Olympiad problems

1969 IMO Longlists, 13

$(CZS 2)$ Let $p$ be a prime odd number. Is it possible to find $p-1$ natural numbers $n + 1, n + 2, . . . , n + p -1$ such that the sum of the squares of these numbers is divisible by the sum of these numbers?

2004 Tournament Of Towns, 5

Tags: modular arithmetic , number theory

How many different ways are there to write 2004 as a sum of one or more positive integers which are all "aproximately equal" to each other? Two numbers are called aproximately equal if their difference is at most 1. The order of terms does not matter: two ways which only differ in the order of terms are not considered different.

1970 AMC 12/AHSME, 26

Tags:

The number of distinct points in the xy-plane common to the graphs of $(x+y-5)(2x-3y+5)=0$ and $(x-y+1)(3x+2y-12)=0$ is $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad \textbf{(E) }4$

2008 Indonesia MO, 4

Tags: function , functional equation , algebra

Find all function $ f: \mathbb{N}\rightarrow\mathbb{N}$ satisfy $ f(mn)\plus{}f(m\plus{}n)\equal{}f(m)f(n)\plus{}1$ for all natural number $ n$

1966 IMO Longlists, 62

Tags: algebra , system of equations

Solve the system of equations \[ |a_1-a_2|x_2+|a_1-a_3|x_3+|a_1-a_4|x_4=1 \] \[ |a_2-a_1|x_1+|a_2-a_3|x_3+|a_2-a_4|x_4=1 \] \[ |a_3-a_1|x_1+|a_3-a_2|x_2+|a_3-a_4|x_4=1 \] \[ |a_4-a_1|x_1+|a_4-a_2|x_2+|a_4-a_3|x_3=1 \] where $a_1, a_2, a_3, a_4$ are four different real numbers.

2016 India Regional Mathematical Olympiad, 2

Tags: inequalities

Let $a,b,c$ be three distinct positive real numbers such that $abc=1$. Prove that $$\dfrac{a^3}{(a-b)(a-c)}+\dfrac{b^3}{(b-c)(b-a)}+\dfrac{c^3}{(c-a)(c-b)} \ge 3$$

2021 Science ON all problems, 2

Tags: combinatorics

There is a football championship with $6$ teams involved, such that for any $2$ teams $A$ and $B$, $A$ plays with $B$ and $B$ plays with $A$ ($2$ such games are distinct). After every match, the winning teams gains $3$ points, the loosing team gains $0$ points and if there is a draw, both teams gain $1$ point each.\\ \\ In the end, the team standing on the last place has $12$ points and there are no $2$ teams that scored the same amount of points.\\ \\ For all the remaining teams, find their final scores and provide an example with the outcomes of all matches for at least one of the possible final situations. $\textit{(Andrei Bâra)}$

2024 LMT Fall, 3

Tags: speed

High schoolers chew a lot of gum. At the supermarket, $15$ packs of $14$ sticks of gum costs $\$10$. If $1400$ high schoolers chew $3$ sticks of gum per day, find the total number of dollars spent by these high schoolers on gum per week.

2011 IMC, 1

Tags: induction

Let $(a_n)\subset (\frac{1}{2},1)$. Define the sequence $x_0=0,\displaystyle x_{n+1}=\frac{a_{n+1}+x_n}{1+a_{n+1}x_n}$. Is this sequence convergent? If yes find the limit.

1966 IMO Shortlist, 34

Tags: number theory , equation

Find all pairs of positive integers $\left( x;\;y\right) $ satisfying the equation $2^{x}=3^{y}+5.$

1951 AMC 12/AHSME, 6

Tags: geometry , 3d geometry

The bottom, side, and front areas of a rectangular box are known. The product of these areas is equal to: $ \textbf{(A)}\ \text{the volume of the box} \qquad\textbf{(B)}\ \text{the square root of the volume} \qquad\textbf{(C)}\ \text{twice the volume}$ $ \textbf{(D)}\ \text{the square of the volume} \qquad\textbf{(E)}\ \text{the cube of the volume}$

2011 Saudi Arabia BMO TST, 2

Tags: algebra , polynomial

Let $n$ be a positive integer. Prove that all roots of the equation $$x(x + 2) (x + 4 )... (x + 2n) + (x +1) (x + 3 )... (x + 2n - 1) = 0$$ are real and irrational.

2006 AMC 12/AHSME, 5

Tags:

Doug and Dave shared a pizza with $ 8$ equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half the pizza. The cost of a plain pizza was $ \$8$, and there was an additional cost of $ \$2$ for putting anchovies on one half. Dave ate all the slices of anchovy pizza and one plain slice. Doug ate the remainder. Each paid for what he had eaten. How many more dollars did Dave pay than Doug? $ \textbf{(A) } 1\qquad \textbf{(B) } 2\qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } 5$

2023 CCA Math Bonanza, I5

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Find the sum of all distinct prime factors of $2023^3 - 2000^3 - 23^3$. [i]Individual #5[/i]

2025 VJIMC, 4

Tags: complex analysis , holomorphic , taylor series , root , function

Let $D = \{z\in \mathbb{C}: |z| < 1\}$ be the open unit disk in the complex plane and let $f : D \to D$ be a holomorphic function such that $\lim_{|z|\to 1}|f(z)| = 1$. Let the Taylor series of $f$ be $f(z) = \sum_{n=0}^{\infty} a_nz^n$. Prove that the number of zeroes of $f$ (counted with multiplicities) equals $\sum_{n=0}^{\infty} n|a_n|^2$.

