Olympiad problems

1988 All Soviet Union Mathematical Olympiad, 472

Tags: trigonometry , inequalities

$A, B, C$ are the angles of a triangle. Show that $2\frac{\sin A}{A} + 2\frac{\sin B}{B} + 2\frac{\sin C}{C} \le \left(\frac{1}{B} + \frac{1}{C}\right) \sin A + \left(\frac{1}{C} + \frac{1}{A}\right) \sin B + \left(\frac{1}{A} + \frac{1}{B}\right) \sin C$

2009 Saint Petersburg Mathematical Olympiad, 5

Tags: combinatorics

Call a set of some cells in infinite chess field as board. Set of rooks on the board call as awesome if no one rook can beat another, but every empty cell is under rook attack. There are awesome set with $2008$ rooks and with $2010$ rooks. Prove, that there are awesome set with $2009$ rooks.

CIME I 2018, 7

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Let $A, B, C, D$ be points, in order, on a straight line such that $AB=BC=CD$. Let $E$ be a point closer to $B$ than $D$ such that $BE=EC=CD$ and let $F$ be the midpoint of $DE$. Let $AF$ intersect $EC$ at $G$ and let $BF$ intersect $EC$ at $H$. If $[BHC]+[GHF]=1$, then $AD^2 = \frac{a\sqrt{b}}{c}$ where $a,b,$ and $c$ are positive integers, $a$ and $c$ are relatively prime, and $b$ is not divisible by the square of any prime. Find $a+b+c$. [i]Proposed by [b]AOPS12142015[/b][/i]

2004 Nicolae Coculescu, 2

Tags: primitive , function , real analysis

Consider a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that admits bounded primitives. Prove that the function $ g:\mathbb{R}\longrightarrow\mathbb{R} $ defined as $$ f(x)=\left\{ \begin{matrix} x, & \quad x\le 0 \\ f(1/x)\cdot\ln x ,& \quad x>0 \end{matrix}\right. $$ admits primitives. [i]Florian Dumitrel[/i]

LMT Team Rounds 2010-20, A11 B20

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Two sequences of nonzero reals $a_1, a_2, a_3, \dots$ and $b_2, b_3, \dots$ are such that $b_n=\prod_{i=1}^{n} a_i$ and $a_n=\frac{b_n^2}{3b_n-3}$ for all integers $n > 1$. Given that $a_1=\frac{1}{2}$, find $\lvert b_{60}\rvert$. [i]Proposed by Andrew Zhao[/i]

2014-2015 SDML (High School), 5

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Tasha and Amy both pick a number, and they notice that Tasha's number is greater than Amy's number by $12$. They each square their numbers to get a new number and see that the sum of these new numbers is half of $169$. Finally, they square their new numbers and note that Tasha's latest number is now greater than Amy's by $5070$. What was the sum of their original numbers? $\text{(A) }-4\qquad\text{(B) }-3\qquad\text{(C) }1\qquad\text{(D) }2\qquad\text{(E) }5$

2013 Math Prize For Girls Problems, 19

Tags: relatively prime , mew , number theory

If $n$ is a positive integer, let $\phi(n)$ be the number of positive integers less than or equal to $n$ that are relatively prime to $n$. Compute the value of the infinite sum \[ \sum_{n=1}^\infty \frac{\phi(n) 2^n}{9^n - 2^n} \, . \]

2017 Polish Junior Math Olympiad Finals, 2.

Tags: geometry

Point $D$ lies on the side $AB$ of triangle $ABC$, and point $E$ lies on the segment $CD$. Prove that if the sum of the areas of triangles $ACE$ and $BDE$ is equal to half the area of triangle $ABC$, then either point $D$ is the midpoint of $AB$ or point $E$ is the midpoint of $CD$.

2000 IMC, 1

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Show that a square may be partitioned into $n$ smaller squares for sufficiently large $n$. Show that for some constant $N(d)$, a $d$-dimensional cube can be partitioned into $n$ smaller cubes if $n \geq N(d) $.

2023 Malaysian Squad Selection Test, 8

Tags: combinatorics

Given two positive integers $m$ and $n$, find the largest $k$ in terms of $m$ and $n$ such that the following condition holds: Any tree graph $G$ with $k$ vertices has two (possibly equal) vertices $u$ and $v$ such that for any other vertex $w$ in $G$, either there is a path of length at most $m$ from $u$ to $w$, or there is a path of length at most $n$ from $v$ to $w$. [i]Proposed by Ivan Chan Kai Chin[/i]

2003 Indonesia MO, 5

Tags: inequalities

For any real numbers $a,b,c$, prove that \[ 5a^2 + 5b^2 + 5c^2 \ge 4ab + 4ac + 4bc \] and determine when equality occurs.

2024 CMIMC Team, 3

Tags: team

Define a function $f: \mathbb{N} \rightarrow \mathbb{N}$ to be $f(x)=(x+1)!-x!$. Find the number of positive integers $x<49$ such that $f(x)$ divides $f(49)$. [i]Proposed by David Tang[/i]

2021 Hong Kong TST, 4

Tags: polynomial , rational roots , algebra

Does there exist a nonzero polynomial $P(x)$ with integer coefficients satisfying both of the following conditions? [list] [*]$P(x)$ has no rational root; [*]For every positive integer $n$, there exists an integer $m$ such that $n$ divides $P(m)$. [/list]

2016 India PRMO, 13

Tags: algebra , number theory

Find the total number of times the digit ‘$2$’ appears in the set of integers $\{1,2,..,1000\}$. For example, the digit ’$2$’ appears twice in the integer $229$.

