question string | answer string | difficulty int64 | problem_id string |
|---|---|---|---|
In the year $2001$, the United States will host the International Mathematical Olympiad. Let $I,M,$ and $O$ be distinct positive integers such that the product $I \cdot M \cdot O = 2001$. What is the largest possible value of the sum $I + M + O$?
\textbf{(A)}\ 23 \qquad \textbf{(B)}\ 55 \qquad \textbf{(C)}\ 99 \qquad ... | E | 2 | 2000A-P1 |
$2000(2000^{2000}) =$
\textbf{(A)}\ 2000^{2001} \qquad \textbf{(B)}\ 4000^{2000} \qquad \textbf{(C)}\ 2000^{4000} \qquad \textbf{(D)}\ 4,000,000^{2000} \qquad \textbf{(E)}\ 2000^{4,000,000} | A | 2 | 2000A-P2 |
Each day, Jenny ate $20\%$ of the jellybeans that were in her jar at the beginning of that day. At the end of the second day, $32$ remained. How many jellybeans were in the jar originally?
\textbf{(A)}\ 40 \qquad \textbf{(B)}\ 50 \qquad \textbf{(C)}\ 55 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 75 | B | 2 | 2000A-P3 |
The Fibonacci sequence $1,1,2,3,5,8,13,21,\ldots$ starts with two $1$'s, and each term afterwards is the sum of its two predecessors. Which one of the ten digits is the last to appear in the units position of a number in the Fibonacci sequence?
\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \text... | C | 2 | 2000A-P4 |
If $|x - 2| = p,$ where $x < 2,$ then $x - p =$
\textbf{(A)}\ -2 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 2-2p \qquad \textbf{(D)}\ 2p-2 \qquad \textbf{(E)}\ |2p-2| | C | 2 | 2000A-P5 |
Two different prime numbers between $4$ and $18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?
\textbf{(A)}\ 22 \qquad \textbf{(B)}\ 60 \qquad \textbf{(C)}\ 119 \qquad \textbf{(D)}\ 194 \qquad \textbf{(E)}\ 231 | C | 2 | 2000A-P6 |
How many positive integers $b$ have the property that $\log_{b} 729$ is a positive integer?
\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4 | E | 2 | 2000A-P7 |
Figures $0$, $1$, $2$, and $3$ consist of $1$, $5$, $13$, and $25$ nonoverlapping unit squares, respectively. If the pattern were continued, how many nonoverlapping unit squares would there be in figure 100?
```asy
unitsize(8);
draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);
draw((9,0)--(10,0)--(10,3)--(9,3)--cycle);
draw((8... | C | 2 | 2000A-P8 |
Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order)... | C | 2 | 2000A-P9 |
The point $P = (1,2,3)$ is reflected in the $xy$-plane, then its image $Q$ is rotated $180^\circ$ about the $x$-axis to produce $R$, and finally, $R$ is translated $5$ units in the positive-$y$ direction to produce $S$. What are the coordinates of $S$?
$\textbf {(A) } (1,7, - 3) \qquad \textbf {(B) } ( - 1,7, - 3) \qq... | E | 2 | 2000A-P10 |
Two non-zero real numbers, $a$ and $b,$ satisfy $ab = a - b$. Which of the following is a possible value of $\frac {a}{b} + \frac {b}{a} - ab$?
\textbf{(A)} \ - 2 \qquad \textbf{(B)} \ \frac {- 1}{2} \qquad \textbf{(C)} \ \frac {1}{3} \qquad \textbf{(D)} \ \frac {1}{2} \qquad \textbf{(E)} \ 2 | E | 3 | 2000A-P11 |
Let $A, M,$ and $C$ be nonnegative integers such that $A + M + C=12$. What is the maximum value of $A \cdot M \cdot C + A \cdot M + M \cdot C + A \cdot C$?
\textbf{(A)}\ 62 \qquad \textbf{(B)}\ 72 \qquad \textbf{(C)}\ 92 \qquad \textbf{(D)}\ 102 \qquad \textbf{(E)}\ 112 | E | 3 | 2000A-P12 |
One morning each member of Angela’s family drank an $8$-ounce mixture of coffee with milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. How many people are in the family?
$\textbf {(A)}\ 3 \qqua... | C | 3 | 2000A-P13 |
When the mean, median, and mode of the list
$$
10,2,5,2,4,2,x
$$
are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of $x$?
$\textbf {(A)}\ 3 \qquad \textbf {(B)}\ 6 \qquad \textbf{(C)}\ 9 \qquad \textbf {(D)}\ 17 \qquad \textbf {(E)}\ 20$ | E | 3 | 2000A-P14 |
Let $f$ be a function for which $f(x/3) = x^2 + x + 1$. Find the sum of all values of $z$ for which $f(3z) = 7$.
$\textbf {(A)}\ -1/3 \qquad \textbf {(B)}\ -1/9 \qquad \textbf {(C)}\ 0 \qquad \textbf {(D)}\ 5/9 \qquad \textbf {(E)}\ 5/3$ | B | 3 | 2000A-P15 |
A checkerboard of $13$ rows and $17$ columns has a number written in each square, beginning in the upper left corner, so that the first row is numbered $1,2,\ldots,17$, the second row $18,19,\ldots,34$, and so on down the board. If the board is renumbered so that the left column, top to bottom, is $1,2,\ldots,13$, the ... | D | 3 | 2000A-P16 |
right
A circle centered at $O$ has radius $1$ and contains the point $A$. The segment $AB$ is tangent to the circle at $A$ and $\angle AOB = \theta$. If point $C$ lies on $\overline{OA}$ and $\overline{BC}$ bisects $\angle ABO$, then $OC =$
$\textbf {(A)}\ \sec^2 \theta - \tan \theta \qquad \textbf {(B)}\ \frac 12 \q... | D | 3 | 2000A-P17 |
In year $N$, the $300$th day of the year is a Tuesday. In year $N+1$, the $200$th day is also a Tuesday. On what day of the week did the $100$th day of year $N-1$ occur?
