Olympiad problems

2020 European Mathematical Cup, 3

Two types of tiles, depicted on the figure below, are given. [img]https://wiki-images.artofproblemsolving.com//2/23/Izrezak.PNG[/img] Find all positive integers $n$ such that an $n\times n$ board consisting of $n^2$ unit squares can be covered without gaps with these two types of tiles (rotations and reflections are allowed) so that no two tiles overlap and no part of any tile covers an area outside the $n\times n$ board. \\ [i]Proposed by Art Waeterschoot[/i]

2022 Puerto Rico Team Selection Test, 3

Tags: ratio , geometry , isosceles

Let $\omega$ be a circle with center $O$ and diameter $AB$. A circle with center at $B$ intersects $\omega$ at C and $AB$ at $D$. The line $CD$ intersects $\omega$ at a point $E$ ($E\ne C$). The intersection of lines $OE$ and $BC$ is $F$. (a) Prove that triangle $OBF$ is isosceles. (b) If $D$ is the midpoint of $OB$, find the value of the ratio $\frac{FB}{BD}$.

2020 Latvia TST, 1.1

Tags: parallelogram , geometry

It is given parallelogram $ABCD$. On it's sides $AB, BC, CD, DA$ are chosen points $E, F, G, H$ such that area of $EFGH$ is half of the area of $ABCD$. Show that at least one of the quadrilaterals $ABFH$ and $AEGD$ is parallelogram.

2006 AIME Problems, 3

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Let $P$ be the product of the first 100 positive odd integers. Find the largest integer $k$ such that $P$ is divisible by $3^k$.

2011 IFYM, Sozopol, 4

Tags: number theory , set , arithmetic progression , arithmetic sequence

Prove that the set $\{1,2,…,12001\}$ can be partitioned into 5 groups so that none of them contains an arithmetic progression with length 11.

1984 Tournament Of Towns, (060) A5

Tags: number theory , prime , consecutive

The two pairs of consecutive natural numbers $(8, 9)$ and $(288, 289)$ have the following property: in each pair, each number contains each of its prime factors to a power not less than $2$. Prove that there are infinitely many such pairs. (A Andjans, Riga)

2015 Sharygin Geometry Olympiad, P11

Tags: geometry , perimeter

Let $H$ be the orthocenter of an acute-angled triangle A$BC$. The perpendicular bisector to segment $BH$ meets $BA$ and $BC$ at points $A_0, C_0$ respectively. Prove that the perimeter of triangle $A_0OC_0$ ($O$ is the circumcenter of triangle $ABC$) is equal to $AC$.

2019 China Team Selection Test, 6

Tags: number theory , combinatorics

Given positive integer $n,k$ such that $2 \le n <2^k$. Prove that there exist a subset $A$ of $\{0,1,\cdots,n\}$ such that for any $x \neq y \in A$, ${y\choose x}$ is even, and $$|A| \ge \frac{{k\choose \lfloor \frac{k}{2} \rfloor}}{2^k} \cdot (n+1)$$

Novosibirsk Oral Geo Oly VIII, 2019.2

Tags: geometry , incircle , semicircle

The circle is inscribed in a triangle, inscribed in a semicircle. Find the marked angle $a$. [img]https://cdn.artofproblemsolving.com/attachments/8/e/334c8662377155086e9211da3589145f460b52.png[/img]

1958 Poland - Second Round, 6

Tags: construction , geometry

In a plane, two circles $ C_1 $ and $ C_2 $ and a line $ m $ are given. Find a point on the line $ m $ from which one can draw tangents to the circles $ C_1 $ and $ C_2 $ with equal inclination to the line $ m $.

2017 Romanian Master of Mathematics Shortlist, G3

Tags: collinear , geometry

Let $ABCD$ be a convex quadrilateral and let $P$ and $Q$ be variable points inside this quadrilateral so that $\angle APB=\angle CPD=\angle AQB=\angle CQD$. Prove that the lines $PQ$ obtained in this way all pass through a fixed point , or they are all parallel.

2016 May Olympiad, 2

Tags: number theory , combinatorics

In a sports competition in which several tests are carried out, only the three athletes $A, B, C$. In each event, the winner receives $x$ points, the second receives $y$ points, and the third receives $z$ points. There are no ties, and the numbers $x, y, z$ are distinct positive integers with $x$ greater than $y$, and $y$ greater than $z$. At the end of the competition it turns out that $A$ has accumulated $20$ points, $B$ has accumulated $10$ points and $C$ has accumulated $9$ points. We know that athlete $A$ was second in the 100-meter event. Determine which of the three athletes he was second in the jumping event.

2016 ASDAN Math Tournament, 13

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Suppose $\{a_n\}_{n=1}^\infty$ is a sequence. The partial sums $\{s_n\}_{n=1}^\infty$ are defined by $$s_n=\sum_{i=1}^na_i.$$ The Cesàro sums are then defined as $\{A_n\}_{n=1}^\infty$, where $$A_n=\frac{1}{n}\cdot\sum_{i=1}^ns_i.$$ Let $a_n=(-1)^{n+1}$. What is the limit of the Cesàro sums of $\{a_n\}_{n=1}^\infty$ as $n$ goes to infinity?

