prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
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The height of a truncated cone is equal to the radius of its base. The perimeter of a regular hexagon circumscribing its top is equal to the perimeter of an equilateral triangle inscribed in its base. Find the angle $\varphi$ between the cone's generating line and its base. Please provide the value of the parameter use... | 4 |
Solve the system $$\left\{\begin{array}{l}x^2+y^2-2z^2=2a^2,\\x+y+2z=4(a^2+1),\\z^2-xy=a^2.\end{array}\right.$$ Please provide the sum of the elements for all solutions when $a=1$. | 12 |
Solve the system \(\left\{\begin{array}{l}x^3-y^3=2b,\\x^2y-xy^2=b.\end{array}\right.\) Determine the value of \(b\) for which there exists a solution \(x, y\). | 0 |
a) How many distinct ways are there of painting the faces of a cube six different colors? (Colorations are considered distinct if they do not coincide when the cube is rotated.) b) How many distinct ways are there of painting the faces of a dodecahedron with 12 different colors? (Colorations are considered distinct if ... | 7983390 |
All rectangles that can be inscribed in an isosceles triangle with two of their vertices on the triangle’s base have the same perimeter. Construct the triangle such that the condition can be expressed as a simple integer calculation, e.g., height = base. Provide the sum of base and height if both are integers. | 10 |
What is the largest number of parts into which $n$ planes can divide space? Please provide the value of the expression when $n = 3$. The answer should be in the format \frac{m}{n}, directly give the value of m + n. | 9 |
Factor the expression $(b-c)^3+(c-a)^3+(a-b)^3$. The original answer is a product of three differences. Please provide the sum of the coefficients in the expression. | 9 |
Given an infinite cone, the measure of its unfolding's angle is equal to $\alpha$. A curve on the cone is represented on any unfolding by the union of line segments. Find the greatest integer $n$ such that $n\cdot \alpha < 180^{\circ }$. Please provide the integer value directly. | 179 |
Solve the equation: $$\begin{aligned}|x+1|-|x|+3|x-1|-2|x-2|=x+2.\end{aligned}$$ Please provide the sum of the minimum and maximum solutions. | 0 |
Find nonzero and nonequal integers $a, b, c$ so that $x( x- a) ( x- b) ( x- c) + 1$ factors into the product of two polynomials with integer coefficients. Provide the sum of $a, b, c$ if they are three consecutive integers. | 6 |
Solve in integers the equation $$x+y=x^2-xy+y^2.$$ Please provide the sum of all x and y values from the solutions. | 12 |
Divide $a^{2^k} − b^{2^k}$ by $(a + b)(a ^2 + b ^2 )(a ^4 + b^ 4 )\cdots (a^ {2 ^{k-1}} + b^ {2 ^{k-1}} )$. If $a=1$ and $b=-1$, what is the result of the division? | 1 |
Find three-digit numbers such that any of its positive integer powers ends with the same three digits and in the same order. Please provide the sum of all such numbers. | 503 |
Calculate (without calculators, tables, etc.) with accuracy to $0.00001$ the product \(\begin{pmatrix}1-\frac{1}{10}\end{pmatrix}\begin{pmatrix}1-\frac{1}{10^2}\end{pmatrix}\cdots\begin{pmatrix}1-\frac{1}{10^{99}}\end{pmatrix}.\) The original answer is a decimal number with five decimal places, please provide the value... | 89001 |
The sum of the reciprocals of three positive integers is equal to 1. What are all the possible such triples? Please provide the sum of all elements in these triples. | 30 |
Find all positive integer solutions of the equation $x^y=y^x\quad(x\neq y).$ Please provide the sum of all elements in the solutions. | 12 |
Find the first three figures after the decimal point in the decimal expression of the number \(\frac{0.123456789101112\ldots495051}{0.515049\ldots121110987654321}\). Please provide the value of the first three digits after the decimal point as a single integer. | 239 |
99 straight lines divide a plane into n parts. Find all possible values of n less than 199. Please provide the sum of all possible values. | 298 |
