Olympiad problems

2012 Kazakhstan National Olympiad, 3

The sequence $a_{n}$ defined as follows: $a_{1}=4, a_{2}=17$ and for any $k\geq1$ true equalities $a_{2k+1}=a_{2}+a_{4}+...+a_{2k}+(k+1)(2^{2k+3}-1)$ $a_{2k+2}=(2^{2k+2}+1)a_{1}+(2^{2k+3}+1)a_{3}+...+(2^{3k+1}+1)a_{2k-1}+k$ Find the smallest $m$ such that $(a_{1}+...a_{m})^{2012^{2012}}-1$ divided $2^{2012^{2012}}$

2022 Israel TST, 3

Tags: combinatorics

A class has 30 students. To celebrate 'Tu BiShvat' each student chose some dried fruits out of $n$ different kinds. Say two students are friends if they both chose from the same type of fruit. Find the minimal $n$ so that it is possible that each student has exactly $6$ friends.

2010 Contests, 1

Tags: algebra

Solve in the real numbers $x, y, z$ a system of the equations: \[ \begin{cases} x^2 - (y+z+yz)x + (y+z)yz = 0 \\ y^2 - (z + x + zx)y + (z+x)zx = 0 \\ z^2 - (x+y+xy)z + (x+y)xy = 0. \\ \end{cases} \]

2011 Princeton University Math Competition, A3 / B6

Tags: algebra

Shirley has a magical machine. If she inputs a positive even integer $n$, the machine will output $n/2$, but if she inputs a positive odd integer $m$, the machine will output $m+3$. The machine keeps going by automatically using its output as a new input, stopping immediately before it obtains a number already processed. Shirley wants to create the longest possible output sequence possible with initial input at most $100$. What number should she input?

2004 IMO Shortlist, 3

Tags: function , number theory , algebra , divisibility , functional equation

Find all functions $ f: \mathbb{N^{*}}\to \mathbb{N^{*}}$ satisfying \[ \left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}\] for any two positive integers $ m$ and $ n$. [i]Remark.[/i] The abbreviation $ \mathbb{N^{*}}$ stands for the set of all positive integers: $ \mathbb{N^{*}}=\left\{1,2,3,...\right\}$. By $ f^{2}\left(m\right)$, we mean $ \left(f\left(m\right)\right)^{2}$ (and not $ f\left(f\left(m\right)\right)$). [i]Proposed by Mohsen Jamali, Iran[/i]

2023 Costa Rica - Final Round, 3.2

Tags: number theory , least common multiple , ordered pairs

Find all ordered pairs of positive integers $(r, s)$ for which there are exactly $35$ ordered pairs of positive integers $(a, b)$ such that the least common multiple of $a$ and $b$ is $2^r \cdot 3^s$.

2005 Sharygin Geometry Olympiad, 7

Tags: equilateral , equilateral triangle , concentric circles , angle , geometry

Two circles with radii $1$ and $2$ have a common center at the point $O$. The vertex $A$ of the regular triangle $ABC$ lies on the larger circle, and the middpoint of the base $CD$ lies on the smaller one. What can the angle $BOC$ be equal to?

2017 Kosovo Team Selection Test, 4

Tags: combinatorics

For every $n \in \mathbb{N}_{0}$, prove that $\sum_{k=0}^{\left[\frac{n}{2} \right]}{2}^{n-2k} \binom{n}{2k}=\frac{3^{n}+1}{2}$

2001 South africa National Olympiad, 4

Tags: induction , combinatorial geometry , extremal principle , combinatorics

$n$ red and $n$ blue points on a plane are given so that no three of the $2n$ points are collinear. Prove that it is always possible to split up the points into $n$ pairs, with one red and one blue point in each pair, so that no two of the $n$ line segments which connect the two members of a pair intersect.

2013 Hanoi Open Mathematics Competitions, 7

Tags: geometry , area of a triangle , equilateral , area

Let $ABC$ be a triangle with $\angle A = 90^o, \angle B = 60^o$ and $BC = 1$ cm. Draw outside of $\vartriangle ABC$ three equilateral triangles $ABD,ACE$ and $BCF$. Determine the area of $\vartriangle DEF$.

2006 AIME Problems, 1

Tags: geometry , perimeter , trigonometry , pythagorean theorem

In quadrilateral $ABCD, \angle B$ is a right angle, diagonal $\overline{AC}$ is perpendicular to $\overline{CD},$ $AB=18, BC=21,$ and $CD=14.$ Find the perimeter of $ABCD$.

