answer stringlengths 1 12 | problem stringlengths 36 2.25k | index int64 8 14.1k | split stringclasses 3
values |
|---|---|---|---|
509 | Determine the smallest prime $p$ such that $2018!$ is divisible by $p^3$, but not divisible by $p^4$. | 8 | train |
1 | Burrito Bear has a white unit square. She inscribes a circle inside the square and paints it black. She then inscribes a square inside the black circle and paints it white. Burrito repeats this process indefinitely. The total black area can be expressed as $\frac{a\pi+b}{c}$. Find $a+b+c$. | 14 | train |
40 | Triangle $\triangle{ABC}$ is isosceles with $AB = AC$. Let the incircle of $\triangle{ABC}$ intersect $BC$ and $AC$ at $D$ and $E$ respectively. Let $F \neq A$ be the point such that $DF = DA$ and $EF = EA$. If $AF = 8$ and the circumradius of $\triangle{AED}$ is $5$, find the area of $\triangle{ABC}$. | 31 | train |
64 | Determine the greatest power of $2$ that is a factor of $3^{15} + 3^{11} + 3^{6} + 1$. | 35 | val |
5 | Let $ABCDEFGHIJ$ be a regular 10-sided polygon inscribed in a circle with center $O$ and radius $5$. The diagonals $AD$ and $BE$ intersect at point $P$, and the diagonals $AH$ and $BI$ intersect at point $Q$. Calculate the length of segment $PQ$. | 41 | val |
45 | There are $2023$ distinct points on a plane, which are colored either white or red. For each white point, there are exactly two red points such that the distance from the white point to each of these red points is $2023$. Find the minimum number of red points. | 47 | train |
152 | The triangle $ABC$ has sides $AB = 137$, $AC = 241$, and $BC = 200$. There is a point $D$ on $BC$ such that both incircles of triangles $ABD$ and $ACD$ touch $AD$ at the same point $E$. Determine the length of $CD$. | 63 | train |
2 | Find all natural numbers $a > 1$ with the property that every prime divisor of $a^6 - 1$ also divides at least one of the numbers $a^3 - 1$, $a^2 - 1$. | 70 | val |
3025 | You have a 4-digit whole number that is a perfect square. Another number is constructed by making the following changes to the digits of the original number:
1. Add 1 to the unit's digit.
2. Subtract 1 from the ten's digit.
3. Add 1 to the hundred's digit.
4. Subtract 1 from the thousand's digit.
If the resulting num... | 99 | train |
18 | Given a regular octagon $ABCDEFGH$ with side length $3$. By drawing the four diagonals $AF$, $BE$, $CH$, and $DG$, the octagon is divided into a square, four triangles, and four rectangles. Find the sum of the areas of the square and the four triangles. | 102 | train |
6 | For a positive integer $n$, let $p(n)$ denote the smallest prime divisor of $n$. Find the maximum number of divisors $m$ can have if $p(m)^4 > m$. | 113 | val |
8 | On a semicircle with diameter $|AB|=d$, there are points $C$ and $D$ such that $|BC|=|CD|=a$ and $|DA|=b$, where $a$, $b$, and $d$ are different positive integers. Find the minimum possible value of $d$. | 119 | test |
726 | A herder has forgotten the number of cows she has and does not want to count all of them. She remembers these four facts about the number of cows:
- It has $3$ digits.
- It is a palindrome.
- The middle digit is a multiple of $4$.
- It is divisible by $11$.