Delete 100 digits from the number $1234567891011 . . . 9899100$ so that the remaining number were as big as possible. Provide the sum of the remaining digits as an integer. | 409 |
Find all rectangles that can be cut into 13 equal squares. Provide the sum of the dimensions of the rectangle. | 14 |
What greatest number of triples of points can be selected from $1955$ given points, so that each two triples have one common point? Please provide the value of \binom{1954}{2}. | 1908081 |
Find all two-digit numbers x the sum of whose digits is the same as that of $2x, 3x,$ etc., $9x$. Please provide the sum of all such two-digit numbers. | 486 |
In the decimal expression of a positive number, a, all decimals beginning with the third after the decimal point, are deleted (i.e., we take an approximation of a with accuracy to 0.01 with deficiency). The number obtained is divided by a and the quotient is similarly approximated with the same accuracy by a number b. ... | 51 |
Find the number of integer values of n for which $N = 20^n + 16^n − 3 ^n − 1$ is divisible by 323. | 323 |
Solve the system: $$\begin{cases}\frac{2x_1^2}{1+x_1^2}=x_2,\\[2ex]\frac{2x_2^2}{1+x_2^2}=x_3,\\[2ex]\frac{2x_3^2}{1+x_3^2}=x_1.\end{cases}$$ Provide the sum of all possible values of x_1 + x_2 + x_3. | 3 |
Represent 1957 as the sum of 12 positive integer summands $a_1, a_2, . . . , a_{12}$ for which the number $a_1! · a_2! · a_3! · · · · · a_{12}!$ is minimal. Please provide the sum of all summands. | 327 |
On a table lies a regular 100-gon whose vertices are numbered consecutively: $1, 2, \cdots , 100$. These numbers were then rewritten, in increasing order, according to the distance of the corresponding vertex from the front edge of the table. If vertices are at the same distance from the edge, the left number is writte... | 20200 |
A circuit of several resistors connects clamps A and B. Each resistor has an input and an output clamp. What is the least number of resistors needed for a circuit design that will not be short or open if any 9 resistors between A and B break? Please provide the sum of the resistors needed for both principal circuit des... | 200 |
Let x and y be unknown digits. The 200-digit number $89 252 525 . . . 2 525$ is multiplied by the number $\overline{444 x18 y27}$. It turns out that the 53-rd digit from the right of the product is 1, and the 54-th digit is 0. Please provide the value of x + y. | 10 |
The decimal expression of a 100-digit number consists of 1’s and 2’s. The Rule allows one to select arbitrarily 10 consecutive digits of which the first five may change places with the second five. Two numbers will be called similar if one can be obtained from the other one in several such operations. What greatest num... | 4084101 |
We have 1985 sets. Each of the sets has 45 elements, the union of any two sets has exactly 89 elements. How many elements has the union of all these 1985 sets? Please provide the final integer value of the total number of elements in the union. | 87341 |
Nick and Pete divide between themselves $2n + 1$ nuts, $n ≥ 2$, and each tries to get the greater share, naturally. According to The Rule there are three ways to divide the nuts. Each way consists of three steps and the 1-st and 2-nd steps are common for all three ways. 1-st step: Pete divides all nuts into two piles, ... | 1 |
Find all positive integers n such that $(n − 1)!$ is not divisible by $n^2$. When p is equal to 1,please provide the sum of all such integers. | 20 |
Denote by $d(N)$ the number of divisors of N (1 and N are also considered as divisors of N). Find all N such that $N/d(N) = p$ is a prime. Please provide the sum of all possible values of N. | 59 |
Number 4 has the following property: when divided by $q^2$, for any q, the remainder is less than $\frac {q^2}{2}$. Find all numbers with the same property and provide the sum of these numbers. | 5 |
