Olympiad problems

ICMC 8, 4

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Let a chain denote a row of positive integers which continue infinitely in both directions, such that for each number $n$, the $n$ numbers directly to the left of $n$ yield $n$ distinct remainders upon division by $n$. (a) If a chain has a maximum integer, what are the possible values of that integer? (b) Does there exist a chain which does not have a maximum integer?

2013 Tournament of Towns, 5

Tags: chessboard , combinatorics

Eight rooks are placed on a chessboard so that no two rooks attack each other. Prove that one can always move all rooks, each by a move of a knight so that in the final position no two rooks attack each other as well. (In intermediate positions several rooks can share the same square).

MOAA Accuracy Rounds, 2023.5

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Let $k$ be a constant such that exactly three real values of $x$ satisfy $$x-|x^2-4x+3| = k$$ The sum of all possible values of $k$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, find $m+n$. [i]Proposed by Andy Xu[/i]

2018 Romanian Master of Mathematics Shortlist, C2

Tags: combinatorics

Fix integers $n\ge k\ge 2$. We call a collection of integral valued coins $n-diverse$ if no value occurs in it more than $n$ times. Given such a collection, a number $S$ is $n-reachable$ if that collection contains $n$ coins whose sum of values equals $S$. Find the least positive integer $D$ such that for any $n$-diverse collection of $D$ coins there are at least $k$ numbers that are $n$-reachable. [I]Proposed by Alexandar Ivanov, Bulgaria.[/i]

1990 Chile National Olympiad, 1

Tags: isosceles triangle , isosceles , geometry

Show that any triangle can be subdivided into isosceles triangles.

2011 India IMO Training Camp, 2

Tags: number theory

Prove that for no integer $ n$ is $ n^7 \plus{} 7$ a perfect square.

2012 Math Prize For Girls Problems, 1

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In the morning, Esther biked from home to school at an average speed of $x$ miles per hour. In the afternoon, having lent her bike to a friend, Esther walked back home along the same route at an average speed of 3 miles per hour. Her average speed for the round trip was 5 miles per hour. What is the value of $x$?

2022 CCA Math Bonanza, T9

Tags: geometry

Equilateral octagon $A_1A_2A_3A_4A_5A_6A_7A_8$ is constructed such that $A_1A_3A_5A_7$ is a square of side length $\sqrt{2}$ and $A_2A_4A_6A_8$ is a square of side length 4/3. For each vertex $A_i$ of the octagon, let $B_i$ be the intersection of lines $A_{i+1}A_{i+2}$ and $A_{i-1}A_{i-2}$, where $A_{i-8} = A_i = A_{i+8}$. Compute $[B_1B_2B_3B_4B_5B_6B_7B_8]^2$. [i]2022 CCA Math Bonanza Team Round #9[/i]

2012 Hanoi Open Mathematics Competitions, 15

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[b]Q15.[/b] Determine the greatest value of the sum $M=xy+yz+zx$, where $x,y,z$ are real numbers satisfying the following condition $x^2+2y^2+5z^2=22.$

2019 Bulgaria EGMO TST, 3

Tags: game , combinatorics

$A$ and $B$ play a game, given an integer $N$, $A$ writes down $1$ first, then every player sees the last number written and if it is $n$ then in his turn he writes $n+1$ or $2n$, but his number cannot be bigger than $N$. The player who writes $N$ wins. For which values of $N$ does $B$ win? [i]Proposed by A. Slinko & S. Marshall, New Zealand[/i]

2004 USAMTS Problems, 4

Tags: arithmetic sequence

The interior angles of a convex polygon form an arithmetic progression with a common difference of $4^\circ$. Determine the number of sides of the polygon if its largest interior angle is $172^\circ.$

2020 Jozsef Wildt International Math Competition, W2

Tags: limit , real analysis , calculus

Let $\left(a_n\right)_{n\geq1}$ be a sequence of nonnegative real numbers which converges to $a \in \mathbb{R}$. [list=1] [*]Calculate$$\lim \limits_{n\to \infty}\sqrt[n]{\int \limits_0^1 \left(1+a_nx^n \right)^ndx}$$ [*]Calculate$$\lim \limits_{n\to \infty}\sqrt[n]{\int \limits_0^1 \left(1+\frac{a_1x+a_3x^3+\cdots+a_{2n-1}x^{2n-1}}{n} \right)^ndx}$$ [/list]

2003 Tuymaada Olympiad, 1

Tags: trigonometry , inequalities

Prove that for every $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}$ in the interval $(0,\pi/2)$ \[\left({1\over \sin \alpha_{1}}+{1\over \sin \alpha_{2}}+\ldots+{1\over \sin \alpha_{n}}\right) \left({1\over \cos \alpha_{1}}+{1\over \cos \alpha_{2}}+\ldots+{1\over \cos \alpha_{n}}\right) \leq\] \[\leq 2 \left({1\over \sin 2\alpha_{1}}+{1\over \sin 2\alpha_{2}}+\ldots+{1\over \sin 2\alpha_{n}}\right)^{2}.\] [i]Proposed by A. Khrabrov[/i]

2012 Indonesia TST, 2

Tags: combinatorics

Let $T$ be the set of all 2-digit numbers whose digits are in $\{1,2,3,4,5,6\}$ and the tens digit is strictly smaller than the units digit. Suppose $S$ is a subset of $T$ such that it contains all six digits and no three numbers in $S$ use all six digits. If the cardinality of $S$ is $n$, find all possible values of $n$.

