problem stringlengths 10 2.73k | answer stringlengths 0 1.22k | solution_hint stringlengths 0 6.85k | solution stringlengths 12 1.23k |
|---|---|---|---|
Given the function \( f(x) = 4 \pi \arcsin x - (\arccos(-x))^2 \), find the difference between its maximum value \( M \) and its minimum value \( m \). Specifically, calculate \( M - m \). | 3\pi^2 | \(\boxed{3\pi^2}\) | |
Two people, A and B, play a guessing game. First, A thinks of a number denoted as $a$, then B guesses the number A thought of, denoting B's guess as $b$. Both $a$ and $b$ belong to the set $\{0,1,2,…,9\}$. If $|a-b|=1$, then A and B are said to have a "telepathic connection". If two people are randomly chosen to play t... | \dfrac {9}{50} | \(\boxed{\dfrac {9}{50}}\) | |
Let \( ABC \) be a right triangle with the hypotenuse \( BC \) measuring \( 4 \) cm. The tangent at \( A \) to the circumcircle of \( ABC \) meets the line \( BC \) at point \( D \). Suppose \( BA = BD \). Let \( S \) be the area of triangle \( ACD \), expressed in square centimeters. Calculate \( S^2 \). | 27 | \(\boxed{27}\) | |
Find the area of a trapezoid with diagonals of 7 cm and 8 cm, and bases of 3 cm and 6 cm. | 12 \sqrt{5} | \(\boxed{12 \sqrt{5}}\) | |
What is the reciprocal of $\frac{3}{4} + \frac{4}{5}$?
A) $\frac{31}{20}$
B) $\frac{20}{31}$
C) $\frac{19}{20}$
D) $\frac{20}{19}$ | \frac{20}{31} | \(\boxed{\frac{20}{31}}\) | |
Last year 100 adult cats, half of whom were female, were brought into the Smallville Animal Shelter. Half of the adult female cats were accompanied by a litter of kittens. The average number of kittens per litter was 4. What was the total number of cats and kittens received by the shelter last year? | 200 | 1. **Determine the number of female cats:**
Given that there are 100 adult cats and half of them are female, we calculate the number of female cats as follows:
\[
\frac{100}{2} = 50 \text{ female cats}
\]
2. **Calculate the number of litters:**
It is stated that half of the adult female cats were accomp... | \(\boxed{200}\) |
Given an ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ ($a>0$, $b>0$) with its left focus $F$ and right vertex $A$, upper vertex $B$. If the distance from point $F$ to line $AB$ is $\frac{5\sqrt{14}}{14}b$, find the eccentricity of the ellipse. | \frac{2}{3} | \(\boxed{\frac{2}{3}}\) | |
If $f(n)$ denotes the number of divisors of $2024^{2024}$ that are either less than $n$ or share at least one prime factor with $n$ , find the remainder when $$ \sum^{2024^{2024}}_{n=1} f(n) $$ is divided by $1000$ . | 224 | \(\boxed{224}\) | |
Given a student measures his steps from one sidewalk to another on a number line where each marking represents 3 meters, calculate the position marking $z$ after taking 5 steps from the starting point, given that the student counts 8 steps and the total distance covered is 48 meters. | 30 | \(\boxed{30}\) | |
What are the first three digits to the right of the decimal point in the decimal representation of $(10^{100} + 1)^{5/3}$? | 666 | \(\boxed{666}\) | |
Let $a$ and $b$ be real numbers. One of the roots of
\[x^3 + ax^2 - x + b = 0\]is $1 - 2i.$ Enter the ordered pair $(a,b).$ | (1,15) | \(\boxed{(1,15)}\) | |
For a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $N$? | 19 | To solve this problem, we need to consider the possible number of intersection points formed by four distinct lines in a plane. Each pair of lines can intersect at most once, and the maximum number of intersection points is determined by the number of ways to choose 2 lines from 4, which is given by the binomial coeffi... | \(\boxed{19}\) |
On the board, two sums are written:
$$
\begin{array}{r}
1+22+333+4444+55555+666666+7777777+ \\
+88888888+999999999
\end{array}
$$
and
$9+98+987+9876+98765+987654+9876543+$
$+98765432+987654321$
Determine which of them is greater (or if they are equal). | 1097393685 | \(\boxed{1097393685}\) | |
For $-25 \le x \le 25,$ find the maximum value of $\sqrt{25 + x} + \sqrt{25 - x}.$ | 10 | \(\boxed{10}\) | |
Let $p, q, r$, and $s$ be positive real numbers such that
\[
\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}
p^2+q^2&=&r^2+s^2&=&2500,\\
pr&=&qs&=&1200.
