prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
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Suppose we have two identical cardboard polygons. We placed one polygon upon the other one and aligned. Then we pierced polygons with a pin at a point. Then we turned one of the polygons around this pin by $25^{\circ}300^{\prime}$ . It turned out that the polygons coincided (aligned again). What is the minimal possible... | 144 |
Ninth and tenth graders participated in a chess tournament. There were ten times as many tenth graders as ninth graders. The total score of tenth graders was 4.5 times that of the ninth graders. What was the ninth graders score? | 10 |
A town has 57 bus routes. How many stops does each route have if it is known that 1) one can reach any stop from any other stop without changing buses; 2) for every pair of routes there is a single stop where one can change buses; 3) each route has three or more stops? | 8 |
In a convex 1950-gon all diagonals are drawn, dividing it into smaller polygons. What is the greatest number of sides can these polygons have? | 1949 |
Dividing $x^{1951}-1$ by $P(x)=x^4+x^3+2x^2+x+1$ one gets a quotient and a remainder. Find the coefficient of $x^{14}$ in the quotient. | -1 |
Define the maximal value of the ratio of a three-digit number to the sum of its digits. | 100 |
Five men play several sets of dominoes (two against two) so that each player has each other player once as a partner and two times as an opponent. Find the number of sets and all ways to arrange the players. | 5 |
What is the least number of points that can be chosen on a circle of length 1956, so that for each of these points there is exactly one chosen point at distance 1, and exactly one chosen point at distance 2 (distances are measured along the circle)? | 1304 |
The distance between towns A and B is 999 km. At every kilometer of the road that connects A and B a sign shows the distances to A and B as follows: $$\boxed{0|999}\quad\boxed{1|998}\quad\boxed{2|997}\quad\ldots\quad\boxed{998|1}\quad\boxed{999|0}$$ How many signs are there, with both distances written with the help of... | 40 |
How many four-digit numbers from 0000 to 9999 (we complete a one-, two-, or three-digit number to a four-digit one by writing zeros in front of it) are there such that the sum of their first two digits is equal to the sum of their last two digits? | 670 |
On a piece of paper, write figures 0 to 9. Observe that if we turn the paper through $180^\circ$ the $0’s, 1’s$ (written as a vertical line segment, not as in the typed texts) and $8’s$ turn into themselves, the $6’s$ and $9’s$ interchange, and the other figures become meaningless. How many 9-digit numbers are there wh... | 1500 |
The sum of digits of integers k and $k +l$ is divisible by 11 and there is no number with similar properties between them. What is the greatest value of l? | 39 |
A factory produces rattles shaped in the form of a ring with 3 red and 7 blue spherical beads on it. Two rattles are said to be of the same type if one can be obtained from the other one by moving a bead along the ring or by flipping the ring over in space. How many different types of rattles can be manufactured? | 8 |
What greatest number of elements can be selected from the set of numbers $1, 2, . . . , 1963$ so that the sum of any two of the selected numbers is divisible by 26? | 76 |
What greatest number of integers can be selected from the set $1, 2, . . . , 1963$ so that no sum of any two selected numbers were divisible by their difference? | 655 |
We drew 11 horizontal and 11 vertical intersecting straight lines on a sheet of paper. We call a segment of one of the straight line drawn that connects two neighboring intersections a “link”. What least number of links must we erase in order for each intersection to be a junction of not more than 3 links? | 41 |
