prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
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Suppose that $N$ is a three-digit number divisible by $7$ such that upon removing its middle digit, the remaining two-digit number is also divisible by $7$. What is the minimum possible value of $N$? | 154 |
Let $a$ and $b$ be real numbers such that \[\begin{cases} a^6 - 3a^2b^4 = 3 \\ b^6 - 3a^4b^2 = 3\sqrt{2}.\end{cases}\] What is the value of $a^4 + b^4$? | 3 |
The number $123454321$ is written on a blackboard. Evan walks by and erases some (but not all) of the digits, and notices that the resulting number (when spaces are removed) is divisible by $9$. What is the fewest number of digits he could have erased? | 2 |
Numbers $\frac{49}{1}, \frac{49}{2}, \ldots , \frac{49}{97}$ are written on a blackboard. Each time, we can replace two numbers (like $a, b$) with $2ab-a-b+1$. After $96$ times doing that action, one number will be left on the board. Find all the possible values for that number. | 1 |
Point $P_1$ is located 600 miles west of point $P_2$. At 7:00 AM, a car departs from $P_1$ and drives east at a speed of 50 mph. At 8:00 AM, another car departs from $P_2$ and drives west at a constant speed of $x$ miles per hour. If the cars meet each other exactly halfway between $P_1$ and $P_2$, what is the value of... | 60 |
There are 1000 rooms in a row along a long corridor. Initially, the first room contains 1000 people, and the remaining rooms are empty. Each minute, the following happens: for each room containing more than one person, someone in that room decides it is too crowded and moves to the next room. All these movements are si... | 31 |
For any positive integer $n$, define a function $f$ by \[f(n)=2n+1-2^{\lfloor\log_2n\rfloor+1}.\] Let $f^m$ denote the function $f$ applied $m$ times. Determine the number of integers $n$ between $1$ and $65535$ inclusive such that $f^n(n)=f^{2015}(2015)." | 8008 |
How many ways can you roll three 20-sided dice such that the sum of the three rolls is exactly $42$? Here the order of the rolls matters. | 190 |
What is the largest integer less than or equal to \( \frac{3^{31}+2^{31}}{3^{29}+2^{29}} \) ? | 8 |
Find the maximal value of $(x^3+1)(y^3+1)$, where $x, y \in \mathbb{R}$, $x+y=1$. | 4 |
If $MATH + WITH = GIRLS$, compute the smallest possible value of $GIRLS$. Here $MATH$ and $WITH$ are 4-digit numbers and $GIRLS$ is a 5-digit number (all with nonzero leading digits). Different letters represent different digits. | 10358 |
Let $S$ be a set of sequences of length 15 formed by using the letters $a$ and $b$, such that every pair of sequences in $S$ differ in at least 3 places. What is the maximum number of sequences in $S$? | 2048 |
Let triangle $ABC$ have $\angle BAC = 45^{\circ}$ and circumcircle $\Gamma$. Let $M$ be the intersection of the angle bisector of $\angle BAC$ with $\Gamma$. Let $\Omega$ be the circle tangent to segments $\overline{AB}$ and $\overline{AC}$ and internally tangent to $\Gamma$ at point $T$. Given that $\angle TMA = 45^{\... | 12 |
Find the remainder when $2^{1990}$ is divided by $1990$. | 1024 |
The number of nonnegative solutions to the equation $2x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 + x_8 + x_9 + x_{10} = 3$ is ________. | 174 |
What is the smallest prime number $p$ such that $p^3 + 4p^2 + 4p$ has exactly 30 positive divisors? | 43 |
Study the convergence of a sequence defined by \( u_0 \ge 0 \) and \( u_{n+1} = \sqrt{u_n} + \frac{1}{n+1} \) for all \( n \in \mathbb{N}_0 \). | 1 |
Let $S$ be the set of monic polynomials in $x$ of degree 6, all of whose roots are members of the set $\{-1, 0, 1\}$. Let $P$ be the sum of the polynomials in $S$. What is the coefficient of $x^4$ in $P(x)$? | 70 |
