Olympiad problems

2006 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: geometry , circumcircle , parallelogram , geometric transformation , reflection , perpendicular bisector , BritishMathematicalOlympiad

The acute triangle $ABC$ with $AB\neq AC$ has circumcircle $\Gamma$, circumcenter $O$, and orthocenter $H$. The midpoint of $BC$ is $M$, and the extension of the median $AM$ intersects $\Gamma$ at $N$. The circle of diameter $AM$ intersects $\Gamma$ again at $A$ and $P$. Show that the lines $AP$, $BC$, and $OH$ are concurrent if and only if $AH = HN$.

Taiwan TST 2015 Round 1, 1

Tags: number theory , Taiwan , Taiwan TST 2015 , BritishMathematicalOlympiad

Find all primes $p,q,r$ such that $qr-1$ is divisible by $p$, $pr-1$ is divisible by $q$, $pq-1$ is divisible by $r$.

2014 BMO TST, 4

Tags: function , algebra solved , algebra , BritishMathematicalOlympiad

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(x)f(y)=f(x+y)+xy$ for all $x,y\in \mathbb{R}$.

2015 Romania Team Selection Tests, 1

Tags: Miquel point , equal angles , geometry , Romanian TST , BritishMathematicalOlympiad , Spiral Similarity

Let $ABC$ be a triangle. Let $P_1$ and $P_2$ be points on the side $AB$ such that $P_2$ lies on the segment $BP_1$ and $AP_1 = BP_2$; similarly, let $Q_1$ and $Q_2$ be points on the side $BC$ such that $Q_2$ lies on the segment $BQ_1$ and $BQ_1 = CQ_2$. The segments $P_1Q_2$ and $P_2Q_1$ meet at $R$, and the circles $P_1P_2R$ and $Q_1Q_2R$ meet again at $S$, situated inside triangle $P_1Q_1R$. Finally, let $M$ be the midpoint of the side $AC$. Prove that the angles $P_1RS$ and $Q_1RM$ are equal.

2004 India IMO Training Camp, 4

Tags: function , arithmetic sequence , number theory solved , number theory , BritishMathematicalOlympiad

Let $f$ be a bijection of the set of all natural numbers on to itself. Prove that there exists positive integers $a < a+d < a+ 2d$ such that $f(a) < f(a+d) <f(a+2d)$

2014 Contests, 4

Tags: function , algebra solved , algebra , BritishMathematicalOlympiad

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(x)f(y)=f(x+y)+xy$ for all $x,y\in \mathbb{R}$.

2015 British Mathematical Olympiad Round 1, 1

Tags: algebra , BritishMathematicalOlympiad

On Thursday 1st January 2015, Anna buys one book and one shelf. For the next two years she buys one book every day and one shelf on alternate Thursdays, so she next buys a shelf on 15th January. On how many days in the period Thursday 1st January 2015 until (and including) Saturday 31st December 2016 is it possible for Anna to put all her books on all her shelves, so that there is an equal number of books on each shelf?

2021 Romania Team Selection Test, 2

Tags: number theory , Romanian TST , TST , BritishMathematicalOlympiad

For any positive integer $n>1$, let $p(n)$ be the greatest prime factor of $n$. Find all the triplets of distinct positive integers $(x,y,z)$ which satisfy the following properties: $x,y$ and $z$ form an arithmetic progression, and $p(xyz)\leq 3.$

2021 ISI Entrance Examination, 7

Tags: algebra , calculus , inequalities , BritishMathematicalOlympiad

Let $a, b, c$ be three real numbers which are roots of a cubic polynomial, and satisfy $a+b+c=6$ and $ab+bc+ca=9$. Suppose $a<b<c$. Show that $$0<a<1<b<3<c<4.$$

2020 Stars of Mathematics, 2

Tags: geometry , romania , BritishMathematicalOlympiad

Let $ABC$ be a triangle, let $I$ be its incentre and let $D$ be the orthogonal projection of $I$ on $BC.$ The circle $\odot(ABC)$ crosses the line $AI$ again at $M,$ and the line $DM$ again at $N.$ Prove that the lines $AN$ and $IN$ are perpendicular. [i]Freddie Illingworth & Dominic Yeo[/i]

2004 India IMO Training Camp, 4

Tags: function , arithmetic sequence , number theory solved , number theory , BritishMathematicalOlympiad

Let $f$ be a bijection of the set of all natural numbers on to itself. Prove that there exists positive integers $a < a+d < a+ 2d$ such that $f(a) < f(a+d) <f(a+2d)$

2020 Switzerland - Final Round, 5

Tags: Diophantine equation , factorial , number theory , diophantine , BritishMathematicalOlympiad

Find all the positive integers $a, b, c$ such that $$a! \cdot b! = a! + b! + c!$$

2021 Winter Stars of Mathematics, 2

Tags: geometry , romania , BritishMathematicalOlympiad

Let $ABC$ be a triangle, let $I$ be its incentre and let $D$ be the orthogonal projection of $I$ on $BC.$ The circle $\odot(ABC)$ crosses the line $AI$ again at $M,$ and the line $DM$ again at $N.$ Prove that the lines $AN$ and $IN$ are perpendicular. [i]Freddie Illingworth & Dominic Yeo[/i]