Olympiad problems

2017 NZMOC Camp Selection Problems, 4

Tags: combinatorics

Ross wants to play solitaire with his deck of $n$ playing cards, but he’s discovered that the deck is “boxed”: some cards are face up, and others are face down. He wants to turn them all face down again, by repeatedly choosing a block of consecutive cards, removing the block from the deck, turning it over, and replacing it back in the deck at the same point. What is the smallest number of such steps Ross needs in order to guarantee that he can turn all the cards face down again, regardless of how they start out?

2018 South East Mathematical Olympiad, 3

Tags: parallelogram , geometry

Let $O$ be the circumcenter of $\triangle ABC$, where $\angle ABC> 90^{\circ}$ and $M$ is the midpoint of $BC$. Point $P$ lies inside $\triangle ABC$ such that $PB\perp PC$. $D,E$ distinct from $P$ lies on the perpendicular to $AP$ through $P$ such that $BD=BP, CE=CP$. If quadrilateral $ADOE$ is a parallelogram, prove that $$\angle OPE = \angle AMB.$$

2007 Pre-Preparation Course Examination, 11

Tags: number theory

Let $p \geq 3$ be a prime and $a_1,a_2,\cdots , a_{p-2}$ be a sequence of positive integers such that for every $k \in \{1,2,\cdots,p-2\}$ neither $a_k$ nor $a_k^k-1$ is divisible by $p$. Prove that product of some of members of this sequence is equivalent to $2$ modulo $p$.

2016 Regional Olympiad of Mexico Southeast, 2

Tags: geometry , trapezoid , circumcircle , cyclic quadrilateral

Let $ABCD$ a trapezium with $AB$ parallel to $CD, \Omega$ the circumcircle of $ABCD$ and $A_1,B_1$ points on segments $AC$ and $BC$ respectively, such that $DA_1B_1C$ is a cyclic cuadrilateral. Let $A_2$ and $B_2$ the symmetric points of $A_1$ and $B_1$ with respect of the midpoint of $AC$ and $BC$, respectively. Prove that points $A, B, A_2, B_2$ are concyclic.

2013 USAJMO, 1

Tags: number theory

Are there integers $a$ and $b$ such that $a^5b+3$ and $ab^5+3$ are both perfect cubes of integers?

2009 Peru MO (ONEM), 4

Tags: domino , combinatorics , tiles , tiling

Let $ n$ be a positive integer. A $4\times n$ rectangular grid is divided in$ 2\times 1$ or $1\times 2$ rectangles (as if it were completely covered with tiles of domino, no overlaps or gaps). Then all the grid points which are vertices of one of the $2\times 1$ or $1\times 2$ rectangles, are painted red. What is the least amount of red points you can get?

1964 IMO Shortlist, 4

Tags: combinatorics , ramsey theory , coloring , graph theory , extremal combinatorics

Seventeen people correspond by mail with one another-each one with all the rest. In their letters only three different topics are discussed. each pair of correspondents deals with only one of these topics. Prove that there are at least three people who write to each other about the same topic.

1995 IMO Shortlist, 5

Tags: function , algebra , functional equation

Let $ \mathbb{R}$ be the set of real numbers. Does there exist a function $ f: \mathbb{R} \mapsto \mathbb{R}$ which simultaneously satisfies the following three conditions? [b](a)[/b] There is a positive number $ M$ such that $ \forall x:$ $ \minus{} M \leq f(x) \leq M.$ [b](b)[/b] The value of $f(1)$ is $1$. [b](c)[/b] If $ x \neq 0,$ then \[ f \left(x \plus{} \frac {1}{x^2} \right) \equal{} f(x) \plus{} \left[ f \left(\frac {1}{x} \right) \right]^2 \]

2009 Brazil Team Selection Test, 1

Tags: combinatorics , permutation , divisibility

Let $n \in \mathbb N$ and $A_n$ set of all permutations $(a_1, \ldots, a_n)$ of the set $\{1, 2, \ldots , n\}$ for which \[k|2(a_1 + \cdots+ a_k), \text{ for all } 1 \leq k \leq n.\] Find the number of elements of the set $A_n$. [i]Proposed by Vidan Govedarica, Serbia[/i]

2004 Vietnam National Olympiad, 1

Tags: trigonometry , limit , algebra

The sequence $ (x_n)^{\infty}_{n\equal{}1}$ is defined by $ x_1 \equal{} 1$ and $ x_{n\plus{}1} \equal{}\frac{(2 \plus{} \cos 2\alpha)x_n \minus{} \cos^2\alpha}{(2 \minus{} 2 \cos 2\alpha)x_n \plus{} 2 \minus{} \cos 2\alpha}$, for all $ n \in\mathbb{N}$, where $ \alpha$ is a given real parameter. Find all values of $ \alpha$ for which the sequence $ (y_n)$ given by $ y_n \equal{} \sum_{k\equal{}1}^{n}\frac{1}{2x_k\plus{}1}$ has a finite limit when $ n \to \plus{}\infty$ and find that limit.

2016 Kosovo Team Selection Test, 2

Tags: number theory

Show that for all positive integers $n\geq 2$ the last digit of the number $2^{2^n}+1$ is $7$ .