Let set $T = \{1, 2, 3, 4, 5, 6, 7, 8\}$. Find the number of all nonempty subsets $A$ of $T$ such that $3 \mid S(A)$ and $5 \nmid S(A)$, where $S(A)$ is the sum of all the elements in $A$. | 70 |
A positive integer $k$ is said to be \textit{good} if there exists a partition of $\{1, 2, 3, \ldots, 20\}$ into disjoint proper subsets such that the sum of the numbers in each subset of the partition is $k$. How many \textit{good} numbers are there? | 6 |
For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is \textit{good} if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$... | 11 |
Let $(a_n)_{n \ge 0}$ and $(b_n)_{n \ge 0}$ be sequences defined by the recurrence relations: \[ a_{n+2} = a_{n+1} + a_n, \quad n = 0, 1, \ldots, \quad a_0 = 1, \quad a_1 = 2,\nb_{n+2} = b_{n+1} + b_n, \quad n = 0, 1, \ldots, \quad b_0 = 2, \quad b_1 = 1. \] Determine how many integers the sequences $(a_n)$ and $(b_n)$... | 3 |
Square $ABCD$ has side length $68$. Let $E$ be the midpoint of segment $\overline{CD}$, and let $F$ be the point on segment $\overline{AB}$ a distance $17$ from point $A$. Point $G$ is on segment $\overline{EF}$ so that $\overline{EF}$ is perpendicular to segment $\overline{GD}$. The length of segment $\overline{BG}$ c... | 46 |
How many positive even numbers have an even number of digits and are less than 10000? | 4545 |
Evan has $10$ cards numbered $1$ through $10$. He chooses some of the cards and takes the product of the numbers on them. When the product is divided by $3$, the remainder is $1$. Find the maximum number of cards he could have chosen. | 6 |
Find the largest positive integer $k$ such that there exists a set $A \subseteq \{1, 2, \ldots, 100\}$ with $k$ elements. For any $a, b \in A$, $a$ divides $b$ if and only if $s(a)$ divides $s(b)$, where $s(k)$ denotes the sum of the digits of $k$. | 17 |
On squared paper, a square is delimited, which comprises $25$ unit squares. How many rectangles are there in this square that contain one or more unit squares? | 225 |
Let $P_1$, $P_2$, $P_3$, $P_4$, $P_5$, and $P_6$ be six parabolas in the plane, each congruent to the parabola $y = \frac{x^2}{16}$. The vertices of the six parabolas are evenly spaced around a circle. The parabolas open outward with their axes being extensions of six of the circle's radii. Parabola $P_1$ is tangent to... | 24 |
What is the largest possible length of an arithmetic progression of positive integers $a_1, a_2, \ldots, a_n$ with a common difference of $2$, such that $a_k^2 + 1$ is a prime number for each $k = 1, 2, \ldots, n$? | 3 |
12 students need to form five study groups. They will form three study groups with 2 students each and two study groups with 3 students each. In how many ways can these groups be formed? | 138600 |
Let $m$ be the smallest positive integer such that $m^2 + (m+1)^2 + \cdots + (m+10)^2$ is the square of a positive integer $n$. Find $m+n$. | 95 |
Determine the smallest integer $k > 1$ such that there exist $k$ distinct primes whose squares sum to a power of $2$. | 5 |
Let $S(r)$ denote the sum of the infinite geometric series $17 + 17r + 17r^2 + 17r^3 + \ldots$ for $-1 < r < 1$. If $S(a) \times S(-a) = 2023$, find $S(a) + S(-a)$. | 238 |
How many unordered triples $A, B, C$ of distinct lattice points in $0 \leq x, y \leq 4$ have the property that $2[ABC]$ is an integer divisible by $5$? | 300 |
Find the smallest number $n$ such that any set of $n$ points in a Cartesian plane, all of them with integer coordinates, contains two points such that the square of their mutual distance is a multiple of $2016$. | 14113 |
Consider real numbers $A$, $B$, \dots, $Z$ such that: \[ EVIL = \frac{5}{31}, \; LOVE = \frac{6}{29}, \text{ and } IMO = \frac{7}{3}. \] If $OMO = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, find the value of $m+n$. | 579 |
