problem string | answer string | source string | domain list | llama8b_solve_rate float64 |
|---|---|---|---|---|
Misha rolls a standard, fair six-sided die until she rolls $1-2-3$ in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is $\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 647 | amc_aime | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.046875 |
Squares $ABCD$ and $EFGH$ have a common center and $\overline{AB} || \overline{EF}$. The area of $ABCD$ is 2016, and the area of $EFGH$ is a smaller positive integer. Square $IJKL$ is constructed so that each of its vertices lies on a side of $ABCD$ and each vertex of $EFGH$ lies on a side of $IJKL$. Find the difference between the largest and smallest positive integer values for the area of $IJKL$. | 0 | amc_aime | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.0625 |
A game of solitaire is played with $R$ red cards, $W$ white cards, and $B$ blue cards. A player plays all the cards one at a time. With each play he accumulates a penalty. If he plays a blue card, then he is charged a penalty which is the number of white cards still in his hand. If he plays a white card, then he is charged a penalty which is twice the number of red cards still in his hand. If he plays a red card, then he is charged a penalty which is three times the number of blue cards still in his hand. Find, as a function of $R, W,$ and $B,$ the minimal total penalty a player can amass and all the ways in which this minimum can be achieved. | \min(BW, 2WR, 3RB) | amc_aime | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 |
During a recent campaign for office, a candidate made a tour of a country which we assume lies in a plane. On the first day of the tour he went east, on the second day he went north, on the third day west, on the fourth day south, on the fifth day east, etc. If the candidate went $n^{2}_{}/2$ miles on the $n^{\mbox{th}}_{}$ day of this tour, how many miles was he from his starting point at the end of the $40^{\mbox{th}}_{}$ day? | 4640 | amc_aime | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 |
The number $n$ can be written in base $14$ as $\underline{a}\text{ }\underline{b}\text{ }\underline{c}$, can be written in base $15$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{b}$, and can be written in base $6$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{a}\text{ }\underline{c}\text{ }$, where $a > 0$. Find the base-$10$ representation of $n$. | 925 | amc_aime | [
"Mathematics -> Number Theory -> Other"
] | 0.015625 |
Find the number of pairs $(m,n)$ of positive integers with $1\le m<n\le 30$ such that there exists a real number $x$ satisfying \[\sin(mx)+\sin(nx)=2.\] | 63 | amc_aime | [
"Mathematics -> Applied Mathematics -> Other"
] | 0 |
Find the number of $7$-tuples of positive integers $(a,b,c,d,e,f,g)$ that satisfy the following systems of equations:
\begin{align*} abc&=70,\\ cde&=71,\\ efg&=72. \end{align*} | 96 | amc_aime | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.015625 |
Let $f(n)$ be the number of ways to write $n$ as a sum of powers of $2$, where we keep track of the order of the summation. For example, $f(4)=6$ because $4$ can be written as $4$, $2+2$, $2+1+1$, $1+2+1$, $1+1+2$, and $1+1+1+1$. Find the smallest $n$ greater than $2013$ for which $f(n)$ is odd. | 2047 | amc_aime | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.09375 |
A right prism with height $h$ has bases that are regular hexagons with sides of length $12$. A vertex $A$ of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and the face of the pyramid that does not contain $A$ measures $60$ degrees. Find $h^2$. | 108 | amc_aime | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.140625 |
Let $x$, $y$ and $z$ all exceed $1$ and let $w$ be a positive number such that $\log_x w = 24$, $\log_y w = 40$ and $\log_{xyz} w = 12$. Find $\log_z w$. | 60 | amc_aime | [
"Mathematics -> Algebra -> Other"
] | 0.046875 |
Let $X_1, X_2, \ldots, X_{100}$ be a sequence of mutually distinct nonempty subsets of a set $S$. Any two sets $X_i$ and $X_{i+1}$ are disjoint and their union is not the whole set $S$, that is, $X_i\cap X_{i+1}=\emptyset$ and $X_i\cup X_{i+1}\neq S$, for all $i\in\{1, \ldots, 99\}$. Find the smallest possible number of elements in $S$. | 8 | amc_aime | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.015625 |
