Olympiad problems

2018 NZMOC Camp Selection Problems, 10

Find all functions $f : R \to R$ such that $$f(x)f(y) = f(xy + 1) + f(x - y) - 2$$ for all $x, y \in R$.

1998 IMO Shortlist, 5

Tags: geometry , reflection , homothety , complex bash

Let $ABC$ be a triangle, $H$ its orthocenter, $O$ its circumcenter, and $R$ its circumradius. Let $D$ be the reflection of the point $A$ across the line $BC$, let $E$ be the reflection of the point $B$ across the line $CA$, and let $F$ be the reflection of the point $C$ across the line $AB$. Prove that the points $D$, $E$ and $F$ are collinear if and only if $OH=2R$.

1984 National High School Mathematics League, 5

Tags: function

If $a>0,a\neq1$, $F(x)$ is an odd function. $G(x)=F(x)\cdot(\frac{1}{a^x-1}+\frac{1}{2})$, then $G(x)$ is $\text{(A)}$ odd function $\text{(B)}$ even function $\text{(C)}$ not odd or even function $\text{(D)}$ not sure

2023 Brazil EGMO Team Selection Test, 3

Tags: geometry , circumcircle , circumcenter , incenter , incircle , concurrency

Let $\Delta ABC$ be a triangle and $L$ be the foot of the bisector of $\angle A$. Let $O_1$ and $O_2$ be the circumcenters of $\triangle ABL$ and $\triangle ACL$ respectively and let $B_1$ and $C_1$ be the projections of $C$ and $B$ through the bisectors of the angles $\angle B$ and $\angle C$ respectively. The incircle of $\Delta ABC$ touches $AC$ and $AB$ at points $B_0$ and $C_0$ respectively and the bisectors of angles $\angle B$ and $\angle C$ meet the perpendicular bisector of $AL$ at points $Q$ and $P$ respectively. Prove that the five lines $PC_0, QB_0, O_1C_1, O_2B_1$ and $BC$ are all concurrent.

2013 Taiwan TST Round 1, 6

Tags: geometry , homothety , incenter

Let $ABCD$ be a convex quadrilateral with non-parallel sides $BC$ and $AD$. Assume that there is a point $E$ on the side $BC$ such that the quadrilaterals $ABED$ and $AECD$ are circumscribed. Prove that there is a point $F$ on the side $AD$ such that the quadrilaterals $ABCF$ and $BCDF$ are circumscribed if and only if $AB$ is parallel to $CD$.

2024 Portugal MO, 2

Tags: geometry

Let $ABC$ be a triangle and $D,E$ and $F$ the midpoints of sides $BC, AC$ and $BC$. Medians $AD$ and $BE$ are perpendicular, $AD = 12$ and $BE = 9$. What is the value of $CF$?

1980 Austrian-Polish Competition, 9

Tags: tangent , geometry , equal segments , circles

Through the endpoints $A$ and $B$ of a diameter $AB$ of a given circle, the tangents $\ell$ and $m$ have been drawn. Let $C\ne A$ be a point on $\ell$ and let $q_1,q_2$ be two rays from $C$. Ray $q_i$ cuts the circle in $D_i$ and $E_i$ with $D_i$ between $C$ and $E_i, i = 1,2$. Rays $AD_1,AD_2,AE_1,AE_2$ meet $m$ in the respective points $M_1,M_2,N_1,N_2$. Prove that $M_1M_2 = N_1N_2$.

1987 China Team Selection Test, 1

Tags: combinatorics , number theoretic functions , set

a.) For all positive integer $k$ find the smallest positive integer $f(k)$ such that $5$ sets $s_1,s_2, \ldots , s_5$ exist satisfying: [b]i.[/b] each has $k$ elements; [b]ii.[/b] $s_i$ and $s_{i+1}$ are disjoint for $i=1,2,...,5$ ($s_6=s_1$) [b]iii.[/b] the union of the $5$ sets has exactly $f(k)$ elements. b.) Generalisation: Consider $n \geq 3$ sets instead of $5$.

2010 Laurențiu Panaitopol, Tulcea, 2

Tags: freshman s dream , frobenius , nilpotence

Let be two $ n\times n $ complex matrices $ A,B $ satisfying the equations $ (A+B)^2=A^2+B^2 $ and $ (A+B)^4=A^4+B^4. $ Show that $ (AB)^2=0. $

2008 Peru IMO TST, 6

Tags: number theory

We say that a positive integer is happy if can expressed in the form $ (a^{2}b)/(a \minus{} b)$ where $ a > b > 0$ are integers. We also say that a positive integer $ m$ is evil if it doesn't a happy integer $ n$ such that $ d(n) \equal{} m$. Prove that all integers happy and evil are a power of $ 4$.

2021 MOAA, 1

Tags: team

The value of \[\frac{1}{20}-\frac{1}{21}+\frac{1}{20\times 21}\] can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Nathan Xiong[/i]

2017 Regional Olympiad of Mexico West, 6

Tags: number theory , divide

A [i]change [/i] in a natural number $n$ consists of adding a pair of zeros between two digits or at the end of the decimal representation of $n$. A [i]countryman [/i] of $n$ is a number that can be obtained from one or more changes in $n$. For example. $40041$, $4410000$ and $4004001$ are all countrymen from $441$. Determine all the natural numbers $n$ for which there is a natural number m with the property that $n$ divides $m$ and all the countrymen of $m$.

