Olympiad problems

2021 Brazil Team Selection Test, 2

Let $ABC$ be an isosceles triangle with $BC=CA$, and let $D$ be a point inside side $AB$ such that $AD< DB$. Let $P$ and $Q$ be two points inside sides $BC$ and $CA$, respectively, such that $\angle DPB = \angle DQA = 90^{\circ}$. Let the perpendicular bisector of $PQ$ meet line segment $CQ$ at $E$, and let the circumcircles of triangles $ABC$ and $CPQ$ meet again at point $F$, different from $C$. Suppose that $P$, $E$, $F$ are collinear. Prove that $\angle ACB = 90^{\circ}$.

2014 Lithuania Team Selection Test, 3

Tags: algebra , linear algebra , matrix , quadratic , system of equations

Given such positive real numbers $a, b$ and $c$, that the system of equations: $ \{\begin{matrix}a^2x+b^2y+c^2z=1&&\\xy+yz+zx=1&&\end{matrix} $ has exactly one solution of real numbers $(x, y, z)$. Prove, that there is a triangle, which borders lengths are equal to $a, b$ and $c$.

2021 Peru PAGMO TST, P4

Tags: combinatorics

A whole number is written on each square of a board of $2019\times 2021$ squares. If the number written in each square is equal to the arithmetic mean of the numbers written in two of its neighboring squares, how many different numbers written on the blackboard can there be at most? Note: Two squares on the board are neighbors when they have a common side.

2017 Korea Winter Program Practice Test, 2

Tags: combinatorial game , combinatorics

Alice and Bob play a game. There are $100$ gold coins, $100$ silver coins, and $100$ bronze coins. Players take turns to take at least one coin, but they cannot take two or more coins of same kind at once. Alice goes first. The player who cannot take any coin loses. Who has a winning strategy?

2023 Ukraine National Mathematical Olympiad, 8.5

Tags: algebra , equality

Do there exist $10$ real numbers, not all of which are equal, each of which is equal to the square of the sum of the remaining $9$ numbers? [i]Proposed by Bogdan Rublov[/i]

2020 MIG, 5

Tags:

What is the side length, in meters, of a square with area $49 \text{ m}^2$? $\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }7$

VI Soros Olympiad 1999 - 2000 (Russia), 10.3

Tags: combinatorics

Some pairs of cities in the country are connected by airlines, and some are not. But every city has an airport, from which you can get to any other city, making no more than one transfer. A tourist who wants to make a round trip through several cities of the country will have to fly around at least five cities. Prove that the same number of airlines depart from each city of the country (If there is an airline from one city to another, then there is also one from the second to the first. A circular trip is a route that passes through at least three cities, starting and ending in same city, other cities are not repeated in it)

1999 VJIMC, Problem 2

Tags: number theory

Find all natural numbers $n\ge1$ such that the implication $$(11\mid a^n+b^n)\implies(11\mid a\wedge11\mid b)$$holds for any two natural numbers $a$ and $b$.

2009 Germany Team Selection Test, 2

Tags: geometry , circumcircle , homothety , trigonometry , quadrilateral , inversion

Let $ ABCD$ be a convex quadrilateral and let $ P$ and $ Q$ be points in $ ABCD$ such that $ PQDA$ and $ QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $ E$ on the line segment $ PQ$ such that $ \angle PAE \equal{} \angle QDE$ and $ \angle PBE \equal{} \angle QCE$. Show that the quadrilateral $ ABCD$ is cyclic. [i]Proposed by John Cuya, Peru[/i]

1994 Bundeswettbewerb Mathematik, 3

Tags: geometry , 3d geometry , sphere , cone , touching

Let $A$ and $B$ be two spheres of different radii, both inscribed in a cone $K$. There are $m$ other, congruent spheres arranged in a ring such that each of them touches $A, B, K$ and two of the other spheres. Prove that this is possible for at most three values of $m.$

Kharkiv City MO Seniors - geometry, 2016.11.5

Tags: geometry , circumcenter , circles , equal segments

The circle $\omega$ passes through the vertices $B$ and $C$ of triangle $ABC$ and intersects its sides $AC,AB$ at points $A,E$, respectively. On the ray $BD$, a point $K$ such that $BK = AC$ is chosen , and on the ray $CE$, a point $L$ such that $CL = AB$ is chosen. Prove that the center $O$ of the circumscribed circle of the triangle $AKL$ lies on the circle $\omega$.