Let $a_1 = 1$ and for $k > 1$ define $a_k=[\sqrt{a_1+a_2+\cdots+a_{k-1}} ],$where $[x]$ denotes the integer part of x. Find $a_{1000}$. | 495 |
On an $11 × 11$ checker-board 22 squares are marked so that exactly two of the marked squares lie in each column and each row. Two arrangements of marked squares are considered equivalent if in any number of permutations of the columns and/or (independent) permutations of rows one arrangement can be obtained from the o... | 14 |
In the city of Fuchs, Ostap Bender organized a distribution of elephants to people. There were present 28 trades union members and 37 non-members. Ostap distributed the elephants so that the shares of all trades union members were equal, and the shares of non-members were also equal. It turned out that there existed on... | 2072 |
Consider integers with the sum of their digits divisible by 7. What is the greatest difference between two consecutive such integers? | 13 |
12 people took part in a chess tournament. After the end of the tournament every participant made 12 lists. The first list consisted of the author; the second list – of the author and of those (s)he has beaten; and so on; the 12-th list consisted of all the people on the 11-th list and those they have beaten. It is kno... | 54 |
Consider a paper scroll of bus tickets numbered 000 000 to 999 999. The tickets with the sum of the digits in the even places equal to the sum of the digits in the odd places are marked blue. What is the greatest difference between the numbers on two consequitive blue tickets? | 990 |
We mark point $O_1$ on a unit circle and, using $O_1$ as the center, we mark (by means of a compass) we mark point $O_2$ on the circle (clockwise starting with $O_1$). Using point $O_2$ as a new center, we repeat the procedure in the same direction with the same radius; and so on. After we had marked point $O_{1968}$ a... | 3 |
A straight line cuts a square piece of paper into two parts. Another straight line cuts one of the parts into two parts. One of the three pieces of paper obtained is again cut into two parts along a line, and so on. What least number of lines must be drawn in order to obtain 100 various (perhaps, (perhaps, identical) 2... | 1699 |
We multiply the number $1234567 . . . 1000$ (juxtaposed are all natural numbers 1 to 1000) by a number from 1 to 9, and strike out all 1’s in the product. We multiply the number obtained by a nonzero one-digit number once again, and strike out the 1’s, and so on, many times over. What is the least number one can obtain... | 0 |
Is there a number whose square begins with the digits $123456789$ and ends with the digits $987654321$? | 111111111 |
We cut a cardboard (perhaps, non-convex) 1000-gon along a straight line. This cut brakes it into several new polygons. What is the greatest possible number of triangles among the new polygons? | 501 |
Given a set of positive integers with the sum of any seven of them less than 15 and the sum of all the numbers in the set equal to 100, determine the least possible number of elements in this set. | 50 |
In the decimal expression of $1^1 + 2^2 + 3^3 + · · · + 999^{999} + 1000^{1000},$ what are its a) first three digits from the left? b) first four digits? | 1000 |
Find the greatest five-digit number A in which the fourth digit is greater than the fifth; the third greater than the sum of the fourth and fifth; the second greater than the sum of the third, fourth and fifth; and the first greater than the sum of the other digits. | 95210 |
Find all positive integers x satisfying the following condition: the product of the digits of x is equal to $44x − 86 868$ and their sum is equal to a cube of a positive integer. | 1989 |
The streets of a city are represented on a map as straight lines that divide a square into 25 smaller squares of side 1. (The borderline of the city is considered to be the union of 4 streets.) There is a snow plow at the bottom right corner of the bottom left square. Find the length of the shortest path for the plow t... | 60 |