Eleven pirates find a treasure chest. When they divide the coins among themselves, there are 5 coins left. They throw one pirate overboard and divide the coins again, only to find that there are 3 coins left over. They throw another pirate overboard and divide the coins again. This time, the coins divide evenly. What i... | 423 |
In $\triangle ABC$, three lines are drawn parallel to side $BC$ dividing the altitude of the triangle into four equal parts. If the area of the second largest part is $35$, what is the area of the whole $\triangle ABC$? | 112 |
Positive integers $a$, $b$, $c$, $d$, and $e$ satisfy the equations: 1. $(a + 1)(3bc + 1) = d + 3e + 1$ 2. $(b + 1)(3ca + 1) = 3d + e + 13$ 3. $(c + 1)(3ab + 1) = 4(26 - d - e) - 1$ Find $d^2 + e^2$. | 146 |
A finite non-empty set of integers is called $3$-good if the sum of its elements is divisible by $3$. Find the number of $3$-good subsets of $\{0,1,2,\ldots,9\}$. | 351 |
Michelle is drawing segments in the plane. She begins from the origin facing up the $y$-axis and draws a segment of length $1$. Now, she rotates her direction by $120^\circ$, with equal probability clockwise or counterclockwise, and draws another segment of length $1$ beginning from the end of the previous segment. She... | 4 |
A wall made of mirrors has the shape of $\triangle ABC$, where $AB = 13$, $BC = 16$, and $CA = 9$. A laser positioned at point $A$ is fired at the midpoint $M$ of $BC$. The shot reflects about $BC$ and then strikes point $P$ on $AB$. If $\frac{AM}{MP} = \frac{m}{n}$ for relatively prime positive integers $m, n$, comput... | 2716 |
A positive integer $n$ is divisible by $3$ and $5$, but not by $2$. If $n > 20$, what is the smallest possible value of $n$? | 45 |
Determine the remainder when \( \prod_{i=1}^{2016} (i^4+5) \) is divided by 2017. | 2013 |
Given that \(x, y,\) and \(z\) are real numbers satisfying \(x + y + z = 10\) and \(x^2 + y^2 + z^2 = 50\), find the maximum possible value of \((x + 2y + 3z)^2 + (y + 2z + 3x)^2 + (z + 2x + 3y)^2\). | 1250 |
Prair has a box with some combination of red and green balls. If she randomly draws two balls out of the box (without replacement), the probability of drawing two balls of the same color is equal to the probability of drawing two balls of different colors. How many possible values between $200$ and $1000$ are there for... | 17 |
There are 47 students in a classroom with seats arranged in 6 rows $\times$ 8 columns, and the seat in the $i$-th row and $j$-th column is denoted by $(i,j)$. Now, an adjustment is made for students’ seats in the new school term. For a student with the original seat $(i,j)$, if his/her new seat is $(m,n)$, we say that ... | 24 |
Nick has a $3 \times 3$ grid and wants to color each square in the grid one of three colors such that no two squares that are adjacent horizontally or vertically are the same color. Compute the number of distinct grids that Nick can create. | 246 |
Find the number of subsets $A \subset M = \{2^0, 2^1, 2^2, \dots, 2^{2005}\}$ such that the equation $x^2 - S(A)x + S(B) = 0$ has integral roots, where $S(M)$ is the sum of all elements of $M$, and $B = M \setminus A$ ($A$ and $B$ are not empty). | 1003 |
If \[ \sum_{n=1}^{\infty}\frac{\frac{1}{1} + \frac{1}{2} + \dots + \frac{1}{n}}{\binom{n+100}{100}} = \frac{p}{q} \] for relatively prime positive integers $p,q$, find $p+q$. | 9901 |
Find the minimum possible value of $\left(\sqrt{x^2+4} + \sqrt{x^2+7\sqrt{3}x + 49}\right)^2$ over all real numbers. | 67 |
Let $a, b, c$ be the roots of the polynomial $x^3 - 9x^2 + 11x - 1 = 0$. Define $s = \sqrt{a} + \sqrt{b} + \sqrt{c}$. Find the value of $s^4 - 18s^2 - 8s$. | -37 |
