Olympiad problems

2020 Sharygin Geometry Olympiad, 13

Let $I$ be the incenter of triangle $ABC$. The excircle with center $I_A$ touches the side $BC$ at point $A'$. The line $l$ passing through $I$ and perpendicular to $BI$ meets $I_AA'$ at point $K$ lying on the medial line parallel to $BC$. Prove that $\angle B \leq 60^\circ$.

1940 Moscow Mathematical Olympiad, 064

Tags: combinatorics , combinatorial geometry , tiling

How does one tile a plane, without gaps or overlappings, with the tiles equal to a given irregular quadrilateral?

1982 Yugoslav Team Selection Test, Problem 3

Tags: combinatorics

Let there be given real numbers $x_i>1~(i=1,2,\ldots,2n)$. Prove that the interval $[0,2]$ contains at most $\binom{2n}n$ sums of the form $\alpha_1x_1+\ldots+\alpha_{2n}x_{2n}$, where $\alpha_i\in\{-1,1\}$ for all $i$.

2010 Estonia Team Selection Test, 6

Tags: combinatorics , coloring , square table

Every unit square of a $n \times n$ board is colored either red or blue so that among all 2 $\times 2$ squares on this board all possible colorings of $2 \times 2$ squares with these two colors are represented (colorings obtained from each other by rotation and reflection are considered different). a) Find the least possible value of $n$. b) For the least possible value of $n$ find the least possible number of red unit squares

2016 Harvard-MIT Mathematics Tournament, 21

Tags:

Tim starts with a number $n$, then repeatedly flips a fair coin. If it lands heads he subtracts 1 from his number and if it lands tails he subtracts 2. Let $E_n$ be the expected number of flips Tim does before his number is zero or negative. Find the pair $(a,b)$ such that \[ \lim_{n \to \infty} (E_n-an-b) = 0. \]

2022 Brazil Team Selection Test, 1

Tags: geometry

Let $ABCD$ be a parallelogram with $AC=BC.$ A point $P$ is chosen on the extension of ray $AB$ past $B.$ The circumcircle of $ACD$ meets the segment $PD$ again at $Q.$ The circumcircle of triangle $APQ$ meets the segment $PC$ at $R.$ Prove that lines $CD,AQ,BR$ are concurrent.

2000 All-Russian Olympiad, 8

Tags: number theory , greatest common divisor , modular arithmetic , relatively prime

One hundred natural numbers whose greatest common divisor is $1$ are arranged around a circle. An allowed operation is to add to a number the greatest common divisor of its two neighhbors. Prove that we can make all the numbers pairwise copirme in a finite number of moves.

2000 Harvard-MIT Mathematics Tournament, 3

Tags: geometry , 3d geometry , geometric transformation , rotation

Using $3$ colors, red, blue and yellow, how many different ways can you color a cube (modulo rigid rotations)?

2015 Regional Olympiad of Mexico Southeast, 4

Tags: number theory

Let $A=\{1,2,4,5,7,8,\dots\}$ the set with naturals not divisible by three. Find all values of $n$ such that exist $2n$ consecutive elements of $A$ which sum it´s $300$.

2021-IMOC, N7

Tags: modular arithmetic , number theory

Let $p$ be a given odd prime. Find the largest integer $k'$ such that it is possible to partition $\{1,2,\cdots,p-1\}$ into two sets $X,Y$ such that for any $k$ with $0 \le k \le k'$, $$\sum_{a \in X}a^k \equiv \sum_{b \in Y}b^k \pmod p$$ [i]houkai[/i]

2013 AMC 12/AHSME, 23

Tags: geometry , geometric transformation , rotation , number theory , greatest common divisor

$ ABCD$ is a square of side length $ \sqrt{3} + 1 $. Point $ P $ is on $ \overline{AC} $ such that $ AP = \sqrt{2} $. The square region bounded by $ ABCD $ is rotated $ 90^{\circ} $ counterclockwise with center $ P $, sweeping out a region whose area is $ \frac{1}{c} (a \pi + b) $, where $a $, $b$, and $ c $ are positive integers and $ \text{gcd}(a,b,c) = 1 $. What is $ a + b + c $? $\textbf{(A)} \ 15 \qquad \textbf{(B)} \ 17 \qquad \textbf{(C)} \ 19 \qquad \textbf{(D)} \ 21 \qquad \textbf{(E)} \ 23 $

2018 Purple Comet Problems, 29

Tags: number theory

Find the three-digit positive integer $n$ for which $\binom n3 \binom n4 \binom n5 \binom n6 $ is a perfect square.

2012 Oral Moscow Geometry Olympiad, 2

Tags: congruent , polygon , interior , geometry

Two equal polygons $F$ and $F'$ are given on the plane. It is known that the vertices of the polygon $F$ belong to $F'$ (may lie inside it or on the border). Is it true that all the vertices of these polygons coincide?

