Olympiad problems

2020 IberoAmerican, 6

Tags: geometry

Let $ABC$ be an acute, scalene triangle. Let $H$ be the orthocenter and $O$ be the circumcenter of triangle $ABC$, and let $P$ be a point interior to the segment $HO.$ The circle with center $P$ and radius $PA$ intersects the lines $AB$ and $AC$ again at $R$ and $S$, respectively. Denote by $Q$ the symmetric point of $P$ with respect to the perpendicular bisector of $BC$. Prove that points $P$, $Q$, $R$ and $S$ lie on the same circle.

1982 Tournament Of Towns, (024) 2

Tags: combinatorics , coloring

A number of objects, each coloured in one of two given colours, are arranged in a line (there is at least one object having each of the given colours). It is known that each two objects, between which there are exactly $10$ or $15$ other objects, are of the same colour. What is the greatest possible number of such pieces?

1984 IMO Longlists, 55

Tags: modular arithmetic , number theory

Let $a, b, c$ be natural numbers such that $a+b+c = 2pq(p^{30}+q^{30}), p > q$ being two given positive integers. $(a)$ Prove that $k = a^3 + b^3 + c^3$ is not a prime number. $(b)$ Prove that if $a\cdot b\cdot c$ is maximum, then $1984$ divides $k$.

2020 USAMTS Problems, 3:

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Given a nonconstant polynomial with real coefficients $f(x),$ let $S(f)$ denote the sum of its roots. Let p and q be nonconstant polynomials with real coefficients such that $S(p) = 7,$ $S(q) = 9,$ and $S(p-q)= 11$. Find, with proof, all possible values for $S(p + q)$.

2017 Online Math Open Problems, 27

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Let $N$ be the number of functions $f: \mathbb{Z}/16\mathbb{Z} \to \mathbb{Z}/16\mathbb{Z}$ such that for all $a,b \in \mathbb{Z}/16\mathbb{Z}$: \[f(a)^2+f(b)^2+f(a+b)^2 \equiv 1+2f(a)f(b)f(a+b) \pmod{16}.\] Find the remainder when $N$ is divided by 2017. [i]Proposed by Zack Chroman[/i]

2014 Contests, 2

Tags: geometry , rectangle , combinatorics

Every cell of a $m \times n$ chess board, $m\ge 2,n\ge 2$, is colored with one of four possible colors, e.g white, green, red, blue. We call such coloring good if the four cells of any $2\times 2$ square of the chessboard are colored with pairwise different colors. Determine the number of all good colorings of the chess board. [i]Proposed by N. Beluhov[/i]

2020 Greece National Olympiad, 3

Tags: combinatorics , number theory

On the board there are written in a row, the integers from $1$ until $2030$ (included that) in an increasing order. We have the right of ''movement'' $K$: [i]We choose any two numbers $a,b$ that are written in consecutive positions and we replace the pair $(a,b)$ by the number $(a-b)^{2020}$.[/i] We repeat the movement $K$, many times until only one number remains written on the board. Determine whether it would be possible, that number to be: (i) $2020^{2020}$ (ii)$2021^{2020}$

2004 Dutch Mathematical Olympiad, 3

Tags: combinatorics

Start with a stack of $100$ cards. Now repeat the following: choose a stack of at least $2$ cards and split them into two smaller piles (at least $1$ card of each). Continue this until there are finally $100$ stacks of $1$ card each. Every time you split a pile into two stacks you get a number of points that is equal to the product of the number of cards in the two new stacks. What is the maximum number of points that you can earn in total?

2008 China Girls Math Olympiad, 3

Tags: ratio , inequalities , geometry

Determine the least real number $ a$ greater than $ 1$ such that for any point $ P$ in the interior of the square $ ABCD$, the area ratio between two of the triangles $ PAB$, $ PBC$, $ PCD$, $ PDA$ lies in the interval $ \left[\frac {1}{a},a\right]$.

PEN A Problems, 73

Tags: number theory , divisibility theory

Determine all pairs $(n,p)$ of positive integers such that [list][*] $p$ is a prime, $n>1$, [*] $(p-1)^{n} + 1$ is divisible by $n^{p-1}$. [/list]

2014 Saint Petersburg Mathematical Olympiad, 4

Tags: algebra , inequalities

$a_1\geq a_2\geq... \geq a_{100n}>0$ If we take from $(a_1,a_2,...,a_{100n})$ some $2n+1$ numbers $b_1\geq b_2 \geq ... \geq b_{2n+1}$ then $b_1+...+b_n > b_{n+1}+...b_{2n+1}$ Prove, that $$(n+1)(a_1+...+a_n)>a_{n+1}+a_{n+2}+...+a_{100n}$$

1986 Greece Junior Math Olympiad, 2

Tags: geometry

Let $ABC$ be a triangle. α) If point $D$ lies on side $BC$, prove that $AD<AB$ or $AD <AC$ β) If point $E$ lies on side $AB$ and point $Z$ lies on side $AC$, prove that line segment is $EZ$ less than largest side of the triangle $ABC$.

