one, and let $k$ denote a circular arc that joins two points $a$, $b$ on $K$ and lies otherwise in the given circular disc. Suppose that $k$ divides the circular disc into two parts of equal area. Prove that the length of $k$ exceeds 2. | If $a$ and $b$ were diametrically opposite on $K$, there would exist no circular arc from $a$ to $b$ that bisects $K$. Hence we may choose coordinates such that $K$ is the unit circle $x^2 + y^2 = 1$ and $a$ and $b$ have coordinates $(c, d)$ and $(c, -d)$, respectively, where $c < 0$.
Now the arc $k$ divides the circu... | 0 |
1946 | 1946_8 | Let $A$, $B$ be variable points on a parabola $P$, such that the tangents at $A$ and $B$ are perpendicular to each other. Show that the locus of the centroid of the triangle formed by $A$, $B$ and the vertex of $P$ is a parabola $P_1$. Apply the same process to $P_1$, obtaining a parabola $P_2$, and repeat the process,... | Since $P$ is a parabola with equation $y^2 = mx$, any point of $P$ has coordinates of the form $(mt^2, mt)$ for some real $t$, and conversely, every such point is on $P$. The slope of the line tangent to $P$ at $(mt^2, mt)$ is $1/2t$.
Let $A$ and $B$ be the points $(ms^2, ms)$ and $(mt^2, mt)$, respectively. The tange... | algebraic | putnam | Geometry Algebra | Let $A$, $B$ be variable points on a parabola $P$, such that the tangents at $A$ and $B$ are perpendicular to each other. Show that the locus of the centroid of the triangle formed by $A$, $B$ and the vertex of $P$ is a parabola $P_1$. Apply the same process to $P_1$, obtaining a parabola $P_2$, and repeat the process,... | Since $P$ is a parabola with equation $y^2 = mx$, any point of $P$ has coordinates of the form $(mt^2, mt)$ for some real $t$, and conversely, every such point is on $P$. The slope of the line tangent to $P$ at $(mt^2, mt)$ is $1/2t$.
Let $A$ and $B$ be the points $(ms^2, ms)$ and $(mt^2, mt)$, respectively. The tange... | 0 |
1946 | 1946_9 | In a solid sphere of radius $R$ the density $\rho$ is a function of $r$, the distance from the center of the sphere. If the magnitude of the gravitational force of attraction due to the sphere at any point inside the sphere is $kr^2$, where $k$ is a constant, find $\rho$ as a function of $r$. Find also the magnitude of... | Let $P$ be a point at distance $r$ from the center. We regard the solid sphere as the union of many thin concentric spherical shells. The force of attraction at $P$ is the sum of the forces of attraction due to the shells. All of these forces act in the same direction so we can simply add their magnitudes.
The shell a... | algebraic | putnam | Calculus Differential Equations | In a solid sphere of radius $R$ the density $\rho$ is a function of $r$, the distance from the center of the sphere. If the magnitude of the gravitational force of attraction due to the sphere at any point inside the sphere is $kr^2$, where $k$ is a constant, find $\rho$ as a function of $r$. Find also the magnitude of... | Let $P$ be a point at distance $r$ from the center. We regard the solid sphere as the union of many thin concentric spherical shells. The force of attraction at $P$ is the sum of the forces of attraction due to the shells. All of these forces act in the same direction so we can simply add their magnitudes.
The shell a... | 0 |
1947 | 1947_1 | If $\{a_n\}$ is a sequence of numbers such that for $n \geq 1$
\[ (2 - a_n)a_{n+1} = 1, \]
prove that $\lim a_n$, as $n \to \infty$, exists and find its value. | We begin by describing a graphical method of great utility in analyzing recursions of the form $a_{n+1} = f(a_n)$.
Draw the graph of $f$ and the line $y = x$ on the same axes. (In this case $f(x) = 1/(2 - x)$.) Then start from the point $(a_1, a_1)$ on the line and move up or down to the point $(a_1, a_2)$ on the grap... | numerical | putnam (modified boxing) | Analysis Calculus | If $\{a_n\}$ is a sequence of numbers such that for $n \geq 1$
\[ (2 - a_n)a_{n+1} = 1, \]
prove that $\lim a_n$, as $n \to \infty$, exists and is equal to one. | We begin by describing a graphical method of great utility in analyzing recursions of the form $a_{n+1} = f(a_n)$.
