Olympiad problems

2016 Online Math Open Problems, 26

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Let $S$ be the set of all pairs $(a, b)$ of integers satisfying $0 \le a, b \le 2014.$ For any pairs $s_1 = (a_1, b_1), s_2 = (a_2, b_2) \in S$, define \[s_1 + s_2 = ((a_1 + a_2)_{2015}, (b_1 + b_2)_{2015}) \\ \text { and } \\ s_1 \times s_2 = ((a_1a_2 + 2b_1b_2)_{2015}, (a_1b_2 + a_2b_1)_{2015}), \] where $n_{2015}$ denotes the remainder when an integer $n$ is divided by $2015.$ Compute the number of functions $f : S \rightarrow S$ satisfying \[ f(s_1 + s_2) = f(s_1) + f(s_2) \text{ and } f(s_1 \times s_2) = f(s_1) \times f(s_2) \] for all $s_1, s_2 \in S.$ [i] Proposed by Yang Liu [/i]

2020 Czech and Slovak Olympiad III A, 6

For each positive integer $k$, denote by $P (k)$ the number of all positive integers $4k$-digit numbers which can be composed of the digits $2, 0$ and which are divisible by the number $2 020$. Prove the inequality $$P (k) \ge \binom{2k - 1}{k}^2$$ and determine all $k$ for which equality occurs. (Note: A positive integer cannot begin with a digit of $0$.) (Jaromir Simsa)

2021 Iran RMM TST, 2

Tags: combinatorics , chess

In a chess board we call a group of queens [i]independant[/i] if no two are threatening each other. In an $n$ by $n$ grid, we put exaxctly one queen in each cell ofa greed. Let us denote by $M_n$ the minimum number of independant groups that hteir union contains all the queens. Let $k$ be a positive integer, prove that $M_{3k+1} \le 3k+2$ Proposed by [i]Alireza Haghi[/i]

2010 China Team Selection Test, 2

Tags: polynomial , function , complex number , algebra

Let $A=\{a_1,a_2,\cdots,a_{2010}\}$ and $B=\{b_1,b_2,\cdots,b_{2010}\}$ be two sets of complex numbers. Suppose \[\sum_{1\leq i<j\leq 2010} (a_i+a_j)^k=\sum_{1\leq i<j\leq 2010}(b_i+b_j)^k\] holds for every $k=1,2,\cdots, 2010$. Prove that $A=B$.

2023 Dutch BxMO TST, 2

Tags: functional equation , functional inequality , algebra

Find all functions $f : \mathbb R \to \mathbb R$ for which \[f(a - b) f(c - d) + f(a - d) f(b - c) \leq (a - c) f(b - d),\] for all real numbers $a, b, c$ and $d$. Note that there is only one occurrence of $f$ on the right hand side!

Russian TST 2015, P4

Tags: induction , combinatorics

Let $G$ be a tournoment such that it's edges are colored either red or blue. Prove that there exists a vertex of $G$ like $v$ with the property that, for every other vertex $u$ there is a mono-color directed path from $v$ to $u$.

2009 All-Russian Olympiad, 5

Tags: absolute value , algebra , system of equations

Let $ a$, $ b$, $ c$ be three real numbers satisfying that \[ \left\{\begin{array}{c c c} \left(a\plus{}b\right)\left(b\plus{}c\right)\left(c\plus{}a\right)&\equal{}&abc\\ \left(a^3\plus{}b^3\right)\left(b^3\plus{}c^3\right)\left(c^3\plus{}a^3\right)&\equal{}&a^3b^3c^3\end{array}\right.\] Prove that $ abc\equal{}0$.

1986 IMO Longlists, 55

Tags: number theory

Given an integer $n \geq 2$, determine all $n$-digit numbers $M_0 = \overline{a_1a_2 \cdots a_n} \ (a_i \neq 0, i = 1, 2, . . ., n)$ divisible by the numbers $M_1 = \overline{a_2a_3 \cdots a_na_1}$ , $M_2 = \overline{a_3a_4 \cdots a_na_1 a_2}$, $\cdots$ , $M_{n-1} = \overline{a_na_1a_2 . . .a_{n-1}}.$

1983 Austrian-Polish Competition, 3

Tags: area , geometry , covering , disc

A bounded planar region of area $S$ is covered by a finite family $F$ of closed discs. Prove that $F$ contains a subfamily consisting of pairwise disjoint discs, of joint area not less than $S/9$.

2013 Princeton University Math Competition, 8

Tags: modular arithmetic , quadratic

Find the number of primes $p$ between $100$ and $200$ for which $x^{11}+y^{16}\equiv 2013\pmod p$ has a solution in integers $x$ and $y$.

2012 IFYM, Sozopol, 8

Tags: combinatorics

On a chess tournament two teams $A$ and $B$ are playing between each other and each consists of $n$ participants. It was noticed that however they arranged them in pairs, there was at least one pair that already played a match. Prove that there can be chosen $a$ chess players from $A$ and $b$ chess players from $B$ so that $a+b>n$ and each from the first chosen group has played a match earlier with each from the second group.