1989 Federal Competition For Advanced Students, P2, 6

Tags: function , polynomial , algebra

Determine all functions $ f: \mathbb{N}_0 \rightarrow \mathbb{N}_0$ such that $ f(f(n))\plus{}f(n)\equal{}2n\plus{}6$ for all $ n \in \mathbb{N}_0$.

2010 Math Prize For Girls Problems, 15

Tags: trigonometry

Compute the value of the sum \begin{align*} \frac{1}{1 + \tan^3 0^\circ} &+ \frac{1}{1 + \tan^3 10^\circ} + \frac{1}{1 + \tan^3 20^\circ} + \frac{1}{1 + \tan^3 30^\circ} + \frac{1}{1 + \tan^3 40^\circ} \\ &+ \frac{1}{1 + \tan^3 50^\circ} + \frac{1}{1 + \tan^3 60^\circ} + \frac{1}{1 + \tan^3 70^\circ} + \frac{1}{1 + \tan^3 80^\circ} \, . \end{align*}

1967 IMO Shortlist, 3

Tags: geometry , 3d geometry , polygon , cube , intersection

Which regular polygon can be obtained (and how) by cutting a cube with a plane ?

2009 Korea Junior Math Olympiad, 4

Tags: combinatorics

There are $n$ clubs composed of $4$ students out of all $9$ students. For two arbitrary clubs, there are no more than $2$ students who are a member of both clubs. Prove that $n\le 18$. Translator’s Note. We can prove $n\le 12$, and we can prove that the bound is tight. (Credits to rkm0959 for translation and document)

2019 ASDAN Math Tournament, 2

Tags:

Consider a triangle $\vartriangle ABC$ with $AB = 5$ and $BC = 4$. Let $G$ be the centroid of the triangle, and let $P$ lie on line $AG$ such that $AG = GP$ and $P\ne A$. Suppose that $P$ lies on the circumcircle of $\vartriangle ABC$. Compute $CA$.

2012 Moldova Team Selection Test, 4

Tags: combinatorics

Points $A_1, A_2,\ldots, A_n$ are found on a circle in this order. Each point $A_i$ has exactly $i$ coins. A move consists in taking two coins from two points (may be the same point) and moving them to adjacent points (one move clockwise and another counter-clockwise). Find all possible values of $n$ for which it is possible after a finite number of moves to obtain a configuration with each point $A_i$ having $n+1-i$ coins.

2021 Macedonian Mathematical Olympiad, Problem 1

Tags: sequence , recursive , inequality , factorial , am-gm , inequalities

Let $(a_n)^{+\infty}_{n=1}$ be a sequence defined recursively as follows: $a_1=1$ and $$a_{n+1}=1 + \sum\limits_{k=1}^{n}ka_k$$ For every $n > 1$, prove that $\sqrt[n]{a_n} < \frac {n+1}{2}$.

2003 Silk Road, 3

Tags: algebra

Let $0<a<b<1$ be reals numbers and \[g(x)=\left\{\begin{array}{cc}x+1-a,&\mbox{ if } 0<x<b\\b-a, & \mbox{ if } x=a \\x-a, & \mbox{ if } a<x<b\\1-a ,&\mbox{ if } x=b \\ x-a ,&\mbox{ if } b<x<1 \end{array}\right.\] Give that there exist $n+1$ reals numbers $0<x_0<x_1<...<x_n<1$, for which $g^{[n]}(x_i)=x_i \ (0 \leq i \leq n)$. Prove that there exists a positive integer $N$, such that $g^{[N]}(x)=x$ for all $0<x<1$. ($g^{[n]}(x)= \underbrace{g(g(....(g(x))....))}_{\text{n times}}$) Official solution [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=125&t=365714&p=2011659#p2011659]here[/url]

2008 Czech-Polish-Slovak Match, 3

Tags: geometry , rectangle , combinatorics

Find all triplets $(k, m, n)$ of positive integers having the following property: Square with side length $m$ can be divided into several rectangles of size $1\times k$ and a square with side length $n$.

2015 Indonesia MO Shortlist, N6

Tags: number theory , divisible , set , subset

Defined as $N_0$ as the set of all non-negative integers. Set $S \subset N_0$ with not so many elements is called beautiful if for every $a, b \in S$ with $a \ge b$ ($a$ and $b$ do not have to be different), exactly one of $a + b$ or $a - b$ is in $S$. Set $T \subset N_0$ with not so many elements is called charming if the largest number $k$ such that up to 3$^k | a$ is the same for each element $a \in T$. Prove that each beautiful set must be charming.