Let $M = m(n+ 9)(m+ 2n ^2 + 3)$. What is the least number of distinct prime divisors the number M can have? | 2 |
To guard a military installation around the clock, a day shift and a night shift are required. A sentry guard may take either a day, or a night shift, or work around the clock. In these cases the guard is given a leave of absence of not less than $1, 1.5$ or 2.5 full days, respectively. What is the least number of guar... | 4 |
A Parliament of $2000 MP$s decided to ratify the state budget of 200 expenditure items. Each $MP$ prepared a draft budget with what (s)he thinks the maximum possible allocation for each item so that the total expenditure does not exceed a given ceiling, S. For each item, the Parliament approves the maximum expenditure ... | 1991 |
In a $9 × 9$ square table, 9 cells are marked: those at the intersection of the 2-nd, 5-th and 8-th rows with the 2-nd, 5-th and 8-th column. In how many ways can one get from the lower left cell to the upper right one moving only upwards and to the right without entering marked cells? | 678 |
What is the least number of weights in the set that can be divided into either $4, 5$, or 6 piles of equal mass? | 11 |
Let $x_1 = 4, x_2 = 6$ and define $x_n$ for $n ≥ 3$ to be the least non-prime greater than $2x_{n−1} − x_{n−2}.$ Find $x_{1000}$. | 501500 |
To every pair of numbers $x, y$ the Rule assigns a number $x ∗ y$. Find $1993 ∗ 1935$ if it is known that $x ∗ x = 0,$ and $x ∗ (y ∗ z) = (x ∗ y) + z$ for any $x, y, z.$ | 58 |
A student did not notice a multiplication sign between two three-digit numbers and wrote one 6-digit number that happened to be 7 times greater than the product of the two three-digit numbers. Find the factors. | 143 |
Professor Oak is feeding his 100 Pokémon. Each Pokémon has a bowl whose capacity is a positive real number of kilograms. These capacities are known to Professor Oak. The total capacity of all the bowls is 100 kilograms. Professor Oak distributes 100 kilograms of food in such a way that each Pokémon receives a non-negat... | 50 |
$\min \{\max \{\sqrt{x},|x-6|\}\}=$ $\qquad$. | 2 |
Let $P(x)=x^{2}-1$. How many distinct real solutions does the following equation have: $$ \underbrace{P(P(\ldots(P}_{2004}(x)) \ldots))=0? \quad[20 \text { points }] $$ | 2005 |
There are $2 n$ students in a school $(n \in \mathbb{N}, n \geq 2)$. Each week $n$ students go on a trip. After several trips the following condition was fulfilled: every two students were together on at least one trip. What is the minimum number of trips needed for this to happen? | 6 |
We colour all the sides and diagonals of a regular polygon $P$ with 43 vertices either red or blue in such a way that every vertex is an endpoint of 20 red segments and 22 blue segments. A triangle formed by vertices of $P$ is called monochromatic if all of its sides have the same colour. Suppose that there are 2022 bl... | 859 |
Find the smallest positive integer $n$ such that for every integer $m$, with $0<m<1993$, there exists an integer $k$ for which $$ \frac{m}{1993}<\frac{k}{n}<\frac{m+1}{1994} $$ | 3987 |
Let $P(x)=x^{2}-1$. How many distinct real solutions does the following equation have: $$ \underbrace{P(P(\ldots(P}_{2004}(x)) \ldots))=0? $$ | 2005 |
Find the number of positive integers $x$ satisfying the following two conditions: 1. $x<10^{2006}$; 2. $x^{2}-x$ is divisible by $10^{2006}$. | 3 |
There are $2 n$ students in a school $(n \in \mathbb{N}, n \geq 2)$. Each week $n$ students go on a trip. After several trips the following condition was fulfilled: every two students were together on at least one trip. What is the minimum number of trips needed for this to happen? | 6 |
Find all prime numbers $p$ for which there exists a unique $a \in\{1,2, \ldots, p\}$ such that $a^{3}-3 a+1$ is divisible by $p$. | 3 |
