Olympiad problems

1989 Romania Team Selection Test, 1

Let $F$ be the set of all functions $f : N \to N$ which satisfy $f(f(x))-2 f(x)+x = 0$ for all $x \in N$. Determine the set $A =\{ f(1989) | f \in F\}$.

1996 USAMO, 5

Tags: isosceles triangle , trigonometry , law of cosines , law of sines , reflection , geometry

Let $ABC$ be a triangle, and $M$ an interior point such that $\angle MAB=10^\circ$, $\angle MBA=20^\circ$, $\angle MAC=40^\circ$ and $\angle MCA=30^\circ$. Prove that the triangle is isosceles.

1989 Brazil National Olympiad, 3

Tags: function , algebra

A function $f$, defined for the set of integers, is such that $f(x)=x-10$ if $x>100$ and $f(x)=f(f(x+11))$ if $x \leq 100$. Determine, justifying your answer, the set of all possible values for $f$.

1988 India National Olympiad, 4

Tags: function , inequalities

If $ a$ and $ b$ are positive and $ a \plus{} b \equal{} 1$, prove that \[ \left(a\plus{}\frac{1}{a}\right)^2\plus{}\left(b\plus{}\frac{1}{b}\right)^2 \geq \frac{25}{2}\]

2003 IMO, 4

Tags: geometry , angle bisector , cyclic quadrilateral , quadrilateral , concurrency

Let $ABCD$ be a cyclic quadrilateral. Let $P$, $Q$, $R$ be the feet of the perpendiculars from $D$ to the lines $BC$, $CA$, $AB$, respectively. Show that $PQ=QR$ if and only if the bisectors of $\angle ABC$ and $\angle ADC$ are concurrent with $AC$.

2024 Sharygin Geometry Olympiad, 10.5

Tags: geo , geometry

The incircle of a right-angled triangle $ABC$ touches the hypothenuse $AB$ at point $T$. The squares $ATMP$ and $BTNQ$ lie outside the triangle. Prove that the areas of triangles $ABC$ and $TPQ$ are equal.

2017 IFYM, Sozopol, 8

Tags: geometry

$k$ is the circumscribed circle of $\Delta ABC$. $M$ and $N$ are arbitrary points on sides $CA$ and $CB$, and $MN$ intersects $k$ in points $U$ and $V$. Prove that the middle points of $BM$,$AN$,$MN$, and $UV$ lie on one circle.

2007 Baltic Way, 20

Tags: geometry , search , function , number theory

Let $a$ and $b$ be positive integers, $b<a$, such that $a^3+b^3+ab$ is divisible by $ab(a-b)$. Prove that $ab$ is a perfect cube.

KoMaL A Problems 2018/2019, A. 754

Tags: geometry

Let $P$ be a point inside the acute triangle $ABC,$ and let $Q$ be the isogonal conjugate of $P.$ Let $L,M$ and $N$ be the midpoints of the shorter arcs $BC,CA$ and $AB$ of the circumcircle of $ABC,$ respectively. Let $X_A$ be the intersection of ray $LQ$ and circle $(PBC),$ let $X_B$ be the intersection of ray $MQ$ and circle $PCA,$ and let $X_C$ be the intersection of ray $NQ$ and circle $(PAB).$ Prove that $P,X_A,X_B$ and $X_C$ are concyclic or coincide. [i]Proposed by Gustavo Cruz (São Paulo)[/i]

1971 IMO, 3

Tags: linear algebra , matrix , algebra , combinatorial inequality , combinatorics

Let $ A \equal{} (a_{ij})$, where $ i,j \equal{} 1,2,\ldots,n$, be a square matrix with all $ a_{ij}$ non-negative integers. For each $ i,j$ such that $ a_{ij} \equal{} 0$, the sum of the elements in the $ i$th row and the $ j$th column is at least $ n$. Prove that the sum of all the elements in the matrix is at least $ \frac {n^2}{2}$.

1981 Bulgaria National Olympiad, Problem 2

Tags: geometry , angle , triangle

Let $ABC$ be a triangle such that the altitude $CH$ and the sides $CA,CB$ are respectively equal to a side and two distinct diagonals of a regular heptagon. Prove that $\angle ACB<120^\circ$.

1992 French Mathematical Olympiad, Problem 5

Tags: number theory , floor function

Determine the number of digits $1$ in the integer part of $\frac{10^{1992}}{10^{83}+7}$.

