Olympiad problems

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

Tags:

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: $a)\:$ $n=14$. $b)\:$ $n=15$.

2021 JHMT HS, 9

Tags: algebra , logarithm

Let $a$ and $b$ be positive real numbers such that $\log_{43}{a} = \log_{47} (3a + 4b) = \log_{2021}b^2$. Then, the value of $\tfrac{b^2}{a^2}$ can be written as $m + \sqrt{n}$, where $m$ and $n$ are integers. Find $m + n$.

2025 CMIMC Team, 2

Tags: team

We are searching for the number $7$ in the following binary tree: [center] [img] https://cdn.artofproblemsolving.com/attachments/8/c/70ad159d239e9fd8dd9775e6391965e1016f03.png [/img] [/center] We use the following algorithm (which terminates with probability $1$): [list=1] [*] Write down the number currently at the root node. [*] If we wrote down $7,$ terminate. [*] Else, pick a random edge, and swap the two numbers at the endpoints of that edge [*] Go back to step $1.$ [/list] Let $p(a)$ be the probability that we ever write down the number $a$ after running the algorithm once. Find $$p(1)+p(2)+p(3)+p(5)+p(6).$$

Kvant 2023, M2743

Tags: geometry

Perimeter of triangle $ABC$ is $1$. Circle $\omega$ touches side $BC$, continuation of side $AB$ at $P$ and continuation of side $AC$ in $Q$. Line through midpoints $AB$ and $AC$ intersects circumcircle of $APQ$ at $X$ and $Y$. Find length of $XY$.

2008 National Olympiad First Round, 28

Tags: geometry

A unit square from one of the corners of a $8\times 8$ chessboard is cut and thrown. At least how many triangles are necessary to divide the new board into triangles with equal areas? $ \textbf{(A)}\ 17 \qquad\textbf{(B)}\ 19 \qquad\textbf{(C)}\ 20 \qquad\textbf{(D)}\ 21 \qquad\textbf{(E)}\ \text{None of the above} $

2017 239 Open Mathematical Olympiad, 4

Tags: polynomial , algebra

A polynomial $f(x)$ with integer coefficients is given. We define $d(a,k)=|f^k(a)-a|.$ It is known that for each integer $a$ and natural number $k$, $d(a,k)$ is positive. Prove that for all such $a,k$, $$d(a,k) \geq \frac{k}{3}.$$ ($f^k(x)=f(f^{k-1}(x)), f^0(x)=x.$)

2012 Bundeswettbewerb Mathematik, 1

Tags: number theory , algebra

given a positive integer $n$. the set $\{ 1,2,..,2n \}$ is partitioned into $a_1<a_2<...<a_n $ and $b_1>b_2>...>b_n$. find the value of : $ \sum_{i=1}^{n}|a_i - b_i| $

2018 AMC 12/AHSME, 4

Tags: geometry

A circle has a chord of length $10$, and the distance from the center of the circle to the chord is $5$. What is the area of the circle? $\textbf{(A) }25\pi\qquad\textbf{(B) }50\pi\qquad\textbf{(C) }75\pi\qquad\textbf{(D) }100\pi\qquad\textbf{(E) }125\pi$