2018 Dutch BxMO TST, 5

Tags: equation , algebra

Let $n$ be a positive integer. Determine all positive real numbers $x$ satisfying $nx^2 +\frac{2^2}{x + 1}+\frac{3^2}{x + 2}+...+\frac{(n + 1)^2}{x + n}= nx + \frac{n(n + 3)}{2}$

2013 Sharygin Geometry Olympiad, 6

Tags: geometry , orthocenter , circles

A line $\ell$ passes through the vertex $B$ of a regular triangle $ABC$. A circle $\omega_a$ centered at $I_a$ is tangent to $BC$ at point $A_1$, and is also tangent to the lines $\ell$ and $AC$. A circle $\omega_c$ centered at $I_c$ is tangent to $BA$ at point $C_1$, and is also tangent to the lines $\ell$ and $AC$. Prove that the orthocenter of triangle $A_1BC_1$ lies on the line $I_aI_c$.

2014 Korea National Olympiad, 1

Tags: geometry

There is a convex quadrilateral $ ABCD $ which satisfies $ \angle A=\angle D $. Let the midpoints of $ AB, AD, CD $ be $ L,M,N $. Let's say the intersection point of $ AC, BD $ be $ E $ . Let's say point $ F $ which lies on $ \overrightarrow{ME} $ satisfies $ \overline{ME}\times \overline{MF}=\overline{MA}^{2} $. Prove that $ \angle LFM=\angle MFN $. :)

2023 Costa Rica - Final Round, 3.3

Tags: geometry , polygon , concurrency

Let $ABCD \dots KLMN$ be a regular polygon with $14$ sides. Show that the diagonals $AE$, $BG$, and $CK$ are concurrent.

2008 Germany Team Selection Test, 1

Tags: inequalities , algebra

Determine $ Q \in \mathbb{R}$ which is so big that a sequence with non-negative reals elements $ a_1 ,a_2, \ldots$ which satisfies the following two conditions: [b](i)[/b] $ \forall m,n \geq 1$ we have $ a_{m \plus{} n} \leq 2 \left(a_m \plus{} a_n \right)$ [b](ii)[/b] $ \forall k \geq 0$ we have $ a_{2^k} \leq \frac {1}{(k \plus{} 1)^{2008}}$ such that for each sequence element we have the inequality $ a_n \leq Q.$

2016 Harvard-MIT Mathematics Tournament, 8

Tags: hmmt

Let $P_1P_2 \ldots P_8$ be a convex octagon. An integer $i$ is chosen uniformly at random from $1$ to $7$, inclusive. For each vertex of the octagon, the line between that vertex and the vertex $i$ vertices to the right is painted red. What is the expected number times two red lines intersect at a point that is not one of the vertices, given that no three diagonals are concurrent?

2021 BMT, 8

Tags: expected value

Consider the randomly generated base 10 real number $r = 0.\overline{p_0p_1p_2\ldots}$, where each $p_i$ is a digit from $0$ to $9$, inclusive, generated as follows: $p_0$ is generated uniformly at random from $0$ to $9$, inclusive, and for all $i \geq 0$, $p_{i + 1}$ is generated uniformly at random from $p_i$ to $9$, inclusive. Compute the expected value of $r$.

1998 Belarus Team Selection Test, 2

Tags: algebra , set

Find all finite sets $M \subset R$ containing at least two elements such that $(2a/3 -b^2) \in M$ for any two different elements $a,b \in M$.

2018 Silk Road, 2

Tags: functional equation , algebra

Find all functions $f:\ \mathbb{R}\rightarrow\mathbb{R}$ such that for any real number $x$ the equalities are true: $f\left(x+1\right)=1+f(x)$ and $f\left(x^4-x^2\right)=f^4(x)-f^2(x).$ [url=http://matol.kz/comments/3373/show]source[/url]

2023 Macedonian Team Selection Test, Problem 3

Tags: function , algebra

Let $f:\mathbb{N} \rightarrow \mathbb{N}$ be a monotonically increasing function over the natural numbers, such that $f(f(n))=n^{2}$. What is the smallest, and what is the largest value that $f(2023)$ can take? [i]Proposed by Ilija Jovcheski[/i]

2022-23 IOQM India, 19

Tags: combinatorics , counting

Consider a string of $n$ $1's$. We wish to place some $+$ signs in between so that the sum is $1000$. For instance, if $n=190$, one may put $+$ signs so as to get $11$ ninety times and $1$ ten times , and get the sum $1000$. If $a$ is the number of positive integers $n$ for which it is possible to place $+$ signs so as to get the sum $1000$, then find the sum of digits of $a$.

2014 Sharygin Geometry Olympiad, 4

Tags: circles , geometry , common tangent

Let $H$ be the orthocenter of a triangle $ABC$. Given that $H$ lies on the incircle of $ABC$ , prove that three circles with centers $A, B, C$ and radii $AH, BH, CH$ have a common tangent. (Mahdi Etesami Fard)

2007 Harvard-MIT Mathematics Tournament, 18

Tags: geometry , circumcircle

Convex quadrilateral $ABCD$ has right angles $\angle A$ and $\angle C$ and is such that $AB=BC$ and $AD=CD$. The diagonals $AC$ and $BD$ intersect at point $M$. Points $P$ and $Q$ lie on the circumcircle of triangle $AMB$ and segment $CD$, respectively, such that points $P$, $M$, and $Q$ are collinear. Suppose that $m\angle ABC=160^\circ$ and $m\angle QMC=40^\circ$. Find $MP\cdot MQ$, given that $MC=6$.