Suppose that the measurement of time during the day is converted to the metric system so that each day has $10$ metric hours, and each metric hour has $100$ metric minutes. Digital clocks would then be produced that would read $\text{9:99}$ just before midnight, $\text{0:00}$ at midnight, $\text{1:25}$ at the former $\text{3:00}$ AM, and $\text{7:50}$ at the former $\text{6:00}$ PM. After the conversion, a person who wanted to wake up at the equivalent of the former $\text{6:36}$ AM would set his new digital alarm clock for $\text{A:BC}$, where $\text{A}$, $\text{B}$, and $\text{C}$ are digits. Find $100\text{A}+10\text{B}+\text{C}$. | 275 | amc_aime | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.1875 |
A circle is circumscribed around an isosceles triangle whose two congruent angles have degree measure $x$. Two points are chosen independently and uniformly at random on the circle, and a chord is drawn between them. The probability that the chord intersects the triangle is $\frac{14}{25}$. Find the difference between the largest and smallest possible values of $x$. | 23.6643 | amc_aime | [
"Mathematics -> Geometry -> Plane Geometry -> Other",
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.015625 |
(Ricky Liu) For what values of $k > 0$ is it possible to dissect a $1 \times k$ rectangle into two similar, but incongruent, polygons? | k \neq 1 | amc_aime | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 |
Find the number of four-element subsets of $\{1,2,3,4,\dots, 20\}$ with the property that two distinct elements of a subset have a sum of $16$, and two distinct elements of a subset have a sum of $24$. For example, $\{3,5,13,19\}$ and $\{6,10,20,18\}$ are two such subsets. | 210 | amc_aime | [
"Mathematics -> Applied Mathematics -> Combinatorics"
] | 0.015625 |
A group of clerks is assigned the task of sorting $1775$ files. Each clerk sorts at a constant rate of $30$ files per hour. At the end of the first hour, some of the clerks are reassigned to another task; at the end of the second hour, the same number of the remaining clerks are also reassigned to another task, and a similar assignment occurs at the end of the third hour. The group finishes the sorting in $3$ hours and $10$ minutes. Find the number of files sorted during the first one and a half hours of sorting. | 945 | amc_aime | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 |
For each positive integer $n$, find the number of $n$-digit positive integers that satisfy both of the following conditions:
$\bullet$ no two consecutive digits are equal, and
$\bullet$ the last digit is a prime. | \frac{2}{5} (9^n + (-1)^{n+1}) | amc_aime | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 |
Let $x$ be a real number such that $\sin^{10}x+\cos^{10} x = \tfrac{11}{36}$. Then $\sin^{12}x+\cos^{12} x = \tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 23 | amc_aime | [
"Mathematics -> Algebra -> Other"
] | 0.015625 |
Call a permutation $a_1, a_2, \ldots, a_n$ of the integers $1, 2, \ldots, n$ quasi-increasing if $a_k \leq a_{k+1} + 2$ for each $1 \leq k \leq n-1$. For example, 53421 and 14253 are quasi-increasing permutations of the integers $1, 2, 3, 4, 5$, but 45123 is not. Find the number of quasi-increasing permutations of the integers $1, 2, \ldots, 7$. | 486 | amc_aime | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.03125 |
Jon and Steve ride their bicycles along a path that parallels two side-by-side train tracks running the east/west direction. Jon rides east at $20$ miles per hour, and Steve rides west at $20$ miles per hour. Two trains of equal length, traveling in opposite directions at constant but different speeds each pass the two riders. Each train takes exactly $1$ minute to go past Jon. The westbound train takes $10$ times as long as the eastbound train to go past Steve. The length of each train is $\tfrac{m}{n}$ miles, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 49 | amc_aime | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 |
Two unit squares are selected at random without replacement from an $n \times n$ grid of unit squares. Find the least positive integer $n$ such that the probability that the two selected unit squares are horizontally or vertically adjacent is less than $\frac{1}{2015}$. | 90 | amc_aime | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.078125 |
An empty $2020 \times 2020 \times 2020$ cube is given, and a $2020 \times 2020$ grid of square unit cells is drawn on each of its six faces. A beam is a $1 \times 1 \times 2020$ rectangular prism. Several beams are placed inside the cube subject to the following conditions:
- The two $1 \times 1$ faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are $3 \cdot {2020}^2$ possible positions for a beam.)
- No two beams have intersecting interiors.
- The interiors of each of the four $1 \times 2020$ faces of each beam touch either a face of the cube or the interior of the face of another beam.