\end{array}
\]
Compute the value of $\lfloor T \rfloor$, where $T=p+q+r+s$. | 140 | \(\boxed{140}\) | |
Calculate the argument of the sum:
\[ e^{5\pi i/36} + e^{11\pi i/36} + e^{17\pi i/36} + e^{23\pi i/36} + e^{29\pi i/36} \]
in the form $r e^{i \theta}$, where $0 \le \theta < 2\pi$. | \frac{17\pi}{36} | \(\boxed{\frac{17\pi}{36}}\) | |
In isosceles $\triangle ABC$, $|AB|=|AC|$, vertex $A$ is the intersection point of line $l: x-y+1=0$ with the y-axis, and $l$ bisects $\angle A$. If $B(1,3)$, find:
(I) The equation of line $BC$;
(II) The area of $\triangle ABC$. | \frac {3}{2} | \(\boxed{\frac {3}{2}}\) | |
Given the function $f(x) = x^3 + ax^2 + bx + a^2$ has an extremum at $x = 1$ with the value of 10, find the values of $a$ and $b$. | -11 | \(\boxed{-11}\) | |
From the 6 finalists, 1 first prize, 2 second prizes, and 3 third prizes are to be awarded. Calculate the total number of possible outcomes. | 60 | \(\boxed{60}\) | |
Calculate $8 \cdot 9\frac{2}{5}$. | 75\frac{1}{5} | \(\boxed{75\frac{1}{5}}\) | |
Find the largest integer less than 2012 all of whose divisors have at most two 1's in their binary representations. | 1536 | Call a number good if all of its positive divisors have at most two 1's in their binary representations. Then, if $p$ is an odd prime divisor of a good number, $p$ must be of the form $2^{k}+1$. The only such primes less than 2012 are $3,5,17$, and 257 , so the only possible prime divisors of $n$ are $2,3,5,17$, and 25... | \(\boxed{1536}\) |
A majority of the $30$ students in Ms. Demeanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than $1$. The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils ... | 11 | 1. **Define Variables:**
Let $s$ be the number of students that bought pencils, $c$ be the cost of each pencil in cents, and $n$ be the number of pencils each student bought.
2. **Analyze Given Information:**
- A majority of the 30 students bought pencils, so $s > \frac{30}{2} = 15$.
- Each student bought mor... | \(\boxed{11}\) |
The distance between two vectors is the magnitude of their difference. Find the value of $t$ for which the vector
\[\bold{v} = \begin{pmatrix} 2 \\ -3 \\ -3 \end{pmatrix} + t \begin{pmatrix} 7 \\ 5 \\ -1 \end{pmatrix}\]is closest to
\[\bold{a} = \begin{pmatrix} 4 \\ 4 \\ 5 \end{pmatrix}.\] | \frac{41}{75} | \(\boxed{\frac{41}{75}}\) | |
For positive integers $a$ and $b$, let $M(a, b)=\frac{\operatorname{lcm}(a, b)}{\operatorname{gcd}(a, b)}$, and for each positive integer $n \geq 2$, define $$x_{n}=M(1, M(2, M(3, \ldots, M(n-2, M(n-1, n)) \ldots)))$$ Compute the number of positive integers $n$ such that $2 \leq n \leq 2021$ and $5 x_{n}^{2}+5 x_{n+1}^... | 20 | The desired condition is that $x_{n}=5 x_{n+1}$ or $x_{n+1}=5 x_{n}$. Note that for any prime $p$, we have $\nu_{p}(M(a, b))=\left|\nu_{p}(a)-\nu_{p}(b)\right|$. Furthermore, $\nu_{p}(M(a, b)) \equiv \nu_{p}(a)+\nu_{p}(b) \bmod 2$. So, we have that $$\nu_{p}\left(x_{n}\right) \equiv \nu_{p}(1)+\nu_{p}(2)+\cdots+\nu_{p}... | \(\boxed{20}\) |
The probability of missing the target at least once in 4 shots is $\frac{1}{81}$, calculate the shooter's hit rate. | \frac{2}{3} | \(\boxed{\frac{2}{3}}\) | |
A circle passes through the three vertices of an isosceles triangle that has two sides of length 5 and a base of length 4. What is the area of this circle? Express your answer in terms of $\pi$. | \frac{13125}{1764}\pi | \(\boxed{\frac{13125}{1764}\pi}\) | |
What is the sum of the six positive integer factors of 30, each multiplied by 2? | 144 | \(\boxed{144}\) | |