2011 Princeton University Math Competition, A7 / B8

Tags: algebra

Let $\alpha_1,\alpha_2,\dots,\alpha_6$ be a fixed labeling of the complex roots of $x^6-1$. Find the number of permutations $\{\alpha_{i_1},\alpha_{i_2},\dots,\alpha_{i_6}\}$ of these roots such that if $P(\alpha_1, \dots, \alpha_6) = 0$, then $P(\alpha_{i_1},\dots,\alpha_{i_6}) = 0$, where $P$ is any polynomial with rational coefficients.

2014 Hanoi Open Mathematics Competitions, 13

Tags: inequalities , system , algebra

Let $a, b,c$ satises the conditions $\begin{cases} 5 \ge a \ge b \ge c \ge 0 \\ a + b \le 8 \\ a + b + c = 10 \end{cases}$ Prove that $a^2 + b^2 + c^2 \le 38$

Gheorghe Țițeica 2024, P3

Tags: ring theory , abstract algebra

Determine all commutative rings $R$ with at least four elements that are not fields, such that for any pairwise distinct and nonzero elements $a,b,c\in R$, $ab+bc+ca$ is invertible. [i]Vlad Matei[/i]

2014 Iran MO (3rd Round), 1

Tags: circumcircle , geometry

We have an equilateral triangle with circumradius $1$. We extend its sides. Determine the point $P$ inside the triangle such that the total lengths of the sides (extended), which lies inside the circle with center $P$ and radius $1$, is maximum. (The total distance of the point P from the sides of an equilateral triangle is fixed ) [i]Proposed by Erfan Salavati[/i]

2014 BMT Spring, 18

Tags: number theory

Suppose the polynomial $f(x) = x^{2014}$ is equal to $f(x) =\sum^{2014}_{k=0} a_k {x \choose k}$ for some real numbers $a_0,... , a_{2014}$. Find the largest integer $m$ such that $2^m$ divides $a_{2013}$.

2016 ASDAN Math Tournament, 2

Tags: team test

Three unit circles are inscribed inside an equilateral triangle such that each circle is tangent to each of the other $2$ circles and to $2$ sides of the triangle. Compute the area of the triangle.

2017 Azerbaijan JBMO TST, 4

Tags: combinatorics , tiles

The central square of the City of Mathematicians is an $n\times n$ rectangular shape, each paved with $1\times 1$ tiles. In order to illuminate the square, night lamps are placed at the corners of the tiles (including the edges of the rectangle) in such a way that each night lamp illuminates all the tiles in its corner. Determine the minimum number of night lamps such that even if one of those night lamps does not work, it is possible to illuminate the entire central square with them.

2014 Dutch BxMO/EGMO TST, 2

Tags: function , algebra

Find all functions $f:\mathbb{R}\backslash\{0\}\rightarrow\mathbb{R}$ for which $xf(xy) + f(-y) = xf(x)$ for all non-zero real numbers $x, y$.

1997 Bosnia and Herzegovina Team Selection Test, 6

Tags: subset , set , number theory

Let $k$, $m$ and $n$ be integers such that $1<n \leq m-1 \leq k$. Find maximum size of subset $S$ of set $\{1,2,...,k\}$ such that sum of any $n$ different elements from $S$ is not: $a)$ equal to $m$, $b)$ exceeding $m$

MIPT Undergraduate Contest 2019, 1.4

Tags: geometry , 3d geometry , sphere , topology

Suppose that in a unit sphere in Euclidean space, there are $2m$ points $x_1, x_2, ..., x_{2m}.$ Prove that it's possible to partition them into two sets of $m$ points in such a way that the centers of mass of these sets are at a distance of at most $\frac{2}{\sqrt{m}}$ from one another.

Kvant 2023, M2769

Tags: geometry , incircle , concurrency

The incircle of the triangle $ABC$ touches the sides $BC, CA$ and $AB{}$ at $D,E$ and $F{}$ respectively. Let the circle $\omega$ touch the segments $CA{}$ and $AB{}$ at $Q{}$ and $R{}$ respectively, and the points $M{}$ and $N{}$ are selected on the segments $AB{}$ and $AC{}$ respectively, so that the segments $CM{}$ and $BN{}$ touch $\omega$. The bisectors of $\angle NBC$ and $\angle MCB$ intersect the segments $DE{}$ and $DF{}$ at $K{}$ and $L{}$ respectively. Prove that the lines $RK{}$ and $QL{}$ intersect on $\omega$. [i]Proposed by Tran Quang Hung[/i]

2016 Postal Coaching, 1

Tags: number theory , rational number

Show that there are infinitely many rational triples $(a, b, c)$ such that $$a + b + c = abc = 6.$$

2018 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: geometry , circumcircle , perpendicular bisector

Let $P$ be a point on circumcircle of triangle $ABC$ on arc $\stackrel{\frown}{BC}$ which does not contain point $A$. Let lines $AB$ and $CP$ intersect at point $E$, and lines $AC$ and $BP$ intersect at $F$. If perpendicular bisector of side $AB$ intersects $AC$ in point $K$, and perpendicular bisector of side $AC$ intersects side $AB$ in point $J$, prove that: ${\left(\frac{CE}{BF}\right)}^2=\frac{AJ\cdot JE}{AK \cdot KF}$

2007 Turkey Junior National Olympiad, 1

Tags: geometry

Let $ABCD$ be a trapezoid such that $AD\parallel BC$ and $|AB|=|BC|$. Let $E$ and $F$ be the midpoints of $[BC]$ and $[AD]$, respectively. If the internal angle bisector of $\triangle ABC$ passes through $F$, find $|BD|/|EF|$.