prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
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Let $ABC$ be a triangle with $AB=20$, $AC=34$, and $BC=42$. Let $\omega_1$ and $\omega_2$ be the semicircles with diameters $\overline{AB}$ and $\overline{AC}$ erected outwards of $\triangle ABC$. Denote by $\ell$ the common external tangent to $\omega_1$ and $\omega_2$. The line through $A$ perpendicular to $\overline... | 962 |
A four-digit positive integer is called *virtual* if it has the form $\overline{abab}$, where $a$ and $b$ are digits and $a \neq 0$. For example, 2020, 2121, and 2222 are virtual numbers, while 2002 and 0202 are not. Find all virtual numbers of the form $n^2+1$, for some positive integer $n$. | 8282 |
How many ordered four-tuples of integers \((a, b, c, d)\) with \(0 < a < b < c < d < 500\) satisfy the conditions: 1. \(a + d = b + c\) 2. \(bc - ad = 93\) | 870 |
Let $a, b, c, d, e$ be real numbers satisfying the following conditions: \[ \begin{align*} & a \le b \le c \le d \le e, \\ & a + e = 1, \\ & b + c + d = 3, \\ & a^2 + b^2 + c^2 + d^2 + e^2 = 14 \end{align*} \] Determine the maximum possible value of $ae$. | -2 |
Let $d(m)$ denote the number of positive integer divisors of a positive integer $m$. If $r$ is the number of integers $n \leq 2023$ for which $\sum_{i=1}^{n} d(i)$ is odd, find the sum of the digits of $r$. | 18 |
Compute the number of positive integers $n \leq 1000$ such that $\text{lcm}(n, 9)$ is a perfect square. (Recall that $\text{lcm}$ denotes the least common multiple.) | 43 |
Let $1 \leq a, b, c, d, e, f, g, h, i \leq 9$ be distinct integers. Denote by $N$ the maximum value of $a \times b \times c$, $d \times e \times f$, $g \times h \times i$. Find the possible minimum value of $N$. | 72 |
Let $a,b$ be positive integers satisfying $$\sqrt{\frac{ab}{2b^2-a}}=\frac{a+2b}{4b}$$. Find $|10(a-5)(b-15)|+8$. | 2018 |
Sinclair starts with the number $1$. Every minute, he either squares his number or adds $1$ to his number, both with equal probability. What is the expected number of minutes until his number is divisible by $3$? | 6 |
Let $a_1, a_2, \dots, a_{1000}$ be positive integers whose sum is $S$. If $a_n!$ divides $n$ for each $n = 1, 2, \dots, 1000$, compute the maximum possible value of $S$. | 1716 |
\(|5x^2 - \frac{2}{5}| \le |x - 8|\) if and only if \(x\) is in the interval \([a, b]\). There are relatively prime positive integers \(m\) and \(n\) such that \(b - a = \frac{m}{n}\). Find \(m + n\). | 18 |
The fraction $\frac{1}{10}$ can be expressed as the sum of two unit fractions in many ways, for example, $\frac{1}{30} + \frac{1}{15}$ and $\frac{1}{60} + \frac{1}{12}$. Find the number of ways that $\frac{1}{2007}$ can be expressed as the sum of two distinct positive unit fractions. | 7 |
How many pairs of integers \((a,b)\) are there such that \(0 \leq a < b \leq 100\) and such that \(\frac{2^b-2^a}{2016}\) is an integer? | 720 |
For an ordered $10$-tuple of nonnegative integers $a_1,a_2,\ldots, a_{10}$, we denote: $$f(a_1,a_2,\ldots,a_{10})=\left(\prod_{i=1}^{10} {\binom{20-(a_1+a_2+\cdots+a_{i-1})}{a_i}}\right) \cdot \left(\sum_{i=1}^{10} {\binom{18+i}{19}}a_i\right).$$ When $i=1$, we take $a_1+a_2+\cdots+a_{i-1}$ to be $0$. Let $N$ be the av... | 462 |
For a function $f: \mathbb{R} \to \mathbb{R}$, it is known that $f(2017) > 0$ and $f(x^2 + yf(z)) = xf(x) + zf(y)$ for all $x, y, z \in \mathbb{R}$. What is the value of $f(-42)$? | -42 |