$\textbf {(A)}\ \text{Thursday} \qquad \textbf {(B)}\ \text{Friday}\qquad \textbf {(C)}\ \text{Saturday}\qquad \textbf {(D)}\ \text{Sunday}\qquad \te... | A | 3 | 2000A-P18 |
In triangle $ABC$, $AB = 13$, $BC = 14$, $AC = 15$. Let $D$ denote the midpoint of $\overline{BC}$ and let $E$ denote the intersection of $\overline{BC}$ with the bisector of angle $BAC$. Which of the following is closest to the area of the triangle $ADE$?
$\textbf {(A)}\ 2 \qquad \textbf {(B)}\ 2.5 \qquad \textbf {(C... | C | 3 | 2000A-P19 |
If $x,y,$ and $z$ are positive numbers satisfying $x + \frac{1}{y} = 4, y + \frac{1}{z} = 1,$ and $z + \frac{1}{x} = \frac73,$ then what is the value of $xyz$ ?
$\textbf {(A)}\ 2/3 \qquad \textbf {(B)}\ 1 \qquad \textbf {(C)}\ 4/3 \qquad \textbf {(D)}\ 2 \qquad \textbf {(E)}\ 7/3$ | B | 3 | 2000A-P20 |
Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the triangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is $m$ times the area of the square. The ratio of the area of the other small right ... | D | 4 | 2000A-P21 |
The graph below shows a portion of the curve defined by the quartic polynomial $P(x) = x^4 + ax^3 + bx^2 + cx + d$. Which of the following is the smallest?
center
\textbf{(A)}\ P(-1)\\ \textbf{(B)}\ \text{The\ product\ of\ the\ zeros\ of\ } P\\ \textbf{(C)}\ \text{The\ product\ of\ the\ non-real\ zeros\ of\ } P \\ ... | C | 4 | 2000A-P22 |
Professor Gamble buys a lottery ticket, which requires that he pick six different integers from $1$ through $46$, inclusive. He chooses his numbers so that the sum of the base-ten logarithms of his six numbers is an integer. It so happens that the integers on the winning ticket have the same property— the sum of the ba... | B | 4 | 2000A-P23 |
right
If circular arcs $AC$ and $BC$ have centers at $B$ and $A$, respectively, then there exists a circle tangent to both $\stackrel{\frown}{AC}$ and $\stackrel{\frown}{BC}$, and to $\overline{AB}$. If the length of $\stackrel{\frown}{BC}$ is $12$, then the circumference of the circle is
$\textbf {(A)}\ 24 \qquad \te... | D | 4 | 2000A-P24 |
Eight congruent equilateral triangles, each of a different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.)
$\textbf {(A)}\ 210 \qquad \textbf {(B)... | E | 4 | 2000A-P25 |
The sum of two numbers is $S$. Suppose $3$ is added to each number and then
each of the resulting numbers is doubled. What is the sum of the final two
numbers?
$\text{(A)}\ 2S + 3\qquad \text{(B)}\ 3S + 2\qquad \text{(C)}\ 3S + 6 \qquad\text{(D)} 2S + 6 \qquad \text{(E)}\ 2S + 12$ | E | 2 | 2001A-P1 |
Let $P(n)$ and $S(n)$ denote the product and the sum, respectively, of the digits
of the integer $n$. For example, $P(23) = 6$ and $S(23) = 5$. Suppose $N$ is a
two-digit number such that $N = P(N)+S(N)$. What is the units digit of $N$?
$\text{(A)}\ 2\qquad \text{(B)}\ 3\qquad \text{(C)}\ 6\qquad \text{(D)}\ 8\qquad \... | E | 2 | 2001A-P2 |
The state income tax where Kristin lives is levied at the rate of $p\%$ of the first
$\textdollar 28000$ of annual income plus $(p + 2)\%$ of any amount above $\textdollar 28000$. Kristin
noticed that the state income tax she paid amounted to $(p + 0.25)\%$ of her
annual income. What was her annual income?
$\text{(A)}... | B | 2 | 2001A-P3 |
The mean of three numbers is $10$ more than the least of the numbers and $15$
less than the greatest. The median of the three numbers is $5$. What is their
sum?
$\text{(A)}\ 5\qquad \text{(B)}\ 20\qquad \text{(C)}\ 25\qquad \text{(D)}\ 30\qquad \text{(E)}\ 36$ | D | 2 | 2001A-P4 |
What is the product of all positive odd integers less than 10000?