Given natural numbers $a$, $b$, and $c$ such that $a, b, c > 100$ and the greatest common divisor $(a, b, c) = 1$, find the minimal value of $b$ under the conditions: 1. $c$ divides $a + b$ (i.e., $c \mid a + b$). 2. $a$ divides $b + c$ (i.e., $a \mid b + c$). | 10099 |
How many 4-digit numbers have exactly $9$ divisors from the set $\{1,2,3,4,5,6,7,8,9,10\}$? | 33 |
Given a right triangle $ABC$ with $AC = 3$, $BC = 4$, and $AB = 5$. Squares are constructed externally on each side of the triangle. Calculate the area of the hexagon $PQRSTU$. | 74 |
In parallelogram $ABCD$, angles $B$ and $D$ are acute while angles $A$ and $C$ are obtuse. The perpendicular from $C$ to $AB$ and the perpendicular from $A$ to $BC$ intersect at a point $P$ inside the parallelogram. If $PB = 700$ and $PD = 821$, what is $AC$? | 429 |
A chess tournament is held with the participation of boys and girls. The number of girls is twice the number of boys. Each player plays against every other player exactly once. By the end of the tournament, there were no draws, and the ratio of girl winnings to boy winnings was $\frac{7}{9}$. How many players took part... | 33 |
Yura placed $2001$ coins, each valued at $1$, $2$, or $3$ kopeykas, in a row. The arrangement follows these conditions: 1. Between any two $1$-kopeyka coins, there is at least one other coin. 2. Between any two $2$-kopeykas coins, there are at least two other coins. 3. Between any two $3$-kopeykas coins, there are at l... | 501 |
Find all natural numbers $n$ such that $n$, $n^2 + 10$, $n^2 - 2$, $n^3 + 6$, and $n^5 + 36$ are all prime numbers. | 7 |
In an enterprise, no two employees have jobs of the same difficulty and no two of them take the same salary. Every employee made the following two claims: 1. Less than $12$ employees have a more difficult job. 2. At least $30$ employees receive a higher salary. Assuming that an employee either always lies or always tel... | 42 |
Find the number of pairs $(a,b)$ of natural numbers such that: 1. $b$ is a 3-digit number. 2. $a+1$ divides $b-1$. 3. $b$ divides $a^{2} + a + 2$. | 16 |
In the figure, a semicircle is folded along the $AN$ string and intersects the $MN$ diameter at $B$. It is given that $MB:BN = 2:3$ and $MN = 10$. If $AN = x$, what is the value of $x^2$? | 80 |
Bob draws the graph of $y = x^3 - 13x^2 + 40x + 25$ and is dismayed to find out that it only has one root. Alice comes to the rescue, translating the axes so that the origin is at the point that used to be $(-20, 16)$. This new graph has three $x$-intercepts; compute their sum. | 73 |
In a triangle $ABC$, the median $AD$ divides $\angle BAC$ in the ratio $1:2$. Extend $AD$ to $E$ such that $EB$ is perpendicular to $AB$. Given that $BE=3$ and $BA=4$, find the integer nearest to $BC^2$. | 29 |
Let $A = (0,0)$, $B=(-1,-1)$, $C=(x,y)$, and $D=(x+1,y)$, where $x > y$ are positive integers. Suppose points $A$, $B$, $C$, $D$ lie on a circle with radius $r$. Denote by $r_1$ and $r_2$ the smallest and second smallest possible values of $r$. Compute $r_1^2 + r_2^2$. | 2523 |
In chess, there are two types of minor pieces, the bishop and the knight. A bishop may move along a diagonal, as long as there are no pieces obstructing its path. A knight may jump to any lattice square $\sqrt{5}$ away as long as it isn't occupied. One day, a bishop and a knight were on squares in the same row of an in... | 102 |
Find the number of ordered pairs $(a, b)$ of positive integers such that $a < b$ and $100 \leq a, b \leq 1000$ satisfying $\gcd(a, b) : \mathrm{lcm}(a, b) = 1 : 495$. | 20 |
The new PUMaC tournament hosts $2020$ students, numbered by the following set of labels $1, 2, \ldots, 2020$. The students are initially divided into $20$ groups of $101$, with each division into groups equally likely. In each group, the contestant with the lowest label wins, and the winners advance to the second round... | 2123 |