We colour all the sides and diagonals of a regular polygon $P$ with 43 vertices either red or blue in such a way that every vertex is an endpoint of 20 red segments and 22 blue segments. A triangle formed by vertices of $P$ is called monochromatic if all of its sides have the same colour. Suppose that there are 2022 bl... | 859 |
Let $V$ be a 10 -dimensional real vector space and $U_{1}$ and $U_{2}$ two linear subspaces such that $U_{1} \subseteq U_{2}, \operatorname{dim}_{\mathbb{R}} U_{1}=3$ and $\operatorname{dim}_{\mathbb{R}} U_{2}=6$. Let $\mathcal{E}$ be the set of all linear maps $T: V \longrightarrow V$ which have $U_{1}$ and $U_{2}$ as... | 67 |
For what integer $a$ does $x^{2}-x+a$ divide $x^{13}+x+90$? | 2 |
Let $p(x)=2+4x+3x^2+5x^3+3x^4+4x^5+2x^6$. For $k$ with $0<k<5$, deline $$I_k=\int_{0}^{\infty}\frac{x^k}{p(x)}dx$$. For which $k$ is $I_k$ smallest. | 2 |
It is well known that the value of the polynomial $(x+1)(x+2) \cdots(x+n)$ is exactly divisible by $n$ for every integer $x$. Given $n$, let $k=k(n)$ be the minimal degree of any monic integral polynomial $$ f(x)=x^{k}+a_{1} x^{k-1}+\cdots+a_{k} $$ (with integer coefficients and leading coefficient 1) such that the val... | 25 |
In the additive group of ordered pairs of integers ( $m, n$ ) [with addition defined componentwise: $\left.(m, n)+\left(m^{\prime}, n^{\prime}\right)=\left(m+m^{\prime}, n+n^{\prime}\right)\right]$ consider the subgroup $H$ generated by the three elements $$ (3,8), \quad(4,-1) $$ Then $H$ has another set of generators ... | 7 |
Call a set of positive integers "conspiratorial" if no three of them are pairwise relatively prime. (A set of integers is "pairwise relatively prime" if no pair of them has a common divisor greater than 1.) What is the largest number of elements in any "conspiratorial" subset of the integers I through 16? | 11 |
Call a set of positive integers "conspiratorial" if no three of them are pairwise relatively prime. (A set of integers is "pairwise relatively prime" if no pair of them has a common divisor greater than 1.) What is the largest number of elements in any "conspiratorial" subset of the integers 1 through 16? | 11 |
Find positive integers $n$ and $a_1, a_2,…,a_n$ such that $$a_1+a_2+…+a_n=1979$$ and the product $a_1a_2…a_n$ is as large as possible. | 660 |
It is well known that the value of the polynomial $(x+1)(x +2) \dots (x + n)$ is exactly divisible by n for every integer x. Given n, let $k=k(n)$ be the minimal degree of any monic integral polynomial. $$f(x)=x^k+a_1x^{k-1}+\dots +a_k$$ (with integer coefficients and leading coefficient 1) such that the value of f(x) ... | 25 |
How many positive integers $n$ are there such that $n$ is an exact divisor of at least one of the numbers $10^{40}, 20^{30}$? | 2301 |
Two distinct squares of the 8 by 8 chessboard C are said to be adjacent if they have a vertex or side in common. Also, g is called a C-gap if for every numbering of the squares of C with all the integers $1,2,\ldots,64$ there exist two adjacent squares whose numbers differ by at least g. Determine the largest C-gap g. | 9 |
Suppose that $G$ is a group generated by elements $A$ and $B$, that is, every element of $G$ can be written as a finite "word" $A{ }^{{ }^{1}} B^{n_{2}} A^{n_{3}} \cdots B^{n_{k}}$, where $n_{1}, \ldots, n_{k}$ are any integers, and $A^{\prime \prime}=B^{\prime \prime}=1$ as usual. Also, suppose that $A^{4}=B^{7}=A B A... | 8 |
Let $V$ be the collection of all quadratic polynomials $P$ with real coefficients such that $$\sup\{\mid P^{\prime}(0)\mid:P\in V\}.$$ | 8 |
Find $$\lim_{n\to\infty}\left[\frac{1}{n^{5}}\sum_{h=1}^{n}\sum_{k=1}^{n}(5h^{4}-18h^{2}k^{2}+5k^{4})\right].$$ | -1 |
Find $$ \lim _{n \rightarrow \infty}\left[\frac{1}{n^{5}} \sum_{h=1}^{n} \sum_{k=1}^{n}\left(5 h^{4}-18 h^{2} k^{2}+5 k^{4}\right)\right]. $$ | -1 |
Consider polynomial forms $ax^2-bx+c$ with integer coefficients which have two distinct zeros in the open interval $0\lt x\lt1$. Exhibit with a proof the least positive integer value of a for which such a polynomial exists. | 5 |