$\text{(A)}\ \dfrac{10000!}{(5000!)^2}\qquad \text{(B)}\ \dfrac{10000!}{2^{5000}}\qquad
\text{(C)}\ \dfrac{9999!}{2^{5000}}\qquad \text{(D)}\ \dfrac{10000!}{2^{5000} \cdot 5000!}\qquad
\text{(E)}\ \dfrac{5000!}{2^{5000}}$ | D | 2 | 2001A-P5 |
A telephone number has the form $\text{ABC-DEF-GHIJ}$, where each letter represents
a different digit. The digits in each part of the number are in decreasing
order; that is, $A > B > C$, $D > E > F$, and $G > H > I > J$. Furthermore,
$D$, $E$, and $F$ are consecutive even digits; $G$, $H$, $I$, and $J$ are consecutive... | E | 2 | 2001A-P6 |
A charity sells $140$ benefit tickets for a total of $\textdollar 2001$. Some tickets sell for full price (a whole dollar amount), and the rest sells for half price. How much money is raised by the full-price tickets?
$\text{(A) }\$782\qquad \text{(B) }\$986\qquad \text{(C) }\$1158\qquad \text{(D) }\$1219\qquad \text{... | A | 2 | 2001A-P7 |
Which of the cones listed below can be formed from a $252^\circ$ sector of a circle of radius $10$ by aligning the two straight sides?
```asy
import graph;
unitsize(1.5cm);
defaultpen(fontsize(8pt));
draw(Arc((0,0),1,-72,180),linewidth(.8pt));
draw(dir(288)--(0,0)--(-1,0),linewidth(.8pt));
label("$10$",(-0.5,0),S);
d... | C | 2 | 2001A-P8 |
Let $f$ be a function satisfying $f(xy) = \frac{f(x)}y$ for all positive real numbers $x$ and $y$. If $f(500) =3$, what is the value of $f(600)$?
$\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ \frac52 \qquad \text{(D)}\ 3 \qquad \text{(E)}\ \frac{18}5$ | C | 2 | 2001A-P9 |
The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to
$\text{(A) }50
\qquad
\text{(B) }52
\qquad
\text{(C) }54
\qquad
\text{(D) }56
\qquad
\text{(E) }58$
```asy
unitsize(3mm);
defaultpen(linewidth(0.8pt));
path p1=(0,0)-... | D | 2 | 2001A-P10 |
A box contains exactly five chips, three red and two white. Chips are randomly removed one at a time without replacement until all the red chips are drawn or all the white chips are drawn. What is the probability that the last chip drawn is white?
$\text{(A) }\frac {3}{10}
\qquad
\text{(B) }\frac {2}{5}
\qquad
\text{(... | D | 3 | 2001A-P11 |
How many positive integers not exceeding $2001$ are multiples of $3$ or $4$ but not $5$?
$\text{(A) }768
\qquad
\text{(B) }801
\qquad
\text{(C) }934
\qquad
\text{(D) }1067
\qquad
\text{(E) }1167$ | B | 3 | 2001A-P12 |
The parabola with equation $y=ax^2+bx+c$ and vertex $(h,k)$ is reflected about the line $y=k$. This results in the parabola with equation $y=dx^2+ex+f$. Which of the following equals $a+b+c+d+e+f$?
$\text{(A) }2b
\qquad
\text{(B) }2c
\qquad
\text{(C) }2a+2b
\qquad
\text{(D) }2h
\qquad
\text{(E) }2k$ | E | 3 | 2001A-P13 |
Given the nine-sided regular polygon $A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8 A_9$, how many distinct equilateral triangles in the plane of the polygon have at least two vertices in the set $\{A_1,A_2,\dots,A_9\}$?
$\text{(A) }30
\qquad
\text{(B) }36
\qquad
\text{(C) }63
\qquad
\text{(D) }66
\qquad
\text{(E) }72$ | D | 3 | 2001A-P14 |
An insect lives on the surface of a regular tetrahedron with edges of length 1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? (Note: Two edges of a tetrahedron are opposite if they have no common en... | B | 3 | 2001A-P15 |
A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe?
$\text{(A) }8!
\qquad
\text{(B) }2^8 \cdot 8!
\qquad
\text{(C) }(8!)^2
\qquad
\text{(D) }\frac {16!}{2^8}
\qquad
\t... | D | 3 | 2001A-P16 |
A point $P$ is selected at random from the interior of the pentagon with vertices $A = (0,2)$, $B = (4,0)$, $C = (2 \pi + 1, 0)$, $D = (2 \pi + 1,4)$, and $E=(0,4)$. What is the probability that $\angle APB$ is obtuse?
$\text{(A) }\frac {1}{5}
\qquad
\text{(B) }\frac {1}{4}
\qquad
\text{(C) }\frac {5}{16}
\qquad
\text... | C | 3 | 2001A-P17 |
A circle centered at $A$ with a radius of 1 and a circle centered at $B$ with a radius of 4 are externally tangent. A third circle is tangent to the first two and to one of their common external tangents as shown. The radius of the third circle is
```asy
unitsize(0.75cm);
pair A=(0,1), B=(4,4);
dot(A); dot(B);
draw( c... | D | 3 | 2001A-P18 |
The polynomial $P(x)=x^3+ax^2+bx+c$ has the property that the mean of its zeros, the product of its zeros, and the sum of its coefficients are all equal. If the $y$-intercept of the graph of $y=P(x)$ is 2, what is $b$?
$\text{(A) }-11
\qquad
\text{(B) }-10
\qquad
\text{(C) }-9
\qquad
\text{(D) }1
\qquad
\text{(E) }5$ | A | 3 | 2001A-P19 |
Points $A = (3,9)$, $B = (1,1)$, $C = (5,3)$, and $D=(a,b)$ lie in the first quadrant and are the vertices of quadrilateral $ABCD$. The quadrilateral formed by joining the midpoints of $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, and $\overline{DA}$ is a square. What is the sum of the coordinates of point $D$?