What is the sum of all possible numbers of cows that the her... | 123 | test |
32 | Vitya cut the chessboard along the borders of the cells into pieces of the same perimeter. It turned out that not all of the received parts are equal. What is the largest possible number of parts that Vitya could get? | 135 | test |
5 | Let $ABCDEFGHIJ$ be a regular 10-sided polygon with all its vertices on a circle centered at $O$ with a radius of 5. The diagonals $AD$ and $BE$ intersect at $P$, and the diagonals $AH$ and $BI$ intersect at $Q$. Calculate the length of the segment $PQ$. | 167 | test |
9 | The difference between the maximal and the minimal diagonals of the regular $n$-gon equals its side ($n > 5$). Find $n$. | 168 | train |
20 | A tram ticket costs $1$ Tug, which is equivalent to $100$ tugriks. There are $20$ passengers, each having only coins in denominations of $2$ and $5$ tugriks. The conductor starts with no money at all. Despite this, all passengers manage to pay the fare and receive the necessary change. What is the smallest total number... | 169 | train |
50 | Given a regular polygon with $n$ sides, it is known that there are 1200 ways to choose three of the vertices of the polygon such that they form the vertices of a right triangle. What is the value of $n$? | 173 | train |
25 | Find the smallest natural number $n$ such that for any coloration of the numbers $1, 2, \dots, n$ with three different colors, there exist two numbers of the same color, whose difference is a perfect square. | 177 | train |
1 | Draw a regular hexagon. Then, construct a square on each edge of the hexagon. Form equilateral triangles by connecting the vertices of neighboring squares. If this pattern is extended symmetrically to infinity, what is the ratio of the number of triangles to the number of squares? | 181 | train |
502 | Find the smallest positive integer $N$ satisfying the following three properties:
- $N$ leaves a remainder of $5$ when divided by $7$.
- $N$ leaves a remainder of $6$ when divided by $8$.
- $N$ leaves a remainder of $7$ when divided by $9$. | 184 | val |
3780 | Five airway companies operate in a country consisting of 36 cities. Between any pair of cities, exactly one company operates two-way flights. If some air company operates between cities $A, B$ and $B, C$, we say that the triple $A, B, C$ is properly-connected. Determine the largest possible value of $k$ such that no ma... | 217 | val |
50 | In each cell, a strip of length $100$ is worth a chip. You can change any $2$ neighboring chips and pay $1$ rouble, and you can also swap any $2$ chips for free, between which there are exactly $4$ chips. What is the smallest amount of rubles you can spend to rearrange the chips in reverse order? | 218 | train |
20 | Let $ABCD$ be a square with side length $10$, and let $P$ be a point on side $BC$. By folding the paper along the line $AP$, point $B$ determines point $Q$, as seen in the figure. The line $PQ$ intersects side $CD$ at point $R$. Calculate the perimeter of triangle $PCR$. | 247 | test |
9 | Solve the equation:
\[
\sin 9^\circ \sin 21^\circ \sin(102^\circ + x^\circ) = \sin 30^\circ \sin 42^\circ \sin x^\circ
\]
for \(x\) where \(0 < x < 90\). | 248 | val |
100 | On a table, there are $100$ red and $k$ white buckets, all initially empty. In each move, a red and a white bucket are selected, and an equal amount of water is added to both. After a certain number of moves, no bucket is empty, and for every pair of buckets that were selected together at least once, the amount of wate... | 280 | train |
1 | Find all integers $n$ such that $n^4 + 8n + 11$ is a product of two or more consecutive integers. | 284 | train |
2 | Three identical circles are packed into a unit square. Each of the three circles is tangent to each other and tangent to at least one side of the square. If $r$ is the maximum possible radius of the circle, what is $(2-\frac{1}{r})^2$? | 286 | train |
52 | What are the last two digits of $2^{3^{4^{\cdots^{2019}}}}$? | 290 | train |
250 | A cross-pentomino is a shape that consists of a unit square and four other unit squares, each sharing a different edge with the first square. If a cross-pentomino is inscribed in a circle of radius $R$, what is $100R^2$? | 299 | test |
6 | How many distinct permutations of the letters in the word REDDER are there that do not contain a palindromic substring of length at least two?
