Olympiad problems

2017 ELMO Shortlist, 2

Tags: inequalities , algebra

Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for all real numbers $a,b,$ and $c$: (i) If $a+b+c\ge 0$ then $f(a^3)+f(b^3)+f(c^3)\ge 3f(abc).$ (ii) If $a+b+c\le 0$ then $f(a^3)+f(b^3)+f(c^3)\le 3f(abc).$ [i]Proposed by Ashwin Sah[/i]

2019 India PRMO, 25

Tags: geometry

Let $ABC$ be an isosceles triangle with $AB=BC$. A trisector of $\angle B$ meets $AC$ at $D$. If $AB,AC$ and $BD$ are integers and $AB-BD$ $=$ $3$, find $AC$.

2024 CCA Math Bonanza, T5

Tags:

Find the number of permutations of the numbers $1,1,2,2,3,3,4,4$ such that no two consecutive numbers are equal. [i]Team #5[/i]

1994 Chile National Olympiad, 1

Tags: algebra , combinatorics

A railway line is divided into ten sections by stations $E_1, E_2,..., E_{11}$. The distance between the first and the last station is $56$ km. A trip through two consecutive stations never exceeds $ 12$ km, and a trip through three consecutive stations is at least $17$ Km. Calculate the distance between $E_2$ and $E_7$.

2017 VTRMC, 6

Tags:

Let $ f ( x ) \in \mathbb { Z } [ x ] $ be a polynomial with integer coefficients such that $ f ( 1 ) = - 1 , f ( 4 ) = 2 $ and $f ( 8 ) = 34 $. Suppose $n\in\mathbb{Z}$ is an integer such that $ f ( n ) = n ^ { 2 } - 4 n - 18 $. Determine all possible values for $n$.

1954 Polish MO Finals, 6

Tags: geometry

Inside a hoop of radius $ 2r $ a disk of radius $ r $ rolls on the hoop without slipping. What line is traced by a point arbitrarily chosen on the edge of the disk?

2010 Indonesia TST, 4

Tags: combinatorics

$300$ parliament members are divided into $3$ chambers, each chamber consists of $100$ members. For every $2$ members, they either know each other or are strangers to each other.Show that no matter how they are divided into these $3$ chambers, it is always possible to choose $2$ members, each from different chamber such that there exist $17$ members from the third chamber so that all of them knows these two members, or all of them are strangers to these two members.

2025 Romania EGMO TST, P4

Tags: binomial coefficient , number theory , least common multiple

How does one show $$\text{lcm}\left(\binom{n}{1},\binom{n}{2},\ldots,\binom{n}{n}\right)=\frac{\text{lcm}(1,2,\ldots,n+1)}{n+1}$$

2020 Peru Cono Sur TST., P6

Tags: number theory

Let $a_1, a_2, a_3, \ldots$ a sequence of positive integers that satisfy the following conditions: $$a_1=1, a_{n+1}=a_n+a_{\lfloor \sqrt{n} \rfloor}, \forall n\ge 1$$ Prove that for every positive integer $k$ there exists a term $a_i$ that is divisible by $k$

2005 Romania National Olympiad, 1

Tags: ratio , quadratic , analytic geometry , vector , gauss , geometry

Let $ABCD$ be a convex quadrilateral with $AD\not\parallel BC$. Define the points $E=AD \cap BC$ and $I = AC\cap BD$. Prove that the triangles $EDC$ and $IAB$ have the same centroid if and only if $AB \parallel CD$ and $IC^{2}= IA \cdot AC$. [i]Virgil Nicula[/i]

2015 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: inequalities , algebra

Let $a$, $b$, $c$ and $d$ be real numbers such that $a+b+c+d=8$. Prove the inequality: $$\frac{a}{\sqrt[3]{8+b-d}}+\frac{b}{\sqrt[3]{8+c-a}}+\frac{c}{\sqrt[3]{8+d-b}}+\frac{d}{\sqrt[3]{8+a-c}} \geq 4$$

2021 Turkey Team Selection Test, 8

Tags: algebra , functional equation , function

Let $c$ be a real number. For all $x$ and $y$ real numbers we have, \[f(x-f(y))=f(x-y)+c(f(x)-f(y))\] and $f(x)$ is not constant. $a)$ Find all possible values of $c$. $b)$ Can $f$ be periodic?

2021 Estonia Team Selection Test, 2

Tags: geometry , right triangle , isosceles triangle

Let $ABC$ be an isosceles triangle with $BC=CA$, and let $D$ be a point inside side $AB$ such that $AD< DB$. Let $P$ and $Q$ be two points inside sides $BC$ and $CA$, respectively, such that $\angle DPB = \angle DQA = 90^{\circ}$. Let the perpendicular bisector of $PQ$ meet line segment $CQ$ at $E$, and let the circumcircles of triangles $ABC$ and $CPQ$ meet again at point $F$, different from $C$. Suppose that $P$, $E$, $F$ are collinear. Prove that $\angle ACB = 90^{\circ}$.