$... | C | 3 | 2001A-P20 |
Four positive integers $a$, $b$, $c$, and $d$ have a product of $8!$ and satisfy:
$$
\begin{align*}
ab + a + b & = 524
\\
bc + b + c & = 146
\\
cd + c + d & = 104
\end{align*}
$$
What is $a-d$?
$\text{(A) }4
\qquad
\text{(B) }6
\qquad
\text{(C) }8
\qquad
\text{(D) }10
\qquad
\text{(E) }12$ | D | 4 | 2001A-P21 |
In rectangle $ABCD$, points $F$ and $G$ lie on $AB$ so that $AF=FG=GB$ and $E$ is the midpoint of $\overline{DC}$. Also, $\overline{AC}$ intersects $\overline{EF}$ at $H$ and $\overline{EG}$ at $J$. The area of the rectangle $ABCD$ is $70$. Find the area of triangle $EHJ$.
$\text{(A) }\frac {5}{2}
\qquad
\text{(B) }\f... | C | 4 | 2001A-P22 |
A polynomial of degree four with leading coefficient 1 and integer coefficients has two real zeros, both of which are integers. Which of the following can also be a zero of the polynomial?
$\text{(A) }\frac {1 + i \sqrt {11}}{2}
\qquad
\text{(B) }\frac {1 + i}{2}
\qquad
\text{(C) }\frac {1}{2} + i
\qquad
\text{(D) }1 ... | A | 4 | 2001A-P23 |
In $\triangle ABC$, $\angle ABC=45^\circ$. Point $D$ is on $\overline{BC}$ so that $2\cdot BD=CD$ and $\angle DAB=15^\circ$. Find $\angle ACB$.
$\text{(A) }54^\circ
\qquad
\text{(B) }60^\circ
\qquad
\text{(C) }72^\circ
\qquad
\text{(D) }75^\circ
\qquad
\text{(E) }90^\circ$ | D | 4 | 2001A-P24 |
Consider sequences of positive real numbers of the form $x, 2000, y, \dots$ in which every term after the first is 1 less than the product of its two immediate neighbors. For how many different values of $x$ does the term $2001$ appear somewhere in the sequence?
$\text{(A) }1
\qquad
\text{(B) }2
\qquad
\text{(C) }3
\q... | D | 4 | 2001A-P25 |
Compute the sum of all the roots of
$(2x+3)(x-4)+(2x+3)(x-6)=0$
$\mathrm{(A) \ } \frac{7}{2}\qquad \mathrm{(B) \ } 4\qquad \mathrm{(C) \ } 5\qquad \mathrm{(D) \ } 7\qquad \mathrm{(E) \ } 13$ | A | 2 | 2002A-P1 |
Cindy was asked by her teacher to subtract 3 from a certain number and then divide the result by 9. Instead, she subtracted 9 and then divided the result by 3, giving an answer of 43. What would her answer have been had she worked the problem correctly?
$\mathrm{(A) \ } 15\qquad \mathrm{(B) \ } 34\qquad \mathrm{(C) \ ... | A | 2 | 2002A-P2 |
According to the standard convention for exponentiation,
$$
2^{2^{2^{2}}} = 2^{\left(2^{\left(2^2\right)}\right)} = 2^{16} = 65536.
$$
If the order in which the exponentiations are performed is changed, how many other values are possible?
$\mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 1\qquad \mathrm{(C) \ } 2\qquad \mathr... | B | 2 | 2002A-P3 |
Find the degree measure of an angle whose complement is 25% of its supplement.
$\mathrm{(A) \ 48 } \qquad \mathrm{(B) \ 60 } \qquad \mathrm{(C) \ 75 } \qquad \mathrm{(D) \ 120 } \qquad \mathrm{(E) \ 150 }$ | B | 2 | 2002A-P4 |
Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region.
```asy
import graph;
unitsize(.3cm);
path c=Circle((0,2),1);
filld... | C | 2 | 2002A-P5 |
For how many positive integers $m$ does there exist at least one positive integer $n$ such that $m \cdot n \le m + n$?
$\mathrm{(A) \ } 4\qquad \mathrm{(B) \ } 6\qquad \mathrm{(C) \ } 9\qquad \mathrm{(D) \ } 12\qquad \mathrm{(E) \ }$ infinitely many | E | 2 | 2002A-P6 |
A $45^\circ$ arc of circle A is equal in length to a $30^\circ$ arc of circle B. What is the ratio of circle A's area and circle B's area?
$\text{(A)}\ 4/9 \qquad \text{(B)}\ 2/3 \qquad \text{(C)}\ 5/6 \qquad \text{(D)}\ 3/2 \qquad \text{(E)}\ 9/4$ | A | 2 | 2002A-P7 |
Betsy designed a flag using blue triangles, small white squares, and a red center square, as shown. Let $B$ be the total area of the blue triangles, $W$ the total area of the white squares, and $R$ the area of the red square. Which of the following is correct?
```asy
unitsize(3mm);
fill((-4,-4)--(-4,4)--(4,4)--(4,-4)-... | A | 2 | 2002A-P8 |
Jamal wants to save 30 files onto disks, each with 1.44 MB space. 3 of the files take up 0.8 MB, 12 of the files take up 0.7 MB, and the rest take up 0.4 MB. It is not possible to split a file onto 2 different disks. What is the smallest number of disks needed to store all 30 files?