(Note: A substring is a continuous block of letters that is part of the string. A string is palindromic if it is the same when read backwards.) | 304 | train |
217 | The digits of a positive integer $n$ are four consecutive integers in decreasing order when read from left to right. What is the sum of the possible remainders when $n$ is divided by $37$? | 311 | train |
2014 | Find the maximum number of permutations of the set $\{1, 2, 3, \ldots, 2014\}$ such that for every two different numbers $a$ and $b$ in this set, there exists at least one permutation where $b$ comes immediately after $a$. | 327 | train |
14499 | For a positive integer $n$, let $S(n)$ be the sum of its decimal digits. Determine the smallest positive integer $n$ for which $4 \cdot S(n) = 3 \cdot S(2n)$. | 339 | train |
625 | A spider is walking on the boundary of an equilateral triangle $\triangle{ABC}$, with vertices labeled in counterclockwise order, starting at vertex $A$. Each second, she moves to one of her two adjacent vertices with equal probability. The windiness of a path that starts and ends at $A$ is the net number of counterclo... | 353 | train |
30 | We take $100$ consecutive natural numbers $a_1, a_2, \ldots, a_{100}$. Determine the last two digits of the number $a_1^8 + a_2^8 + \ldots + a_{100}^8$. | 357 | train |
118 | Five girls and five boys randomly sit in ten seats that are equally spaced around a circle. The probability that there is at least one diameter of the circle with two girls sitting on opposite ends of the diameter is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | 360 | train |
84 | Regular hexagon $ABCDEF$ has side length $2$. Points $M$ and $N$ lie on $BC$ and $DE$, respectively. Find the minimum possible value of $(AM + MN + NA)^2$. | 379 | train |
27 | Suppose $w, x, y, z$ satisfy the following equations:
\[
\begin{align*}
w + x + y + z &= 25, \\
wx + wy + wz + xy + xz + yz &= 2y + 2z + 193
\end{align*}
\]
The largest possible value of $w$ can be expressed in lowest terms as $\frac{w_1}{w_2}$ for some integers $w_1, w_2 > 0$. Find $w_1 + w_2$. | 385 | train |
1033 | How many integers of the form $n^{2023-n}$ are perfect squares, where $n$ is a positive integer between $1$ and $2023$ inclusive? | 388 | train |
660 | The function $f(n)$ is defined on the positive integers and takes non-negative integer values. It is given that $f(2)=0$, $f(3)>0$, and $f(9999)=3333$. For all integers $m$ and $n$, the following condition holds:
\[
f(m+n)-f(m)-f(n)=0 \text{ or } 1.
\]
Determine the value of $f(1982)$. | 394 | val |
10 | Sets $A$, $B$, and $C$ satisfy the following conditions:
- $|A| = 92$
- $|B| = 35$
- $|C| = 63$
- $|A \cap B| = 16$
- $|A \cap C| = 51$
- $|B \cap C| = 19$
Compute the number of possible values of $|A \cap B \cap C|$. | 399 | train |
47 | Three spheres are all externally tangent to a plane and to each other. Suppose that the radii of these spheres are $6$, $8$, and $10$. The tangency points of these spheres with the plane form the vertices of a triangle. Determine the largest integer that is smaller than the perimeter of this triangle. | 410 | train |
87 | The sides $x$ and $y$ of a scalene triangle satisfy the equation $x + \frac{2\Delta}{x} = y + \frac{2\Delta}{y}$, where $\Delta$ is the area of the triangle. Given that $x = 60$ and $y = 63$, determine the length of the largest side of the triangle. | 411 | test |
5 | Determine the number of positive integral values of $p$ for which there exists a triangle with sides $a$, $b$, and $c$ that satisfy the equation: $$a^2 + (p^2 + 9)b^2 + 9c^2 - 6ab - 6pbc = 0.$$ | 431 | test |