Towers grow at points along a line. All towers start with height $0$ and grow at the rate of $1$ meter per second. As soon as any two adjacent towers are each at least $1$ meter tall, a new tower begins to grow at a point along the line exactly halfway between those two adjacent towers. Before time $0$, there are no to... | 1033 |
Find the minimum value of $m$ such that any $m$-element subset of the set of integers \( \{1, 2, \ldots, 2016\} \) contains at least two distinct numbers \( a \) and \( b \) which satisfy \(|a - b| \le 3\). | 505 |
Suppose real numbers $a, b, c, d$ satisfy the following conditions: - $a + b + c + d = 17$ - $ab + bc + cd + da = 46$ Determine the minimum possible value of $a^2 + b^2 + c^2 + d^2$. If this minimum value can be expressed as a rational number $\frac{p}{q}$ in its simplest form, find $p + q$. | 199 |
Anders is solving a math problem and encounters the expression $\sqrt{15!}$. He attempts to simplify this radical as $a\sqrt{b}$, where $a$ and $b$ are positive integers. The sum of all possible values of $ab$ can be expressed in the form $q \cdot 15!$ for some rational number $q$. Find $q$. | 4 |
For some complex number $\omega$ with $|\omega| = 2016$, there is some real $\lambda>1$ such that $\omega$, $\omega^{2}$, and $\lambda \omega$ form an equilateral triangle in the complex plane. Then, $\lambda$ can be written in the form $\frac{a + \sqrt{b}}{c}$, where $a,b,$ and $c$ are positive integers and $b$ is squ... | 4032 |
Calculate the product of all positive integers less than $100$ that have exactly three positive divisors. Show that this product is a square. | 44100 |
Connor is thinking of a two-digit number $n$, which satisfies the following properties: - If $n > 70$, then $n$ is a perfect square. - If $n > 40$, then $n$ is prime. - If $n < 80$, then the sum of the digits of $n$ is $14$. What is Connor's number? | 59 |
Find the sum of the decimal digits of the number $(10^{2021} + 2021)^2$? | 36 |
Given a regular 2007-gon, find the minimal number $k$ such that among every $k$ vertices of the polygon, there always exist 4 vertices forming a convex quadrilateral where 3 sides of the quadrilateral are also sides of the polygon. | 1506 |
An isosceles triangle has a base with length $12$ and the altitude to the base has length $18$. Find the area of the region of points inside the triangle that are a distance of at most $3$ from that altitude. | 81 |
Jacob likes to watch Mickey Mouse Clubhouse! One day, he decides to create his own Mickey Mouse head with two circles $\omega_1$ and $\omega_2$ and a circle $\omega$, with centers $O_1$, $O_2$, and $O$, respectively. Let $\omega_1$ and $\omega$ meet at points $P_1$ and $Q_1$, and let $\omega_2$ and $\omega$ meet at poi... | 4 |
Suppose $a_1, a_2, a_3, \dots$ is an increasing arithmetic progression of positive integers. Given that $a_3 = 13$, compute the maximum possible value of $a_{a_1} + a_{a_2} + a_{a_3} + a_{a_4} + a_{a_5}$. | 365 |
Define $f(x) = 2x + 3$ and suppose that $g(x+2) = f(f(x-1) \cdot f(x+1) + f(x))$. Find $g(6)$. | 259 |
A list of positive integers is called *good* if the maximum element of the list appears exactly once. A *sublist* is a list formed by one or more consecutive elements of a list. For example, in the list $10, 34, 34, 22, 30, 22$, the sublist $22, 30, 22$ is good, while $10, 34, 34, 22$ is not. A list is *very good* if a... | 11 |
Rectangle $ABCD$ has $AB = 24$ and $BC = 7$. Let $d$ be the distance between the centers of the incircles of $\triangle ABC$ and $\triangle CDA$. Find $d^2$. | 325 |
A large rectangle is tiled by some $1 \times 1$ tiles. In the center, there is a small rectangle tiled by some white tiles. The small rectangle is surrounded by a red border which is five tiles wide. That red border is surrounded by a white border which is five tiles wide. Finally, the white border is surrounded by a r... | 350 |