22 football players took part in the football training. They were divided into teams of equal size for each game (11 vs 11). It is known that each football player played with each other at least once in opposing teams. What is the smallest possible number of games they played during the training? | 5 |
For nonnegative integer $n$, the following are true: - $f(0) = 0$ - $f(1) = 1$ - $f(n) = f\left(n - \frac{m(m-1)}{2}\right) - f\left(\frac{m(m+1)}{2} - n\right)$ for integer $m$ satisfying $m \geq 2$ and $\frac{m(m-1)}{2} < n \leq \frac{m(m+1)}{2}$. Find the smallest integer $n$ such that $f(n) = 4$. | 646 |
Define the binary operation $a \Delta b = ab + a - 1$. Compute the following expression: \[ 10 \Delta (10 \Delta (10 \Delta (10 \Delta (10 \Delta (10 \Delta (10 \Delta (10 \Delta (10 \Delta 10)))))))) \] where $10$ is written $10$ times. | 10999999999 |
Let $ABCD$ be a regular tetrahedron with side length $12$. Select points $E$, $F$, $G$, $H$ on $AC$, $BC$, $BD$, and $AD$, respectively, such that $AE = BF = BG = AH = 3$. Compute the area of quadrilateral $EFGH$. | 27 |
If $\alpha$, $\beta$, and $\gamma$ are the roots of the equation $x^3 - x - 1 = 0$, compute the expression $\frac{1+\alpha}{1-\alpha} + \frac{1+\beta}{1-\beta} + \frac{1+\gamma}{1-\gamma}$. | -7 |
The side lengths of a trapezoid are $\sqrt[4]{3}$, $\sqrt[4]{3}$, $\sqrt[4]{3}$, and $2 \cdot \sqrt[4]{3}$. Its area is expressed as a ratio of two relatively prime positive integers, $m$ and $n$. Find $m + n$. | 13 |
Acute triangle $ABC$ satisfies $AB=2AC$ and $AB^4+BC^4+CA^4 = 2006 \cdot 10^{10}$. A tetrahedron $DEFP$ is formed by choosing points $D$, $E$, and $F$ on the segments $BC$, $CA$, and $AB$ (respectively) and folding $A$, $B$, $C$ over $EF$, $FD$, and $DE$ (respectively) to the common point $P$. Let $R$ denote the circum... | 6 |
In a sequence of numbers, a term is called "golden" if it is divisible by the term immediately before it. What is the maximum possible number of golden terms in a permutation of $1, 2, 3, \ldots, 2021$? | 1010 |
The expression \( \frac{(1+2+\cdots + 10)(1^3+2^3+\cdots + 10^3)}{(1^2+2^2+\cdots + 10^2)^2} \) reduces to \( \frac{m}{n} \), where \( m \) and \( n \) are relatively prime positive integers. Find \( m+n \). | 104 |
Let $\triangle ABC$ be a right triangle with right angle $C$. Let $I$ be the incenter of $\triangle ABC$, and let $M$ lie on $AC$ and $N$ on $BC$, respectively, such that $M, I, N$ are collinear and $\overline{MN}$ is parallel to $AB$. If $AB = 36$ and the perimeter of $\triangle CMN$ is $48$, find the area of $\triang... | 252 |
How many positive integers less than 500 have exactly 15 positive integer factors? | 3 |
A sequence of vertices $v_1, v_2, \ldots, v_k$ in a graph, where $v_i = v_j$ only if $i = j$ and $k$ can be any positive integer, is called a \textit{cycle} if $v_1$ is attached by an edge to $v_2$, $v_2$ to $v_3$, and so on to $v_k$ connected to $v_1$. Rotations and reflections are distinct: $A, B, C$ is distinct from... | 1001 |
For every $n \in \mathbb{N}$, let $d(n)$ denote the sum of digits of $n$. It is easy to see that the sequence $d(n), d(d(n)), d(d(d(n))), \ldots$ will eventually become a constant integer between $1$ and $9$ (both inclusive). This number is called the digital root of $n$, denoted by $b(n)$. Determine for how many natur... | 444 |