2018 239 Open Mathematical Olympiad, 10-11.8

Tags: combinatorics , graph theory

Graph $G$ becomes planar when any vertex is removed. Prove that its vertices can be properly colored with 5 colors. (Using the four-color theorem without proof is not allowed!) [i]Proposed by D. Karpov[/i]

2022 IMO Shortlist, A5

Tags: algebra , geometric sequence

Find all positive integers $n \geqslant 2$ for which there exist $n$ real numbers $a_1<\cdots<a_n$ and a real number $r>0$ such that the $\tfrac{1}{2}n(n-1)$ differences $a_j-a_i$ for $1 \leqslant i<j \leqslant n$ are equal, in some order, to the numbers $r^1,r^2,\ldots,r^{\frac{1}{2}n(n-1)}$.

2008 IMO Shortlist, 4

Tags: geometry , circumcircle , reflection

In an acute triangle $ ABC$ segments $ BE$ and $ CF$ are altitudes. Two circles passing through the point $ A$ and $ F$ and tangent to the line $ BC$ at the points $ P$ and $ Q$ so that $ B$ lies between $ C$ and $ Q$. Prove that lines $ PE$ and $ QF$ intersect on the circumcircle of triangle $ AEF$. [i]Proposed by Davood Vakili, Iran[/i]

2003 Germany Team Selection Test, 3

Tags: combinatorics , tiling , dissection

For $n$ an odd positive integer, the unit squares of an $n\times n$ chessboard are coloured alternately black and white, with the four corners coloured black. A it tromino is an $L$-shape formed by three connected unit squares. For which values of $n$ is it possible to cover all the black squares with non-overlapping trominos? When it is possible, what is the minimum number of trominos needed?

PEN M Problems, 3

Tags: floor function , recursive sequences

Let $f(n)=n+\lfloor \sqrt{n}\rfloor$. Prove that, for every positive integer $m$, the sequence \[m, f(m), f(f(m)), f(f(f(m))), \cdots\] contains at least one square of an integer.

2018 European Mathematical Cup, 3

Tags: geometry

Let $ABC$ be an acute triangle with $ |AB | < |AC |$and orthocenter $H$. The circle with center A and radius$ |AC |$ intersects the circumcircle of $\triangle ABC$ at point $D$ and the circle with center $A$ and radius$ |AB |$ intersects the segment $\overline{AD}$ at point $K. $ The line through $K$ parallel to $CD $ intersects $BC$ at the point $ L.$ If $M$ is the midpoint of $\overline{BC}$ and N is the foot of the perpendicular from $H$ to $AL, $ prove that the line $ MN $ bisects the segment $\overline{AH}$.

2012 India National Olympiad, 4

Tags: ratio , geometry , analytic geometry , geometric transformation , homothety , calculus , integration

Let $ABC$ be a triangle. An interior point $P$ of $ABC$ is said to be [i]good [/i]if we can find exactly $27$ rays emanating from $P$ intersecting the sides of the triangle $ABC$ such that the triangle is divided by these rays into $27$ [i]smaller triangles of equal area.[/i] Determine the number of good points for a given triangle $ABC$.

2010 Sharygin Geometry Olympiad, 12

Tags: geometric transformation , reflection , geometry

Let $AC$ be the greatest leg of a right triangle $ABC,$ and $CH$ be the altitude to its hypotenuse. The circle of radius $CH$ centered at $H$ intersects $AC$ in point $M.$ Let a point $B'$ be the reflection of $B$ with respect to the point $H.$ The perpendicular to $AB$ erected at $B'$ meets the circle in a point $K$. Prove that [b]a)[/b] $B'M \parallel BC$ [b]b)[/b] $AK$ is tangent to the circle.

2003 Estonia National Olympiad, 3

Tags: angle , right triangle , geometry

Let $ABC$ be a triangle with $\angle C = 90^o$ and $D$ a point on the ray $CB$ such that $|AC| \cdot |CD| = |BC|^2$. A parallel line to $AB$ through $D$ intersects the ray $CA$ at $E$. Find $\angle BEC$.

1999 Slovenia National Olympiad, Problem 4

Tags: game , combinatorics

Three integers are written on a blackboard. At every step one of them is erased and the sum of the other two decreased by $1$ is written instead. Is it possible to obtain the numbers $17,75,91$ if the three initial numbers were: $\textbf{(a)}~2,2,2$; $\textbf{(b)}~3,3,3$?

2017 Saudi Arabia BMO TST, 3

Tags: geometry , similar triangles , cyclic quadrilateral , perpendicular

Let $ABCD$ be a cyclic quadrilateral and triangles $ACD, BCD$ are acute. Suppose that the lines $AB$ and $CD$ meet at $S$. Denote by $E$ the intersection of $AC, BD$. The circles $(ADE)$ and $(BC E)$ meet again at $F$. a) Prove that $SF \perp EF.$ b) The point $G$ is taken out side of the quadrilateral $ABCD$ such that triangle $GAB$ and $FDC$ are similar. Prove that $GA+ FB = GB + FA$

2016 Iran MO (3rd Round), 1

Tags: inequalities , algebra , sequence

The sequence $(a_n)$ is defined as: $$a_1=1007$$ $$a_{i+1}\geq a_i+1$$ Prove the inequality: $$\frac{1}{2016}>\sum_{i=1}^{2016}\frac{1}{a_{i+1}^{2}+a_{i+2}^2}$$

1997 India Regional Mathematical Olympiad, 4

Tags: geometry , inequalities

In a quadrilateral $ABCD$, it is given that $AB$ is parallel to $CD$ and the diagonals $AC$ and $BD$ are perpendicular to each other. Show that (a) $AD \cdot BC \geq AB \cdot CD$ (b) $AD + BC \geq AB + CD.$