2011 Benelux, 2

Tags: geometry , incenter , circumcircle , perpendicular bisector , cyclic quadrilateral , angle bisector

Let $ABC$ be a triangle with incentre $I$. The angle bisectors $AI$, $BI$ and $CI$ meet $[BC]$, $[CA]$ and $[AB]$ at $D$, $E$ and $F$, respectively. The perpendicular bisector of $[AD]$ intersects the lines $BI$ and $CI$ at $M$ and $N$, respectively. Show that $A$, $I$, $M$ and $N$ lie on a circle.

2007 AMC 10, 8

Tags:

Triangles $ ABC$ and $ ADC$ are isosceles with $ AB \equal{} BC$ and $ AD \equal{} DC$. Point D is inside $ \triangle ABC$. $ \angle ABC \equal{} 40^\circ$, and $ \angle ADC \equal{} 140^\circ$. What is the degree measure of $ \angle BAD$? $ \textbf{(A)}\ 20 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 40 \qquad \textbf{(D)}\ 50 \qquad \textbf{(E)}\ 60$

2018 CHMMC (Fall), 3

Tags: sum , algebra

Compute $$\sum^{\infty}_{n=1} \left( \frac{1}{n^2 + 3n} - \frac{1}{n^2 + 3n + 2}\right)$$

2005 AMC 10, 14

Tags: calculus , integration

How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits? $ \textbf{(A)}\ 41\qquad \textbf{(B)}\ 42\qquad \textbf{(C)}\ 43\qquad \textbf{(D)}\ 44\qquad \textbf{(E)}\ 45$

2020 Polish Junior MO Second Round, 2.

Tags: geometry , parallelogram , perpendicular bisector

Let $ABCD$ be the parallelogram, such that angle at vertex $A$ is acute. Perpendicular bisector of the segment $AB$ intersects the segment $CD$ in the point $X$. Let $E$ be the intersection point of the diagonals of the parallelogram $ABCD$. Prove that $XE = \frac{1}{2}AD$.

2007 Belarusian National Olympiad, 2

Tags: geometry

Circles $S_1$ and $S_2$ with centers $O_1$ and $O_2$, respectively, pass through the centers of each other. Let $A$ be one of their intersection points. Two points $M_1$ and $M_2$ begin to move simultaneously starting from $A$. Point $M_1$ moves along $S_1$ and point $M_2$ moves along $S_2$. Both points move in clockwise direction and have the same linear velocity $v$. (a) Prove that all triangles $AM_1M_2$ are equilateral. (b) Determine the trajectory of the movement of the center of the triangle $AM_1M_2$ and find its linear velocity.

2013 Stars Of Mathematics, 2

Tags: geometry , rectangle

Three points inside a rectangle determine a triangle. A fourth point is taken inside the triangle. i) Prove at least one of the three concave quadrilaterals formed by these four points has perimeter lesser than that of the rectangle. ii) Assuming the three points inside the rectangle are three corners of it, prove at least two of the three concave quadrilaterals formed by these four points have perimeters lesser than that of the rectangle. [i](Dan Schwarz)[/i]

1961 Putnam, B3

Tags: geometry , circumcircle

Consider four points in the plane, no three of which are collinear, and such that the circle through three of them does not pass through the fourth. Prove that one of the four points can be selected having the property that it lies inside the circle determined by the other three.

2017 Azerbaijan EGMO TST, 4

Tags: calculus , integration , induction , modular arithmetic , number theory

Find all natural numbers a, b such that $ a^{n}\plus{} b^{n} \equal{} c^{n\plus{}1}$ where c and n are naturals.

2003 Vietnam National Olympiad, 2

Tags: geometry , geometric transformation , homothety , invariant , trigonometry , ratio

The circles $ C_{1}$ and $ C_{2}$ touch externally at $ M$ and the radius of $ C_{2}$ is larger than that of $ C_{1}$. $ A$ is any point on $ C_{2}$ which does not lie on the line joining the centers of the circles. $ B$ and $ C$ are points on $ C_{1}$ such that $ AB$ and $ AC$ are tangent to $ C_{1}$. The lines $ BM$, $ CM$ intersect $ C_{2}$ again at $ E$, $ F$ respectively. $ D$ is the intersection of the tangent at $ A$ and the line $ EF$. Show that the locus of $ D$ as $ A$ varies is a straight line.

2018 JHMT, 9

Tags: geometry

In a trapezoid $ABCD$, $AD \parallel BC$ and $\angle A = 60^o$. Let $E$ be a point on $AB$, and let $O_1$ and $O_2$ be circumcenters of $\vartriangle AED$ and $\vartriangle BEC$, respectively. Let $\frac{\overline{O_1O_2}}{\overline{DC}}$ be $x$. $x^2$ is in the form $\frac{p}{q}$ where $p, q$ are relatively prime positive integers. Find $p + q$.

1985 IMO Longlists, 82

Tags: diophantine equation , algebra , polynomial

Find all cubic polynomials $x^3 +ax^2 +bx+c$ admitting the rational numbers $a$, $b$ and $c$ as roots.