There are real numbers $x$, $y$, and $z$ such that the value of \(x + y + z - \left(\frac{x^2}{5} + \frac{y^2}{6} + \frac{z^2}{7}\right)\) reaches its maximum of \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Find \(m + n + x + y + z\). | 20 |
In triangle $\triangle ABC$, point $D$ lies between $A$ and $C$ such that $AC = 3AD$. Point $E$ lies between $B$ and $C$ such that $BC = 4EC$. Points $B$, $G$, $F$, and $D$ are on a straight line in that order, and $BD = 5GF = 5FD$. Given that the area of $\triangle ABC$ is $900$, find the area of triangle $\triangle E... | 90 |
There exist real numbers $a$, $b$, $c$, $d$, and $e$ such that for all positive integers $n$, we have: \[ \sqrt{n} = \sum_{i=0}^{n-1} \sqrt[5]{\sqrt{ai^5 + bi^4 + ci^3 + di^2 + ei + 1} - \sqrt{ai^5 + bi^4 + ci^3 + di^2 + ei}}. \] Find the sum $a + b + c + d$. | 1656 |
A **binary palindrome** is a positive integer whose standard base 2 (binary) representation is a palindrome (reads the same backward or forward). Leading zeroes are not permitted in the standard representation. For example, 2015 is a binary palindrome because in base 2 it is \(11111011111\). How many positive integers ... | 92 |
Alexis and Joshua are walking along the beach when they decide to draw symbols in the sand. Alex draws only stars and only draws them in pairs, while Joshua draws only squares in trios. "Let's see how many rows of $15$ adjacent symbols we can make this way," suggests Josh. Alexis is game for the task and the two get bu... | 28 |
Let $a_1, a_2, \ldots, a_{2017}$ be the 2017 distinct complex numbers which satisfy $a_i^{2017} = a_i + 1$ for $i = 1, 2, \ldots, 2017$. Compute $$\sum_{i=1}^{2017} \frac{a_i}{a_i^2 + 1}.$$ | 2016 |
A function $f(x)$ satisfies $f(x) = f\left(\frac{c}{x}\right)$ for some real number $c > 1$ and all positive numbers $x$. If $\int_1^{\sqrt{c}} \frac{f(x)}{x} \, dx = 3$, evaluate $\int_1^c \frac{f(x)}{x} \, dx$. | 6 |
Find the smallest positive integer $m$ satisfying the following condition: for all prime numbers $p$ such that $p > 3$, we have $105 \mid 9^{p^2} - 29^p + m$. | 20 |
Let $P = (-2, 0)$. Points $P$, $Q$, $R$ lie on the graph of the function $y = x^3 - 3x + 2$ such that $Q$ is the midpoint of segment $PR$. Compute $PR^2$. | 32 |
Evaluate the expression: \[ \frac{121 \left( \frac{1}{13} - \frac{1}{17} \right) + 169 \left( \frac{1}{17} - \frac{1}{11} \right) + 289 \left( \frac{1}{11} - \frac{1}{13} \right)}{ 11 \left( \frac{1}{13} - \frac{1}{17} \right) + 13 \left( \frac{1}{17} - \frac{1}{11} \right) + 17 \left( \frac{1}{11} - \frac{1}{13} \righ... | 41 |
Frist Campus Center is located $1$ mile north and $1$ mile west of Fine Hall. The area within $5$ miles of Fine Hall that is located north and east of Frist can be expressed in the form $\frac{a}{b} \pi - c$, where $a$, $b$, $c$ are positive integers and $a$ and $b$ are relatively prime. Find $a + b + c$. | 30 |
For real numbers $x$ and $y$, let $M$ be the maximum value of the expression $x^4y + x^3y + x^2y + xy + xy^2 + xy^3 + xy^4$, subject to $x + y = 3$. Find $\lfloor M \rfloor$. | 36 |
As shown in the picture, the triangles $ABC$, $ADE$, and $EFG$ are all isosceles right triangles, such that $\angle ACB = \angle AED = \angle EGF = 90^{\circ}$. Suppose the area of the pentagon $ABDGE$ is $23$, and $AB = 8$, with $FD > DG$. Find the length of the segment $FD$. | 3 |