How may 50 cities be connected by the least possible number of airlines so that from any city one could get to any other by changing airplanes not more than once (i.e., using two planes)? Please provide the number of airlines needed. | 49 |
Is there a quadrilateral $ABCD$ of area 1 such that for any point O inside it the area of at least one of the triangles $\triangle OAB, \triangle OBC, \triangle OCD,$ or $\triangle OAD$ is an irrational number? The original answer involves a trapezoid with bases m and $\sqrt[n]{p}$. Please provide the sum of m，n and p ... | 6 |
Is there a real number h such that $[h\cdot1969^n]$ is not divisible by $[h\cdot1969^{n-1}]$ for any positive integer n? The original answer is in \frac{m^3}{n} format, please give the value of m + n. | 3937 |
Each pan of a balance has k weights, numbered from 1 to k. The left pan is heavier. It turns out that if we interchange the places of any two weights with the same number, then either the right pan becomes heavier or the two pans reach an equilibrium. For what k this is possible? Please provide the sum of all possible ... | 3 |
A closed non-selfintersecting broken line is drawn on an infinite chessboard along the sides of its squares. There are K black squares inside the broken line. What is the greatest area of the figure bounded by the broken line when $K=4$? | 17 |
In a soccer tournament, n teams take part. Every team plays with the other one only once. Find the greatest difference between the final scores of the team with neighboring final positions, and express this difference as a function of n. Please provide the value of this difference when n = 10. | 10 |
Solve in positive integers $3 \cdot 2^x + 1 = y^2$. Please provide the sum of all integers from the solutions in the form (x, y). | 21 |
Given a collection of stones. The mass of each stone is $≤ 2 kg$ and their total mass is equal to $100 kg$. We selected a set of stones whose total mass differs from $10 kg$ by the least possible for this set number d. What is the greatest value of d for every admissible collection of stones? The original answer is in ... | 21 |
On an $m × n$ rectangular piece of graph paper there are several squares whose sides are on the vertical and horizontal lines of the paper. It is known that no two squares coincide and no square is situated inside another one. What is the maximal number of such squares? Please provide the value when $m=3,n=4$. | 12 |
Consider a triangle. The radius of the inscribed circle is equal to $\frac 43$ ; the lengths of the triangle’s heights are integers whose sum is equal to 13. Find the lengths of the triangle’s sides. If the sum of the squares of all sides is in the form of $\frac{m}{n}$, find the value of $m + n$. | 1871 |
$$\frac{2}{\sqrt{4-3\sqrt[4]{5}+2\sqrt{5}-\sqrt[4]{125}}}=a+\sqrt[b]{c}$$ Find the value of $a+b+c$. | 10 |
Find all positive integers n for which both $\frac{1}{n}$ and $\frac{1}{n+1}$ are finite decimal fractions. Provide the sum of all such n values. | 5 |
Find the total number of pairs of integers $(x, y)$ that satisfy the equation $x ^2 = y ^2 + 2y + 13$. | 4 |
The numbers $1, 2, . . . , 1983$ stand at the vertices of a regular 1983-gon. Any of the axes of symmetry of the 1983-gon divides the numbers which do not stand at the vertices through which the axis passes (if any) into two sets: on either side of the axis. Let us call an arrangement of numbers good with respect to a ... | 1 |
Three circles of radii $3, 4, 5$ are externally tangent to one another. The common tangent to the first two circles is drawn through the point at which they are tangent to each other. Find the length of this tangent contained inside the circle of radius 5. The original answer is in $\frac{m\sqrt{3}}{n}$ format, please ... | 47 |
Given the equation $\begin{aligned}\frac{x-49}{50}+\frac{x-50}{49}=\frac{49}{x-50}+\frac{50}{x-49}\end{aligned}$, if the sum of all possible values of $x$ is $\frac{b}{a}$, find the value of $a+b$. | 14801 |