Boys and girls are standing in a circle (there are both), a total of 20 children. It is known that each boy's neighbor in the clockwise direction is a child in a blue T-shirt, and each girl's neighbor in the counterclockwise direction is a child in a red T-shirt. Can you uniquely determine how many boys are in the circ... | 10 | \(\boxed{10}\) | |
Given that all faces of the tetrahedron P-ABC are right triangles, and the longest edge PC equals $2\sqrt{3}$, the surface area of the circumscribed sphere of this tetrahedron is \_\_\_\_\_\_. | 12\pi | \(\boxed{12\pi}\) | |
Determine the largest natural number $r$ with the property that among any five subsets with $500$ elements of the set $\{1,2,\ldots,1000\}$ there exist two of them which share at least $r$ elements. | 200 | \(\boxed{200}\) | |
Given a sample with a sample size of $7$, an average of $5$, and a variance of $2$. If a new data point of $5$ is added to the sample, what will be the variance of the sample? | \frac{7}{4} | \(\boxed{\frac{7}{4}}\) | |
Place 5 balls, numbered 1, 2, 3, 4, 5, into three different boxes, with two boxes each containing 2 balls and the other box containing 1 ball. How many different arrangements are there? (Answer with a number). | 90 | \(\boxed{90}\) | |
If the orthocenter of \( \triangle OAB \) is exactly the focus of the parabola \( y^2 = 4x \), where \( O \) is the origin and points \( A \) and \( B \) lie on the parabola, then the area of \( \triangle OAB \) is equal to ____. | 10\sqrt{5} | \(\boxed{10\sqrt{5}}\) | |
Parallelogram $PQRS$ has vertices $P(4,4)$, $Q(-2,-2)$, $R(-8,-2)$, and $S(-2,4)$. If a point is chosen at random from the region defined by the parallelogram, what is the probability that the point lies below or on the line $y = -1$? Express your answer as a common fraction. | \frac{1}{6} | \(\boxed{\frac{1}{6}}\) | |
Mary and Pat play the following number game. Mary picks an initial integer greater than $2017$ . She then multiplies this number by $2017$ and adds $2$ to the result. Pat will add $2019$ to this new number and it will again be Mary’s turn. Both players will continue to take alternating turns. Mary will always mu... | 2022 | \(\boxed{2022}\) | |
The coefficient of the $x^3$ term in the expansion of $(2-\sqrt{x})^8$ is $1120x^3$. | 112 | \(\boxed{112}\) | |
Compute
\[\sum_{n = 2}^\infty \frac{4n^3 - n^2 - n + 1}{n^6 - n^5 + n^4 - n^3 + n^2 - n}.\] | 1 | \(\boxed{1}\) | |
In $\triangle ABC$, point $D$ is the midpoint of side $BC$. Point $E$ is on $AC$ such that $AE:EC =1:2$. Point $F$ is on $AD$ such that $AF:FD=3:1$. If the area of $\triangle DEF$ is 17, determine the area of $\triangle ABC$. [asy]
size(6cm);defaultpen(fontsize(11));
pair b =(0,0);pair c = (10, 0);pair a=(4, 6);
pair... | 408 | \(\boxed{408}\) | |
For how many integer values of $n$ between 1 and 180 inclusive does the decimal representation of $\frac{n}{180}$ terminate? | 20 | \(\boxed{20}\) | |
If the internal angles of $\triangle ABC$ satisfy $\sin A + 2\sin B = 3\sin C$, then the minimum value of $\cos C$ is | \frac{2 \sqrt{10} - 2}{9} | \(\boxed{\frac{2 \sqrt{10} - 2}{9}}\) | |
In a certain number quiz, the test score of a student with seat number $n$ ($n=1,2,3,4$) is denoted as $f(n)$. If $f(n) \in \{70,85,88,90,98,100\}$ and it satisfies $f(1)<f(2) \leq f(3)<f(4)$, then the total number of possible combinations of test scores for these 4 students is \_\_\_\_\_\_\_\_. | 35 | \(\boxed{35}\) | |
Let $a,b,c,d$ be real numbers such that $a^2+b^2+c^2+d^2=1$. Determine the minimum value of $(a-b)(b-c)(c-d)(d-a)$ and determine all values of $(a,b,c,d)$ such that the minimum value is achived. | -\frac{1}{8} |
Let \(a, b, c, d\) be real numbers such that \(a^2 + b^2 + c^2 + d^2 = 1\). We want to determine the minimum value of the expression \((a-b)(b-c)(c-d)(d-a)\).