$\text{(A)}\ 12 \qquad \text{(B)}\ ... | B | 2 | 2002A-P9 |
Sarah places four ounces of coffee into an eight-ounce cup and four ounces of cream into a second cup of the same size. She then pours half the coffee from the first cup to the second and, after stirring thoroughly, pours half the liquid in the second cup back to the first. What fraction of the liquid in the first cup ... | D | 2 | 2002A-P10 |
Mr. Earl E. Bird gets up every day at 8:00 AM to go to work. If he drives at an average speed of 40 miles per hour, he will be late by 3 minutes. If he drives at an average speed of 60 miles per hour, he will be early by 3 minutes. How many miles per hour does Mr. Bird need to drive to get to work exactly on time?
$\t... | B | 3 | 2002A-P11 |
Both roots of the quadratic equation $x^2 - 63x + k = 0$ are prime numbers. The number of possible values of $k$ is
$\text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 4 \qquad \text{(E) more than 4}$ | B | 3 | 2002A-P12 |
Two different positive numbers $a$ and $b$ each differ from their reciprocals by $1$. What is $a+b$?
$\text{(A) }1
\qquad
\text{(B) }2
\qquad
\text{(C) }\sqrt 5
\qquad
\text{(D) }\sqrt 6
\qquad
\text{(E) }3$ | C | 3 | 2002A-P13 |
For all positive integers $n$, let $f(n)=\log_{2002} n^2$. Let $N=f(11)+f(13)+f(14)$. Which of the following relations is true?
$\text{(A) }N<1
\qquad
\text{(B) }N=1
\qquad
\text{(C) }1<N<2
\qquad
\text{(D) }N=2
\qquad
\text{(E) }N>2$ | D | 3 | 2002A-P14 |
The mean, median, unique mode, and range of a collection of eight integers are all equal to 8. The largest integer that can be an element of this collection is
$\text{(A) }11
\qquad
\text{(B) }12
\qquad
\text{(C) }13
\qquad
\text{(D) }14
\qquad
\text{(E) }15$ | D | 3 | 2002A-P15 |
Tina randomly selects two distinct numbers from the set $\{1, 2, 3, 4, 5\}$, and Sergio randomly selects a number from the set $\{1, 2, \ldots, 10\}$. What is the probability that Sergio's number is larger than the sum of the two numbers chosen by Tina?
$\text{(A)}\ 2/5 \qquad \text{(B)}\ 9/20 \qquad \text{(C)}\ 1/2 \... | A | 3 | 2002A-P16 |
Several sets of prime numbers, such as $\{7,83,421,659\}$ use each of the nine nonzero digits exactly once. What is the smallest possible sum such a set of primes could have?
$\text{(A) }193
\qquad
\text{(B) }207
\qquad
\text{(C) }225
\qquad
\text{(D) }252
\qquad
\text{(E) }447$ | B | 3 | 2002A-P17 |
Let $C_1$ and $C_2$ be circles defined by $(x-10)^2 + y^2 = 36$ and $(x+15)^2 + y^2 = 81$
respectively. What is the length of the shortest line segment $PQ$ that is tangent to $C_1$ at $P$ and to $C_2$ at $Q$?
$\text{(A) }15
\qquad
\text{(B) }18
\qquad
\text{(C) }20
\qquad
\text{(D) }21
\qquad
\text{(E) }24$ | C | 3 | 2002A-P18 |
The graph of the function $f$ is shown below. How many solutions does the equation $f(f(x))=6$ have?
```asy
import graph;
size(200);
defaultpen(fontsize(10pt)+linewidth(.8pt));
dotfactor=4;
pair P1=(-7,-4), P2=(-2,6), P3=(0,0), P4=(1,6), P5=(5,-6);
real[] xticks={-7,-6,-5,-4,-3,-2,-1,1,2,3,4,5,6};
real[] yticks={-6,-... | D | 3 | 2002A-P19 |
Suppose that $a$ and $b$ are digits, not both nine and not both zero, and the repeating decimal $0.\overline{ab}$ is expressed as a fraction in lowest terms. How many different denominators are possible?
$\text{(A) }3
\qquad
\text{(B) }4
\qquad
\text{(C) }5
\qquad
\text{(D) }8
\qquad
\text{(E) }9$ | C | 3 | 2002A-P20 |
Consider the sequence of numbers: $4,7,1,8,9,7,6,\dots$ For $n>2$, the $n$-th term of the sequence is the units digit of the sum of the two previous terms. Let $S_n$ denote the sum of the first $n$ terms of this sequence. The smallest value of $n$ for which $S_n>10,000$ is:
$\text{(A) }1992
\qquad
\text{(B) }1999
\qqu... | B | 4 | 2002A-P21 |
Triangle $ABC$ is a right triangle with $\angle ACB$ as its right angle, $m\angle ABC = 60^{\circ}$, and $AB = 10$. Let $P$ be randomly chosen inside $\triangle ABC$, and extend $\overline{BP}$ to meet $\overline{AC}$ at $D$. What is the probability that $BD > 5\sqrt{2}$?
\textbf{(A)}\ \frac{2-\sqrt2}{2}\qquad\textbf{... | C | 4 | 2002A-P22 |
In triangle $ABC$, side $AC$ and the perpendicular bisector of $BC$ meet in point $D$, and $BD$ bisects $\angle ABC$. If $AD = 9$ and $DC = 7$, what is the area of triangle $ABD$?
