Olympiad problems

2025 Bulgarian Spring Mathematical Competition, 9.2

Let $ABC$ be an acute scalene triangle inscribed in a circle $ \Gamma $. The angle bisector of $ \angle BAC $ intersects $ BC $ at $ L $ and $ \Gamma $ at $ S $. The point $ M $ is the midpoint of $ AL $. Let $ AD $ be the altitude in $ \triangle ABC $, and the circumcircle of $ \triangle DSL $ intersects $ \Gamma $ again at $ P $. Let $ N $ be the midpoint of $ BC $, and let $ K $ be the reflection of $ D $ with respect to $ N $. Prove that the triangles $ \triangle MPS $ and $ \triangle ADK $ are similar.

2012 India Regional Mathematical Olympiad, 5

Tags: ratio , geometry , trigonometry , area of a triangle

Let $ABC$ be a triangle. Let $D,E$ be points on the segment $BC$ such that $BD=DE=EC$. Let $F$ be the mid-point of $AC$. Let $BF$ intersect $AD$ in $P$ and $AE$ in $Q$ respectively. Determine the ratio of the area of the triangle $APQ$ to that of the quadrilateral $PDEQ$.

2011 AMC 12/AHSME, 21

Tags: quadratic , algebra , difference of squares , special factorizations

The arithmetic mean of two distinct positive integers $x$ and $y$ is a two-digit integer. The geometric mean of $x$ and $y$ is obtained by reversing the digits of the arithmetic mean. What is $|x-y|$? $ \textbf{(A)}\ 24 \qquad \textbf{(B)}\ 48 \qquad \textbf{(C)}\ 54 \qquad \textbf{(D)}\ 66 \qquad \textbf{(E)}\ 70 $

1997 Turkey Junior National Olympiad, 2

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

Let $ABC$ be a triangle with $|AB|=|AC|=26$, $|BC|=20$. The altitudes of $\triangle ABC$ from $A$ and $B$ cut the opposite sides at $D$ and $E$, respectively. Calculate the radius of the circle passing through $D$ and tangent to $AC$ at $E$.

2020 Belarusian National Olympiad, 11.7

Tags: geometry

Line $AL$ is an angle bisector in the triangle $ABC$ ($L \in BC$), and $\omega$ is its circumcircle. Chords $X_1X_2$ and $Y_1Y_2$ pass through $L$ such that points $X_1,Y_1$ and $A$ lie in the same half-plane with respect to $BC$. Lines $X_1Y_2$ and $Y_1X_2$ intersect side $BC$ in points $Z_1$ and $Z_2$ respectively. Prove that $\angle BAZ_1=\angle CAZ_2$.

2018 Malaysia National Olympiad, A1

Tags: geometry , area

Quadrilateral $ABCD$ is neither a kite nor a rectangle. It is known that its sidelengths are integers, $AB = 6$, $BC = 7$, and $\angle B = \angle D = 90^o$. Find the area of$ ABCD$.

2010 Nordic, 1

Tags: inequalities , function , modular arithmetic , number theory , relatively prime , algebra

A function $f : \mathbb{Z}_+ \to \mathbb{Z}_+$, where $\mathbb{Z}_+$ is the set of positive integers, is non-decreasing and satisfies $f(mn) = f(m)f(n)$ for all relatively prime positive integers $m$ and $n$. Prove that $f(8)f(13) \ge (f(10))^2$.

1991 National High School Mathematics League, 9

Tags:

Devide all odd numbers from small to large into groups: there are $2n-1$ numbers in the $n$th group. For example: the first group is $\{1\}$, the second group is $\{3,5,7\}$, the third group is $\{9,11,13,15,17\}$. Then, $1991$ is in group________.

2002 Greece Junior Math Olympiad, 2

Tags: ratio

In the Mathematical Competition of HMS (Hellenic Mathematical Society) take part boys and girls who are divided into two groups : [i]Juniors[/i] and [i]seniors.[/i]The number of the boys taking part of this year competition is 55% of the number of all participants. The ratio of the number of juniors boys to the number of senior boys is equal to the ratio of the number of juniors to the number of seniors. Find the ratio of the number of junior boys to the number of junior girls.

2014 Hanoi Open Mathematics Competitions, 4

Tags: number theory , diophantine equation , diophantine , prime

If $p$ is a prime number such that there exist positive integers $a$ and $b$ such that $\frac{1}{p}=\frac{1}{a^2}+\frac{1}{b^2}$ then $p$ is (A): $3$, (B): $5$, (C): $11$, (D): $7$, (E) None of the above.

2007 Singapore Senior Math Olympiad, 5

Tags: inequalities , min , max , algebra

Find the maximum and minimum of $x + y$ such that $x + y = \sqrt{2x-1}+\sqrt{4y+3}$

2018 Harvard-MIT Mathematics Tournament, 6

Tags: combinatorics

Farmer James invents a new currency, such that for every positive integer $n\le 6$, there exists an $n$-coin worth $n!$ cents. Furthermore, he has exactly $n$ copies of each $n$-coin. An integer $k$ is said to be [i]nice[/i] if Farmer James can make $k$ cents using at least one copy of each type of coin. How many positive integers less than 2018 are nice?