2013 All-Russian Olympiad, 1

Tags: algebra

Given three distinct real numbers $a$, $b$, and $c$, show that at least two of the three following equations \[(x-a)(x-b)=x-c\] \[(x-c)(x-b)=x-a\] \[(x-c)(x-a)=x-b\] have real solutions.

2014 Harvard-MIT Mathematics Tournament, 1

Tags: hmmt , geometry , analytic geometry , symmetry , pythagorean theorem , power of a point

Let $O_1$ and $O_2$ be concentric circles with radii 4 and 6, respectively. A chord $AB$ is drawn in $O_1$ with length $2$. Extend $AB$ to intersect $O_2$ in points $C$ and $D$. Find $CD$.

2007 Hungary-Israel Binational, 2

Tags: ellipse , geometry , 3d geometry , sphere , geometric transformation , rotation , conic

Given is an ellipse $ e$ in the plane. Find the locus of all points $ P$ in space such that the cone of apex $ P$ and directrix $ e$ is a right circular cone.

Ukrainian TYM Qualifying - geometry, I.10

Tags: ratio , max , geometric inequality , geometry

Given a circle of radius $R$. Find the ratio of the largest area of the circumscribed quadrilateral to the smallest area of the inscribed one.

2023 China Girls Math Olympiad, 7

Tags: number theory

Let $p$ be an odd prime. Suppose that positive integers $a,b,m,r$ satisfy $p\nmid ab$ and $ab > m^2$. Prove that there exists at most one pair of coprime positive integers $(x,y)$ such that $ax^2+by^2=mp^r$.

2017 Iran MO (3rd round), 2

Tags: combinatorics

Two persons are playing the following game on a $n\times m$ table, with drawn lines: Person $\#1$ starts the game. Each person in their move, folds the table on one of its lines. The one that could not fold the table on their turn loses the game. Who has a winning strategy?

2004 Tournament Of Towns, 1

Tags: arithmetic progression , sum , power of 2 , number theory

The sum of all terms of a finite arithmetical progression of integers is a power of two. Prove that the number of terms is also a power of two.

2024-IMOC, A1

Tags: algebra , inequalities , series

Given a positive integer $N$. Prove that \[\sum_{m=1}^N \sum_{n=1}^N \frac{1}{mn^2+m^2n+2mn}<\frac{7}{4}.\] [i]Proposed by tan-1[/i]

2016 Romania Team Selection Tests, 4

Tags: number theory

Determine the integers $k\geq 2$ for which the sequence $\Big\{ \binom{2n}{n} \pmod{k}\Big\}_{n\in \mathbb{Z}_{\geq 0}}$ is eventually periodic.

2012 Online Math Open Problems, 8

Tags: trigonometry , trig identities , law of cosines , geometry

In triangle $ABC$ let $D$ be the foot of the altitude from $A$. Suppose that $AD = 4$, $BD = 3$, $CD = 2$, and $AB$ is extended past $B$ to a point $E$ such that $BE = 5$. Determine the value of $CE^2$. [i]Ray Li.[/i] [hide="Clarifications"][list=1][*]Triangle $ABC$ is acute.[/list][/hide]

2024 Bulgarian Autumn Math Competition, 9.2

Tags: geometry

Let $ABC$ be an acute scalene triangle with altitudes $AE$ $(E \in BC)$ and $BD$ $(D \in AC)$. Point $M$ lies on $AC$, such that $AM = AE$ and $C,A$ and $M$ lie in this order. Point $L$ lies on $BC$, such that $BL=BD$ and $C,B$ and $L$ lie in this order. Let $P$ be the midpoint of $DE$. Prove that $EM,DL$ and the perpendicular from $P$ to $AB$ are concurrent.

2023 NMTC Junior, P7

Tags: number theory

Let $n$ be a positive integer; and $S(n)$ denote the sum of all digits in the decimal representation of $n$. A positive integer obtained by removing one or several digits from the right hand end of the decimal representation of $n$ is called the [i]truncation[/i] of $n$. The sum of all truncations of $n$ is denoted as $T(n)$. Prove that $S(n)+9T(n)=n$

MOAA Team Rounds, 2023.12

Tags:

Let $N$ be the number of $105$-digit positive integers that contain the digit 1 an odd number of times. Find the remainder when $N$ is divided by $1000$. [i]Proposed by Harry Kim[/i]