$\text{(A)}\ 14 \qquad \text{(B)}\ 21 \qquad \text{(C)}\ 28 \qquad \text{(D)}\ 14\sqrt5 \qquad \text{(E)}\ 28\sqrt5$ | D | 4 | 2002A-P23 |
Find the number of ordered pairs of real numbers $(a,b)$ such that $(a+bi)^{2002} = a-bi$.
$\text{(A) }1001
\qquad
\text{(B) }1002
\qquad
\text{(C) }2001
\qquad
\text{(D) }2002
\qquad
\text{(E) }2004$ | E | 4 | 2002A-P24 |
The nonzero coefficients of a polynomial $P$ with real coefficients are all replaced by their mean to form a polynomial $Q$. Which of the following could be a graph of $y = P(x)$ and $y = Q(x)$ over the interval $-4\leq x \leq 4$?
File:2002AMC12A25.png | B | 4 | 2002A-P25 |
The arithmetic mean of the nine numbers in the set $\{9, 99, 999, 9999, \ldots, 999999999\}$ is a $9$-digit number $M$, all of whose digits are distinct. The number $M$ does not contain the digit
$\mathrm{(A)}\ 0
\qquad\mathrm{(B)}\ 2
\qquad\mathrm{(C)}\ 4
\qquad\mathrm{(D)}\ 6
\qquad\mathrm{(E)}\ 8$ | A | 2 | 2002B-P1 |
What is the value of
$$
(3x - 2)(4x + 1) - (3x - 2)4x + 1
$$
when $x=4$?
$\mathrm{(A)}\ 0
\qquad\mathrm{(B)}\ 1
\qquad\mathrm{(C)}\ 10
\qquad\mathrm{(D)}\ 11
\qquad\mathrm{(E)}\ 12$ | D | 2 | 2002B-P2 |
For how many positive integers $n$ is $n^2 - 3n + 2$ a prime number?
$\mathrm{(A)}\ \text{none}
\qquad\mathrm{(B)}\ \text{one}
\qquad\mathrm{(C)}\ \text{two}
\qquad\mathrm{(D)}\ \text{more\ than\ two,\ but\ finitely\ many}
\qquad\mathrm{(E)}\ \text{infinitely\ many}$ | B | 2 | 2002B-P3 |
Let $n$ be a positive integer such that $\frac 12 + \frac 13 + \frac 17 + \frac 1n$ is an integer. Which of the following statements is '''not''' true:
$\mathrm{(A)}\ 2\ \text{divides\ }n
\qquad\mathrm{(B)}\ 3\ \text{divides\ }n
\qquad\mathrm{(C)}\ 6\ \text{divides\ }n
\qquad\mathrm{(D)}\ 7\ \text{divides\ }n
\qquad\m... | E | 2 | 2002B-P4 |
Let $v, w, x, y,$ and $z$ be the degree measures of the five angles of a pentagon. Suppose that $v < w < x < y < z$ and $v, w, x, y,$ and $z$ form an arithmetic sequence. Find the value of $x$.
$\mathrm{(A)}\ 72
\qquad\mathrm{(B)}\ 84
\qquad\mathrm{(C)}\ 90
\qquad\mathrm{(D)}\ 108
\qquad\mathrm{(E)}\ 120$ | D | 2 | 2002B-P5 |
Suppose that $a$ and $b$ are nonzero real numbers, and that the equation $x^2 + ax + b = 0$ has solutions $a$ and $b$. Then the pair $(a,b)$ is
$\mathrm{(A)}\ (-2,1)
\qquad\mathrm{(B)}\ (-1,2)
\qquad\mathrm{(C)}\ (1,-2)
\qquad\mathrm{(D)}\ (2,-1)
\qquad\mathrm{(E)}\ (4,4)$ | C | 2 | 2002B-P6 |
The product of three consecutive positive integers is $8$ times their sum. What is the sum of their squares?
$\mathrm{(A)}\ 50
\qquad\mathrm{(B)}\ 77
\qquad\mathrm{(C)}\ 110
\qquad\mathrm{(D)}\ 149
\qquad\mathrm{(E)}\ 194$ | B | 2 | 2002B-P7 |
Suppose July of year $N$ has five Mondays. Which of the following must occur five times in August of year $N$? (Note: Both months have 31 days.)
$\mathrm{(A)}\ \text{Monday}
\qquad\mathrm{(B)}\ \text{Tuesday}
\qquad\mathrm{(C)}\ \text{Wednesday}
\qquad\mathrm{(D)}\ \text{Thursday}
\qquad\mathrm{(E)}\ \text{Friday}$ | D | 2 | 2002B-P8 |
If $a,b,c,d$ are positive real numbers such that $a,b,c,d$ form an increasing arithmetic sequence and $a,b,d$ form a geometric sequence, then $\frac ad$ is
$\mathrm{(A)}\ \frac 1{12}
\qquad\mathrm{(B)}\ \frac 16
\qquad\mathrm{(C)}\ \frac 14
\qquad\mathrm{(D)}\ \frac 13
\qquad\mathrm{(E)}\ \frac 12$ | C | 2 | 2002B-P9 |
How many different integers can be expressed as the sum of three distinct members of the set $\{1,4,7,10,13,16,19\}$?