2128 | A circular track with diameter $500$ is externally tangent at a point $A$ to a second circular track with diameter $1700$. Two runners start at point $A$ at the same time and run at the same speed. The first runner runs clockwise along the smaller track while the second runner runs clockwise along the larger track. The... | 450 | train |
1 | Let $n$ be an arbitrary positive integer. Calculate \( S_n = \sum_{r=0}^n 2^{r-2n} \binom{2n-r}{n} \). | 455 | train |
7 | Let $S$ be the set of all positive integers $n$ such that each of the numbers $n + 1$, $n + 3$, $n + 4$, $n + 5$, $n + 6$, and $n + 8$ is composite. Determine the largest integer $k$ with the following property: For each $n \in S$, there exist at least $k$ consecutive composite integers in the set \{$n, n + 1, n + 2, n... | 457 | train |
0 | Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(x + y) = f(x) + f(y) + f(xy)$ for all $x, y \in \mathbb{R}$. | 465 | val |
35 | Compute the smallest positive integer $x$ which satisfies the following congruences:
1. $x^2 - 8x + 1 \equiv 0 \pmod{22}$
2. $x^2 - 22x + 1 \equiv 0 \pmod{8}$ | 468 | train |
21 | Find the number of prime numbers $p$ between $100$ and $200$ for which the congruence equation $x^{11} + y^{16} \equiv 2013 \pmod{p}$ has a solution in integers $x$ and $y$. | 473 | train |
3 | Let $a$ be a positive number. Find the number of integer solutions $x$ of the inequality $2 < xa < 3$, given that the inequality $1 < xa < 2$ has exactly 3 integer solutions. Consider all possible cases. | 476 | train |
210 | The area of the region in the $xy$-plane satisfying the inequality
\[\min_{1 \le n \le 10} \max\left(\frac{x^2+y^2}{4n^2}, \, 2 - \frac{x^2+y^2}{4n^2-4n+1}\right) \le 1\]
is $k\pi$, for some integer $k$. Find $k$. | 487 | train |
511 | Let $x_1, x_2, \dots, x_n$ be a sequence of integers such that $-1 \leq x_i \leq 2$ for $i = 1, 2, \dots, n$. It is given that:
\[ x_1 + x_2 + \dots + x_n = 7 \]
and
\[ x_1^8 + x_2^8 + \dots + x_n^8 = 2009. \]
Let $m$ and $M$ be the minimal and maximal possible values of $x_1^9 + x_2^9 + \dots + x_n^9$, respectively. F... | 511 | test |
23 | Freddy, the king of flavortext, has an infinite chest of coins. For each number \( p \) in the interval \([0, 1]\), Freddy has a coin that has a probability \( p \) of coming up heads. Jenny the Joyous pulls out a random coin from the chest and flips it 10 times, and it comes up heads every time. She then flips the coi... | 518 | train |
5 | We select a real number $\alpha$ uniformly and at random from the interval $(0,500)$. Define
\[ S = \frac{1}{\alpha} \sum_{m=1}^{1000} \sum_{n=m}^{1000} \left\lfloor \frac{m+\alpha}{n} \right\rfloor. \]
Let $p$ denote the probability that $S \ge 1200$. Compute $1000p$. | 533 | train |
12 | Three vertices of a parallelogram are \((2, -4), (-2, 8),\) and \((12, 7)\). Determine the sum of the three possible x-coordinates of the fourth vertex. | 538 | val |
60 | Point $A$ lies on the circumference of a circle $\Omega$ with radius $78$. Point $B$ is placed such that $AB$ is tangent to the circle and $AB=65$, while point $C$ is located on $\Omega$ such that $BC=25$. Compute the length of $\overline{AC}$. | 542 | train |
8 | In a right trapezoid $ABCD$ where $AB \parallel CD$, the angle at vertex $B$ measures $75^\circ$. Point $H$ is the foot of the perpendicular from point $A$ to the line $BC$. If $BH = DC$ and $AD + AH = 8$, find the area of $ABCD$. | 586 | train |
1112 | Suppose $x$ is a random real number between $1$ and $4$, and $y$ is a random real number between $1$ and $9$. If the expected value of \[ \left\lceil \log_2 x \right\rceil - \left\lfloor \log_3 y \right\rfloor \] can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, compute $100m +... | 592 | val |
