Olympiad problems

1997 Moldova Team Selection Test, 2

In a convex pentagon every diagonal is parallel to one side. Show that the ratios between the lengths of diagonals and the sides parallel to them are equal and find their value.

2012 Czech-Polish-Slovak Match, 1

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

Let $ABC$ be a right angled triangle with hypotenuse $AB$ and $P$ be a point on the shorter arc $AC$ of the circumcircle of triangle $ABC$. The line, perpendicuar to $CP$ and passing through $C$, intersects $AP$, $BP$ at points $K$ and $L$ respectively. Prove that the ratio of area of triangles $BKL$ and $ACP$ is independent of the position of point $P$.

2001 Junior Balkan MO, 4

Tags: geometry , perimeter , trigonometry , combinatorics

Let $N$ be a convex polygon with 1415 vertices and perimeter 2001. Prove that we can find 3 vertices of $N$ which form a triangle of area smaller than 1.

2015 Silk Road, 1 (original)

Tags: inequalities , algebra

Given positive real numbers $a,b,c,d$ such that $ \frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}=6 \quad \text{and} \quad \frac{b}{a}+\frac{c}{b}+\frac{d}{c}+\frac{a}{d}=36.$ Prove the inequality ${{a}^{2}}+{{b}^{2}}+{{c}^{2}}+{{d}^{2}}>ab+ac+ad+bc+bd+cd.$

1998 Flanders Math Olympiad, 3

Tags: linear algebra , matrix , geometry , geometric transformation , rotation

a magical $3\times3$ square is a $3\times3$ matrix containing all number from 1 to 9, and of which the sum of every row, every column, every diagonal, are all equal. Determine all magical $3\times3$ square

2010 Balkan MO Shortlist, G6

Tags: geometry , circumcircle

In a triangle $ABC$ the excircle at the side $BC$ touches $BC$ in point $D$ and the lines $AB$ and $AC$ in points $E$ and $F$ respectively. Let $P$ be the projection of $D$ on $EF$. Prove that the circumcircle $k$ of the triangle $ABC$ passes through $P$ if and only if $k$ passes through the midpoint $M$ of the segment $EF$.

2013 China Team Selection Test, 1

Tags: modular arithmetic , geometry , geometric transformation , real analysis , number theory

Let $p$ be a prime number and $a, k$ be positive integers such that $p^a<k<2p^a$. Prove that there exists a positive integer $n$ such that \[n<p^{2a}, C_n^k\equiv n\equiv k\pmod {p^a}.\]

2013 BMT Spring, 2

Tags: algebra

Find the sum of all positive integers $N$ such that $s =\sqrt[3]{2 + \sqrt{N}} + \sqrt[3]{2 - \sqrt{N}}$ is also a positive integer

VI Soros Olympiad 1999 - 2000 (Russia), 10.2

Tags: geometry , angle

In the triangle $ABC$, the point $X$ is the projection of the touchpoint of the inscribed circle to the side $BC$ on the middle line parallel to $BC$. It is known that $\angle BAC \ge 60^o$. Prove that the angle $BXC$ is obtuse.

1999 All-Russian Olympiad Regional Round, 11.5

Tags: algebra , inequalities

Are there real numbers $a, b$ and $c$ such that for all real $x$ and $y$ the following inequality holds: $$|x + a| + |x + y + b| + |y + c| > |x| + |x + y| + |y|?$$

2022 Brazil Team Selection Test, 3

Tags: number theory

Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $(d_1,d_2,\ldots,d_k)$ such that for $i=1,2,\ldots,k$, the number $d_1+d_2+\cdots+d_i$ is a perfect square.

2021 BMT, 25

Tags: geometry

Let $\vartriangle BMT$ be a triangle with $BT = 1$ and height $1$. Let $O_0$ be the centroid of $\vartriangle BMT$, and let $\overline{BO_0}$ and $\overline{TO_0}$ intersect $\overline{MT}$ and $\overline{BM}$ at $B_1$ and $T_1$, respectively. Similarly, let $O_1$ be the centroid of $\vartriangle B_1MT_1$, and in the same way, denote the centroid of $\vartriangle B_nMT_n$ by $O_n$, the intersection of $\overline{BO_n}$ with $\overline{MT}$ by $B_{n+1}$, and the intersection of $\overline{TO_n}$ with $\overline{BM}$ by $T_{n+1}$. Compute the area of quadrilateral $MBO_{2021}T$.

2014 Chile TST IMO, 1

Tags: inequalities , jensen

Given positive real numbers $a$, $b$, and $c$ such that $a+b+c \leq \frac{3}{2}$, find the minimum of \[ a+b+c + \frac{1}{a} + \frac{1}{b} + \frac{1}{c}. \]

Champions Tournament Seniors - geometry, 2004.2

Tags: geometry , circles , concurrency

Two different circles $\omega_1$ ,$\omega_2$, with centers $O_1, O_2$ respectively intersect at the points $A, B$. The line $O_1B$ intersects $\omega_2$ at the point $F (F \ne B)$, and the line $O_2B$ intersects $\omega_1$ at the point $E (E\ne B)$. A line was drawn through the point $B$, parallel to the $EF$, which intersects $\omega_1$ at the point $M (M \ne B)$, and $\omega_2$ at the point $N (N\ne B)$. Prove that the lines $ME, AB$ and $NF$ intersect at one point.

