Olympiad problems

2020 Czech-Austrian-Polish-Slovak Match, 2

Tags: combinatorics , algebra

Given a positive integer $n$, we say that a real number $x$ is $n$-good if there exist $n$ positive integers $a_1,...,a_n$ such that $$x=\frac{1}{a_1}+...+\frac{1}{a_n}.$$ Find all positive integers $k$ for which the following assertion is true: if $a,b$ are real numbers such that the closed interval $[a,b]$ contains infinitely many $2020$-good numbers, then the interval $[a,b]$ contains at least one $k$-good number. (Josef Tkadlec, Czech Republic)

2013 Sharygin Geometry Olympiad, 2

Tags: circumcircle , geometry

Let $ABC$ be an isosceles triangle ($AC = BC$) with $\angle C = 20^\circ$. The bisectors of angles $A$ and $B$ meet the opposite sides at points $A_1$ and $B_1$ respectively. Prove that the triangle $A_1OB_1$ (where $O$ is the circumcenter of $ABC$) is regular.

2021 HMNT, 8

Tags: combinatorics , geometry

Let $n$ be the answer to this problem. Find the number of distinct (i.e. non-congruent), non-degenerate triangles with integer side lengths and perimeter $n$.

2013 Irish Math Olympiad, 1

Tags: exponent , number theory

Find the smallest positive integer $m$ such that $5m$ is an exact 5th power, $6m$ is an exact 6th power, and $7m$ is an exact 7th power.

Indonesia MO Shortlist - geometry, g1

Tags: geometry , geometric transformation , reflection

The inscribed circle of the $ABC$ triangle has center $I$ and touches to $BC$ at $X$. Suppose the $AI$ and $BC$ lines intersect at $L$, and $D$ is the reflection of $L$ wrt $X$. Points $E$ and $F$ respectively are the result of a reflection of $D$ wrt to lines $CI$ and $BI$ respectively. Show that quadrilateral $BCEF$ is cyclic .

1988 AMC 12/AHSME, 6

Tags: geometry , parallelogram , rectangle , rhombus , trapezoid

A figure is an equiangular parallelogram if and only if it is a $ \textbf{(A)}\ \text{rectangle}\qquad\textbf{(B)}\ \text{regular polygon}\qquad\textbf{(C)}\ \text{rhombus}\qquad\textbf{(D)}\ \text{square}\qquad\textbf{(E)}\ \text{trapezoid} $

2004 CHKMO, 3

Tags: parallelogram , analytic geometry , cyclic quadrilateral , geometry

Let $K, L, M, N$ be the midpoints of sides $AB, BC, CD, DA$ of a cyclic quadrilateral $ABCD$. Prove that the orthocentres of triangles $ANK, BKL, CLM, DMN$ are the vertices of a parallelogram.

PEN A Problems, 94

Tags: algorithm , divisibility theory

Find all $n \in \mathbb{N}$ such that $3^{n}-n$ is divisible by $17$.

2023 Thailand TST, 3

Tags: algebra , polynomial

For a positive integer $n$ we denote by $s(n)$ the sum of the digits of $n$. Let $P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ be a polynomial, where $n \geqslant 2$ and $a_i$ is a positive integer for all $0 \leqslant i \leqslant n-1$. Could it be the case that, for all positive integers $k$, $s(k)$ and $s(P(k))$ have the same parity?

2007 Rioplatense Mathematical Olympiad, Level 3, 4

Tags: functional equation , functional equation in z , algebra

Find all functions $ f:Z\to Z$ with the following property: if $x+y+z=0$, then $f(x)+f(y)+f(z)=xyz.$

2008 Estonia Team Selection Test, 1

Tags: combinatorics

There are $2008$ participants in a programming competition. In every round, all programmers are divided into two equal-sized teams. Find the minimal number of rounds after which there can be a situation in which every two programmers have been in different teams at least once.

2023 Junior Balkan Team Selection Tests - Moldova, 3

Tags: number theory

Prove that the number $A=2024^{n+1}-2023n-2024$ has at least $15$ different positive divisors for every nonnegative integer $ n $.

2009 Croatia Team Selection Test, 2

Tags: combinatorics

In each field of 2009*2009 table you can write either 1 or -1. Denote Ak multiple of all numbers in k-th row and Bj the multiple of all numbers in j-th column. Is it possible to write the numbers in such a way that $ \sum_{i\equal{}1}^{2009}{Ai}\plus{} \sum_{i\equal{}1}^{2009}{Bi}\equal{}0$?