2012 Korea National Olympiad, 1

Tags: induction , modular arithmetic , number theory

$ p >3 $ is a prime number such that $ p | 2^{p-1} -1 $ and $ p \not | 2^x - 1 $ for $ x = 1, 2, \cdots , p-2 $. Let $ p = 2k+3 $. Now we define sequence $ \{ a_n \} $ as \[ a_i = a_{i+k}= 2^i ( 1 \le i \le k ) , \ a_{j+2k} = a_j a_{j+k} \ ( j \ge 1 ) \] Prove that there exist $2k$ consecutive terms of sequence $ a_{x+1} , a_{x+2} , \cdots , a_{x+2k} $ such that for all $ 1 \le i < j \le 2k $, $ a_{x+i} \not \equiv a_{x+j} \ (mod \ p) $.

2016-2017 SDML (Middle School), 9

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Let $N$ be the product of all odd primes less than $2^4$. What remainder does $N$ leave when divided by $2^4$? $\text{(A) }5\qquad\text{(B) }7\qquad\text{(C) }9\qquad\text{(D) }11\qquad\text{(E) }13$

2017 Czech-Polish-Slovak Junior Match, 3

Tags: digit , divisible , number theory

How many $8$-digit numbers are $*2*0*1*7$, where four unknown numbers are replaced by stars, which are divisible by $7$?

1989 IMO Longlists, 8

Tags: algebra , polynomial , root , equation , diophantine equation

Find the roots $ r_i \in \mathbb{R}$ of the polynomial \[ p(x) \equal{} x^n \plus{} n \cdot x^{n\minus{}1} \plus{} a_2 \cdot x^{n\minus{}2} \plus{} \ldots \plus{} a_n\] satisfying \[ \sum^{16}_{k\equal{}1} r^{16}_k \equal{} n.\]

2008 AIME Problems, 1

Tags: algebra , difference of squares , special factorizations , arithmetic series

Let $ N\equal{}100^2\plus{}99^2\minus{}98^2\minus{}97^2\plus{}96^2\plus{}\cdots\plus{}4^2\plus{}3^2\minus{}2^2\minus{}1^2$, where the additions and subtractions alternate in pairs. Find the remainder when $ N$ is divided by $ 1000$.

1984 Miklós Schweitzer, 2

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[b]2.[/b] Show that threre exist a compact set $K \subset \mathbb{R}$ and a set $A \subset \mathbb{R}$ of type $F_{\sigma}$ such that the set $\{ x\in \mathbb{R} : K+x \subset A\}$ is not Borel-measurable (here $K+x = \{y+x : y \in K\}$). ([b]M.16[/b]) [M. Laczkovich]

1950 Moscow Mathematical Olympiad, 187

Tags: combinatorics

Is it possible to draw $10$ bus routes with stops such that for any $8$ routes there is a stop that does not belong to any of the routes, but any $9$ routes pass through all the stops?

2019 IMO Shortlist, N6

Tags: number theory , floor function

Let $H = \{ \lfloor i\sqrt{2}\rfloor : i \in \mathbb Z_{>0}\} = \{1,2,4,5,7,\dots \}$ and let $n$ be a positive integer. Prove that there exists a constant $C$ such that, if $A\subseteq \{1,2,\dots, n\}$ satisfies $|A| \ge C\sqrt{n}$, then there exist $a,b\in A$ such that $a-b\in H$. (Here $\mathbb Z_{>0}$ is the set of positive integers, and $\lfloor z\rfloor$ denotes the greatest integer less than or equal to $z$.)

2012 All-Russian Olympiad, 4

Tags: graph theory , set theory , combinatorics

In a city's bus route system, any two routes share exactly one stop, and every route includes at least four stops. Prove that the stops can be classified into two groups such that each route includes stops from each group.

2020 Ukrainian Geometry Olympiad - April, 3

Tags: circumcircle , parallel , perpendicular , equal segments , geometry

Triangle $ABC$. Let $B_1$ and $C_1$ be such points, that $AB= BB_1, AC=CC_1$ and $B_1, C_1$ lie on the circumscribed circle $\Gamma$ of $\vartriangle ABC$. Perpendiculars drawn from from points $B_1$ and $C_1$ on the lines $AB$ and $AC$ intersect $\Gamma$ at points $B_2$ and $C_2$ respectively, these points lie on smaller arcs $AB$ and $AC$ of circle $\Gamma$ respectively, Prove that $BB_2 \parallel CC_2$.

2018 Israel Olympic Revenge, 4

Tags: function , functional equation , algebra

Let $F:\mathbb R^{\mathbb R}\to\mathbb R^{\mathbb R}$ be a function (from the set of real-valued functions to itself) such that $$F(F(f)\circ g+g)=f\circ F(g)+F(F(F(g)))$$ for all $f,g:\mathbb R\to\mathbb R$. Prove that there exists a function $\sigma:\mathbb R\to\mathbb R$ such that $$F(f)=\sigma\circ f\circ\sigma$$ for all $f:\mathbb R\to\mathbb R$.