Suppose the "complexity" of a number is the minimum possible length of a sequence (if any) that begins with 0 and ends with that number, with each subsequent term either equal to half of the previous term or equal to the sum of the previous term and the previous term, which is equal to 1. (The length of the empty seque... | 6 |
Solve the equation: $$(1 + x + x^ 2 )(1 + x + . . . + x^{10}) = (1 + x + . . . + x^6 ) ^2.$$ Please write the sum of all solutions for $x$. | -1 |
Given that in a convex quadrilateral $ABCD$, $\angle BAC=30^\circ, \angle ACD=40^\circ,\angle ADB=50^\circ,\angle CBD=60^\circ$ and $\angle ABC+\angle ADC>180^\circ.$, if $\angle A=\angle C=a^\circ, \angle B=b^\circ, \angle D=c^\circ.$, find the value of $a+b+c$. | 290 |
In the representation of numbers A and B as decimal fractions, the lengths of their minimal periods are equal to 6 and 12, respectively. What might the length of the minimal period in the similar representation of $A + B$ be? Find all answers and provide the sum of these possible lengths. | 16 |
Given n points on a plane, no three of them lie on a straight line. There is a straight line through each pair of points. When $n=5$, what is the minimum number of pairs of non-parallel lines among these lines? | 5 |
A fly is flying inside a regular tetrahedron with side a. The shortest distance for the fly to fly over every face and return to the initial position is $\frac{a}{\sqrt{m}}$. Find the value of $m$. | 10 |
In a round goblet whose section is the graph of the function $y = x^4$, a cherry — a ball of radius r — is dropped. What is the largest r for which the ball can touch the lowest point of the bottom? (In plain math words: what is the maximal radius of the disc lying in the domain $y ≥ x^4$ and containing the origin?) Th... | 7 |
Let us refer to a rectangular of size$1 × k$ for any natural k a strip.For $n <= a$ and for $n > 2 · b + 1$can cut a $1995 × n$ rectangle into pairwise different strips, Find the value of $a+b$. | 2993 |
Determine the maximal length $L$ of a sequence $a_{1}, \\ldots, a_{L}$ of positive integers satisfying both the following properties: - every term in the sequence is less than or equal to $2^{2023}$, and - there does not exist a consecutive subsequence $a_{i}, a_{i+1}, \\ldots, a_{j}$ (where $1 \\leqslant i \\leqslant ... | 2026 |
Let $n$ be a positive integer. We arrange $1+2+\cdots+n$ circles in a triangle with $n$ rows, such that the $i^{\text {th }}$ row contains exactly $i$ circles. The following figure shows the case $n=6$. In this triangle, a ninja-path is a sequence of circles obtained by repeatedly going from a circle to one of the two ... | 1024 |
Determine all pairs $(a, p)$ of positive integers with $p$ prime such that $p^{a}+a^{4}$ is a perfect square. Please provide the sum of all elements in these pairs. | 30 |
Given integers $m, n \geqslant 2$. Color each cell of an $m$-row by $n$-column grid $S$ with either red or blue such that the following condition is satisfied: For any two cells in the same row, if both are colored red, then one column must have all cells colored red, and the other column must have at least one cell co... | 34 |
Let $\left(a_{n}\right)_{n \in \mathbb{N}}$ be the sequence defined by $$ a_{0}=1, a_{n+1}=\frac{1}{n+1} \sum_{k=0}^{n} \frac{a_{k}}{n-k+2} $$ Find the limit $$ \lim _{n \rightarrow \infty} \sum_{k=0}^{n} \frac{a_{k}}{2^{k}} $$ if it exists. The original answer is in the form $f\left(\frac{a}{b}\right)=\frac{e}{c}$, pl... | 5 |
In the linear space of all real $n \times n$ matrices, find the maximum possible dimension of a linear subspace $V$ such that $$ \forall X, Y \in V \quad \operatorname{trace}(X Y)=0 $$ (The trace of a matrix is the sum of the diagonal entries.) Please provide the value of the answer when $n = 8$. The answer should be i... | 58 |
Today, Ivan the Confessor prefers continuous functions $f:[0,1] \rightarrow \mathbb{R}$ satisfying $f(x)+f(y) \geq|x-y|$ for all pairs $x, y \in[0,1]$. Find the minimum of $\int_{0}^{1} f$ over all preferred functions. The original answer is in \frac{m}{n} format, please give the value of m + n. | 5 |