2011 Pre-Preparation Course Examination, 7

Tags: combinatorics

prove or disprove: in a connected graph $G$, every three longest paths have a vertex in common.

2010 Contests, 1

Tags: geometry

Let $CD$ be a chord of a circle $\Gamma_1$ and $AB$ a diameter of $\Gamma_1$ perpendicular to $CD$ at $N$ with $AN > NB$. A circle $\Gamma_2$ centered at $C$ with radius $CN$ intersects $\Gamma_1$ at points $P$ and $Q$. The line $PQ$ intersects $CD$ at $M$ and $AC$ at $K$; and the extension of $NK$ meets $\Gamma_2$ at $L$. Prove that $PQ$ is perpendicular to $AL$

2004 Thailand Mathematical Olympiad, 3

Tags: algebra , polynomial , cubic

Let $u, v, w$ be the roots of $x^3 -5x^2 + 4x-3 = 0$. Find a cubic polynomial having $u^3, v^3, w^3$ as roots.

2020 Paraguay Mathematical Olympiad, 3

Tags: geometry , area

In triangle $ABC$, side $AC$ is $8$ cm. Two segments are drawn parallel to $AC$ that have their ends on $AB$ and $BC$ and that divide the triangle into three parts of equal area. What is the length of the parallel segment closest to $AC$?

2025 India National Olympiad, P2

Tags: combinatorics , casework

Let $n\ge 2$ be a positive integer. The integers $1,2,\cdots,n$ are written on a board. In a move, Alice can pick two integers written on the board $a\neq b$ such that $a+b$ is an even number, erase both $a$ and $b$ from the board and write the number $\frac{a+b}{2}$ on the board instead. Find all $n$ for which Alice can make a sequence of moves so that she ends up with only one number remaining on the board. [b]Note.[/b] When $n=3$, Alice changes $(1,2,3)$ to $(2,2)$ and can't make any further moves. [i]Proposed by Rohan Goyal[/i]

1947 Moscow Mathematical Olympiad, 127

Tags: equal circles , circles , geometry , circumcircle

Point $O$ is the intersection point of the heights of an acute triangle $\vartriangle ABC$. Prove that the three circles which pass: a) through $O, A, B$, b) through $O, B, C$, and c) through $O, C, A$, are equal

2009 Sharygin Geometry Olympiad, 22

Tags: search , quadratic , trigonometry , circumcircle , 3d geometry , prism , geometry

Construct a quadrilateral which is inscribed and circumscribed, given the radii of the respective circles and the angle between the diagonals of quadrilateral.

2011 China Second Round Olympiad, 10

Tags: integration , algebra

A sequence $a_n$ satisfies $a_1 =2t-3$ ($t \ne 1,-1$), and $a_{n+1}=\dfrac{(2t^{n+1}-3)a_n+2(t-1)t^n-1}{a_n+2t^n-1}$. [list] [b][i]i)[/i][/b] Find $a_n$, [b][i]ii)[/i][/b] If $t>0$, compare $a_{n+1}$ with $a_n$.[/list]

2008 Princeton University Math Competition, 2

Tags: algebra

Find $\log_2 3 * \log_3 4 * \log_4 5 * ... * \log_{62} 63 * \log_{63} 64$ .

1992 Putnam, B6

Tags: linear algebra , matrix

Let $M$ be a set of real $n \times n$ matrices such that i) $I_{n} \in M$, where $I_n$ is the identity matrix. ii) If $A\in M$ and $B\in M$, then either $AB\in M$ or $-AB\in M$, but not both iii) If $A\in M$ and $B \in M$, then either $AB=BA$ or $AB=-BA$. iv) If $A\in M$ and $A \ne I_n$, there is at least one $B\in M$ such that $AB=-BA$. Prove that $M$ contains at most $n^2 $ matrices.

1982 IMO Longlists, 9

Tags: function , floor function , number theory

Given any two real numbers $\alpha$ and $\beta , 0 \leq \alpha < \beta \leq 1$, prove that there exists a natural number $m$ such that \[\alpha < \frac{\phi(m)}{m} < \beta.\]

2011 Mathcenter Contest + Longlist, 10

Tags: algebra , inequalities

Let $p,q,r\in R $ with $pqr=1$. Prove that $$\left(\frac{1}{1-p}\right)^2+\left(\frac{1}{1-q}\right)^2+\left(\frac{1}{1-r}\right)^2\ge 1$$ [i](Real Matrik)[/i]

2019 JBMO Shortlist, C1

Tags: combinatorics

Let $S$ be a set of $100$ positive integer numbers having the following property: “Among every four numbers of $S$, there is a number which divides each of the other three or there is a number which is equal to the sum of the other three.” Prove that the set $S$ contains a number which divides all other $99$ numbers of $S$. [i]Proposed by Tajikistan[/i]

2016 AMC 8, 15

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What is the largest power of 2 that is a divisor of $13^4-11^4$? $\textbf{(A) } 8\qquad\textbf{(B) } 16\qquad\textbf{(C) } 32\qquad\textbf{(D) } 64\qquad \textbf{(E) } 128$