To find the minimum value of \((a-b)(b-c)(c-d)(d-a)\), we first recognize the symmetry and potential simplifications. The key is to find a particular symmetric... | \(\boxed{-\frac{1}{8}}\) |
The average value of all the pennies, nickels, dimes, and quarters in Paula's purse is 20 cents. If she had one more quarter, the average value would be 21 cents. How many dimes does she have in her purse? | 0 | \(\boxed{0}\) | |
Square A has side lengths each measuring $x$ inches. Square B has side lengths each measuring $5x$ inches. Square C has side lengths each measuring $2x$ inches. What is the ratio of the area of Square A to the area of Square B, and what is the ratio of the area of Square C to the area of Square B? Express each answer a... | \frac{4}{25} | \(\boxed{\frac{4}{25}}\) | |
Find the smallest natural number ending with the digit 2, which doubles if this digit is moved to the beginning. | 105263157894736842 | \(\boxed{105263157894736842}\) | |
Simplify first, then evaluate: $(1-\frac{m}{{m+3}})÷\frac{{{m^2}-9}}{{{m^2}+6m+9}}$, where $m=\sqrt{3}+3$. | \sqrt{3} | \(\boxed{\sqrt{3}}\) | |
In Theresa's first $8$ basketball games, she scored $7, 4, 3, 6, 8, 3, 1$ and $5$ points. In her ninth game, she scored fewer than $10$ points and her points-per-game average for the nine games was an integer. Similarly in her tenth game, she scored fewer than $10$ points and her points-per-game average for the $10$ ga... | 40 | 1. **Calculate the total points from the first 8 games:**
\[
7 + 4 + 3 + 6 + 8 + 3 + 1 + 5 = 37
\]
2. **Determine the points needed in the ninth game for an integer average:**
- The total points after 9 games must be a multiple of 9 for the average to be an integer.
- The closest multiple of 9 that is g... | \(\boxed{40}\) |
A local community group sells 180 event tickets for a total of $2652. Some tickets are sold at full price, while others are sold at a discounted rate of half price. Determine the total revenue generated from the full-price tickets.
A) $960
B) $984
C) $1008
D) $1032 | 984 | \(\boxed{984}\) | |
In the production of a certain item, its weight \( X \) is subject to random fluctuations. The standard weight of the item is 30 g, its standard deviation is 0.7, and the random variable \( X \) follows a normal distribution. Find the probability that the weight of a randomly selected item is within the range from 28 t... | 0.9215 | \(\boxed{0.9215}\) | |
Christine must buy at least $45$ fluid ounces of milk at the store. The store only sells milk in $200$ milliliter bottles. If there are $33.8$ fluid ounces in $1$ liter, then what is the smallest number of bottles that Christine could buy? (You may use a calculator on this problem.) | 7 | \(\boxed{7}\) | |
If $a-1=b+2=c-3=d+4$, which of the four quantities $a,b,c,d$ is the largest? | $c$ | Given the equation $a-1=b+2=c-3=d+4$, we can express each variable in terms of $a$:
1. From $a-1 = b+2$, solve for $b$:
\[
b = a - 1 - 2 = a - 3
\]
2. From $a-1 = c-3$, solve for $c$:
\[
c = a - 1 + 3 = a + 2
\]
3. From $a-1 = d+4$, solve for $d$:
\[
d = a - 1 - 4 = a - 5
\]
Now, we compa... | \(\boxed{$c$}\) |
Let $m$ be the product of all positive integers less than $4!$ which are invertible modulo $4!$. Find the remainder when $m$ is divided by $4!$.