69 | Let $N$ denote the number of all natural numbers $n$ such that $n$ is divisible by a prime $p > \sqrt{n}$ and $p < 20$. What is the value of $N$? | 605 | train |
67 | Daniel had a string that formed the perimeter of a square with area $98$. Daniel cut the string into two pieces. With one piece, he formed the perimeter of a rectangle whose width and length are in the ratio $2 : 3$. With the other piece, he formed the perimeter of a rectangle whose width and length are in the ratio $3... | 608 | train |
67 | $p, q, r$ are real numbers satisfying the following equations:
\[ \frac{(p+q)(q+r)(r+p)}{pqr} = 24 \]
\[ \frac{(p-2q)(q-2r)(r-2p)}{pqr} = 10. \]
Given that \( \frac{p}{q} + \frac{q}{r} + \frac{r}{p} \) can be expressed in the form \( \frac{m}{n} \), where \( m, n \) are relatively prime positive integers, compute... | 624 | train |
1234 | Does there exist a four-digit positive integer with different non-zero digits, which has the following property: if we add the same number written in the reverse order, then we get a number divisible by $101$? | 657 | test |
1170 | Given that $x$ and $y$ are positive real numbers such that $\frac{5}{x} = \frac{y}{13} = \frac{x}{y}$, find the value of $x^3 + y^3$. | 696 | val |
48 | Consider the grid of points $X = \{(m,n) \mid 0 \leq m,n \leq 4 \}$. We say a pair of points $\{(a,b),(c,d)\}$ in $X$ is a knight-move pair if $(c = a \pm 2$ and $d = b \pm 1)$ or $(c = a \pm 1$ and $d = b \pm 2)$. The number of knight-move pairs in $X$ is: | 708 | train |
2 | Let $a_1$, $a_2$, $a_3$, $a_4$, $a_5$ be real numbers satisfying the following system of equations:
\[
\begin{align*}
2a_1 + a_2 + a_3 + a_4 + a_5 &= 1 + \frac{1}{8}a_4, \\
2a_2 + a_3 + a_4 + a_5 &= 2 + \frac{1}{4}a_3, \\
2a_3 + a_4 + a_5 &= 4 + \frac{1}{2}a_2, \\
2a_4 + a_5 &= 6 + a_1.
\end{align*}
\]
Compute ... | 720 | test |
1225 | How many ways are there to choose distinct positive integers $a, b, c, d$ dividing $15^6$ such that none of $a, b, c,$ or $d$ divide each other? (Order does not matter.) | 732 | train |
600 | Point $P$ is in the interior of $\triangle ABC$. The side lengths of $ABC$ are $AB = 7$, $BC = 8$, $CA = 9$. The three feet of perpendicular lines from $P$ to sides $BC$, $CA$, $AB$ are $D$, $E$, $F$ respectively. Suppose the minimal value of $\frac{BC}{PD} + \frac{CA}{PE} + \frac{AB}{PF}$ can be written as $\frac{a}{b... | 735 | train |
9765 | The **doubling sum** function is defined by:
\[ D(a,n) = a + 2a + 4a + 8a + \ldots \]
This sum contains \( n \) terms. For example, we have:
\[ D(5,3) = 5 + 10 + 20 = 35 \]
and
\[ D(11,5) = 11 + 22 + 44 + 88 + 176 = 341. \]
Determine the smallest positive integer \( n \) such that for every integer \( i \) between 1 a... | 736 | train |
23606 | Find the number of all unordered pairs $\{A,B \}$ of subsets of an $8$-element set, such that $A \cap B \neq \emptyset$ and $|A| \neq |B|$. | 747 | train |
23520 | Jonathan discovers that his ideal match is Sara Lark. To increase his chances of finding a girlfriend, he is open to dating any girl whose name is an anagram of "Sara Lark," provided that the name includes both a first and last name with at least one letter each. How many such anagrams are possible? | 749 | train |
17 | Determine the sum of all possible positive integers $n$, such that the product of its digits equals $n^2 - 15n - 27$. | 772 | train |
6 | Determine the smallest positive integer $M$ with the following property: For every choice of integers $a, b, c$, there exists a polynomial $P(x)$ with integer coefficients such that $P(1) = aM$, $P(2) = bM$, and $P(4) = cM$. | 794 | train |