What is the smallest positive number of beams that can be placed to satisfy these conditions? | 3030 | amc_aime | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.015625 |
Find all solutions to $(m^2+n)(m + n^2)= (m - n)^3$, where m and n are non-zero integers. | (-1, -1), (8, -10), (9, -6), (9, -21) | amc_aime | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.03125 |
Before starting to paint, Bill had $130$ ounces of blue paint, $164$ ounces of red paint, and $188$ ounces of white paint. Bill painted four equally sized stripes on a wall, making a blue stripe, a red stripe, a white stripe, and a pink stripe. Pink is a mixture of red and white, not necessarily in equal amounts. When Bill finished, he had equal amounts of blue, red, and white paint left. Find the total number of ounces of paint Bill had left. | 114 | amc_aime | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 |
A hexagon that is inscribed in a circle has side lengths $22$, $22$, $20$, $22$, $22$, and $20$ in that order. The radius of the circle can be written as $p+\sqrt{q}$, where $p$ and $q$ are positive integers. Find $p+q$. | 272 | amc_aime | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.015625 |
What is the smallest integer $n$, greater than one, for which the root-mean-square of the first $n$ positive integers is an integer?
$\mathbf{Note.}$ The root-mean-square of $n$ numbers $a_1, a_2, \cdots, a_n$ is defined to be
\[\left[\frac{a_1^2 + a_2^2 + \cdots + a_n^2}n\right]^{1/2}\] | 337 | amc_aime | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.03125 |
Anh read a book. On the first day she read $n$ pages in $t$ minutes, where $n$ and $t$ are positive integers. On the second day Anh read $n + 1$ pages in $t + 1$ minutes. Each day thereafter Anh read one more page than she read on the previous day, and it took her one more minute than on the previous day until she completely read the $374$ page book. It took her a total of $319$ minutes to read the book. Find $n + t$. | 53 | amc_aime | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 |
Initially Alex, Betty, and Charlie had a total of $444$ peanuts. Charlie had the most peanuts, and Alex had the least. The three numbers of peanuts that each person had formed a geometric progression. Alex eats $5$ of his peanuts, Betty eats $9$ of her peanuts, and Charlie eats $25$ of his peanuts. Now the three numbers of peanuts each person has forms an arithmetic progression. Find the number of peanuts Alex had initially. | 108 | amc_aime | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.046875 |
Find the number of ordered pairs of positive integer solutions $(m, n)$ to the equation $20m + 12n = 2012$. | 34 | amc_aime | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.015625 |
Find the number of positive integers with three not necessarily distinct digits, $abc$, with $a \neq 0$ and $c \neq 0$ such that both $abc$ and $cba$ are multiples of $4$. | 36 | amc_aime | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.015625 |
For positive integers $n$ and $k$, let $f(n, k)$ be the remainder when $n$ is divided by $k$, and for $n > 1$ let $F(n) = \max_{\substack{1\le k\le \frac{n}{2}}} f(n, k)$. Find the remainder when $\sum\limits_{n=20}^{100} F(n)$ is divided by $1000$. | 512 | amc_aime | [
"Mathematics -> Applied Mathematics -> Other"
] | 0 |
Points $A$, $B$, and $C$ lie in that order along a straight path where the distance from $A$ to $C$ is $1800$ meters. Ina runs twice as fast as Eve, and Paul runs twice as fast as Ina. The three runners start running at the same time with Ina starting at $A$ and running toward $C$, Paul starting at $B$ and running toward $C$, and Eve starting at $C$ and running toward $A$. When Paul meets Eve, he turns around and runs toward $A$. Paul and Ina both arrive at $B$ at the same time. Find the number of meters from $A$ to $B$. | 800 | amc_aime | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 |
For any positive integer $k$, let $f_1(k)$ denote the square of the sum of the digits of $k$. For $n \ge 2$, let $f_n(k) = f_1(f_{n - 1}(k))$. Find $f_{1988}(11)$. | 169 | amc_aime | [
"Mathematics -> Other -> Other"
] | 0.3125 |