2008 Princeton University Math Competition, A1/B2

Tags: number theory

How many zeros are there at the end of $792!$ when written in base $10$?

2012 Mathcenter Contest + Longlist, 1 sl8

Tags: matrix , matrices , linear algebra , algebra

For matrices $A=[a_{ij}]_{m \times m}$ and $B=[b_{ij}]_{m \times m}$ where $A,B \in \mathbb{Z} ^{m \times m}$ let $A \equiv B \pmod{n}$ only if $a_{ij} \equiv b_{ij} \pmod{n}$ for every $i,j \in \{ 1,2,...,m \}$, that's $A-B=nZ$ for some $Z \in \mathbb{Z}^{m \times m}$. (The symbol $A \in \mathbb{Z} ^{m \times m}$ means that every element in $A$ is an integer.) Prove that for $A \in \mathbb{Z} ^{m \times m}$ there is $B \in \mathbb{Z} ^{m \times m}$ , where $AB \equiv I \pmod{n }$ only if $(\det (A),n)=1$ and find the value of $B$ in the form of $A$ where $I$ represents the dimensional identity matrix $m \times m$. [i](PP-nine)[/i]

1998 Italy TST, 2

Tags: median , geometry , equilateral , angle bisector , altitude

In a triangle $ABC$, points $H,M,L$ are the feet of the altitude from $C$, the median from $A$, and the angle bisector from $B$, respectively. Show that if triangle $HML$ is equilateral, then so is triangle $ABC$.

2018 CCA Math Bonanza, T10

Tags: quadratic

The irrational number $\alpha>1$ satisfies $\alpha^2-3\alpha-1=0$. Given that there is a fraction $\frac{m}{n}$ such that $n<500$ and $\left|\alpha-\frac{m}{n}\right|<3\cdot10^{-6}$, find $m$. [i]2018 CCA Math Bonanza Team Round #10[/i]

ICMC 7, 1

Tags: number theory , fibonacci

Let $F_n{}$ denote the $n{}$-th Fibonacci number. Prove that $3^{2023}$ divides \[3^2\cdot F_4+3^3\cdot F_6+3^4\cdot F_8+\dots+3^{2023}F_{4046}.\][i]Proposed by Dylan Toh[/i]

2010 Brazil Team Selection Test, 4

Tags: function , algebra , functional equation

Find all functions $f$ from the set of real numbers into the set of real numbers which satisfy for all $x$, $y$ the identity \[ f\left(xf(x+y)\right) = f\left(yf(x)\right) +x^2\] [i]Proposed by Japan[/i]

2011 Morocco National Olympiad, 2

Tags: geometry , perimeter , trigonometry , inequalities

Let $\alpha , \beta ,\gamma$ be the angles of a triangle $ABC$ of perimeter $ 2p $ and $R$ is the radius of its circumscribed circle. $(a)$ Prove that \[\cot^{2}\alpha +\cot^{2}\beta+\cot^{2}\gamma\geq 3\left(9\cdot \frac{R^{2}}{p^{2}} - 1\right).\] $(b)$ When do we have equality?

2001 Korea Junior Math Olympiad, 4

Tags: combinatorics , graph theory

Some $n \geq 3$ cities are connected with railways, so that you can travel from one city to every other, not necessarily directly. However, the railways are structured in such a way that there is only one way to get from one city to another, assuming you don't pass through the same city again. Let $A$ be the set of these cities and railways. Show that there exists a Subset of $A$, let's say $C$, such that (1) $C$ has at least $[(n+1)/2]$ cities as its element. (2) No two elements of $C$ are directly connected with railways.

2006 Pan African, 2

Tags: vieta , algebra , polynomial , rational root theorem , number theory

Let $a, b, c$ be three non-zero integers. It is known that the sums $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ and $\frac{b}{a}+\frac{c}{b}+\frac{a}{c}$ are integers. Find these sums.

2007 Kazakhstan National Olympiad, 3

Tags: modular arithmetic , number theory

Let $p$ be a prime such that $2^{p-1}\equiv 1 \pmod{p^2}$. Show that $(p-1)(p!+2^n)$ has at least three distinct prime divisors for each $n\in \mathbb{N}$ .

1986 IMO Longlists, 48

Tags: geometry

Let $P$ be a convex $1986$-gon in the plane. Let $A,D$ be interior points of two distinct sides of P and let $B,C$ be two distinct interior points of the line segment $AD$. Starting with an arbitrary point $Q_1$ on the boundary of $P$, define recursively a sequence of points $Q_n$ as follows: given $Q_n$ extend the directed line segment $Q_nB$ to meet the boundary of $P$ in a point $R_n$ and then extend $R_nC$ to meet the boundary of $P$ again in a point, which is defined to be $Q_{n+1}$. Prove that for all $n$ large enough the points $Q_n$ are on one of the sides of $P$ containing $A$ or $D$.

2013 VTRMC, Problem 2

Tags: geometry , triangle

Let $ABC$ be a right-angled triangle with $\angle ABC=90^\circ$, and let $D$ be on $AB$ such that $AD=2DB$. What is the maximum possible value of $\angle ACD$?