Olympiad problems

Estonia Open Senior - geometry, 2003.2.4

Consider the points $D, E$ and $F$ on the respective sides $BC, CA$ and $AB$ of the triangle $ABC$ in a way that the segments $AD, BE$ and $CF$ have a common point $P$. Let $\frac{|AP|}{|PD|}= x,$ $\frac{|BP|}{|PE|}= y$ and $\frac{|CP|}{|PF|}= z$. Prove that $xyz - (x + y + z) = 2$.

PEN M Problems, 26

Tags: modular arithmetic , function , inequalities , floor function , recursive sequences

Let $p$ be an odd prime $p$ such that $2h \neq 1 \; \pmod{p}$ for all $h \in \mathbb{N}$ with $h< p-1$, and let $a$ be an even integer with $a \in] \tfrac{p}{2}, p [$. The sequence $\{a_n\}_{n \ge 0}$ is defined by $a_{0}=a$, $a_{n+1}=p -b_{n}$ \; $(n \ge 0)$, where $b_{n}$ is the greatest odd divisor of $a_n$. Show that the sequence $\{a_n\}_{n \ge 0}$ is periodic and find its minimal (positive) period.

2017 Israel National Olympiad, 1

Tags: ratio , geometry , area

[list=a] [*] In the right picture there is a square with four congruent circles inside it. Each circle is tangent to two others, and to two of the edges of the square. Evaluate the ratio between the blue part and white part of the square's area. [*] In the left picture there is a regular hexagon with six congruent circles inside it. Each circle is tangent to two others, and to one of the edges on the hexagon in its midpoint. Evaluate the ratio between the blue part and white part of the hexagon's area. [/list] [img]https://i.imgur.com/fAuxoc9.png[/img]

2015 Harvard-MIT Mathematics Tournament, 6

Tags:

In triangle $ABC$, $AB=2$, $AC=1+\sqrt{5}$, and $\angle CAB=54^{\circ}$. Suppose $D$ lies on the extension of $AC$ through $C$ such that $CD=\sqrt{5}-1$. If $M$ is the midpoint of $BD$, determine the measure of $\angle ACM$, in degrees.

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 $2009.$