For a sequence $a_{1}<a_{2}<\cdots<a_{n}$ of integers, a pair ( $a_{i}, a_{j}$ ) with $1 \leq i<j \leq n$ is called interesting if there exists a pair $\left(a_{k}, a_{l}\right)$ of integers with $1 \leq k<l \leq n$ such that $$ \frac{a_{l}-a_{k}}{a_{j}-a_{i}}=2. $$ For each $n \geq 3$, find the largest possible number... | 2 |
Let $n = 10$. There are $n$ points in the plane, no three of them collinear. Each day, Tom erases one of the points, until there are three points left. On the $i$-th day, for $1 \leqslant i \leqslant n-3$, before erasing that day's point, Tom writes down the positive integer $v(i)$ such that the convex hull of the poin... | 12 |
Given the function $f(x)=\left\{\begin{array}{ll}x^{a} \sin \left(\frac{1}{x^{3}}\right) & x>0, a \in N_{+} \\ 0 & x \leq 0\end{array}\right.$, determine the minimum value of $a$ such that: (1) $f(x)$ is differentiable everywhere; (2) $f(x)$ is continuously differentiable and $f^{\prime}(x)$ is continuous everywhere. P... | 7 |
Compute the limit: $\lim \limits_{n \rightarrow \infty} n \int_{0}^{1}(\sin x+2) x^{n-1} d x$. Provide the final integer value of the expression when simplified.The answer is \sin(m)+n,please give the sum of m and n. | 3 |
Let the *taxicab distance* between points $(x_1, y_1)$ and $(x_2, y_2)$ in the coordinate plane be given by $|x_1 - x_2| + |y_1 - y_2|$. For how many points $P$ with integer coordinates is the taxicab distance between $P$ and the origin less than or equal to $20$? | 841 |
How many digits are in the base-ten representation of $8^5 \cdot 5^{10} \cdot 15^5$? | 18 |
Let \(ABC\) be an equilateral triangle. \(D\) and \(E\) are midpoints of \([AB]\) and \([AC]\). The ray \([DE\) cuts the circumcircle of \(\triangle ABC\) at \(F\). The original answer is in the form \(\frac{\sqrt{k} - m}{n}\). Please find the value of \(k + m + n\). | 8 |
The equiangular convex hexagon $ABCDEF$ has $AB = 1$, $BC = 4$, $CD = 2$, and $DE = 4$. Find the area of the hexagon. The original answer is in the form \frac{m}{n}\sqrt{k}, where \frac{m}{n} is a simplified fraction. Please provide the value of m + n + k. | 50 |
Given \( 1 \le n \le 455 \) and \( n^3 \equiv 1 \pmod{455} \), find the number of solutions. | 9 |
The numbers from 1 to 1996 are written sequentially as follows: 12345678910111213... up to 1996. Find how many zeros appear in this sequence. | 489 |
For how many positive integers $n$ is $n^3 - 8n^2 + 20n - 13$ a prime number? Provide your answer as an integer. | 3 |
A big $L$ is formed as shown. Find its area. \[ \text{[asy]} \text{unitsize}(4\text{mm}); \\ \text{defaultpen}(\text{linewidth}(0.8\text{pt})+\text{fontsize}(12\text{pt})); \\ \text{draw}((0,0)--(5,0)--(5,2)--(2,2)--(2,8)--(0,8)--\text{cycle}); \\ \text{label}("5",(2.5,0),S); \\ \text{label}("2",(5,1),E); \\ \text{labe... | 22 |
For all positive integers $n$ less than $2002$, let \[ a_n = \begin{cases} 11 & \text{if } n \text{ is divisible by } 13 \text{ and } 14 \\ 13 & \text{if } n \text{ is divisible by } 11 \text{ and } 14 \\ 14 & \text{if } n \text{ is divisible by } 11 \text{ and } 13 \\ 0 & \text{otherwise} \end{cases} \] Calculate $\su... | 448 |
Given distinct straight lines $OA$ and $OB$. From a point on $OA$, a perpendicular is drawn to $OB$; from the foot of this perpendicular, a line is drawn perpendicular to $OA$. From the foot of this second perpendicular, a line is drawn perpendicular to $OB$; and so on indefinitely. The lengths of the first and second ... | 2 |
The sum of the base-\( 10 \) logarithms of the divisors of \( 10^n \) is \( 792 \). What is the value of \( n \)? | 11 |
Chloé chooses a real number uniformly at random from the interval \([0, 2017]\). Independently, Laurent chooses a real number uniformly at random from the interval \([0, 4034]\). Find the probability that Laurent's number is greater than Chloé's number. The original answer is in \(\frac{k}{m}\) format, please give the ... | 7 |
