Olympiad problems

2021 Saudi Arabia Training Tests, 39

Determine if there exists pairwise distinct positive integers $a_1$, $a_2$,$ ...$, $a_{101}$, $b_1$, $b_2$,$ ...$, $b_{101}$ satisfying the following property: for each non-empty subset $S$ of $\{1, 2, ..., 101\}$ the sum $\sum_{i \in S} a_i$ divides $100! + \sum_{i \in S} b_i$.

2004 Switzerland Team Selection Test, 4

Tags: inequalities

[i]Second Test, May 16[/i] Let $a$, $b$, and $c$ be positive real numbers such that $abc = 1$. Prove that $\frac{ab}{a^{5}+b^{5}+ab}+\frac{bc}{b^{5}+c^{5}+bc}+\frac{ca}{c^{5}+a^{5}+ca}\le 1$ . When does equality hold?

1993 Romania Team Selection Test, 1

Tags: arithmetic sequence , function , increasing , algebra

Let $f : R^+ \to R$ be a strictly increasing function such that $f\left(\frac{x+y}{2}\right) < \frac{f(x)+ f(y)}{2}$ for all $x,y > 0$. Prove that the sequence $a_n = f(n)$ ($n \in N$) does not contain an infinite arithmetic progression.

2017 USAMTS Problems, 2

Tags:

After each Goober ride, the driver rates the passenger as $1$, $2$, $3$, $4$, or $5$ stars. The passenger's overall rating is determined as the average of all of the ratings given to him or her by drivers so far. Noah had been on several rides, and his rating was neither $1$ nor $5$. Then he got a $1$ star on a ride because he barfed on the driver. Show that the number of $5$ stars that Noah needs in order to climb back to at least his overall rating before barng is independent of the number of rides that he had taken.

Kvant 2021, M2646

Tags: combinatorics , number theory

Koshchey opened an account at the bank. Initially, it had 0 rubles. On the first day, Koshchey puts $k>0$ rubles in, and every next day adds one ruble more there than the day before. Each time after Koshchey deposits money into the account, the total amount in the account is divided by two by the bank. Find all such $k{}$ for which the amount on the account will always be an integer number of rubles. [i]Proposed by S. Berlov[/i]

2010 Stanford Mathematics Tournament, 10

Tags: algebra , binomial theorem

Compute the base 10 value of $14641_{99}$

2025 SEEMOUS, P1

Tags: linear algebra , matrix , vector

Let $A$ be an $n\times n$ matrix with strictly positive elements and two vectors $u,v\in\mathbb{R}^n$, also with strictly positive elements, such that $$Au=v\text{ and }Av=u.$$ Prove that $u=v$.

2005 VTRMC, Problem 1

Tags: number theory

Find the largest positive integer $n$ with the property that $n+6(p^3+1)$ is prime whenever $p$ is a prime number such that $2\le p<n$. Justify your answer.

1962 Putnam, A1

Tags: point , geometry , convex

Consider $5$ points in the plane, such that there are no $3$ of them collinear. Prove that there is a convex quadrilateral with vertices at $4$ points.

1966 IMO Shortlist, 5

Tags: inequalities , trigonometry , geometric inequality

Prove the inequality \[\tan \frac{\pi \sin x}{4\sin \alpha} + \tan \frac{\pi \cos x}{4\cos \alpha} >1\] for any $x, \alpha$ with $0 \leq x \leq \frac{\pi }{2}$ and $\frac{\pi}{6} < \alpha < \frac{\pi}{3}.$

2020 Moldova Team Selection Test, 3

Tags: combinatorics

Let $n$, $(n \geq 3)$ be a positive integer and the set $A$={$1,2,...,n$}. All the elements of $A$ are randomly arranged in a sequence $(a_1,a_2,...,a_n)$. The pair $(a_i,a_j)$ forms an $inversion$ if $1 \leq i \leq j \leq n$ and $a_i > a_j$. In how many different ways all the elements of the set $A$ can be arranged in a sequence that contains exactly $3$ inversions?

2004 IMO, 6

Tags: induction , modular arithmetic , number theory , decimal representation

We call a positive integer [i]alternating[/i] if every two consecutive digits in its decimal representation are of different parity. Find all positive integers $n$ such that $n$ has a multiple which is alternating.

2014 Iranian Geometry Olympiad (junior), P1

Tags: geometry

ABC is a triangle with A=90 and C=30.Let M be the midpoint of BC. Let W be a circle passing through A tangent in M to BC. Let P be the circumcircle of ABC. W is intersecting AC in N and P in M. prove that MN is perpendicular to BC.

1989 Putnam, A3

Tags: search , algebra

Prove that all roots of $ 11z^{10} \plus{} 10iz^9 \plus{} 10iz \minus{}11 \equal{} 0$ have unit modulus (or equivalent $ |z| \equal{} 1$).