2006 Cezar Ivănescu, 3

Tags: inequalities , algebra

[b]a)[/b] Given two positive reals $ x,y, $ prove that $ \min\left( x,1/x+y,1/y \right)\le\sqrt 2. $ and determine when equality holds. [b]b)[/b] Find all triplets of real numbers $ (a,b,c) $ having the property that for every triplet of real numbers $ (x,y,z) , $ the following equality holds: $$ |ax+by+cz|+|bx+cy+az|+|cx+ay+bz|=|x|+|y|+|z| $$

2018 Peru IMO TST, 6

Tags: string , sorting , distance , extremal combinatorics , radius , combinatorics

Let $n$ be a positive integer. Define a chameleon to be any sequence of $3n$ letters, with exactly $n$ occurrences of each of the letters $a, b,$ and $c$. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon $X$ , there exists a chameleon $Y$ such that $X$ cannot be changed to $Y$ using fewer than $3n^2/2$ swaps.

2011 239 Open Mathematical Olympiad, 2

Tags: combinatorics

There are $100$ people in the group. Is it possible that for each pair of people exist at least $50$ others, so every in that group knows exactly one person from the pair?

1951 Moscow Mathematical Olympiad, 204

Tags: least common multiple , reciprocal , sum , number theory

* Given several numbers each of which is less than $1951$ and the least common multiple of any two of which is greater than $1951$. Prove that the sum of their reciprocals is less than $2$.

1997 Moldova Team Selection Test, 5

Tags: polynomial , algebra

Let $P(x)\in\mathbb{Z}[x]$ with deg $P=2015$. Let $Q(x)=(P(x))^2-9$. Prove that: the number of distinct roots of $Q(x)$ can not bigger than $2015$

2023 India IMO Training Camp, 2

Tags: modular arithmetic , number theory , logarithm

For a positive integer $k$, let $s(k)$ denote the sum of the digits of $k$. Show that there are infinitely many natural numbers $n$ such that $s(2^n) > s(2^{n+1})$.

2020 Princeton University Math Competition, A3/B5

Tags: geometry

Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$ and radius $10$. Let sides $AB$, $BC$, $CD$, and $DA$ have midpoints $M, N, P$, and $Q$, respectively. If $MP = NQ$ and $OM + OP = 16$, then what is the area of triangle $\vartriangle OAB$?

Dumbest FE I ever created, 3.

Tags: function

Let $c_1,c_2 \in \mathbb{R^+}$. Find all $f : \mathbb{R^+} \rightarrow \mathbb{R^+}$ such that for all $x,y \in \mathbb{R^+}$ $$f(x+c_1f(y))=f(x)+c_2f(y)$$

1985 All Soviet Union Mathematical Olympiad, 403

Tags: trigonometry , inequalities , algebra

Find all the pairs $(x,y)$ such that $|\sin x-\sin y| + \sin x \sin y \le 0$.

2014 Taiwan TST Round 3, 2

Tags: polynomial , algebra

Let $m \neq 0 $ be an integer. Find all polynomials $P(x) $ with real coefficients such that \[ (x^3 - mx^2 +1 ) P(x+1) + (x^3+mx^2+1) P(x-1) =2(x^3 - mx +1 ) P(x) \] for all real number $x$.

1994 Taiwan National Olympiad, 1

Tags: cyclic quadrilateral , geometry

Let $ABCD$ be a quadrilateral with $AD=BC$ and $\widehat{A}+\widehat{B}=120^{0}$. Let us draw equilateral $ACP,DCQ,DBR$ away from $AB$ . Prove that the points $P,Q,R$ are collinear.

2021 Sharygin Geometry Olympiad, 22

Tags: geometry , 3d geometry

A convex polyhedron and a point $K$ outside it are given. For each point $M$ of a polyhedron construct a ball with diameter $MK$. Prove that there exists a unique point on a polyhedron which belongs to all such balls.

2011 Brazil National Olympiad, 1

Tags: algebra

We call a number [i]pal[/i] if it doesn't have a zero digit and the sum of the squares of the digits is a perfect square. For example, $122$ and $34$ are pal but $304$ and $12$ are not pal. Prove that there exists a pal number with $n$ digits, $n > 1$.

2023 Novosibirsk Oral Olympiad in Geometry, 6

Tags: geometry , angle

Two quarter-circles touch as shown. Find the angle $x$. [img]https://cdn.artofproblemsolving.com/attachments/b/4/e70d5d69e46d6d40368f143cb83cf10b7d6d98.png[/img]