Triangle $ABC$ with $AB=50$ and $AC=10$ has area $120$. Let $D$ be the midpoint of $\overline{AB}$, and let $E$ be the midpoint of $\overline{AC}$. The angle bisector of $\angle BAC$ intersects $\overline{DE}$ and $\overline{BC}$ at $F$ and $G$, respectively. What is the area of quadrilateral $FDBG$? | 75 |
Each of 2010 boxes in a line contains a single red marble, and for $1 \le k \le 2010$, the box in the $k^{th}$ position also contains $k$ white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let $P(n)$ be t... | 45 |
Jane and Jena sit at non-adjacent chairs of a four-chair circular table. In a turn, one person can move to an adjacent chair without a person. Jane moves in the first turn, and alternates with Jena afterwards. After nine moves, in how many ways can Jena be adjacent to Jane? | 32 |
A rectangle with perimeter $176$ is divided into five congruent rectangles as shown in the diagram. What is the perimeter of one of the five congruent rectangles? | 80 |
If $\sin 2x \sin 3x = \cos 2x \cos 3x$, then one value for $x$ is $x^\circ$. What is the value of $x$? | 18 |
What is the sum of the last two digits of $7^{42} + 7^{43}$ in base $10$? | 11 |
Let $N = 123456789101112\dots4344$ be the $79$-digit number obtained by writing the integers from $1$ to $44$ in order, one after the other. Find the remainder when $N$ is divided by $45$. | 9 |
A baseball is dropped on top of a basketball. The basketball hits the ground, rebounds with a speed of $4.0 \, \text{m/s}$, and collides with the baseball as it is moving downward at $4.0 \, \text{m/s}$. After the collision, the baseball moves upward as shown in the figure, and the basketball is instantaneously at rest... | 6 |
How many distinct four-tuples $(a, b, c, d)$ of rational numbers are there such that: $$a \log_{10} 2 + b \log_{10} 3 + c \log_{10} 5 + d \log_{10} 7 = 2005$$ Find the number of such distinct four-tuples. | 1 |
A circle is inscribed in a triangle with side lengths $8$, $13$, and $17$. Let the segments of the side of length $8$, made by a point of tangency, be $r$ and $s$, with $r < s$. What is the ratio $r:s$? Express your answer in the form of $x:y$ where $x$ and $y$ are integers. Please give the value of x + y. | 4 |
What is the sum of all real numbers $x$ for which the median of the numbers $4, 6, 8, 17,$ and $x$ is equal to the mean of those five numbers? Provide your answer as an integer. | -5 |
In triangle $ABC$, point $F$ divides side $AC$ in the ratio $1:2$. Let $E$ be the point of intersection of side $BC$ and $AG$, where $G$ is the midpoint of $BF$. Find the ratio in which point $E$ divides side $BC$. The answer is in the form a:b, please give the value of a + b. | 4 |
Consider the set of complex numbers $z$ satisfying $|1+z+z^{2}|=4$. The maximum value of the imaginary part of $z$ can be written in the form $\frac{\sqrt{m}}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the value of $m+n$. | 21 |
In triangle $\triangle ABC$ with $|BC| > |BA|$, $D$ is a point inside the triangle such that $\angle ABD = \angle DBC$, $\angle BDC = 150^\circ$, and $\angle DAC = 60^\circ$. If the measure of $\angle BAD$ is $x^\circ$, what is the value of $x$? | 60 |
Quadrilateral $ABCD$ is inscribed in a circle with the following properties: - $\angle BAC = 70^{\circ}$ - $\angle ADB = 40^{\circ}$ - $AD = 4$ - $BC = 6$ What is the length of $AC$? | 6 |
Find the number of real numbers which satisfy the equation $x|x-1|-4|x|+3=0$. | 3 |
Triangle $ABC$ is inscribed in a circle. The measure of the non-overlapping minor arcs $AB$, $BC$, and $CA$ are, respectively, $x+75^\circ$, $2x+25^\circ$, $3x-22^\circ$. Find one interior angle of the triangle, in degrees, as an integer. | 61 |
$A, B, C$ are playing a backgammon tournament. Initially, $A$ plays against $B$. The winner of this game then plays against $C$. The tournament continues such that the last winner plays against the player who did not play in the previous game. The tournament concludes when a player wins two consecutive games. If each p... | 9 |
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