Draw the graph of $f$ and the line $y = x$ on the same axes. (In this case $f(x) = 1/(2 - x)$.) Then start from the point $(a_1, a_1)$ on the line and move up or down to the point $(a_1, a_2)$ on the grap... | 0 |
1947 | 1947_2 | A real valued continuous function satisfies for all real $x$ and $y$ the functional equation
\[ f(\sqrt{x^2 + y^2}) = f(x)f(y). \]
Find the value of f(x) in terms of f(1). \] | A slight qualification in the statement of the problem is needed since the real valued continuous function $f(x) \equiv 0$ satisfies the functional equation for all real $x$ and $y$, but does not satisfy the relation $f(0) = [f(1)]^0$, since $0^0$ is undefined.
Assume then that for some $y_0$, $f(y_0) \neq 0$. Since
\... | algebraic | putnam (modified boxing) | Analysis Algebra | A real valued continuous function satisfies for all real $x$ and $y$ the functional equation
\[ f(\sqrt{x^2 + y^2}) = f(x)f(y). \]
Prove that
\[ f(x) = [f(1)]^{x^2}. \] | A slight qualification in the statement of the problem is needed since the real valued continuous function $f(x) \equiv 0$ satisfies the functional equation for all real $x$ and $y$, but does not satisfy the relation $f(0) = [f(1)]^0$, since $0^0$ is undefined.
Assume then that for some $y_0$, $f(y_0) \neq 0$. Since
\... | 0 |
1947 | 1947_5 | $a_1, b_1, c_1$ are positive numbers whose sum is 1, and for $n = 1, 2, \dots$ we define
\[ a_{n+1} = a_n^2 + 2b_nc_n, \quad b_{n+1} = b_n^2 + 2c_na_n, \quad c_{n+1} = c_n^2 + 2a_nb_n. \]
Show that $a_n, b_n, c_n$ approach limits as $n \to \infty$ and find these limits. | First note that
\[ a_{n+1} + b_{n+1} + c_{n+1} = (a_n + b_n + c_n)^2 \]
so $a_k + b_k + c_k = 1$ for all $k$ by induction. Also it is clear that the $a_n$'s, $b_n$'s, and $c_n$'s are all positive.
Define $E_n = \max (a_n, b_n, c_n)$ and $F_n = \min (a_n, b_n, c_n)$. We will show
\[ F_1 \leq F_2 \leq F_3 \leq \cdots... | numerical | putnam | Algebra Analysis | $a_1, b_1, c_1$ are positive numbers whose sum is 1, and for $n = 1, 2, \dots$ we define
\[ a_{n+1} = a_n^2 + 2b_nc_n, \quad b_{n+1} = b_n^2 + 2c_na_n, \quad c_{n+1} = c_n^2 + 2a_nb_n. \]
Show that $a_n, b_n, c_n$ approach limits as $n \to \infty$ and find these limits. | First note that
\[ a_{n+1} + b_{n+1} + c_{n+1} = (a_n + b_n + c_n)^2 \]
so $a_k + b_k + c_k = 1$ for all $k$ by induction. Also it is clear that the $a_n$'s, $b_n$'s, and $c_n$'s are all positive.