(Here $n!$ denotes $1\times\cdots\times n$ for each positive integer $n$.) | 1 | \(\boxed{1}\) | |
Let $T_1$ and $T_2$ be the points of tangency of the excircles of a triangle $ABC$ with its sides $BC$ and $AC$ respectively. It is known that the reflection of the incenter of $ABC$ across the midpoint of $AB$ lies on the circumcircle of triangle $CT_1T_2$ . Find $\angle BCA$ . | 90 | \(\boxed{90}\) | |
Let $f(x) = |x+1| - |x-4|$.
(1) Find the range of real number $m$ such that $f(x) \leq -m^2 + 6m$ always holds;
(2) Let $m_0$ be the maximum value of $m$. Suppose $a$, $b$, and $c$ are all positive real numbers and $3a + 4b + 5c = m_0$. Find the minimum value of $a^2 + b^2 + c^2$. | \frac{1}{2} | \(\boxed{\frac{1}{2}}\) | |
Given that $n$ is a positive integer, find the minimum value of $|n-1| + |n-2| + \cdots + |n-100|$. | 2500 | \(\boxed{2500}\) | |
The side \( AD \) of the rectangle \( ABCD \) is equal to 2. On the extension of the side \( AD \) beyond point \( A \), a point \( E \) is taken such that \( EA = 1 \), and \( \angle BEC = 30^\circ \). Find \( BE \). | 2\sqrt{3} | \(\boxed{2\sqrt{3}}\) | |
Let \( s(n) \) denote the sum of the digits of the natural number \( n \). Solve the equation \( n + s(n) = 2018 \). | 2008 | \(\boxed{2008}\) | |
Given the complex number $z=a^{2}-1+(a+1)i (a \in \mathbb{R})$ is a purely imaginary number, find the imaginary part of $\dfrac{1}{z+a}$. | -\dfrac{2}{5} | \(\boxed{-\dfrac{2}{5}}\) | |
A box contains one hundred multicolored balls: 28 red, 20 green, 13 yellow, 19 blue, 11 white, and 9 black. What is the minimum number of balls that must be drawn from the box, without looking, to ensure that at least 15 balls of one color are among them? | 76 | \(\boxed{76}\) | |
Consider the function $g(x)=3x-4$. For what value of $a$ is $g(a)=0$? | \frac{4}{3} | \(\boxed{\frac{4}{3}}\) | |
In a polar coordinate system, the equation of curve C<sub>1</sub> is given by $\rho^2 - 2\rho(\cos\theta - 2\sin\theta) + 4 = 0$. With the pole as the origin and the polar axis in the direction of the positive x-axis, a Cartesian coordinate system is established using the same unit length. The parametric equation of cu... | \frac{7}{8} | \(\boxed{\frac{7}{8}}\) | |
Given real numbers \(a, b, c\), the polynomial
$$
g(x) = x^{3} + a x^{2} + x + 10
$$
has three distinct roots, and these three roots are also roots of the polynomial
$$
f(x) = x^{4} + x^{3} + b x^{2} + 100 x + c.
$$
Find the value of \(f(1)\). | -7007 | \(\boxed{-7007}\) | |
Given 5 differently colored balls and 3 different boxes, with the requirement that no box is empty, calculate the total number of different ways to place 4 balls into the boxes. | 180 | \(\boxed{180}\) | |
ABCD is a rectangle, D is the center of the circle, and B is on the circle. If AD=4 and CD=3, then the area of the shaded region is between | 7 and 8 | 1. **Identify the given information**: We are given a rectangle ABCD with D as the center of a circle and B on the circle. The lengths AD and CD are given as 4 and 3 respectively.
