Olympiad problems

2004 Nicolae Coculescu, 2

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 $Z[X]$.

2012 AMC 12/AHSME, 7

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Small lights are hung on a string 6 inches apart in the order red, red, green, green, green, red, red, green, green, green, and so on continuing this pattern of 2 red lights followed by 3 green lights. How many feet separate the third red light and the 21st red light? [b]Note:[/b] 1 foot is equal to 12 inches. $\textbf{(A)}\ 18 \qquad\textbf{(B)}\ 18.5 \qquad\textbf{(C)}\ 20 \qquad\textbf{(D)}\ 20.5 \qquad\textbf{(E)}\ 22.5 $

2015 Romania Team Selection Tests, 1

Tags: equal angles , perimeter , geometry

Let $ABC$ and $ABD$ be coplanar triangles with equal perimeters. The lines of support of the internal bisectrices of the angles $CAD$ and $CBD$ meet at $P$. Show that the angles $APC$ and $BPD$ are congruent.

2018 Peru Cono Sur TST, 6

Tags: queen , combinatorics , chessboard , chess

Let $n$ be a positive integer. In an $n \times n$ board, two opposite sides have been joined, forming a cylinder. Determine whether it is possible to place $n$ queens on the board such that no two threaten each other when: $a)\:$ $n=14$. $b)\:$ $n=15$.