Define $E_n = \max (a_n, b_n, c_n)$ and $F_n = \min (a_n, b_n, c_n)$. We will show
\[ F_1 \leq F_2 \leq F_3 \leq \cdots... | 0 |
1947 | 1947_6 | A three-by-three matrix has determinant zero, and has the further property that the cofactor of any element is equal to the square of that element. (The cofactor of $a_{ij}$ is $(-1)^{i+j}$ multiplied by the determinant obtained by striking out the $i$th row and $j$th column.) Find the maximum value across each element... | Let $A$ be an $n \times n$ matrix ($n > 1$), and let $B$ be the transpose of the matrix of its cofactors. Classically $B$ is called the adjoint of $A$. Then
\[ AB = BA = (\det A)I, \]
where $I$ is the identity matrix. Furthermore, the adjoint of $B$ is $(\det A)^{n-2}A$. (For the case $n = 3$, this can be verified quit... | numerical | putnam (modified boxing) | Linear Algebra | A three-by-three matrix has determinant zero, and has the further property that the cofactor of any element is equal to the square of that element. (The cofactor of $a_{ij}$ is $(-1)^{i+j}$ multiplied by the determinant obtained by striking out the $i$th row and $j$th column.) Show that every element in the matrix is z... | Let $A$ be an $n \times n$ matrix ($n > 1$), and let $B$ be the transpose of the matrix of its cofactors. Classically $B$ is called the adjoint of $A$. Then
\[ AB = BA = (\det A)I, \]
where $I$ is the identity matrix. Furthermore, the adjoint of $B$ is $(\det A)^{n-2}A$. (For the case $n = 3$, this can be verified quit... | 0 |
1947 | 1947_7 | Let $f(x)$ be a function such that $f(1) = 1$ and for $x \geq 1$
\[ f'(x) = \frac{1}{x^2 + f^2(x)}. \]
Prove that
\[ \lim_{x \to \infty} f(x) \]
exists and find its upper bound. | Since $f'$ is everywhere positive, $f$ is strictly increasing and therefore
\[ f(t) > f(1) = 1 \quad \text{for } t > 1. \]
Therefore
\[ f'(t) = \frac{1}{t^2 + f^2(t)} < \frac{1}{t^2 + 1} \quad \text{for } t > 1. \]
So
\[ f(x) = 1 + \int_1^x f'(t) dt \]
\[ < 1 + \int_1^x \frac{1}{1 + t^2} dt < 1 + \int_1^\infty \frac{... | numerical | putnam (modified boxing) | Calculus Analysis | Let $f(x)$ be a function such that $f(1) = 1$ and for $x \geq 1$
\[ f'(x) = \frac{1}{x^2 + f^2(x)}. \]
Prove that
\[ \lim_{x \to \infty} f(x) \]
exists and is less than $1 + \pi/4$. | Since $f'$ is everywhere positive, $f$ is strictly increasing and therefore
\[ f(t) > f(1) = 1 \quad \text{for } t > 1. \]
Therefore
\[ f'(t) = \frac{1}{t^2 + f^2(t)} < \frac{1}{t^2 + 1} \quad \text{for } t > 1. \]
So
\[ f(x) = 1 + \int_1^x f'(t) dt \]
\[ < 1 + \int_1^x \frac{1}{1 + t^2} dt < 1 + \int_1^\infty \frac{... | 0 |
1947 | 1947_8 | Let $f(x)$ be a differentiable function defined in the closed interval $(0, 1)$ and such that
\[ |f'(x)| \leq M, \quad 0 < x < 1. \]
Find the upper bound of
\[ \left| \int_0^1 f(x) \, dx - \frac{1}{n} \sum_{k=1}^n f\left(\frac{k}{n}\right) \right| in terms of M. \] | Let
\[ E_k = \int_{(k-1)/n}^{k/n} f(x)dx - \frac{1}{n} f\left(\frac{k}{n}\right) \]
for $k = 1, 2, \ldots, n$. Since $f$ is differentiable, it is continuous, and therefore by the mean value theorem for integrals there exists a number $\eta_k$ such that
\[ \frac{k-1}{n} < \eta_k < \frac{k}{n} \]
and
\[ \int_{(k-1)/n}^{k... | algebraic | putnam (modified boxing) | Calculus Analysis | Let $f(x)$ be a differentiable function defined in the closed interval $(0, 1)$ and such that
\[ |f'(x)| \leq M, \quad 0 < x < 1. \]
Prove that
\[ \left| \int_0^1 f(x) \, dx - \frac{1}{n} \sum_{k=1}^n f\left(\frac{k}{n}\right) \right| \leq \frac{M}{n}. \] | Let
\[ E_k = \int_{(k-1)/n}^{k/n} f(x)dx - \frac{1}{n} f\left(\frac{k}{n}\right) \]
for $k = 1, 2, \ldots, n$. Since $f$ is differentiable, it is continuous, and therefore by the mean value theorem for integrals there exists a number $\eta_k$ such that
\[ \frac{k-1}{n} < \eta_k < \frac{k}{n} \]
and
\[ \int_{(k-1)/n}^{k... | 0 |
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Putnam AXIOM Dataset (ICML 2025 Version)
Note: for questions, feedback, bugs, etc. please open a Huggingface discussion here.