Fifteen distinct points are designated on $\triangle ABC$: the 3 vertices $A$, $B$, and $C$; $3$ other points on side $\overline{AB}$; $4$ other points on side $\overline{BC}$; and $5$ other points on side $\overline{CA}$. Find the number of triangles with positive area whose vertices are among these $15$ points. | 390 | amc_aime | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.203125 |
Call a set $S$ product-free if there do not exist $a, b, c \in S$ (not necessarily distinct) such that $a b = c$. For example, the empty set and the set $\{16, 20\}$ are product-free, whereas the sets $\{4, 16\}$ and $\{2, 8, 16\}$ are not product-free. Find the number of product-free subsets of the set $\{1, 2, 3, 4,..., 7, 8, 9, 10\}$. | 252 | amc_aime | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0 |
There exist unique positive integers $x$ and $y$ that satisfy the equation $x^2 + 84x + 2008 = y^2$. Find $x + y$. | 80 | amc_aime | [
"Mathematics -> Number Theory -> Other"
] | 0 |
Let the set $S = \{P_1, P_2, \dots, P_{12}\}$ consist of the twelve vertices of a regular $12$-gon. A subset $Q$ of $S$ is called "communal" if there is a circle such that all points of $Q$ are inside the circle, and all points of $S$ not in $Q$ are outside of the circle. How many communal subsets are there? (Note that the empty set is a communal subset.) | 134 | amc_aime | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 |
If this path is to continue in the same pattern:
then which sequence of arrows goes from point $425$ to point $427$? | (A) | amc_aime | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.015625 |
Triangle $ABC$ has side lengths $AB = 9$, $BC =$ $5\sqrt{3}$, and $AC = 12$. Points $A = P_{0}, P_{1}, P_{2}, ... , P_{2450} = B$ are on segment $\overline{AB}$ with $P_{k}$ between $P_{k-1}$ and $P_{k+1}$ for $k = 1, 2, ..., 2449$, and points $A = Q_{0}, Q_{1}, Q_{2}, ... , Q_{2450} = C$ are on segment $\overline{AC}$ with $Q_{k}$ between $Q_{k-1}$ and $Q_{k+1}$ for $k = 1, 2, ..., 2449$. Furthermore, each segment $\overline{P_{k}Q_{k}}$, $k = 1, 2, ..., 2449$, is parallel to $\overline{BC}$. The segments cut the triangle into $2450$ regions, consisting of $2449$ trapezoids and $1$ triangle. Each of the $2450$ regions has the same area. Find the number of segments $\overline{P_{k}Q_{k}}$, $k = 1, 2, ..., 2450$, that have rational length. | 28 | amc_aime | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.015625 |
For any positive integer $a, \sigma(a)$ denotes the sum of the positive integer divisors of $a$. Let $n$ be the least positive integer such that $\sigma(a^n)-1$ is divisible by $2021$ for all positive integers $a$. Find the sum of the prime factors in the prime factorization of $n$. | 125 | amc_aime | [
"Mathematics -> Number Theory -> Other"
] | 0 |
Equilateral triangle $ABC$ has side length $840$. Point $D$ lies on the same side of line $BC$ as $A$ such that $\overline{BD} \perp \overline{BC}$. The line $\ell$ through $D$ parallel to line $BC$ intersects sides $\overline{AB}$ and $\overline{AC}$ at points $E$ and $F$, respectively. Point $G$ lies on $\ell$ such that $F$ is between $E$ and $G$, $\triangle AFG$ is isosceles, and the ratio of the area of $\triangle AFG$ to the area of $\triangle BED$ is $8:9$. Find $AF$. | 336 | amc_aime | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.03125 |
A bug walks all day and sleeps all night. On the first day, it starts at point $O$, faces east, and walks a distance of $5$ units due east. Each night the bug rotates $60^\circ$ counterclockwise. Each day it walks in this new direction half as far as it walked the previous day. The bug gets arbitrarily close to the point $P$. Then $OP^2=\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 103 | amc_aime | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.0625 |
Let $m$ be the number of five-element subsets that can be chosen from the set of the first $14$ natural numbers so that at least two of the five numbers are consecutive. Find the remainder when $m$ is divided by $1000$. | 750 | amc_aime | [
"Mathematics -> Applied Mathematics -> Statistics -> Combinations"
] | 0.078125 |
There is a unique positive real number $x$ such that the three numbers $\log_8{2x}$, $\log_4{x}$, and $\log_2{x}$, in that order, form a geometric progression with positive common ratio. The number $x$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | 17 | amc_aime | [
"Mathematics -> Algebra -> Other"