On a $5 \times 5$ board, pieces made up of $4$ squares are placed, as seen in the figure, each covering exactly $4$ squares of the board. The pieces can be rotated or turned over. They can also overlap, but they cannot protrude from the board. Suppose that each square on the board is covered by at most two pieces. Find... | 24 |
On a circle, 2022 points are chosen such that the distance between any two adjacent points is the same. There are $k$ arcs, each having endpoints on chosen points, with different lengths. The arcs do not contain each other. What is the maximum possible number of $k$? | 1011 |
The equation $x^3 - 9x^2 + 8x + 2 = 0$ has three real roots $p$, $q$, $r$. Find $\frac{1}{p^2} + \frac{1}{q^2} + \frac{1}{r^2}$. | 25 |
How many 3-digit positive integers have the property that the sum of their digits is greater than the product of their digits? | 199 |
A math contest consists of $9$ objective type questions and $6$ fill in the blanks questions. From a school, some number of students took the test, and it was noticed that all students had attempted exactly $14$ out of $15$ questions. Let $O_1, O_2, \dots , O_9$ be the nine objective questions and $F_1, F_2, \dots , F_... | 21 |
Sofia has forgotten the passcode of her phone. She only remembers that it has four digits and that the product of its digits is $18$. How many passcodes satisfy these conditions? | 36 |
Let $S$ be the sum of all the real coefficients of the expansion of $(1+ix)^{2009}$. What is $\log_2(S)$? | 1004 |
Find the largest prime factor of $8001$. | 127 |
How many positive integers less than $1998$ are relatively prime to $1547$? Two integers are relatively prime if they have no common factors besides 1. | 1487 |
What is the largest $n$ such that there exists a non-degenerate convex $n$-gon where each of its angles is an integer number of degrees, and all angles are distinct? | 26 |
In a simple graph, there are two disjoint sets of vertices $A$ and $B$ where $A$ has $k$ vertices and $B$ has $2016$ vertices. Four numbers are written on each vertex using the colors red, green, blue, and black. Initially, there are no edges. For each vertex in $A$, a color is chosen, and edges are drawn from this ver... | 6 |
Compute the number of ways to completely fill a $3 \times 15$ rectangle with non-overlapping $1 \times 3$ rectangles. | 189 |
Compute the number of subsets $S$ with at least two elements of $\{2^2, 3^3, \dots, 216^{216}\}$ such that the product of the elements of $S$ has exactly $216$ positive divisors. | 8 |
A caretaker is giving candy to his two babies. Every minute, he gives a candy to one of his two babies at random. The five possible moods for the babies to be in, from saddest to happiest, are "upset," "sad," "okay," "happy," and "delighted." A baby gets happier by one mood when they get a candy and gets sadder by one ... | 337 |
For positive real numbers $p$ and $q$, define the remainder when dividing $p$ by $q$ as the smallest nonnegative real $r$ such that $\frac{p-r}{q}$ is an integer. For an ordered pair $(a, b)$ of positive integers, let $r_1$ and $r_2$ be the remainders when $a\sqrt{2} + b\sqrt{3}$ is divided by $\sqrt{2}$ and $\sqrt{3}$... | 16 |
Define a number to be \( \text{boring} \) if all the digits of the number are the same. How many positive integers less than \( 10000 \) are both prime and boring? | 5 |