Two distinct squares of the 8 by 8 chessboard $C$ are said to be adjacent if they have a vertex or side in common. Also, $g$ is called a $C$-gap if for every numbering of the squares of $C$ with all the integers $1,2, \ldots, 64$ there exist two adjacent squares whose numbers differ by at least $g$. Determine the large... | 9 |
Find the largest $A$ for which there exists a polynomial $$ P(x)=A x^{4}+B x^{3}+C x^{2}+D x+E $$ with real coefficients, which satisfies $$ 0<P(x)<1 \text { for }-1<x<1 $$ | 4 |
How many positive integers n are there such that n is an exact divisor of at least one of the numbers $10^{40}$, $20^{30}$? | 2301 |
Find the length of the longest sequence of equal nonzero digits in which an integral square can terminate (in base 10) and find the smallest square which terminates in such a sequence. | 1444 |
Consider polynomial forms $ax^2-bx+c$ with integer coefficients which have two distinct zeros in the open interval $0<x<1$. Exhibit with a proof the least positive integer value of a for which such a polynomial exists. | 5 |
What is the units (i.e., rightmost) digit of $\left[\frac{10^{20000}}{10^{100}+3}\right]$ ? Here $[x]$ is the greatest integer $\leqslant x$. | 3 |
The sequence of digits $1~2~3~4~5~6~7~8~9~1~0~1~1~1~2~1~3~1~4~1~5~1~6~1~7~1~8~1~9~2~0~2~1 \ldots$ is obtained by writing the positive integers in order. If the $10^{n}$ th digit in this sequence occurs in the part of the sequence in which the $m$-digit numbers are placed, define $f(n)$ to be $m$. For example, $f(2)=2$ ... | 1984 |
Find the smallest positive integer $j$ such that for every polynomial $p(x)$ with integer coefficients and for every integer $k$, the integer $$ p^{(j)}(k)=\left.\frac{d^j}{d x^j} p(x)\right|_{x=k} $$ (the $j$-th derivative of $p(x)$ at $k$ ) is divisible by 2016. | 8 |
How many positive integers $N$ satisfy all of the following three conditions? (i) $N$ is divisible by 2020 . (ii) $N$ has at most 2020 decimal digits. (iii) The decimal digits of $N$ are a string of consecutive ones followed by a string of consecutive zeros. | 508536 |
A grasshopper starts at the origin in the coordinate plane and makes a sequence of hops. Each hop has length 5, and after each hop the grasshopper is at a point whose coordinates are both integers; thus, there are 12 possible locations for the grasshopper after the first hop. What is the smallest number of hops needed ... | 578 |
Let $a_0=1, a_1=2$, and $a_n=4 a_{n-1}-a_{n-2}$ for $n \geq 2$. Find an odd prime factor of $a_{2015}$. | 181 |
Compute $$\log_2\left(\prod\limits_{a=1}^{2015}\prod\limits_{b=1}^{2015}(1+e^{2\pi iab/2015})\right)$$ Here $i$ is the imaginary unit (that is, $i^2=-1)$ | 13725 |
Find the smallest positive integer $n$ such that for every integer $m$, with $0<m<1993$, there exists an integer $k$ for which $$\frac{m}{1993}<\frac{k}{n}<\frac{m+1}{1994}.$$ | 3987 |
$\operatorname{Let}\left(x_{1},y_{1}\right)=\left(0.8,0.6\right)$and let $x_{n+1}=x_{n}\cos y_{n}-y_{n}\sin y_{n}$ and $y_{n+1}=x_{n}\sin y_{n}+$ $y_{n}$cos$y_n$ for $n=1,2,3,\ldots.$ For each of $\lim_{n\to\infty} x_n$ and $\lim_{n\to\infty }y_n$, prove that the limit exists and find it or prove that the limit does no... | -1 |
Let $p(x)$ be a nonzero polynomial of degree less than 1992 having no noncon- stant factor in common with $x^3-x.$ Let $$\frac{d^{1992}}{dx^{1992}}\left(\frac{p(x)}{x^3-x}\right)=\frac{f(x)}{g(x)}$$ for polynomials $f(x)$ and $g(x).$ Find the smallest possible degree of $f(x)$. | 3984 |
For a positive integer $n$, define $d(n)$ to be the sum of the digits of $n$ when written in binary (for example, $d(13)=1+1+0+1=3).$Let $$S=\sum_{k=1}^{2020}(-1)^{d(k)}k^3.$$ Determine $S$ modulo 2020. | 1990 |