2023 District Olympiad, P3

Tags: complex number , inequalities

Let $n\geqslant 2$ be an integer. Determine all complex numbers $z{}$ which satisfy \[|z^{n+1}-z^n|\geqslant|z^{n+1}-1|+|z^{n+1}-z|.\]

1990 India Regional Mathematical Olympiad, 8

Tags: geometry , circumcircle , analytic geometry , perpendicular bisector

If the circumcenter and centroid of a triangle coincide, prove that it must be equilateral.

2023 India Regional Mathematical Olympiad, 5

Tags: inequalities , am-gm

Let $n>k>1$ be positive integers. Determine all positive real numbers $a_1, a_2, \ldots, a_n$ which satisfy $$ \sum_{i=1}^n \sqrt{\frac{k a_i^k}{(k-1) a_i^k+1}}=\sum_{i=1}^n a_i=n . $$

2021 Czech-Polish-Slovak Junior Match, 3

Tags: number theory , greatest common divisor , least common multiple , lcm , gcd

Find the number of pairs $(a, b)$ of positive integers with the property that the greatest common divisor of $a$ and $ b$ is equal to $1\cdot 2 \cdot 3\cdot ... \cdot50$, and the least common multiple of $a$ and $ b$ is $1^2 \cdot 2^2 \cdot 3^2\cdot ... \cdot 50^2$.

1999 All-Russian Olympiad, 5

Tags: modular arithmetic , number theory

Four natural numbers are such that the square of the sum of any two of them is divisible by the product of the other two numbers. Prove that at least three of these numbers are equal.

ICMC 3, 1

Tags: group theory

An [I]automorphism[/i] of a group $\left(G,*\right)$ is a bijective function $f:G\to G$ satisfying $f(x*y)=f(x)*f(y)$ for all $x,y\in G$. Find a group $(G,*)$ with fewer than $(201.6)^2=40642.56$ unique elements and exactly $2016^2$ unique automorphisms. [i]Proposed by the ICMC Problem Committee[/i]

2018 Saudi Arabia IMO TST, 2

Tags: combinatorics , perfect square , number theory

Let $n$ be an even positive integer. We fill in a number on each cell of a rectangle table of $n$ columns and multiple rows as following: i. Each row is assigned to some positive integer $a$ and its cells are filled by $0$ or $a$ (in any order); ii. The sum of all numbers in each row is $n$. Note that we cannot add any more row to the table such that the conditions (i) and (ii) still hold. Prove that if the number of $0$’s on the table is odd then the maximum odd number on the table is a perfect square.

1992 IMO Longlists, 24

Tags: function , algebra , functional equation

[i](a)[/i] Show that there exists exactly one function $ f : \mathbb Q^+ \to \mathbb Q^+$ satisfying the following conditions: [b](i)[/b] if $0 < q < \frac 12$, then $f(q)=1+f \left( \frac{q}{1-2q} \right);$ [b](ii)[/b] if $1 < q \leq 2$, then $f(q) = 1+f(q + 1);$ [b](iii)[/b] $f(q)f(1/q) = 1$ for all $q \in \mathbb Q^+.$ [i](b)[/i] Find the smallest rational number $q \in \mathbb Q^+$ such that $f(q) = \frac{19}{92}.$

1993 All-Russian Olympiad Regional Round, 9.5

Tags: number theory , diophantine

Show that the equation $x^3 +y^3 = 4(x^2y+xy^2 +1)$ has no integer solutions.

2022 Estonia Team Selection Test, 4

Tags: geometry

Let $ABCD$ be a cyclic quadrilateral whose center of the circumscribed circle is inside this quadrilateral, and its diagonals intersect in point $S{}$. Let $P{}$ and $Q{}$ be the centers of the curcimuscribed circles of triangles $ABS$ and $BCS$. The lines through the points $P{}$ and $Q{}$, which are parallel to the sides $AD$ and $CD$, respectively, intersect at the point $R$. Prove that the point $R$ lies on the line $BD$.

2018 Sharygin Geometry Olympiad, 4

Tags: geometry , combinatorics

We say that a finite set $S$ of red and green points in the plane is [i]separable[/i] if there exists a triangle $\delta$ such that all points of one colour lie strictly inside $\delta$ and all points of the other colour lie strictly outside of $\delta$. Let $A$ be a finite set of red and green points in the plane, in general position. Is it always true that if every $1000$ points in $A$ form a separable set then $A$ is also separable?