2. **Calculate the diagonal AC using the Pythagorean Theorem**: Since ABCD is a rectangle, the diagonal AC can be calculated using the Pyth... | \(\boxed{7 and 8}\) |
Given that the sum of the first $n$ terms of the arithmetic sequences $\{a_n\}$ and $\{b_n\}$ are $(S_n)$ and $(T_n)$, respectively. If for any positive integer $n$, $\frac{S_n}{T_n}=\frac{2n-5}{3n-5}$, determine the value of $\frac{a_7}{b_2+b_8}+\frac{a_3}{b_4+b_6}$. | \frac{13}{22} | \(\boxed{\frac{13}{22}}\) | |
Let $a$ and $b$ be angles such that $\sin (a + b) = \frac{3}{4}$ and $\sin (a - b) = \frac{1}{2}.$ Find $\frac{\tan a}{\tan b}.$ | 5 | \(\boxed{5}\) | |
Find a four-digit number that is a perfect square if its first two digits are the same as its last two digits. | 7744 | \(\boxed{7744}\) | |
Given a trapezoid \(ABCD\) with bases \(AB\) and \(CD\), and angles \(\angle C = 30^\circ\) and \(\angle D = 80^\circ\). Find \(\angle ACB\), given that \(DB\) is the bisector of \(\angle D\). | 10 | \(\boxed{10}\) | |
Suppose \(a\), \(b\), and \(c\) are real numbers such that:
\[
\frac{ac}{a + b} + \frac{ba}{b + c} + \frac{cb}{c + a} = -12
\]
and
\[
\frac{bc}{a + b} + \frac{ca}{b + c} + \frac{ab}{c + a} = 15.
\]
Compute the value of:
\[
\frac{a}{a + b} + \frac{b}{b + c} + \frac{c}{c + a}.
\] | -12 | \(\boxed{-12}\) | |
The function $f$ satisfies \[
f(x) + f(2x+y) + 5xy = f(3x - y) + 2x^2 + 1
\]for all real numbers $x,y$. Determine the value of $f(10)$. | -49 | \(\boxed{-49}\) | |
The diagonals of a regular hexagon have two possible lengths. What is the ratio of the shorter length to the longer length? Express your answer as a common fraction in simplest radical form. | \frac{\sqrt{3}}{2} | \(\boxed{\frac{\sqrt{3}}{2}}\) | |
A certain regular tetrahedron has three of its vertices at the points $(0,1,2),$ $(4,2,1),$ and $(3,1,5).$ Find the coordinates of the fourth vertex, given that they are also all integers. | (3,-2,2) | \(\boxed{(3,-2,2)}\) | |
Given $\tan \theta =2$, find $\cos 2\theta =$____ and $\tan (\theta -\frac{π}{4})=$____. | \frac{1}{3} | \(\boxed{\frac{1}{3}}\) | |
What is the smallest value that the ratio of the areas of two isosceles right triangles can have, given that three vertices of one of the triangles lie on three different sides of the other triangle? | 1/5 | \(\boxed{1/5}\) | |
There are three novel series Peter wishes to read. Each consists of 4 volumes that must be read in order, but not necessarily one after the other. Let \( N \) be the number of ways Peter can finish reading all the volumes. Find the sum of the digits of \( N \). (Assume that he must finish a volume before reading a new ... | 18 | \(\boxed{18}\) | |
A high school math team received 5 college students for a teaching internship, who are about to graduate. They need to be assigned to three freshman classes: 1, 2, and 3, with at least one and at most two interns per class. Calculate the number of different allocation schemes. | 90 | \(\boxed{90}\) | |
In a triangle $ABC$, points $M$ and $N$ are on sides $AB$ and $AC$, respectively, such that $MB = BC = CN$. Let $R$ and $r$ denote the circumradius and the inradius of the triangle $ABC$, respectively. Express the ratio $MN/BC$ in terms of $R$ and $r$. | \sqrt{1 - \frac{2r}{R}} |
In this problem, we analyze a triangle \( \triangle ABC \) where points \( M \) and \( N \) lie on sides \( AB \) and \( AC \) respectively, satisfying \( MB = BC = CN \).
### Step 1: Set up the problem using geometric principles
Consider the given conditions:
- \( MB = BC = CN \).