2023 Puerto Rico Team Selection Test, 2

Tags: geometry , perpendicular , perpendicularity

Let $I$ be the incenter of a triangle $ABC$ and let $D$ and $E$ be the touchpoints of the incircle with sides $BC$ and $AC$, respectively. The lines $DE$ and $BI$ intersect at point $P$. Prove that $AP$ is perpendicular to $BP$.

2009 Baltic Way, 5

Tags: function , algebra

Let $f_0=f_1=1$ and $f_{i+2}=f_{i+1}+f_i$ for all $n\ge 0$. Find all real solutions to the equation \[x^{2010}=f_{2009}\cdot x+f_{2008}\]

1996 Flanders Math Olympiad, 2

Tags: greatest common divisor , number theory

Determine the gcd of all numbers of the form $p^8-1$, with p a prime above 5.

1985 Polish MO Finals, 2

Tags: combinatorics , combinatorial geometry , tiling

Given a square side $1$ and $2n$ positive reals $a_1, b_1, ... , a_n, b_n$ each $\le 1$ and satisfying $\sum a_ib_i \ge 100$. Show that the square can be covered with rectangles $R_i$ with sides length $(a_i, b_i)$ parallel to the square sides.

1988 National High School Mathematics League, 2

Tags: ellipse , conic

If the coordinate origin is inside the ellipse $k^2x^2+y^2-4kx+2ky+k^2-1=0$, then the range value of $k$ is $\text{(A)}|k|>1\qquad\text{(B)}|k|\neq1\qquad\text{(C)}-1<k<1\qquad\text{(D)}0<|k|<1$

1951 Moscow Mathematical Olympiad, 190

Tags: algebra , compare

Which number is greater: $\frac{2.00 000 000 004}{(1.00 000 000 004)^2 + 2.00 000 000 004}$ or $\frac{2.00 000 000 002}{(1.00 000 000 002)^2 + 2.00 000 000 002}$ ?

2009 Tuymaada Olympiad, 4

Tags: floor function , inequalities , function , algebra

The sum of several non-negative numbers is not greater than 200, while the sum of their squares is not less than 2500. Prove that among them there are four numbers whose sum is not less than 50. [i]Proposed by A. Khabrov[/i]

2019 Bulgaria National Olympiad, 5

Tags: combinatorics , combinatorial geometry , geometry

Let $P$ be a $2019-$gon, such that no three of its diagonals concur at an internal point. We will call each internal intersection point of diagonals of $P$ a knot. What is the greatest number of knots one can choose, such that there doesn't exist a cycle of chosen knots? ( Every two adjacent knots in a cycle must be on the same diagonal and on every diagonal there are at most two knots from a cycle.)

2008 IMO Shortlist, 4

Tags: binomial coefficient , number theory , modular arithmetic

Let $ n$ be a positive integer. Show that the numbers \[ \binom{2^n \minus{} 1}{0},\; \binom{2^n \minus{} 1}{1},\; \binom{2^n \minus{} 1}{2},\; \ldots,\; \binom{2^n \minus{} 1}{2^{n \minus{} 1} \minus{} 1}\] are congruent modulo $ 2^n$ to $ 1$, $ 3$, $ 5$, $ \ldots$, $ 2^n \minus{} 1$ in some order. [i]Proposed by Duskan Dukic, Serbia[/i]

2014 Contests, 3

Tags: circumcircle , trapezoid , geometry

Let $ABC$ be an acute,non-isosceles triangle with $AB<AC<BC$.Let $D,E,Z$ be the midpoints of $BC,AC,AB$ respectively and segments $BK,CL$ are altitudes.In the extension of $DZ$ we take a point $M$ such that the parallel from $M$ to $KL$ crosses the extensions of $CA,BA,DE$ at $S,T,N$ respectively (we extend $CA$ to $A$-side and $BA$ to $A$-side and $DE$ to $E$-side).If the circumcirle $(c_{1})$ of $\triangle{MBD}$ crosses the line $DN$ at $R$ and the circumcirle $(c_{2})$ of $\triangle{NCD}$ crosses the line $DM$ at $P$ prove that $ST\parallel PR$.