] | 0.09375 |
Find the number of positive integers less than or equal to $2017$ whose base-three representation contains no digit equal to $0$. | 222 | amc_aime | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 |
Let $n \geq 4$ be an integer. Find all positive real solutions to the following system of $2n$ equations:\begin{align*} a_{1} &=\frac{1}{a_{2 n}}+\frac{1}{a_{2}}, & a_{2}&=a_{1}+a_{3}, \\ a_{3}&=\frac{1}{a_{2}}+\frac{1}{a_{4}}, & a_{4}&=a_{3}+a_{5}, \\ a_{5}&=\frac{1}{a_{4}}+\frac{1}{a_{6}}, & a_{6}&=a_{5}+a_{7}, \\ &\vdots \\ a_{2 n-1}&=\frac{1}{a_{2 n-2}}+\frac{1}{a_{2 n}}, & a_{2 n}&=a_{2 n-1}+a_{1} \end{align*} | a_{2k+1} = 1, \quad a_{2k+2} = 2 \quad \text{for all } k \text{ such that } 0 \leq k < n | amc_aime | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.046875 |
In a Martian civilization, all logarithms whose bases are not specified are assumed to be base $b$, for some fixed $b\ge2$. A Martian student writes down
\[3\log(\sqrt{x}\log x)=56\]
\[\log_{\log x}(x)=54\]
and finds that this system of equations has a single real number solution $x>1$. Find $b$. | 216 | amc_aime | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0 |
Four ambassadors and one advisor for each of them are to be seated at a round table with $12$ chairs numbered in order $1$ to $12$. Each ambassador must sit in an even-numbered chair. Each advisor must sit in a chair adjacent to his or her ambassador. There are $N$ ways for the $8$ people to be seated at the table under these conditions. Find the remainder when $N$ is divided by $1000$. | 520 | amc_aime | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.03125 |
Jar A contains four liters of a solution that is 45% acid. Jar B contains five liters of a solution that is 48% acid. Jar C contains one liter of a solution that is $k\%$ acid. From jar C, $\frac{m}{n}$ liters of the solution is added to jar A, and the remainder of the solution in jar C is added to jar B. At the end both jar A and jar B contain solutions that are 50% acid. Given that $m$ and $n$ are relatively prime positive integers, find $k + m + n$. | 085 | amc_aime | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 |
Let $ABCD$ be a square, and let $E$ and $F$ be points on $\overline{AB}$ and $\overline{BC},$ respectively. The line through $E$ parallel to $\overline{BC}$ and the line through $F$ parallel to $\overline{AB}$ divide $ABCD$ into two squares and two nonsquare rectangles. The sum of the areas of the two squares is $\frac{9}{10}$ of the area of square $ABCD.$ Find $\frac{AE}{EB} + \frac{EB}{AE}.$ | 18 | amc_aime | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.015625 |
(Zuming Feng) Determine all composite positive integers $n$ for which it is possible to arrange all divisors of $n$ that are greater than 1 in a circle so that no two adjacent divisors are relatively prime. | \text{The composite integers } n \text{ for which the arrangement is possible are all except } n = pq \text{ where } p \text{ and } q \text{ are distinct primes.} | amc_aime | [
"Mathematics -> Number Theory -> Other"
] | 0.015625 |
Let $a_1,a_2,a_3,\cdots$ be a non-decreasing sequence of positive integers. For $m\ge1$, define $b_m=\min\{n: a_n \ge m\}$, that is, $b_m$ is the minimum value of $n$ such that $a_n\ge m$. If $a_{19}=85$, determine the maximum value of $a_1+a_2+\cdots+a_{19}+b_1+b_2+\cdots+b_{85}$. | 1700 | amc_aime | [
"Mathematics -> Algebra -> Other"
] | 0 |
If the sum of the lengths of the six edges of a trirectangular tetrahedron $PABC$ (i.e., $\angle APB=\angle BPC=\angle CPA=90^o$) is $S$, determine its maximum volume. | \frac{S^3(\sqrt{2}-1)^3}{162} | amc_aime | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0 |
Let the sum of a set of numbers be the sum of its elements. Let $S$ be a set of positive integers, none greater than 15. Suppose no two disjoint subsets of $S$ have the same sum. What is the largest sum a set $S$ with these properties can have? | 61 | amc_aime | [
"Mathematics -> Number Theory -> Other"
] | 0.015625 |