Let $T = TNFTPP$. As $n$ ranges over the integers, the expression $n^4 - 898n^2 + T - 2160$ evaluates to just one prime number. Find this prime. Note: For those who want to try the problem separately, here's the value of $T$: $T=2161$. | 1801 |
Ava and Tiffany participate in a knockout tournament consisting of a total of $32$ players. In each of the $5$ rounds, the remaining players are paired uniformly at random. In each pair, both players are equally likely to win, and the loser is knocked out of the tournament. The probability that Ava and Tiffany play eac... | 116 |
Suppose that \[ \prod_{n=1}^{\infty}\left(\frac{1+i\cot\left(\frac{n\pi}{2n+1}\right)}{1-i\cot\left(\frac{n\pi}{2n+1}\right)}\right)^{\frac{1}{n}} = \left(\frac{p}{q}\right)^{i \pi}, \] where $p$ and $q$ are relatively prime positive integers. Find $p+q$. | 5 |
Let $a, b, c$ be the roots of the polynomial $x^3 - 20x^2 + 22$. Find \[ \frac{bc}{a^2} + \frac{ac}{b^2} + \frac{ab}{c^2}. \] | -3 |
Isosceles triangle $\triangle ABC$ has $\angle ABC = \angle ACB = 72^\circ$ and $BC = 1$. If the angle bisector of $\angle ABC$ meets $AC$ at $D$, what is the positive difference between the perimeters of $\triangle ABD$ and $\triangle BCD$? | 1 |
Find the least prime factor of the number \( \frac{2016^{2016}-3}{3} \). | 11 |
Consider the sum: \[ S_n = \sum_{k = 1}^n \frac{1}{\sqrt{2k-1}} \] Determine \( \lfloor S_{4901} \rfloor \). Recall that if \( x \) is a real number, then \( \lfloor x \rfloor \) (the floor of \( x \)) is the greatest integer that is less than or equal to \( x \). | 98 |
Let $\mathcal{P}$ be a parallelepiped with side lengths $x$, $y$, and $z$. Suppose that the four space diagonals of $\mathcal{P}$ have lengths $15$, $17$, $21$, and $23$. Compute $x^2 + y^2 + z^2$. | 371 |
Alice and Bob live on the same road. At time $t$, they both decide to walk to each other's houses at constant speed. However, they were busy thinking about math so that they didn't realize passing each other. Alice arrived at Bob's house at $3:19\text{pm}$, and Bob arrived at Alice's house at $3:29\text{pm}$. Charlie, ... | 179 |
Let $AXYZB$ be a convex pentagon inscribed in a semicircle with diameter $AB$. Suppose that $AZ - AX = 6$, $BX - BZ = 9$, $AY = 12$, and $BY = 5$. Find the greatest integer not exceeding the perimeter of quadrilateral $OXYZ$, where $O$ is the midpoint of $AB$. | 23 |
For primes $a$, $b$, and $c$ that satisfy the following conditions, calculate $abc$: 1. $b + 8$ is a multiple of $a$. 2. $b^2 - 1$ is a multiple of both $a$ and $c$. 3. $b + c = a^2 - 1$. | 2009 |
Let $ABCDEF$ be a regular hexagon with side length $6$, and construct squares $ABGH$, $BCIJ$, $CDKL$, $DEMN$, $EFOP$, and $FAQR$ outside the hexagon. Find the perimeter of the dodecagon $HGJILKNMPORQ$. | 72 |
The remainder of $1991^{2000}$ modulo $10^6$ is \underline{\hspace{2cm}}. | 1 |
Let $x$ and $y$ be positive real numbers and $\theta$ an angle such that $\theta \neq \frac{\pi}{2}n$ for any integer $n$. Suppose: \[ \frac{\sin\theta}{x}=\frac{\cos\theta}{y} \] and \[ \frac{\cos^4 \theta}{x^4}+\frac{\sin^4\theta}{y^4}=\frac{97\sin2\theta}{x^3y+y^3x}. \] Compute $\frac{x}{y}+\frac{y}{x}$. | 4 |
Find the greatest integer less than or equal to \( \sum_{k=1}^{2^{1983}} k^{\frac{1}{1983} -1} \). | 1983 |
A large collection of congruent right triangles is given, each with side lengths $3$, $4$, and $5$. Find the maximal number of such triangles you can place inside a $20 \times 20$ square, with no two triangles intersecting in their interiors. | 66 |
