Olympiad problems

2011 IFYM, Sozopol, 4

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

Tags:

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]

2016 Math Prize for Girls Problems, 6

Tags:

The largest term in the binomial expansion of $(1 + \tfrac{1}{2})^{31}$ is of the form $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. What is the value of $b$? As an example of a binomial expansion, the binomial expansion of an expression of the form $(x + y)^3$ is the sum of four terms \[ x^3 + 3x^2y + 3xy^2 + y^3. \]

2004 AMC 12/AHSME, 12

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In the sequence $ 2001, 2002, 2003, \ldots$, each term after the third is found by subtracting the previous term from the sum of the two terms that precede that term. For example, the fourth term is $ 2001 \plus{} 2002 \minus{} 2003 \equal{} 2000$. What is the $ 2004^\text{th}$ term in this sequence? $ \textbf{(A)} \minus{} \! 2004 \qquad \textbf{(B)} \minus{} \! 2 \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ 4003 \qquad \textbf{(E)}\ 6007$

TNO 2008 Senior, 3

Tags: geometry

Luis' friends decided to play a prank on him in his geometry homework. They erased most of a triangle and, instead, drew an equivalent triangle with the sum of its three side lengths. Help Luis complete his homework by reconstructing the original triangle using only a straightedge and compass. Since Luis' method involves measurements, prove that his method results in a triangle longer than the sum of its three sides.

2000 Estonia National Olympiad, 4

Tags: parallelogram , geometry , isosceles , midpoint

Let $E$ be the midpoint of the side $AB$ of the parallelogram $ABCD$. Let $F$ be the projection of $B$ on $AC$. Prove that the triangle $ABF$ is isosceles