2024 Olimphíada, 4

Tags: geometry

Let $ABC$ be a triangle, $I$ its incenter and $I_a$ its $A$-excenter. Let $\omega$ be its circuncircle and $D$ be the intersection of $AI$ and $\omega$. Let some line $r$ through $D$ cut $BC$ in $E$ and $\omega$ in $F$. The lines $IE$ and $I_aE$ intersect $I_aF$ and $IF$ in $P$ and $Q$, respectively. Furthermore, the circles $PII_a$ and $QII_a$ intersect $I_aE$ and $IE$ in $R$ and $S$, respectively. Prove that there is a circle passing through $F,E,R$ and $S$.

2001 IMO Shortlist, 6

Tags: inequalities

Prove that for all positive real numbers $a,b,c$, \[ \frac{a}{\sqrt{a^2 + 8bc}} + \frac{b}{\sqrt{b^2 + 8ca}} + \frac{c}{\sqrt{c^2 + 8ab}} \geq 1. \]

2007 Baltic Way, 5

Tags: function , algebra

A function $f$ is defined on the set of all real numbers except $0$ and takes all real values except $1$. It is also known that $\color{white}\ . \ \color{black}\ \quad f(xy)=f(x)f(-y)-f(x)+f(y)$ for any $x,y\not= 0$ and that $\color{white}\ . \ \color{black}\ \quad f(f(x))=\frac{1}{f(\frac{1}{x})}$ for any $x\not\in\{ 0,1\}$. Determine all such functions $f$.

2019 Saudi Arabia JBMO TST, 1

Tags: combinatorics

A set $S$ is called perfect if it has the following two properties: a) $S$ has exactly four elements b) for every element $x$ of $S$, at least one of the numbers $x - 1$ or $x+1$ belongs to $S$. Find the number of all perfect subsets of the set $\{1,2,... ,n\}$

2011 Kyrgyzstan National Olympiad, 3

Tags: inequalities

Given positive numbers ${a_1},{a_2},...,{a_n}$ with ${a_1} + {a_2} + ... + {a_n} = 1$. Prove that $\left( {\frac{1}{{a_1^2}} - 1} \right)\left( {\frac{1}{{a_2^2}} - 1} \right)...\left( {\frac{1}{{a_n^2}} - 1} \right) \geqslant {({n^2} - 1)^n}$.

2001 Hungary-Israel Binational, 1

Tags: induction , graph theory , combinatorics

Here $G_{n}$ denotes a simple undirected graph with $n$ vertices, $K_{n}$ denotes the complete graph with $n$ vertices, $K_{n,m}$ the complete bipartite graph whose components have $m$ and $n$ vertices, and $C_{n}$ a circuit with $n$ vertices. The number of edges in the graph $G_{n}$ is denoted $e(G_{n})$. The edges of $K_{n}(n \geq 3)$ are colored with $n$ colors, and every color is used. Show that there is a triangle whose sides have different colors.

2011 Sharygin Geometry Olympiad, 16

Tags: reflection , geometry , concurrency

Given are triangle $ABC$ and line $\ell$. The reflections of $\ell$ in $AB$ and $AC$ meet at point $A_1$. Points $B_1, C_1$ are defined similarly. Prove that a) lines $AA_1, BB_1, CC_1$ concur, b) their common point lies on the circumcircle of $ABC$ c) two points constructed in this way for two perpendicular lines are opposite.

2013 BMT Spring, 8

Tags: combinatorics , expected value

Let $f(n)$ take in a nonnegative integer $n$ and return an integer between $0$ and $n-1$ at random (with the exception being $f(0)=0$ always). What is the expected value of $f(f(22))$?

1998 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 4

Tags: pigeonhole principle , ceiling function

In a store, there are 7 cases containing 128 apples altogether. Let $ N$ be the greatest number such that one can be certain to find a case with at least $ N$ apples. Then, the last digit of $ N$ is $ \text{(A)}\ 0 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 9$

2009 Korea Junior Math Olympiad, 1

Tags: number theory , multiple , product , prime

For primes $a, b,c$ that satisfy the following, calculate $abc$. $\bullet$ $b + 8$ is a multiple of $a$, $\bullet$ $b^2 - 1$ is a multiple of $a$ and $c$ $\bullet$ $b + c = a^2 - 1$.

2012 JBMO ShortLists, 1

Tags: algebra , inequality

Let $a,b,c$ be positive real numbers such that $a+b+c=1$. Prove that \[\frac {a}{b} + \frac {a}{c} + \frac {c}{b} + \frac {c}{a} + \frac {b}{c} + \frac {b}{a} + 6 \geq 2\sqrt{2}\left (\sqrt{\frac{1-a}{a}} + \sqrt{\frac{1-b}{b}} + \sqrt{\frac{1-c}{c}}\right ).\] When does equality hold?