At the 4 PM show, all the seats in the theater were taken, and 65\% of the audience was children. At the 6 PM show, again, all the seats were taken, but this time only 50\% of the audience was children. Of all the people who attended either of the shows, 57\% were children, although there were 12 adults and 28 children... | 520 |
In triangle $ABC$, where $AB=2$, $AC=1+\sqrt{5}$, and $\angle CAB=54^{\circ}$, suppose point $D$ lies on the extension of $AC$ through $C$ such that $CD=\sqrt{5}-1$. If $M$ is the midpoint of $BD$, determine the measure of $\angle ACM$ in degrees. | 63 |
For any integer $n \ge 2$, let $b_n$ be the least positive integer such that, for any integer $N$, $m$ divides $N$ whenever $m$ divides the digit sum of $N$ written in base $b_n$, for $2 \le m \le n$. Find the integer nearest to $\frac{b_{36}}{b_{25}}$. | 5394 |
Determine the minimum value of the expression $2x^4 - 2x^2y^2 + y^4 - 8x^2 + 18$ where $x, y \in \mathbb{R}$. | 2 |
Compute the number of values of $x$ in the interval $[-11\pi, -2\pi]$ that satisfy \[ \frac{5\cos(x) + 4}{5\sin(x) + 3} = 0. \] | 4 |
A rhombus with side length $1$ has an inscribed circle with radius $\frac{1}{3}$. If the area of the rhombus can be expressed as $\frac{a}{b}$ for relatively prime, positive integers $a$ and $b$, evaluate $a+b$. | 5 |
Given that $\frac{a}{c}=\frac{b}{d}=\frac{3}{4}$ and $\sqrt{a^2+c^2}-\sqrt{b^2+d^2}=15$, find the value of $ac+bd-ad-bc$. | 108 |
What is the value of \( \sum_{1 \le i < j \le 10}(i+j) \) for odd \( i+j \) minus \( \sum_{1 \le i < j \le 10}(i+j) \) for even \( i+j \)? | 55 |
Using only the digits $2$, $3$, and $9$, how many six-digit numbers can be formed that are divisible by $6$? | 81 |
For a positive integer $k$, let $d(k)$ denote the number of divisors of $k$, and let $s(k)$ denote the digit sum of $k$. A positive integer $n$ is said to be amusing if there exists a positive integer $k$ such that $d(k) = s(k) = n$. What is the smallest amusing odd integer greater than 1? | 9 |
Let $D, E, F$ be points on the sides $BC, CA, AB$ of a triangle $ABC$, respectively. Suppose $AD, BE, CF$ are concurrent at $P$. If $\frac{PF}{PC} = \frac{2}{3}$, $\frac{PE}{PB} = \frac{2}{7}$ and $\frac{PD}{PA} = \frac{m}{n}$, where $m, n$ are positive integers with $\gcd(m, n) = 1$, find $m + n$. | 45 |
In an acute triangle $ABC$, the lines tangent to the circumcircle of $ABC$ at $A$ and $B$ intersect at point $D$. Let $E$ and $F$ be points on $CA$ and $CB$ such that $DECF$ forms a parallelogram. Given that $AB = 20$, $CA=25$, and $\tan C = \frac{4\sqrt{21}}{17}$, the value of $EF$ may be expressed as $\frac{m}{n}$ fo... | 267 |
Let $D$, $E$, and $F$ be the feet of the altitudes from vertices $A$, $B$, and $C$ of an acute triangle $\triangle ABC$, respectively. Given that $AF = 28$, $FB = 35$, and $BD = 45$. Let $P$ be a point on segment $BE$ such that $AP = 42$. Determine the length of $CP$. | 14 |
fantasticbobob is proctoring a room for the SiSiEyMB with $841$ seats arranged in $29$ rows and $29$ columns. The contestants sit down, take part 1 of the contest, go outside for a break, and come back to take part 2 of the contest. fantasticbobob sits among the contestants during part 1, also goes outside during the b... | 421 |