Dataset Summary
The Putnam AXIOM dataset is designed for evaluating large language models (LLMs) on advanced mathematical reasoning skills. It is based on challenging problems from the Putnam Mathematical Competition. This version contains 522 original problems prepared for the ICML 2025 submission.
The ICML 2025 paper is available on OpenReview: https://openreview.net/forum?id=kqj2Cn3Sxr
The dataset includes:
- Full Evaluation Set (522 problems): Complete set of original problems
- Originals for Generating Variations (100 problems): A subset of problems used to create variations
- Variations (500 problems): Variations generated from the original problems
Each problem includes:
- Problem statement
- Solution
- Original problem (where applicable)
- Answer type (e.g., numerical, proof)
- Source and type of problem (e.g., Algebra, Calculus, Geometry)
- Year (extracted from problem ID)
- Variation flag (0 for original problems, 1 for variations)
Note About Splits
For experimental purposes, validation and test splits derived from this dataset are available in a separate repository:
- ZIP-FIT experiments splits - Contains validation/test splits used for ZIP-FIT methodology research
Supported Tasks and Leaderboards
- Mathematical Reasoning: Evaluate mathematical reasoning and problem-solving skills.
- Language Model Benchmarking: Use this dataset to benchmark performance of language models on advanced mathematical questions.
Languages
The dataset is presented in English.
Dataset Structure
Data Fields
- year: The year of the competition (extracted from the problem ID).
- id: Unique identifier for each problem.
- problem: The problem statement.
- solution: The solution or explanation for the problem.
- answer_type: The expected type of answer (e.g., numerical, proof).
- source: The origin of the problem (Putnam).
- type: A description of the problem's mathematical topic (e.g., "Algebra Geometry").
- original_problem: Original form of the problem, where applicable.
- original_solution: Original solution to the problem, where applicable.
- variation: Flag for variations (0 for original problems, 1 for generated variations).
Splits
| Split | Description | Number of Problems |
|---|---|---|
full_eval |
Complete set of 522 original problems | 522 |
originals_for_generating_vars |
Original problems used to create variations | 100 |
variations |
Generated variations of the original problems | 500 |
Variations
The variations split contains problems that were algorithmically generated as variations of problems in the originals_for_generating_vars split. These variations maintain the core mathematical concepts of the original problems but present them with different contexts, numbers, or phrasings. The variation field is set to 1 for these problems to distinguish them from the original problems.
Dataset Usage
from datasets import load_dataset
# Load the dataset
dataset = load_dataset("Putnam-AXIOM/putnam-axiom-dataset-ICML-2025-522")
# Access each split
full_eval = dataset["full_eval"] # Original problems
originals = dataset["originals_for_generating_vars"] # Original problems used for variations
variations = dataset["variations"] # Generated variations
# Filter for original problems only (variation = 0)
original_problems = [p for p in full_eval if p["variation"] == 0]
# Filter for variation problems (variation = 1)
variation_problems = [p for p in variations if p["variation"] == 1]
# Example usage: print the first original problem
print(full_eval[0])
Citation
If you use this dataset, please cite it as follows:
@article{putnam_axiom2025,
title={Putnam-AXIOM: A Functional and Static Benchmark for Measuring Higher Level Mathematical Reasoning},
author={Aryan Gulati and Brando Miranda and Eric Chen and Emily Xia and Kai Fronsdal and Bruno de Moraes Dumont and Sanmi Koyejo},
journal={39th International Conference on Machine Learning (ICML 2025)},
year={2025},
note={Preprint available at: https://openreview.net/pdf?id=YXnwlZe0yf, ICML paper: https://openreview.net/forum?id=kqj2Cn3Sxr}
}
License
This dataset is licensed under the Apache 2.0.
Last updated: May 22, 2024
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