$\mathrm{(A)}\ 13
\qquad\mathrm{(B)}\ 16
\qquad\mathrm{(C)}\ 24
\qquad\mathrm{(D)}\ 30
\qquad\mathrm{(E)}\ 35$ | A | 2 | 2002B-P10 |
The positive integers $A, B, A-B,$ and $A+B$ are all prime numbers. The sum of these four primes is
$\mathrm{(A)}\ \mathrm{even}
\qquad\mathrm{(B)}\ \mathrm{divisible\ by\ }3
\qquad\mathrm{(C)}\ \mathrm{divisible\ by\ }5
\qquad\mathrm{(D)}\ \mathrm{divisible\ by\ }7
\qquad\mathrm{(E)}\ \mathrm{prime}$ | E | 3 | 2002B-P11 |
For how many integers $n$ is $\dfrac n{20-n}$ the square of an integer?
$\mathrm{(A)}\ 1
\qquad\mathrm{(B)}\ 2
\qquad\mathrm{(C)}\ 3
\qquad\mathrm{(D)}\ 4
\qquad\mathrm{(E)}\ 10$ | D | 3 | 2002B-P12 |
The sum of $18$ consecutive positive integers is a perfect square. The smallest possible value of this sum is
$\mathrm{(A)}\ 169
\qquad\mathrm{(B)}\ 225
\qquad\mathrm{(C)}\ 289
\qquad\mathrm{(D)}\ 361
\qquad\mathrm{(E)}\ 441$ | B | 3 | 2002B-P13 |
Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect?
$\mathrm{(A)}\ 8
\qquad\mathrm{(B)}\ 9
\qquad\mathrm{(C)}\ 10
\qquad\mathrm{(D)}\ 12
\qquad\mathrm{(E)}\ 16$ | D | 3 | 2002B-P14 |
How many four-digit numbers $N$ have the property that the three-digit number obtained by removing the leftmost digit is one ninth of $N$?
$\mathrm{(A)}\ 4
\qquad\mathrm{(B)}\ 5
\qquad\mathrm{(C)}\ 6
\qquad\mathrm{(D)}\ 7
\qquad\mathrm{(E)}\ 8$ | D | 3 | 2002B-P15 |
Juan rolls a fair regular octahedral die marked with the numbers $1$ through $8$. Then Amal rolls a fair six-sided die. What is the probability that the product of the two rolls is a multiple of 3?
$\mathrm{(A)}\ \frac1{12}
\qquad\mathrm{(B)}\ \frac 13
\qquad\mathrm{(C)}\ \frac 12
\qquad\mathrm{(D)}\ \frac 7{12}
\qqua... | C | 3 | 2002B-P16 |
Andy’s lawn has twice as much area as Beth’s lawn and three times as much area as Carlos’ lawn. Carlos’ lawn mower cuts half as fast as Beth’s mower and one third as fast as Andy’s mower. If they all start to mow their lawns at the same time, who will finish first?
$\mathrm{(A)}\ \text{Andy}
\qquad\mathrm{(B)}\ \text{... | B | 3 | 2002B-P17 |
A point $P$ is randomly selected from the rectangular region with vertices $(0,0),(2,0),(2,1),(0,1)$. What is the probability that $P$ is closer to the origin than it is to the point $(3,1)$?
$\mathrm{(A)}\ \frac 12
\qquad\mathrm{(B)}\ \frac 23
\qquad\mathrm{(C)}\ \frac 34
\qquad\mathrm{(D)}\ \frac 45
\qquad\mathrm{(E... | C | 3 | 2002B-P18 |
If $a,b,$ and $c$ are positive real numbers such that $a(b+c) = 152, b(c+a) = 162,$ and $c(a+b) = 170$, then $abc$ is
$\mathrm{(A)}\ 672
\qquad\mathrm{(B)}\ 688
\qquad\mathrm{(C)}\ 704
\qquad\mathrm{(D)}\ 720
\qquad\mathrm{(E)}\ 750$ | D | 3 | 2002B-P19 |
Let $\triangle XOY$ be a right-angled triangle with $\angle XOY = 90^{\circ}$. Let $M$ and $N$ be the midpoints of legs $OX$ and $OY$, respectively. Given that $XN = 19$ and $YM = 22$, find $XY$.
$\mathrm{(A)}\ 24
\qquad\mathrm{(B)}\ 26
\qquad\mathrm{(C)}\ 28
\qquad\mathrm{(D)}\ 30
\qquad\mathrm{(E)}\ 32$ | B | 3 | 2002B-P20 |
For all positive integers $n$ less than $2002$, let
$$
\begin{eqnarray*}
a_n =\left\{
\begin{array}{lr}
11, & \text{if\ }n\ \text{is\ divisible\ by\ }13\ \text{and\ }14;\\
13, & \text{if\ }n\ \text{is\ divisible\ by\ }14\ \text{and\ }11;\\
14, & \text{if\ }n\ \text{is\ divisible\ by\ }11\ \text{and\ }13;\\
0, & \text{... | A | 4 | 2002B-P21 |
For all integers $n$ greater than $1$, define $a_n = \frac{1}{\log_n 2002}$. Let $b = a_2 + a_3 + a_4 + a_5$ and $c = a_{10} + a_{11} + a_{12} + a_{13} + a_{14}$. Then $b- c$ equals
$\mathrm{(A)}\ -2
\qquad\mathrm{(B)}\ -1
\qquad\mathrm{(C)}\ \frac{1}{2002}
\qquad\mathrm{(D)}\ \frac{1}{1001}
\qquad\mathrm{(E)}\ \frac ... | B | 4 | 2002B-P22 |
In $\triangle ABC$, we have $AB = 1$ and $AC = 2$. Side $\overline{BC}$ and the median from $A$ to $\overline{BC}$ have the same length. What is $BC$?
