Olympiad problems

2019 May Olympiad, 2

More than five competitors participated in a chess tournament. Each competitor played exactly once against each of the other competitors. Five of the competitors they each lost exactly two games. All other competitors each won exactly three games. There were no draws in the tournament. Determine how many competitors there were and show a tournament that verifies all conditions.

2013 ELMO Shortlist, 7

Tags: function , induction , 3n-3 theorem , continuity , algebra

Consider a function $f: \mathbb Z \to \mathbb Z$ such that for every integer $n \ge 0$, there are at most $0.001n^2$ pairs of integers $(x,y)$ for which $f(x+y) \neq f(x)+f(y)$ and $\max\{ \lvert x \rvert, \lvert y \rvert \} \le n$. Is it possible that for some integer $n \ge 0$, there are more than $n$ integers $a$ such that $f(a) \neq a \cdot f(1)$ and $\lvert a \rvert \le n$? [i]Proposed by David Yang[/i]

2012 Online Math Open Problems, 42

Tags: trigonometry , geometry , circumcircle , vector , analytic geometry

In triangle $ABC,$ $\sin \angle A=\frac{4}{5}$ and $\angle A<90^\circ$ Let $D$ be a point outside triangle $ABC$ such that $\angle BAD=\angle DAC$ and $\angle BDC = 90^{\circ}.$ Suppose that $AD=1$ and that $\frac{BD} {CD} = \frac{3}{2}.$ If $AB+AC$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ where $a,b,c$ are pairwise relatively prime integers, find $a+b+c$. [i]Author: Ray Li[/i]

2014 IMO Shortlist, N7

Tags: number theory , sequence , prime divisor

Let $c \ge 1$ be an integer. Define a sequence of positive integers by $a_1 = c$ and \[a_{n+1}=a_n^3-4c\cdot a_n^2+5c^2\cdot a_n+c\] for all $n\ge 1$. Prove that for each integer $n \ge 2$ there exists a prime number $p$ dividing $a_n$ but none of the numbers $a_1 , \ldots , a_{n -1}$ . [i]Proposed by Austria[/i]

2019 Pan-African Shortlist, N6

Tags: binomial coefficient , number theory

Find the $2019$th strictly positive integer $n$ such that $\binom{2n}{n}$ is not divisible by $5$.

2017 Harvard-MIT Mathematics Tournament, 3

Tags:

There are $2017$ jars in a row on a table, initially empty. Each day, a nice man picks ten consecutive jars and deposits one coin in each of the ten jars. Later, Kelvin the Frog comes back to see that $N$ of the jars all contain the same positive integer number of coins (i.e. there is an integer $d>0$ such that $N$ of the jars have exactly $d$ coins). What is the maximum possible value of $N$?

2021 IMO Shortlist, G2

Tags: geometry

Let $\Gamma$ be a circle with centre $I$, and $A B C D$ a convex quadrilateral such that each of the segments $A B, B C, C D$ and $D A$ is tangent to $\Gamma$. Let $\Omega$ be the circumcircle of the triangle $A I C$. The extension of $B A$ beyond $A$ meets $\Omega$ at $X$, and the extension of $B C$ beyond $C$ meets $\Omega$ at $Z$. The extensions of $A D$ and $C D$ beyond $D$ meet $\Omega$ at $Y$ and $T$, respectively. Prove that \[A D+D T+T X+X A=C D+D Y+Y Z+Z C.\] [i]Proposed by Dominik Burek, Poland and Tomasz Ciesla, Poland[/i]

2022 Brazil National Olympiad, 4

Tags: geometry

Let $ABC$ a triangle with $AB=BC$ and incircle $\omega$. Let $M$ the mindpoint of $BC$; $P, Q$ points in the sides $AB, AC$ such that $PQ\parallel BC$, $PQ$ is tangent to $\omega$ and $\angle CQM=\angle PQM$. Find the perimeter of triangle $ABC$ knowing that $AQ=1$.

2024 Chile TST Ibero., 2

Tags: algebra

A collection of regular polygons with sides of equal length is said to "fit" if, when arranged around a common vertex, they exactly complete the surrounding area of the point on the plane. For example, a square fits with two octagons. Determine all possible collections of regular polygons that fit.

Kvant 2020, M2618

Tags: algebra , floor function

For a given number $\alpha{}$ let $f_\alpha$ be a function defined as \[f_\alpha(x)=\left\lfloor\alpha x+\frac{1}{2}\right\rfloor.\]Let $\alpha>1$ and $\beta=1/\alpha$. Prove that for any natural $n{}$ the relation $f_\beta(f_\alpha(n))=n$ holds. [i]Proposed by I. Dorofeev[/i]

2008 Iran MO (3rd Round), 8

Tags: inequalities , number theory

In an old script found in ruins of Perspolis is written: [code] This script has been finished in a year whose 13th power is 258145266804692077858261512663 You should know that if you are skilled in Arithmetics you will know the year this script is finished easily.[/code] Find the year the script is finished. Give a reason for your answer.