128 | Let $S = \left\{ 1,2, \dots, 2014 \right\}$. Suppose that \[ \sum_{T \subseteq S} i^{\left\lvert T \right\rvert} = p + qi \] where $p$ and $q$ are integers, $i = \sqrt{-1}$, and the summation runs over all $2^{2014}$ subsets of $S$. Find the remainder when $\left\lvert p\right\rvert + \left\lvert q \right\rvert$ is div... | 795 | train |
15 | Let circle $O$ have radius $5$ with diameter $\overline{AE}$. Point $F$ is outside circle $O$ such that lines $\overline{FA}$ and $\overline{FE}$ intersect circle $O$ at points $B$ and $D$, respectively. If $FA = 10$ and $m \angle FAE = 30^\circ$, then the perimeter of quadrilateral $ABDE$ can be expressed as $a + b\sq... | 811 | train |
5 | Benji has a $2 \times 2$ grid, which he proceeds to place chips on. One by one, he places a chip on one of the unit squares of the grid at random. However, if at any point there is more than one chip on the same square, Benji moves two chips on that square to the two adjacent squares, which he calls a chip-fire. He kee... | 833 | test |
640 | Let $\ell$ be a line with negative slope passing through the point $(20,16)$. What is the minimum possible area of a triangle that is bounded by the $x$-axis, $y$-axis, and $\ell$? | 836 | train |
30 | 100 schoolchildren participated in the Mathematical Olympiad where 4 tasks were given. The first problem was solved by 90 students, the second by 80, the third by 70, and the fourth by 60. No student solved all the problems. Students who solved both the third and fourth questions received an award. How many students we... | 862 | train |
540 | Compute the sum of the two smallest positive integers $b$ with the following property: there are at least ten integers $0 \leq n < b$ such that $n^2$ and $n$ end in the same digit in base $b$. | 868 | val |
72 | Let $A = \frac{1}{6}((\log_2(3))^3-(\log_2(6))^3-(\log_2(12))^3+(\log_2(24))^3)$. Compute $2^A$. | 872 | test |
2013 | While taking the SAT, you become distracted by your own answer sheet. Because you are not bound to the College Board's limiting rules, you realize that there are actually $32$ ways to mark your answer for each question, because you could fight the system and bubble in multiple letters at once: for example, you could ma... | 884 | train |
401 | The numbers in the sequence 101, 104, 109, 116, $\dots$ are of the form $a_n = 100 + n^2$, where $n = 1, 2, 3, \dots$. For each $n$, let $d_n$ be the greatest common divisor of $a_n$ and $a_{n + 1}$. Find the maximum value of $d_n$ as $n$ ranges through the positive integers. | 900 | val |
6 | A sequence $\{a_n\}$ satisfies the following conditions:
- $a_0 = 0$
- $a_1 = a_2 = 1$
- $a_{3n} = a_n$
- $a_{3n+1} = a_{3n+2} = a_n + 1$
for all $n \geq 1$. Compute $a_{2021}$. | 902 | train |
250 | Euler's formula states that for a convex polyhedron with $V$ vertices, $E$ edges, and $F$ faces, $V - E + F = 2$. A particular convex polyhedron has 32 faces, each of which is either a triangle or a pentagon. At each of its $V$ vertices, $T$ triangular faces and $P$ pentagonal faces meet. What is the value of $100P + 1... | 903 | train |
251 | Let $ABC$ be a triangle with side lengths $AB=6$, $AC=7$, and $BC=8$. Let $H$ be the orthocenter of $\triangle ABC$ and $H'$ be the reflection of $H$ across the midpoint $M$ of $BC$. The expression $\frac{[ABH']}{[ACH']}$ can be written as $\frac{p}{q}$. Find $p+q$. | 927 | train |
3225 | Let $ABCD$ be a square with side length $2$. Let $M$ and $N$ be the midpoints of $\overline{BC}$ and $\overline{CD}$ respectively, and let $X$ and $Y$ be the feet of the perpendiculars from $A$ to $\overline{MD}$ and $\overline{NB}$, also respectively. The square of the length of segment $\overline{XY}$ can be written ... | 937 | test |