$\mathrm{(A)}\ \frac{1+\sqrt{2}}{2}
\qquad\mathrm{(B)}\ \frac{1+\sqrt{3}}2
\qquad\mathrm{(C)}\ \sqrt{2}
\qquad\mathrm{(D)}\ \frac 32
\qquad\mathrm{(E)}\ \sqrt{3}$ | C | 4 | 2002B-P23 |
A convex quadrilateral $ABCD$ with area $2002$ contains a point $P$ in its interior such that $PA = 24, PB = 32, PC = 28, PD = 45$. Find the perimeter of $ABCD$.
$\mathrm{(A)}\ 4\sqrt{2002}
\qquad\mathrm{(B)}\ 2\sqrt{8465}
\qquad\mathrm{(C)}\ 2(48+$ $\sqrt{2002})
\qquad\mathrm{(D)}\ 2\sqrt{8633}
\qquad\mathrm{(E)}\ 4(... | E | 4 | 2002B-P24 |
Let $f(x) = x^2 + 6x + 1$, and let $R$ denote the set of points $(x,y)$ in the coordinate plane such that
$$
f(x) + f(y) \le 0 \qquad \text{and} \qquad f(x)-f(y) \le 0
$$
The area of $R$ is closest to
$\mathrm{(A)}\ 21
\qquad\mathrm{(B)}\ 22
\qquad\mathrm{(C)}\ 23
\qquad\mathrm{(D)}\ 24
\qquad\mathrm{(E)}\ 25$ | E | 4 | 2002B-P25 |
End of preview. Expand in Data Studio
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Check out the documentation for more information.
AMC12 Dataset (Research-Oriented)
A structured dataset derived from the AMC 12 (American Mathematics Competitions), designed for LLM training, evaluation, and reinforcement learning (RL) on mathematical reasoning tasks.
This repository contains all AMC 12 problems from 2000–2025, making it one of the most complete AMC12 datasets available for research.
📘 Introduction
The AMC 12 is a 25-question, 75-minute multiple-choice examination aimed at high school students. Problems are designed to increase in difficulty progressively, requiring a combination of algebra, geometry, combinatorics, and number theory reasoning.
Format: 25 multiple-choice questions (A–E)
Duration: 75 minutes
Difficulty progression: Problems 1 → 25 increase in complexity
Calculator policy:
- Since 2008, calculators are not permitted
- Problems are designed to be solvable without computational aids
Top-performing students (~top 6%) are invited to participate in the AIME, making AMC 12 a strong proxy for high-level mathematical reasoning ability.
📦 Dataset Overview
Each sample corresponds to a single AMC 12 problem.
Example (JSONL)
{
"year": 2019,
"problem_id": "2019A-15",
"question": "...",
"answer": "D",
"difficulty": 3
}
🧱 Schema
Each entry in the dataset follows this structure:
| Field | Type | Description |
|---|---|---|
problem_id |
string | Unique identifier in the format {year}{A/B}-{problem_number} (e.g., 2019A-15) |
year |
int | Competition year (2000–2025) |
question |
string | Full problem statement (including choices) |
answer |
string | Correct answer option (A–E) |
difficulty |
int | Difficulty level derived from problem order |
🔑 Problem ID Definition
The problem_id encodes the full provenance of each problem:
{year}{A/B}-{problem_number}
{year}→ competition year{A/B}→ AMC12A or AMC12B{problem_number}→ position in the exam (1–25)
Examples
2007B-5→ AMC 12B, 2007, Problem 52019A-15→ AMC 12A, 2019, Problem 15
Difficulty Structure
We adopt a coarse-grained difficulty approximation aligned with problem order:
| Problem Range | Difficulty |
|---|---|
| 1–10 | Easy–Medium (2) |
| 11–20 | Medium–Hard (3) |
| 21–25 | Hard (4) |
This structure enables:
- Curriculum learning
- Difficulty-aware evaluation
- Model capability stratification
🚀 Why This Dataset?
Compared to Other Math Datasets
Not heavily pretrained Unlike datasets such as GSM8K, AMC-style problems are less likely to be memorized by models
Higher reasoning complexity Problems typically require multi-step, structured reasoning, often exceeding datasets like MATH500
Clean evaluation signal
- Multiple-choice format eliminates ambiguity
- No unit mismatch issues (e.g., “8 months vs 240 days”)
Fully verifiable Every problem has a unique, discrete answer, ideal for RL reward design
Compared to Other AMC Datasets
Complete coverage (2000–2025) Includes all AMC12A and AMC12B problems across 25 years
Fully indexed & traceable Each problem maps directly to its original contest and position
Structured for ML pipelines Ready for:
- RL training (PPO / GRPO)
- Pass@k evaluation
- Verifier-based reward systems
📚 Data Source & Attribution
This dataset is curated from publicly available resources, with primary reference to:
- Art of Problem Solving
All AMC problems are copyrighted by the Mathematical Association of America (MAA) under the American Mathematics Competitions program.
This repository does not claim ownership of the original problem statements and provides them solely for research and educational purposes.
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