2012 EGMO, 4

Tags: algorithm , induction , absolute value , combinatorics

A set $A$ of integers is called [i]sum-full[/i] if $A \subseteq A + A$, i.e. each element $a \in A$ is the sum of some pair of (not necessarily different) elements $b,c \in A$. A set $A$ of integers is said to be [i]zero-sum-free[/i] if $0$ is the only integer that cannot be expressed as the sum of the elements of a finite nonempty subset of $A$. Does there exist a sum-full zero-sum-free set of integers? [i]Romania (Dan Schwarz)[/i]

2004 India National Olympiad, 3

Tags: algebra

If $a$ is a real root of $x^5 - x^3 + x - 2 = 0$, show that $[a^6] =3$

2006 CHKMO, 2

Tags: inequalities , triangle inequality , geometry

Suppose there are $4n$ line segments of unit length inside a circle of radius $n$. Furthermore, a straight line $L$ is given. Prove that there exists a straight line $L'$ that is either parallel or perpendicular to $L$ and that $L'$ cuts at least two of the given line segments.

2007 iTest Tournament of Champions, 4

Tags:

Let $x_1,x_2,\ldots, x_{2007}$ be real numbers such that $-1\leq x_i\leq 1$ for $1\leq i\leq 2007$, and \[\sum_{i=1}^{2007}x_i^3 = 0.\] Find the maximum possible value of $\Big\lfloor\sum_{i=1}^{2007}x_i\Big\rfloor$.

2009 Princeton University Math Competition, 5

Tags: geometry , floor function

Lines $l$ and $m$ are perpendicular. Line $l$ partitions a convex polygon into two parts of equal area, and partitions the projection of the polygon onto $m$ into two line segments of length $a$ and $b$ respectively. Determine the maximum value of $\left\lfloor \frac{1000a}{b} \right\rfloor$. (The floor notation $\lfloor x \rfloor$ denotes largest integer not exceeding $x$)

2022 Stanford Mathematics Tournament, 5

Tags:

A net for hexagonal pyramid is constructed by placing a triangle with side lengths $x$, $x$, and $y$ on each side of a regular hexagon with side length $y$. What is the maximum volume of the pyramid formed by the net if $x+y=20$?

2011 All-Russian Olympiad, 1

Tags: floor function , combinatorics , divisibility

For some 2011 natural numbers, all the $\frac{2010\cdot 2011}{2}$ possible sums were written out on a board. Could it have happened that exactly one third of the written numbers were divisible by three and also exactly one third of them give a remainder of one when divided by three?

2023 LMT Fall, 1C

Tags: theme , geo

How many distinct triangles are there with prime side lengths and perimeter $100$? [i]Proposed by Muztaba Syed[/i] [hide=Solution][i]Solution.[/i] $\boxed{0}$ As the perimeter is even, $1$ of the sides must be $2$. Thus, the other $2$ sides are congruent by Triangle Inequality. Thus, for the perimeter to be $100$, both of the other sides must be $49$, but as $49$ is obviously composite, the answer is thus $\boxed{0}$.[/hide]

1983 IMO Longlists, 44

Tags: combinatorics

We are given twelve coins, one of which is a fake with a different mass from the other eleven. Determine that coin with three weighings and whether it is heavier or lighter than the others.

I Soros Olympiad 1994-95 (Rus + Ukr), 11.1

Tags: algebra , inequalities

Without using a calculator, prove that $$2^{1995} > 5^{856}$$

2013 Stanford Mathematics Tournament, 5

Tags: geometry , 3d geometry , prism

A polygonal prism is made from a flexible material such that the two bases are regular $2^n$-gons $(n>1)$ of the same size. The prism is bent to join the two bases together without twisting, giving a figure with $2^n$ faces. The prism is then repeatedly twisted so that each edge of one base becomes aligned with each edge of the base exactly once. For an arbitrary $n$, what is the sum of the number of faces over all of these configurations (including the non-twisted case)?

2023 Portugal MO, 4

Tags: geometry , equilateral , area

Let $[ABC]$ be an equilateral triangle and $P$ be a point on $AC$ such that $\overline{PC}= 7$. The straight line that passes through $P$ and is perpendicular to $AC$, intersects $CB$ at point $M$ and intersects $AB$ at point $Q$. The midpoint $N$ of $[MQ]$ is such that $\overline{BN} = 14$. Determine the side of the triangle $[ABC]$.

2021 Taiwan Mathematics Olympiad, 4.

Tags: geometry , incenter , geometric transformation , reflection

Let $I$ be the incenter of triangle $ABC$ and let $D$ the foot of altitude from $I$ to $BC$. Suppose the reflection point $D’$ of $D$ with respect to $I$ satisfying $\overline{AD’} = \overline{ID’}$. Let $\Gamma$ be the circle centered at $D’$ that passing through $A$ and $I$, and let $X$, $Y\neq A$ be the intersection of $\Gamma$ and $AB$, $AC$, respectively. Suppose $Z$ is a point on $\Gamma$ so that $AZ$ is perpendicular to $BC$. Prove that $AD$, $D’Z$, $XY$ concurrent at a point.

2024/2025 TOURNAMENT OF TOWNS, P4

Tags: geometry

In an equilateral triangle ${ABC}$ the segments ${ED}$ and ${GF}$ are drawn to obtain two equilateral triangles ${ADE}$ and ${GFC}$ with sides 1 and 100 (points $E$ and $G$ are on the side ${AC}$ ). The segments ${EF}$ and ${DG}$ meet at point $O$ so that the angle ${EOG}$ is equal to ${120}^{ \circ }$ . What is the length of the side of the triangle ${ABC}$ ? Mikhail Evdokimov