This means that \( M \) and \( N ... | \(\boxed{\sqrt{1 - \frac{2r}{R}}}\) |
Given that a four-digit integer $MMMM$, with all identical digits, is multiplied by the one-digit integer $M$, the result is the five-digit integer $NPMPP$. Assuming $M$ is the largest possible single-digit integer that maintains the units digit property of $M^2$, find the greatest possible value of $NPMPP$. | 89991 | \(\boxed{89991}\) | |
Determine the sum of all distinct real values of $x$ such that $|||\cdots||x|+x|\cdots|+x|+x|=1$ where there are 2017 $x$ 's in the equation. | -\frac{2016}{2017} | Note that $|x+| x||=2 x$ when $x$ is nonnegative, and is equal to 0 otherwise. Thus, when there are 2017 $x$ 's, the expression equals $2017 x$ when $x \geq 0$ and $-x$ otherwise, so the two solutions to the equation are $x=-1$ and $\frac{1}{2017}$, and their sum is $-\frac{2016}{2017}$. | \(\boxed{-\frac{2016}{2017}}\) |
Positive integers \( d, e, \) and \( f \) are chosen such that \( d < e < f \), and the system of equations
\[ 2x + y = 2010 \quad \text{and} \quad y = |x-d| + |x-e| + |x-f| \]
has exactly one solution. What is the minimum value of \( f \)? | 1006 | \(\boxed{1006}\) | |
Three of the four vertices of a square are $(2, 8)$, $(13, 8)$, and $(13, -3)$. What is the area of the intersection of this square region and the region inside the graph of the equation $(x - 2)^2 + (y + 3)^2 = 16$? | 4\pi | \(\boxed{4\pi}\) | |
The workers in a factory produce widgets and whoosits. For each product, production time is constant and identical for all workers, but not necessarily equal for the two products. In one hour, 100 workers can produce 300 widgets and 200 whoosits. In two hours, 60 workers can produce 240 widgets and 300 whoosits. In thr... | 450 | \(\boxed{450}\) | |
In a regular tetrahedron embedded in 3-dimensional space, the centers of the four faces are the vertices of a smaller tetrahedron. If the vertices of the larger tetrahedron are located on the surface of a sphere of radius \(r\), find the ratio of the volume of the smaller tetrahedron to that of the larger tetrahedron. ... | \frac{1}{27} | \(\boxed{\frac{1}{27}}\) | |
In triangle $ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$, respectively. If the function $f(x)=\frac{1}{3} x^{3}+bx^{2}+(a^{2}+c^{2}-ac)x+1$ has no extreme points, then the maximum value of angle $B$ is \_\_\_\_\_ | \frac{\pi}{3} | \(\boxed{\frac{\pi}{3}}\) | |
Rhombus $ABCD$ is inscribed in rectangle $WXYZ$ such that vertices $A$, $B$, $C$, and $D$ are on sides $\overline{WX}$, $\overline{XY}$, $\overline{YZ}$, and $\overline{ZW}$, respectively. It is given that $WA=12$, $XB=9$, $BD=15$, and the diagonal $AC$ of rhombus equals side $XY$ of the rectangle. Calculate the perime... | 66 | \(\boxed{66}\) | |
Two dice are rolled consecutively, and the numbers obtained are denoted as $a$ and $b$.
(Ⅰ) Find the probability that the point $(a, b)$ lies on the graph of the function $y=2^x$.
(Ⅱ) Using the values of $a$, $b$, and $4$ as the lengths of three line segments, find the probability that these three segments can form... | \frac{7}{18} | \(\boxed{\frac{7}{18}}\) | |
Triangle $ABC$ has vertices $A(0, 10)$, $B(3, 0)$, $C(9, 0)$. A horizontal line with equation $y=s$ intersects line segment $\overline{AB}$ at $P$ and line segment $\overline{AC}$ at $Q$, forming $\triangle APQ$ with area 18. Compute $s$. | 10 - 2\sqrt{15} | \(\boxed{10 - 2\sqrt{15}}\) | |
The rails on a railroad are $30$ feet long. As the train passes over the point where the rails are joined, there is an audible click.
The speed of the train in miles per hour is approximately the number of clicks heard in: | 20 seconds | 1. **Convert the train's speed from miles per hour to feet per minute:**
Given the speed of the train is $x$ miles per hour, we first convert this speed to feet per minute. We know:
\[
1 \text{ mile} = 5280 \text{ feet} \quad \text{and} \quad 1 \text{ hour} = 60 \text{ minutes}.