Suppose that a positive integer $N$ can be expressed as the sum of $k$ consecutive positive integers $$N=a+(a+1)+(a+2)+\cdots+(a+k-1)$$ for $k=2017$ but for no other values of $k>1.$ Considering all positive integers $N$ with this property, what is the smallest positive integer $a$ that occurs in any of these expressio... | 16 |
For each positive integer $n$, let $k(n)$ be the number of ones in the binary representation of 2023·n. What is the minimum value of $k(n)?$ | 3 |
For a nonnegative integer $n$ and a strictly increasing sequence of real numbers $t_0,t_1,\ldots,t_n$, let $f(t)$ be the corresponding real-valued function defined for $t\geq t_0$ by the following properties: (a) $f(t)$ is continuous for $t\geq t_0$,and is twice differentiable for all $t>t_0$ other than $t_1,\ldots,t_n... | 29 |
Let $P_n$ be the number of permutations $\pi$ of $\{1,2,\ldots,n\}$ such that $|i-j|=1$ implies $|\pi(i)-\pi(j)|\leq2$ for all $i,j$ in $\{1,2,\ldots,n\}.$ Show that for $n\geq2$,the quantity $$P_{n+5}-P_{n+4}-P_{n+3}+P_n$$ does not depend on $n$, and find its value. | 4 |
How many primes among the positive integers, written as usual in base 10, are such that their digits are alternating 1's and 0's, beginning and ending with 1? | 0 |
The sequence of digits ## $123456789101112131415161718192021 \ldots$ is obtained by writing the positive integers in order. If the $10^{n}$ th digit in this sequence occurs in the part of the sequence in which the $m$-digit numbers are placed, define $f(n)$ to be $m$. For example, $f(2)=2$ because the 100 th digit ente... | 5 |
Define $C(\alpha)$ to be the coefficient of $x^{1992}$ in the power series expansion about $x=0$ of $(1+x)^{\alpha}$. Evaluate $$ \int_{0}^{1} C(-y-1)\left(\frac{1}{y+1}+\frac{1}{y+2}+\frac{1}{y+3}+\cdots+\frac{1}{y+1992}\right) d y. $$ | 1992 |
Let $f$ be an infinitely differentiable real-valued function defined on the real numbers. If $$ f\left(\frac{1}{n}\right)=\frac{n^{2}}{n^{2}+1}, \quad n=1,2,3, \ldots $$ compute the values of the derivatives $f^{(k)}(0), k=1,2,3, \ldots$. | 0 |
Let $p(x)$ be a nonzero polynomial of degree less than 1992 having no nonconstant factor in common with $x^{3}-x$. Let $$ \frac{d^{1992}}{d x^{1992}}\left(\frac{p(x)}{x^{3}-x}\right)=\frac{f(x)}{g(x)} $$ for polynomials $f(x)$ and $g(x)$. Find the smallest possible degree of $f(x)$. | 3984 |
$\cos18^{\circ}+2\cos36^{\circ}+\cdots+20\cos360^{\circ}$ =_______。 | 10 |
In the additive group of ordered pairs of integers $(m,n) [$with addition defined componentwise: $\left.(m,n)+\left(m^{\prime},n^{\prime}\right)=\left(m+m^{\prime}, n+n^{\prime}\right)\right]$ consider the subgroup $H$ generated by the three elements$$(3,8),\quad(4,-1),\quad(5,4).$$Then $H$ has another set of generator... | 7 |
Find positive integers $n$ and $a_{1}, a_{2}, \ldots, a_{n}$ such that $$ a_{1}+a_{2}+\cdots+a_{n}=1979 $$ and the product $a_{1} a_{2} \cdots a_{n}$ is as large as possible. | 660 |
Let $p(x)=2+4x+3x^{2}+5x^{3}+3x^{4}+4x^{5}+2x^{6}$. For $k$ with $0<k<5$, define $$I_{k}=\int_{0}^{\infty} \frac{x^{k}}{p(x)} d x$$ For which $k$ is $I_{k}$ smallest? | 2 |
What is the units (i.e., rightmost) digit of $\left[\frac{10^{20000}}{10^{100}+3}\right]$? Here $[x]$ is the greatest integer $\leqslant x$. | 3 |
The sequence of digits $1~2~3~4~5~6~7~8~9~1~0~1~1~1~2~1~3~1~4~1~5~1~6~1~7~1~8~1~9~2~0~2~1 \ldots$ is obtained by writing the positive integers in order. If the $10^{n}$ th digit in this sequence occurs in the part of the sequence in which the $m$-digit numbers are placed, define $f(n)$ to be $m$. For example, $f(2)=2$ ... | 1984 |
How many primes among the positive integers, written as usual in base $10$, are such that their digits are alternating $1'$s and $0'$s, beginning and ending with $1$? | 1 |
Define a sequence $\left\{a_{1}\right\}$ by $a_{1}=3$ and $a_{t+1}=3^{a_{i}}$ for $i \geqslant 1$. Which integers between $00$ and $99$ inclusive occur as the last two digits in the decimal expansion of infinitely many $a_{t}$? | 87 |
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