2020 Taiwan TST Round 3, 6

Tags: combinatorics

Alice has a map of Wonderland, a country consisting of $n \geq 2$ towns. For every pair of towns, there is a narrow road going from one town to the other. One day, all the roads are declared to be “one way” only. Alice has no information on the direction of the roads, but the King of Hearts has offered to help her. She is allowed to ask him a number of questions. For each question in turn, Alice chooses a pair of towns and the King of Hearts tells her the direction of the road connecting those two towns. Alice wants to know whether there is at least one town in Wonderland with at most one outgoing road. Prove that she can always find out by asking at most $4n$ questions.

1969 Spain Mathematical Olympiad, 1

Tags: geometry , locus , inversion

Find the locus of the centers of the inversions that transform two points $A, B$ of a given circle $\gamma$ , at diametrically opposite points of the inverse circles of $\gamma$ .

2007 International Zhautykov Olympiad, 2

Tags: trigonometry , geometric transformation , reflection , geometry

Let $ABCD$ be a convex quadrilateral, with $\angle BAC=\angle DAC$ and $M$ a point inside such that $\angle MBA=\angle MCD$ and $\angle MBC=\angle MDC$. Show that the angle $\angle ADC$ is equal to $\angle BMC$ or $\angle AMB$.

2005 France Team Selection Test, 2

Tags: geometry , circumcircle , perimeter , inradius , inequalities , trigonometry , pythagorean theorem

Two right angled triangles are given, such that the incircle of the first one is equal to the circumcircle of the second one. Let $S$ (respectively $S'$) be the area of the first triangle (respectively of the second triangle). Prove that $\frac{S}{S'}\geq 3+2\sqrt{2}$.

2016 LMT, 2

Tags:

Mater is confused and starts going around the track in the wrong direction. He can go around 7 times in an hour. Lightning and Chick start in the same place at Mater and at the same time, both going the correct direction. Lightning can go around 91 times per hour, while Chick can go around 84 times per hour. When Lightning passes Chick for the third time, how many times will he have passed Mater (if Lightning is passing Mater just as he passes Chick for the third time, count this as passing Mater)? [i]Proposed by Matthew Weiss

1980 Kurschak Competition, 1

Tags: combinatorics , coloring , combinatorial geometry

The points of space are coloured with five colours, with all colours being used. Prove that some plane contains four points of different colours.

2010 IMO Shortlist, 4

Tags: geometry , inversion , p2

Given a triangle $ABC$, with $I$ as its incenter and $\Gamma$ as its circumcircle, $AI$ intersects $\Gamma$ again at $D$. Let $E$ be a point on the arc $BDC$, and $F$ a point on the segment $BC$, such that $\angle BAF=\angle CAE < \dfrac12\angle BAC$. If $G$ is the midpoint of $IF$, prove that the meeting point of the lines $EI$ and $DG$ lies on $\Gamma$. [i]Proposed by Tai Wai Ming and Wang Chongli, Hong Kong[/i]

1976 Vietnam National Olympiad, 3

Tags: geometry , geometric inequality , tetrahedron , area of a triangle , distance

$P$ is a point inside the triangle $ABC$. The perpendicular distances from $P$ to the three sides have product $p$. Show that $p \le \frac{ 8 S^3}{27abc}$, where $S =$ area $ABC$ and $a, b, c$ are the sides. Prove a similar result for a tetrahedron.

2025 Harvard-MIT Mathematics Tournament, 2

Tags: geometry

In a two-dimensional cave with a parallel floor and ceiling, two stalactites of lengths $16$ and $36$ hang perpendicularly from the ceiling, while two stalagmites of heights $25$ and $49$ grow perpendicularly from the ground. If the tips of these four structures form the vertices of a square in some order, compute the height of the cave.

PEN H Problems, 72

Tags: diophantine equation , induction , number theory , relatively prime , absolute value

Find all pairs $(x, y)$ of positive rational numbers such that $x^{y}=y^{x}$.

1992 Nordic, 2

Tags: integer polynomial , polynomial division , algebra

Let $n > 1$ be an integer and let $a_1, a_2,... , a_n$ be $n$ different integers. Show that the polynomial $f(x) = (x -a_1)(x - a_2)\cdot ... \cdot (x -a_n) - 1$ is not divisible by any polynomial with integer coefficients and of degree greater than zero but less than $n$ and such that the highest power of $x$ has coefficient $1$.

2013 Turkmenistan National Math Olympiad, 2

Tags: induction , algebra

Sequence $x_1 , x_2 , ..., $ with $x_1=20$ ; $x_2=12$ for all $n\geq 1$ such that $x_{n+2}=x_n+x_{n+1}+2\sqrt{x_{n}*x_{n+1}+121} $then prove that $x_{2013}$ is an integer number.

1997 Moldova Team Selection Test, 4

Tags:

Let $A=\{1,2,\ldots,1997\}$ be a set. Find the samllest integer $k>1$ such that in each subset $M{}$ of $A{}$, which cointain $k{}$ elements, there is a multiple of the smallest element from $M{}$, different from itself.