210 | We want to paint some identically-sized cubes so that each face of each cube is painted a solid color, and each cube is painted with six different colors. If we have seven different colors to choose from, how many distinguishable cubes can we produce? | 970 | train |
186 | Find the sum of all two-digit prime numbers whose digits are also both prime numbers. | 977 | train |
56 | Jerry and Hannah Kubik live in Jupiter Falls with their five children. Jerry works as a Renewable Energy Engineer for the Southern Company, and Hannah runs a lab at Jupiter Falls University where she researches biomass (renewable fuel) conversion rates. Michael is their oldest child, and Wendy their oldest daughter. To... | 993 | train |
2 | For an integer $k$, let $T_k$ denote the number of $k$-tuples of integers $(x_1, x_2, \ldots, x_k)$ with $0 \le x_i < 73$ for each $i$, such that $73 \mid x_1^2 + x_2^2 + \ldots + x_k^2 - 1$. Compute the remainder when $T_1 + T_2 + \ldots + T_{2017}$ is divided by $2017$. | 1,002 | test |
2046 | A particle is moving randomly around a plane. It starts at $(0,0)$. Every second, it moves one unit randomly in a direction parallel to the $x$ or $y$ axis. At some time in the first hour, the particle was at the point $(2023,23)$. After $4092$ seconds, the particle is at $(x,y)$. Find the expected value of $x+y$. | 1,006 | train |
40 | Find $x$ such that $2^{2^{3^{2^{2}}}} = 4^{4^{x}}$. | 1,010 | train |
33 | In rectangle $ABCD$, point $M$ is the midpoint of $AB$, and $P$ is a point on side $BC$. The perpendicular bisector of $MP$ intersects side $DA$ at point $X$. Given that $AB = 33$ and $BC = 56$, find the least possible value of $MX$. | 1,022 | train |
115 | Points $A$, $B$, $C$, and $D$ lie on a circle such that chords $\overline{AC}$ and $\overline{BD}$ intersect at a point $E$ inside the circle. Suppose that $\angle ADE = \angle CBE = 75^\circ$, $BE=4$, and $DE=8$. The value of $AB^2$ can be written in the form $a+b\sqrt{c}$ for positive integers $a$, $b$, and $c$ such ... | 1,026 | test |
12 | A right triangle $ABC$ is inscribed in the circular base of a cone. If two of the side lengths of $ABC$ are $3$ and $4$, and the distance from the vertex of the cone to any point on the circumference of the base is $3$, then the minimum possible volume of the cone can be written as $\frac{m\pi\sqrt{n}}{p}$, where $m$, ... | 1,034 | train |
6 | Let $E$ and $F$ be the respective midpoints of $BC$ and $CD$ in a convex quadrilateral $ABCD$. Segments $AE$, $AF$, and $EF$ divide the quadrilateral into four triangles whose areas are four consecutive integers. Find the maximum possible area of $\Delta BAD$. | 1,039 | train |
816 | Find the number of ways a series of $+$ and $-$ signs can be inserted between the numbers $0, 1, 2, \ldots, 12$ such that the value of the resulting expression is divisible by 5. | 1,053 | train |
4028 | The USAMO is a $6$ question test. For each question, you submit a positive integer number $p$ of pages on which your solution is written. On the $i$th page of this question, you write the fraction $\frac{i}{p}$ to denote that this is the $i$th page out of $p$ for this question. When you turned in your submissions for t... | 1,061 | train |
109 | Compute the least possible value of $ABCD - AB \times CD$, where $ABCD$ is a 4-digit positive integer, and $AB$ and $CD$ are 2-digit positive integers. Here $A$, $B$, $C$, and $D$ are digits, possibly equal. Neither $A$ nor $C$ can be zero. | 1,078 | train |
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