2016 Harvard-MIT Mathematics Tournament, 8

Tags: hmmt

Let $S = \{1, 2, \ldots, 2016\}$, and let $f$ be a randomly chosen bijection from $S$ to itself. Let $n$ be the smallest positive integer such that $f^{(n)}(1) = 1$, where $f^{(i)}(x) = f(f^{(i-1)}(x))$. What is the expected value of $n$?

1997 Switzerland Team Selection Test, 2

Tags: geometry

2. Let ABCD be a convex quadrilateral. Find the necessary and sufficient condition for the existence of point P inside the quadrilateral such that the triangles ABP,BCP,CDP,DAP have the same area

2006 Estonia Math Open Junior Contests, 3

Tags: parallelogram , angle bisector , geometry

Let ABCD be a parallelogram, M the midpoint of AB and N the intersection of CD and the angle bisector of ABC. Prove that CM and BN are perpendicular iff AN is the angle bisector of DAB.

2020 Thailand Mathematical Olympiad, 5

Tags: combinatorics

You have an $n\times n$ grid and want to remove all edges of the grid by the sequence of the following moves. In each move, you can select a cell and remove exactly three edges surrounding that cell; in particular, that cell must have at least three remaining edges for the operation to be valid. For which positive integers $n$ is this possible?

2020 LIMIT Category 1, 7

Tags: algebra , polynomial , limit , vieta

Let $P(x)=x^6-x^5-x^3-x^2-x$ and $a,b,c$ and $d$ be the roots of the equation $x^4-x^3-x^2-1=0$, then determine the value of $P(a)+P(b)+P(c)+P(d)$ (A)$5$ (B)$6$ (C)$7$ (D)$8$

2011 Albania Team Selection Test, 2

Tags: perimeter , calculus , integration , inradius , trigonometry , algorithm , geometry

The area and the perimeter of the triangle with sides $10,8,6$ are equal. Find all the triangles with integral sides whose area and perimeter are equal.

2023 Chile TST Ibero., 3

Tags: number theory

Determine the smallest positive integer $ n $ with the following property: for every triple of positive integers $ x, y, z $, with $ x $ dividing $ y^3 $, $ y $ dividing $ z^3 $, and $ z $ dividing $ x^3 $, it also holds that $ (xyz) $ divides $ (x + y + z)^n $.

2009 Chile National Olympiad, 4

Tags: divide , divisible , number theory

Find a positive integer $x$, with $x> 1$ such that all numbers in the sequence $$x + 1,x^x + 1,x^{x^x}+1,...$$ are divisible by $2009.$

2017 CMIMC Geometry, 9

Tags: cyclic quadrilateral , geometry

Let $\triangle ABC$ be an acute triangle with circumcenter $O$, and let $Q\neq A$ denote the point on $\odot (ABC)$ for which $AQ\perp BC$. The circumcircle of $\triangle BOC$ intersects lines $AC$ and $AB$ for the second time at $D$ and $E$ respectively. Suppose that $AQ$, $BC$, and $DE$ are concurrent. If $OD=3$ and $OE=7$, compute $AQ$.

2010 Middle European Mathematical Olympiad, 11

Tags: geometry , 3d geometry , number theory

For a nonnegative integer $n$, define $a_n$ to be the positive integer with decimal representation \[1\underbrace{0\ldots0}_{n}2\underbrace{0\ldots0}_{n}2\underbrace{0\ldots0}_{n}1\mbox{.}\] Prove that $\frac{a_n}{3}$ is always the sum of two positive perfect cubes but never the sum of two perfect squares. [i](4th Middle European Mathematical Olympiad, Team Competition, Problem 7)[/i]

2005 Chile National Olympiad, 5

Tags: number theory , functional

Compute $g(2005)$ where $g$ is a function defined on the natural numbers that has the following properties: i) $g(1) = 0$ ii) $g(nm) = g(n) + g(m) + g(n)g(m)$ for any pair of integers $n, m$. iii) $g(n^2 + 1) = (g(n) + 1)^2$ for every integer $n$.

2021 Science ON all problems, 1

Tags: functional equation , integration , differentiable function , calculus

Find all differentiable functions $f, g:[0,\infty) \to \mathbb{R}$ and the real constant $k\geq 0$ such that \begin{align*} f(x) &=k+ \int_0^x \frac{g(t)}{f(t)}dt \\ g(x) &= -k-\int_0^x f(t)g(t) dt \end{align*} and $f(0)=k, f'(0)=-k^2/3$ and also $f(x)\neq 0$ for all $x\geq 0$.\\ \\ [i] (Nora Gavrea)[/i]