An $a \times b \times c$ rectangular box is built from $a \cdot b \cdot c$ unit cubes. Each unit cube is colored red, green, or yellow. Each of the $a$ layers of size $1 \times b \times c$ parallel to the $(b \times c)$ faces of the box contains exactly $9$ red cubes, exactly $12$ green cubes, and some yellow cubes. Each of the $b$ layers of size $a \times 1 \times c$ parallel to the $(a \times c)$ faces of the box contains exactly $20$ green cubes, exactly $25$ yellow cubes, and some red cubes. Find the smallest possible volume of the box. | 180 | amc_aime | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 |
Let $B$ be the set of all binary integers that can be written using exactly $5$ zeros and $8$ ones where leading zeros are allowed. If all possible subtractions are performed in which one element of $B$ is subtracted from another, find the number of times the answer $1$ is obtained. | 330 | amc_aime | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.015625 |
(Ricky Liu) Find all positive integers $n$ such that there are $k\ge 2$ positive rational numbers $a_1, a_2, \ldots, a_k$ satisfying $a_1 + a_2 + \cdots + a_k = a_1\cdot a_2\cdots a_k = n$. | n \in \{4, 6, 7, 8, 9, 10, \ldots\} \text{ and } n \neq 5 | amc_aime | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.046875 |
Let $ABC$ be a triangle. Find all points $P$ on segment $BC$ satisfying the following property: If $X$ and $Y$ are the intersections of line $PA$ with the common external tangent lines of the circumcircles of triangles $PAB$ and $PAC$, then \[\left(\frac{PA}{XY}\right)^2+\frac{PB\cdot PC}{AB\cdot AC}=1.\] | P \text{ such that } PB = \frac{ab}{b+c} \text{ or } PB = \frac{ac}{b+c} | amc_aime | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.015625 |
In an isosceles trapezoid, the parallel bases have lengths $\log 3$ and $\log 192$, and the altitude to these bases has length $\log 16$. The perimeter of the trapezoid can be written in the form $\log 2^p 3^q$, where $p$ and $q$ are positive integers. Find $p + q$. | 18 | amc_aime | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 |
In the diagram below, $ABCD$ is a rectangle with side lengths $AB=3$ and $BC=11$, and $AECF$ is a rectangle with side lengths $AF=7$ and $FC=9,$ as shown. The area of the shaded region common to the interiors of both rectangles is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 109 | amc_aime | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.015625 |
Determine each real root of
$x^4-(2\cdot10^{10}+1)x^2-x+10^{20}+10^{10}-1=0$
correct to four decimal places. | x_1 \approx 99999.9997, \quad x_2 \approx 100000.0003. | amc_aime | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.046875 |
Let $n$ be the least positive integer for which $149^n-2^n$ is divisible by $3^3\cdot5^5\cdot7^7.$ Find the number of positive integer divisors of $n.$ | 270 | amc_aime | [
"Mathematics -> Number Theory -> Other"
] | 0.015625 |
Triangle $ABC$ has $AB=40,AC=31,$ and $\sin{A}=\frac{1}{5}$. This triangle is inscribed in rectangle $AQRS$ with $B$ on $\overline{QR}$ and $C$ on $\overline{RS}$. Find the maximum possible area of $AQRS$. | 744 | amc_aime | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.015625 |
A game uses a deck of $n$ different cards, where $n$ is an integer and $n \geq 6.$ The number of possible sets of 6 cards that can be drawn from the deck is 6 times the number of possible sets of 3 cards that can be drawn. Find $n.$ | 13 | amc_aime | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.109375 |
Circles $\omega_1$ and $\omega_2$ with radii $961$ and $625$, respectively, intersect at distinct points $A$ and $B$. A third circle $\omega$ is externally tangent to both $\omega_1$ and $\omega_2$. Suppose line $AB$ intersects $\omega$ at two points $P$ and $Q$ such that the measure of minor arc $\widehat{PQ}$ is $120^{\circ}$. Find the distance between the centers of $\omega_1$ and $\omega_2$. | 672 | amc_aime | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0 |
A rational number written in base eight is $\underline{ab} . \underline{cd}$, where all digits are nonzero. The same number in base twelve is $\underline{bb} . \underline{ba}$. Find the base-ten number $\underline{abc}$. | 321 | amc_aime | [
"Mathematics -> Number Theory -> Other"
] | 0.03125 |
The polynomial $f(z)=az^{2018}+bz^{2017}+cz^{2016}$ has real coefficients not exceeding $2019$, and $f\left(\tfrac{1+\sqrt{3}i}{2}\right)=2015+2019\sqrt{3}i$. Find the remainder when $f(1)$ is divided by $1000$. | 053 | amc_aime | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0 |