Mark and William are playing a game with a stored value. On his turn, a player may either multiply the stored value by $2$ and add $1$ or he may multiply the stored value by $4$ and add $3$. The first player to make the stored value exceed $2^{100}$ wins. The stored value starts at $1$ and Mark goes first. Assuming bot... | 33 |
Let $(a,b,c,d)$ be a solution to the system: \[ \begin{align*} a + b &= 15,\\ ab + c + d &= 78,\\ ad + bc &= 160,\\ cd &= 96. \end{align*} \] Find the greatest possible value of $a^2 + b^2 + c^2 + d^2$. | 717 |
Let $ABCD$ be a square. Let $E$ and $Z$ be points on the sides $AB$ and $CD$ of the square, respectively, such that $DE \parallel BZ$. Assume that the triangles $\triangle EAD$, $\triangle ZCB$, and the parallelogram $BEDZ$ have the same area. If the distance between the parallel lines $DE$ and $BZ$ is equal to $1$, de... | 13 |
We have a group of $n$ kids. For each pair of kids, at least one has sent a message to the other one. For each kid $A$, among the kids to whom $A$ has sent a message, exactly $25\%$ have sent a message to $A$. How many possible two-digit values of $n$ are there? | 26 |
Find the greatest positive integer $A$ with the following property: For every permutation of \{1001, 1002, \ldots, 2000\}, the sum of some ten consecutive terms is greater than or equal to $A$. | 15005 |
The numbers $1, 2, \ldots, 49, 50$ are written on the blackboard. Ann performs the following operation: she chooses three arbitrary numbers $a, b, c$ from the board, replaces them by their sum $a + b + c$, and writes $(a + b)(b + c)(c + a)$ in her notebook. Ann performs such operations until only two numbers remain on ... | 4 |
Balls of $3$ colours — red, blue, and white — are placed in two boxes. If you take out $3$ balls from the first box, there would definitely be a blue one among them. If you take out $4$ balls from the second box, there would definitely be a red one among them. If you take out any $5$ balls (only from the first, only fr... | 9 |
King William is located at $(1, 1)$ on the coordinate plane. Every day, he chooses one of the eight lattice points closest to him and moves to one of them with equal probability. When he exits the region bounded by the $x, y$ axes and $x+y = 4$, he stops moving and remains there forever. Given that after an arbitrarily... | 65 |
There are $14$ players participating in a chess tournament, each playing one game with every other player. After the tournament, players are ranked in descending order based on their points. The sum of the points of the first three players is equal to the sum of the points of the last nine players. What is the highest ... | 40 |
In how many ways can we change the sign $*$ with $+$ or $-$, such that the following equation is true? \[ 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 = 29 \] | 8 |
Determine the largest odd positive integer $n$ such that every odd integer $k$ with $1 < k < n$ and $\gcd(k, n) = 1$ is a prime. | 105 |
Let $A = \{1,2,3,4,5,6,7,8\}$, $B = \{9,10,11,12,13,14,15,16\}$, and $C =\{17,18,19,20,21,22,23,24\}$. Find the number of triples $(x, y, z)$ such that $x \in A$, $y \in B$, $z \in C$, and $x + y + z = 36$. | 46 |
Jeffrey rolls three fair six-sided dice and records their results. The probability that the mean of these three numbers is greater than the median of these three numbers can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. | 101 |
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