2013 NIMO Problems, 11

Tags: number theory , relatively prime , geometric series

Find $100m+n$ if $m$ and $n$ are relatively prime positive integers such that \[ \sum_{\substack{i,j \ge 0 \\ i+j \text{ odd}}} \frac{1}{2^i3^j} = \frac{m}{n}. \][i]Proposed by Aaron Lin[/i]

2014 BMT Spring, 12

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

Suppose four coplanar points $A, B, C$, and $D$ satisfy $AB = 3$, $BC = 4$, $CA = 5$, and $BD = 6$. Determine the maximal possible area of $\vartriangle ACD$.

2013 India National Olympiad, 1

Tags: geometry , circumcircle

Let $\Gamma_1$ and $\Gamma_2$ be two circles touching each other externally at $R.$ Let $O_1$ and $O_2$ be the centres of $\Gamma_1$ and $\Gamma_2,$ respectively. Let $\ell_1$ be a line which is tangent to $\Gamma_2$ at $P$ and passing through $O_1,$ and let $\ell_2$ be the line tangent to $\Gamma_1$ at $Q$ and passing through $O_2.$ Let $K=\ell_1\cap \ell_2.$ If $KP=KQ$ then prove that the triangle $PQR$ is equilateral.

2010 China Team Selection Test, 3

Tags: number theory , p-adic

Given positive integer $k$, prove that there exists a positive integer $N$ depending only on $k$ such that for any integer $n\geq N$, $\binom{n}{k}$ has at least $k$ different prime divisors.

2019 APMO, 2

Tags: number theory

Let $m$ be a fixed positive integer. The infinite sequence $\{a_n\}_{n\geq 1}$ is defined in the following way: $a_1$ is a positive integer, and for every integer $n\geq 1$ we have $$a_{n+1} = \begin{cases}a_n^2+2^m & \text{if } a_n< 2^m \\ a_n/2 &\text{if } a_n\geq 2^m\end{cases}$$ For each $m$, determine all possible values of $a_1$ such that every term in the sequence is an integer.

2012 HMNT, 9

Tags: geometry

Consider triangle $ABC$ where $BC = 7$, $CA = 8$, and $AB = 9$. $D$ and $E$ are the midpoints of $BC$ and $CA$, respectively, and $AD$ and $BE$ meet at $G$. The reflection of $G$ across $D$ is $G'$, and $G'E$ meets $CG$ at $P$. Find the length $PG$.

2011 Indonesia TST, 2

Tags: floor function , graph , graph theory , combinatorics

A graph $G$ with $n$ vertex is called [i]good [/i] if every vertex could be labelled with distinct positive integers which are less than or equal $\lfloor \frac{n^2}{4} \rfloor$ such that there exists a set of nonnegative integers $D$ with the following property: there exists an edge between $2$ vertices if and only if the difference of their labels is in $D$. Show that there exists a positive integer $N$ such that for every $n \ge N$, there exist a not-good graph with $n$ vertices.

1996 Greece Junior Math Olympiad, 4a

Tags: fraction , number theory , simplify , even

If the fraction $\frac{an + b}{cn + d}$ may be simplified using $2$ (as a common divisor ), show that the number $ad - bc$ is even. ($a, b, c, d, n$ are natural numbers and the $cn + d$ different from zero).

2016 District Olympiad, 2

Tags: linear algebra , matrix

Let A,B,C,D four matrices of order n with complex entries, n>=2 and let k real number such that AC+kBD=I and AD=BC. Prove that CA+kDB=I and DA=CB.

2011 Saudi Arabia Pre-TST, 3.3

Tags: isosceles , angle , geometry

In the isosceles triangle $ABC$, with $AB = AC$, the angle bisector of $\angle B$ intersects side $AC$ at $B'$. Suppose that $ B B' + B'A = BC$. Find the angles of the triangle.

2011 Mongolia Team Selection Test, 3

Tags: circumcircle , power of a point , radical axis , geometry

We are given an acute triangle $ABC$. Let $(w,I)$ be the inscribed circle of $ABC$, $(\Omega,O)$ be the circumscribed circle of $ABC$, and $A_0$ be the midpoint of altitude $AH$. $w$ touches $BC$ at point $D$. $A_0 D$ and $w$ intersect at point $P$, and the perpendicular from $I$ to $A_0 D$ intersects $BC$ at the point $M$. $MR$ and $MS$ lines touch $\Omega$ at $R$ and $S$ respectively [note: I am not entirely sure of what is meant by this, but I am pretty sure it means draw the tangents to $\Omega$ from $M$]. Prove that the points $R,P,D,S$ are concyclic. (proposed by E. Enkzaya, inspired by Vietnamese olympiad problem)

2014 Online Math Open Problems, 19

Tags: geometry , geometric transformation , reflection , circumcircle , incenter , trigonometry

In triangle $ABC$, $AB=3$, $AC=5$, and $BC=7$. Let $E$ be the reflection of $A$ over $\overline{BC}$, and let line $BE$ meet the circumcircle of $ABC$ again at $D$. Let $I$ be the incenter of $\triangle ABD$. Given that $\cos ^2 \angle AEI = \frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers, determine $m+n$. [i]Proposed by Ray Li[/i]