In a right triangle $ABC$ (where $\angle C = 90^\circ$), it is given that $AC = 4$ cm and $BC = 3$ cm. The points $A_1$, $B_1$, and $C_1$ are positioned such that $AA_1 \parallel BC$, $BB_1 \parallel A_1C$, and $CC_1 \parallel A_1B_1$. Additionally, $\angle A_1B_1C_1 = 90^\circ$ and $A_1B_1 = 1$ cm. Find the length of ... | 12 |
Let $(x_n)$ be a sequence of positive integers defined by $x_1=2$ and $x_{n+1}=2x_n^3+x_n$ for all integers $n \ge 1$. Determine the largest power of $5$ that divides $x_{2014}^2 + 1$. | 2014 |
In the figure, points $A$, $B$, $C$, and $D$ are on the same line and are the centers of four tangent circles at the same point. Segment $AB$ measures $8$, and segment $CD$ measures $4$. The circumferences with centers $A$ and $C$ are of equal size. We know that the sum of the areas of the two medium circles is equival... | 24 |
Compute the sum of the roots of the equation \((2 - x)^{2005} + x^{2005} = 0\). | 2004 |
What is the smallest value that the sum of the digits of the number $3n^2 + n + 1$, where $n \in \mathbb{N}$, can take? | 3 |
A number is considered *interesting* if it meets the following criteria: 1. It is a $6$-digit integer. 2. It contains no zeros. 3. The first $3$ digits are strictly increasing. 4. The last $3$ digits are non-increasing. What is the average of all interesting numbers? | 308253 |
Among $100$ points in the plane, no three are collinear, exactly $4026$ pairs are connected by line segments. Each point is then randomly assigned an integer from $1$ to $100$ inclusive, each equally likely, such that no integer appears more than once. Find the expected value of the number of segments which join two po... | 1037 |
The triangle $ABC$ has side lengths $AB = 39$, $BC = 57$, and $CA = 70$. The median $\overline{AD}$ is divided into three congruent segments by points $E$ and $F$. Lines $BE$ and $BF$ intersect side $\overline{AC}$ at points $G$ and $H$, respectively. Find the distance from $G$ to $H$. | 21 |
What is the maximal number of checkers that can be placed on an $8 \times 8$ checkerboard so that each checker stands on the middle one of three squares in a row diagonally, with exactly one of the other two squares occupied by another checker? | 32 |
How many 4-digit numbers \( \overline{abcd} \) are there such that \( a < b < c < d \) and \( b-a < c-b < d-c \)? | 7 |
For how many numbers $n$ does $2017$ divided by $n$ have a remainder of either $1$ or $2$? | 42 |
Mustafa bought a big rug. The seller measured the rug with a ruler that was supposed to measure one meter. As it turned out to be $30$ meters long by $20$ meters wide, he charged Rs $120,000$. When Mustafa arrived home, he measured the rug again and realized that the seller had overcharged him by $9,408$ Rs. How many c... | 96 |
Determine the smallest integer $n > 1$ such that $n^2(n - 1)$ is divisible by 2009. | 42 |
Tyler has two calculators, both of which initially display zero. The first calculator has only two buttons, $[+1]$ and $[\times 2]$. The second has only the buttons $[+1]$ and $[\times 4]$. Both calculators update their displays immediately after each keystroke. A positive integer $n$ is called \textit{ambivalent} if t... | 34776 |
A square $ABCD$ has a side length of $1$. A circle passes through the vertices of the square. Let $P, Q, R, S$ be the midpoints of the arcs which are symmetrical to the arcs $AB$, $BC$, $CD$, $DA$ when reflected on sides $AB$, $BC$, $CD$, $DA$, respectively. The area of square $PQRS$ is $a + b\sqrt{2}$, where $a$ and $... | 1 |
What is the greatest integer not exceeding the sum $\sum_{n=1}^{1599} \frac{1}{\sqrt{n}}$? | 78 |
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