Zou and Chou are practicing their $100$-meter sprints by running $6$ races against each other. Zou wins the first race, and after that, the probability that one of them wins a race is $\frac23$ if they won the previous race but only $\frac13$ if they lost the previous race. The probability that Zou will win exactly $5$ of the $6$ races is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ | 323 | amc_aime | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.03125 |
A long thin strip of paper is $1024$ units in length, $1$ unit in width, and is divided into $1024$ unit squares. The paper is folded in half repeatedly. For the first fold, the right end of the paper is folded over to coincide with and lie on top of the left end. The result is a $512$ by $1$ strip of double thickness. Next, the right end of this strip is folded over to coincide with and lie on top of the left end, resulting in a $256$ by $1$ strip of quadruple thickness. This process is repeated $8$ more times. After the last fold, the strip has become a stack of $1024$ unit squares. How many of these squares lie below the square that was originally the $942$nd square counting from the left? | 1 | amc_aime | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.015625 |
Let $a > 1$ and $x > 1$ satisfy $\log_a(\log_a(\log_a 2) + \log_a 24 - 128) = 128$ and $\log_a(\log_a x) = 256$. Find the remainder when $x$ is divided by $1000$. | 896 | amc_aime | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0 |
Given that $A_k = \frac {k(k - 1)}2\cos\frac {k(k - 1)\pi}2,$ find $|A_{19} + A_{20} + \cdots + A_{98}|.$ | 040 | amc_aime | [
"Mathematics -> Algebra -> Other"
] | 0.171875 |
Let $N$ be the least positive integer that is both $22$ percent less than one integer and $16$ percent greater than another integer. Find the remainder when $N$ is divided by $1000$. | 262 | amc_aime | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 |
For every subset $T$ of $U = \{ 1,2,3,\ldots,18 \}$, let $s(T)$ be the sum of the elements of $T$, with $s(\emptyset)$ defined to be $0$. If $T$ is chosen at random among all subsets of $U$, the probability that $s(T)$ is divisible by $3$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m$. | 683 | amc_aime | [
"Mathematics -> Applied Mathematics -> Probability"
] | 0 |
Consider arrangements of the $9$ numbers $1, 2, 3, \dots, 9$ in a $3 \times 3$ array. For each such arrangement, let $a_1$, $a_2$, and $a_3$ be the medians of the numbers in rows $1$, $2$, and $3$ respectively, and let $m$ be the median of $\{a_1, a_2, a_3\}$. Let $Q$ be the number of arrangements for which $m = 5$. Find the remainder when $Q$ is divided by $1000$. | 360 | amc_aime | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.015625 |
Segments $\overline{AB}, \overline{AC},$ and $\overline{AD}$ are edges of a cube and $\overline{AG}$ is a diagonal through the center of the cube. Point $P$ satisfies $BP=60\sqrt{10}$, $CP=60\sqrt{5}$, $DP=120\sqrt{2}$, and $GP=36\sqrt{7}$. Find $AP.$ | 192 | amc_aime | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.015625 |
Let $S$ be a list of positive integers--not necessarily distinct--in which the number $68$ appears. The average (arithmetic mean) of the numbers in $S$ is $56$. However, if $68$ is removed, the average of the remaining numbers drops to $55$. What is the largest number that can appear in $S$? | 649 | amc_aime | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 |
Let $S$ be the set of positive integers $N$ with the property that the last four digits of $N$ are $2020,$ and when the last four digits are removed, the result is a divisor of $N.$ For example, $42,020$ is in $S$ because $4$ is a divisor of $42,020.$ Find the sum of all the digits of all the numbers in $S.$ For example, the number $42,020$ contributes $4+2+0+2+0=8$ to this total. | 93 | amc_aime | [
"Mathematics -> Number Theory -> Divisibility"
] | 0.0625 |
Let $n$ be a positive integer. Determine the size of the largest subset of $\{ - n, - n + 1, \ldots , n - 1, n\}$ which does not contain three elements $a, b, c$ (not necessarily distinct) satisfying $a + b + c = 0$. | 2\left\lceil \frac{n}{2} \right\rceil | amc_aime | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.0625 |
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