\]
Therefore, the speed i... | \(\boxed{20 seconds}\) |
Let $\alpha, \beta$, and $\gamma$ be three real numbers. Suppose that $\cos \alpha+\cos \beta+\cos \gamma =1$ and $\sin \alpha+\sin \beta+\sin \gamma =1$. Find the smallest possible value of $\cos \alpha$. | \frac{-1-\sqrt{7}}{4} | Let $a=\cos \alpha+i \sin \alpha, b=\cos \beta+i \sin \beta$, and $c=\cos \gamma+i \sin \gamma$. We then have $a+b+c=1+i$ where $a, b, c$ are complex numbers on the unit circle. Now, to minimize $\cos \alpha=\operatorname{Re}[a]$, consider a triangle with vertices $a, 1+i$, and the origin. We want $a$ as far away from ... | \(\boxed{\frac{-1-\sqrt{7}}{4}}\) |
Given that the domain of the function $f(x)$ is $R$, $f(2x+2)$ is an even function, $f(x+1)$ is an odd function, and when $x\in [0,1]$, $f(x)=ax+b$. If $f(4)=1$, find the value of $\sum_{i=1}^3f(i+\frac{1}{2})$. | -\frac{1}{2} | \(\boxed{-\frac{1}{2}}\) | |
In an isosceles triangle \(ABC\) (\(AB = BC\)), the angle bisectors \(AM\) and \(BK\) intersect at point \(O\). The areas of triangles \(BOM\) and \(COM\) are 25 and 30, respectively. Find the area of triangle \(ABC\). | 110 | \(\boxed{110}\) | |
What is the sum of the distinct prime factors of 315? | 15 | \(\boxed{15}\) | |
Suppose $ n$ is a product of four distinct primes $ a,b,c,d$ such that:
$ (i)$ $ a\plus{}c\equal{}d;$
$ (ii)$ $ a(a\plus{}b\plus{}c\plus{}d)\equal{}c(d\minus{}b);$
$ (iii)$ $ 1\plus{}bc\plus{}d\equal{}bd$ .
Determine $ n$ . | 2002 | \(\boxed{2002}\) | |
Six students participate in an apple eating contest. The graph shows the number of apples eaten by each participating student. Aaron ate the most apples and Zeb ate the fewest. How many more apples than Zeb did Aaron eat?
[asy]
defaultpen(linewidth(1pt)+fontsize(10pt));
pair[] yaxis = new pair[8];
for( int i = 0 ; i <... | 5 | \(\boxed{5}\) | |
A new dump truck delivered sand to a construction site, forming a conical pile with a diameter of $12$ feet. The height of the cone was $50\%$ of its diameter. However, the pile was too large, causing some sand to spill, forming a cylindrical layer directly around the base of the cone. The height of this cylindrical la... | 98\pi | \(\boxed{98\pi}\) | |
Find the sum of all positive integers such that their expression in base $5$ digits is the reverse of their expression in base $11$ digits. Express your answer in base $10$. | 10 | \(\boxed{10}\) | |
Let $P$ be the maximum possible value of $x_1x_2 + x_2x_3 + \cdots + x_6x_1$ where $x_1, x_2, \dots, x_6$ is a permutation of $(1,2,3,4,5,6)$ and let $Q$ be the number of permutations for which this maximum is achieved, given the additional condition that $x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 21$. Evaluate $P + Q$. | 83 | \(\boxed{83}\) | |
In quadrilateral \(ABCD\), \(\angle ABD = 70^\circ\), \(\angle CAD = 20^\circ\), \(\angle BAC = 48^\circ\), \(\angle CBD = 40^\circ\). Find \(\angle ACD\). | 22 | \(\boxed{22}\) | |
How many integers are greater than $\sqrt{15}$ and less than $\sqrt{50}$? | 4 | Using a calculator, $\sqrt{15} \approx 3.87$ and $\sqrt{50} \approx 7.07$. The integers between these real numbers are $4,5,6,7$, of which there are 4 . Alternatively, we could note that integers between $\sqrt{15}$ and $\sqrt{50}$ correspond to values of $\sqrt{n}$ where $n$ is a perfect square and $n$ is between 15 a... | \(\boxed{4}\) |
Find the number of different complex numbers $z$ such that $|z|=1$ and $z^{7!}-z^{6!}$ is a real number. | 7200 | \(\boxed{7200}\) |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.