1986 IMO Shortlist, 11

Tags: point set , combinatorial geometry , straight lines , geometry

Let $f(n)$ be the least number of distinct points in the plane such that for each $k = 1, 2, \cdots, n$ there exists a straight line containing exactly $k$ of these points. Find an explicit expression for $f(n).$ [i]Simplified version.[/i] Show that $f(n)=\left[\frac{n+1}{2}\right]\left[\frac{n+2}{2}\right].$ Where $[x]$ denoting the greatest integer not exceeding $x.$

2003 Spain Mathematical Olympiad, Problem 1

Tags: number theory , prime number

Prove that for any prime ${p}$, different than ${2}$ and ${5}$, there exists such a multiple of ${p}$ whose digits are all nines. For example, if ${p = 13}$, such a multiple is ${999999 = 13 * 76923}$.

1991 Canada National Olympiad, 1

Tags: number theory

Show that the equation $x^2+y^5=z^3$ has infinitely many solutions in integers $x, y,z$ for which $xyz \neq 0$.

2024 Bundeswettbewerb Mathematik, 4

Tags: combinatorics

For positive integers $p$, $q$ and $r$ we are given $p \cdot q \cdot r$ unit cubes. We drill a hole along the space diagonal of each of these cubes and then tie them to a very thin thread of length $p \cdot q \cdot r \cdot \sqrt{3}$ like a string of pearls. We now want to construct a cuboid of side lengths $p$, $q$ and $r$ out of the cubes, without tearing the thread. a) For which numbers $p$, $q$ and $r$ is this possible? b) For which numbers $p$, $q$ and $r$ is this possible in a way such that both ends of the thread coincide?

2001 District Olympiad, 1

Tags: superior algebra , group theory , abstract algebra , pigeonhole principle

For any $n\in \mathbb{N}^*$, let $H_n=\left\{\frac{k}{n!}\ |\ k\in \mathbb{Z}\right\}$. a) Prove that $H_n$ is a subgroup of the group $(Q,+)$ and that $Q=\bigcup_{n\in \mathbb{N}^*} H_n$; b) Prove that if $G_1,G_2,\ldots, G_m$ are subgroups of the group $(Q,+)$ and $G_i\neq Q,\ (\forall) 1\le i\le m$, then $G_1\cup G_2\cup \ldots \cup G_m\neq Q$ [i]Marian Andronache & Ion Savu[/i]

Kvant 2024, M2791

Tags: combinatorics

A number is written in each cell of the $N \times N$ square. Let's call cell $C$ [i]good[/i] if in one of the cells adjacent to $C$ on the side, there is a number $1$ more than in $C$, and in some other of the cells adjacent to $C$ on the side, there is a number $3$ more than in $C$. What is the largest possible number of good cells? [i] Proposed by A. Chebotarev [/i]

2022 Vietnam National Olympiad, 4

Tags: number theory , divisibility

For every pair of positive integers $(n,m)$ with $n<m$, denote $s(n,m)$ be the number of positive integers such that the number is in the range $[n,m]$ and the number is coprime with $m$. Find all positive integers $m\ge 2$ such that $m$ satisfy these condition: i) $\frac{s(n,m)}{m-n} \ge \frac{s(1,m)}{m}$ for all $n=1,2,...,m-1$; ii) $2022^m+1$ is divisible by $m^2$

2015 Sharygin Geometry Olympiad, 2

Tags: geometry , orthocenter , angle

A circle passing through $A, B$ and the orthocenter of triangle $ABC$ meets sides $AC, BC$ at their inner points. Prove that $60^o < \angle C < 90^o$ . (A. Blinkov)

1993 Italy TST, 4

Tags: domino , tiling , graph theory , combinatorics

An $m \times n$ chessboard with $m,n \ge 2$ is given. Some dominoes are placed on the chessboard so that the following conditions are satisfied: (i) Each domino occupies two adjacent squares of the chessboard, (ii) It is not possible to put another domino onto the chessboard without overlapping, (iii) It is not possible to slide a domino horizontally or vertically without overlapping. Prove that the number of squares that are not covered by a domino is less than $\frac15 mn$.

2001 Spain Mathematical Olympiad, Problem 3

Tags: triangle inequality , geometry

You have five segments of lengths $a_1, a_2, a_3, a_4,$ and $a_5$ such that it is possible to form a triangle with any three of them. Demonstrate that at least one of those triangles has angles that are all acute.

MathLinks Contest 4th, 7.3

Tags: algebra

Let $\{f_n\}_{n \ge 0}$ be the Fibonacci sequence, given by $f_0 = f_1 = 1$, and for all positive integers $n$ the recurrence $f_{n+1} = f_n + f_{n-1}$. Let $a_n = f_{n+1}f_n$ for any non-negative integer $n$, and let $$P_n(X) = X^n + a_{n-1}X^{n-1} + ... + a_1X + a_0.$$ Prove that for all positive integers $n \ge 3$ the polynomial $P_n(X)$ is irreducible in $Z[X]$.