Olympiad problems

2016 Azerbaijan National Mathematical Olympiad, 1

Tags: geometry , perimeter

Find the perimeter of the convex polygon whose coordinates of the vertices are the set of pairs of the integer solutions of the equation $x^2+xy = x + 2y + 9$.

2007 Junior Balkan Team Selection Tests - Romania, 3

Tags: modular arithmetic , combinatorics , induction

Consider a $n$x$n$ table such that the unit squares are colored arbitrary in black and white, such that exactly three of the squares placed in the corners of the table are white, and the other one is black. Prove that there exists a $2$x$2$ square which contains an odd number of unit squares white colored.

1973 Spain Mathematical Olympiad, 2

Tags: inequalities , system of equations , system , algebra

Determine all solutions of the system $$\begin{cases} 2x - 5y + 11z - 6 = 0 \\ -x + 3y - 16z + 8 = 0 \\ 4x - 5y - 83z + 38 = 0 \\ 3x + 11y - z + 9 > 0 \end{cases}$$ in which the first three are equations and the last one is a linear inequality.

2006 Junior Tuymaada Olympiad, 7

Tags: geometry , circumcircle , median , geometric inequality

The median $ BM $ of a triangle $ ABC $ intersects the circumscribed circle at point $ K $. The circumcircle of the triangle $ KMC $ intersects the segment $ BC $ at point $ P $, and the circumcircle of $ AMK $ intersects the extension of $ BA $ at $ Q $. Prove that $ PQ> AC $.

2010 Switzerland - Final Round, 5

Tags: combinatorics

Some sides and diagonals of a regular $ n$-gon form a connected path that visits each vertex exactly once. A [i]parallel pair[/i] of edges is a pair of two different parallel edges of the path. Prove that (a) if $ n$ is even, there is at least one [i]parallel pair[/i]. (b) if $ n$ is odd, there can't be one single [i]parallel pair[/i].

2007 Estonia National Olympiad, 4

Tags: combinatorics , table , square table

The figure shows a figure of $5$ unit squares, a Greek cross. What is the largest number of Greek crosses that can be placed on a grid of dimensions $8 \times 8$ without any overlaps, with each unit square covering just one square in a grid?

2019 AMC 12/AHSME, 17

Tags: polynomial , algebra

Let $s_k$ denote the sum of the $\textit{k}$th powers of the roots of the polynomial $x^3-5x^2+8x-13$. In particular, $s_0=3$, $s_1=5$, and $s_2=9$. Let $a$, $b$, and $c$ be real numbers such that $s_{k+1} = a \, s_k + b \, s_{k-1} + c \, s_{k-2}$ for $k = 2$, $3$, $....$ What is $a+b+c$? $\textbf{(A)} \; -6 \qquad \textbf{(B)} \; 0 \qquad \textbf{(C)} \; 6 \qquad \textbf{(D)} \; 10 \qquad \textbf{(E)} \; 26$

2024-IMOC, N1

Tags: number theory , prime , perfect square

Proof that for every primes $p$, $q$ \[p^{q^2-q+1}+q^{p^2-p+1}-p-q\] is never a perfect square. [i]Proposed by chengbilly[/i]

2011 ISI B.Stat Entrance Exam, 7

Tags: arithmetic sequence , number theory , prime number

[b](i)[/b] Show that there cannot exists three peime numbers, each greater than $3$, which are in arithmetic progression with a common difference less than $5$. [b](ii)[/b] Let $k > 3$ be an integer. Show that it is not possible for $k$ prime numbers, each greater than $k$, to be in an arithmetic progression with a common difference less than or equal to $k+1$.

2005 Turkey Team Selection Test, 3

Tags: combinatorics

We are given 5040 balls in k different colors, where the number of balls of each color is the same. The balls are put into 2520 bags so that each bag contains two balls of different colors. Find the smallest k such that, however the balls are distributed into the bags, we can arrange the bags around a circle so that no two balls of the same color are in two neighboring bags.

2017 NIMO Problems, 3

Tags:

How many triples of integers $(a,b,c)$ with $-10\leq a,b,c\leq 10$ satisfy \[a^2+b^2+c^2=(a+b+c)^2?\] [i]Proposed by David Altizio

PEN A Problems, 41

Tags: modular arithmetic , divisibility theory

Show that there are infinitely many composite numbers $n$ such that $3^{n-1}-2^{n-1}$ is divisible by $n$.

2005 All-Russian Olympiad, 4

Tags: combinatorics , algorithm

100 people from 25 countries, four from each countries, stay on a circle. Prove that one may partition them onto 4 groups in such way that neither no two countrymans, nor two neighbours will be in the same group.

2014 India IMO Training Camp, 3

Tags: function , number theory

For integers $a,b$ we define $f((a,b))=(2a,b-a)$ if $a<b$ and $f((a,b))=(a-b,2b)$ if $a\geq b$. Given a natural number $n>1$ show that there exist natural numbers $m,k$ with $m<n$ such that $f^{k}((n,m))=(m,n)$,where $f^{k}(x)=f(f(f(...f(x))))$,$f$ being composed with itself $k$ times.

2003 Austrian-Polish Competition, 7

Tags: number theory , divide , factorial , divisible

Put $f(n) = \frac{n^n - 1}{n - 1}$. Show that $n!^{f(n)}$ divides $(n^n)! $. Find as many positive integers as possible for which $n!^{f(n)+1}$ does not divide $(n^n)!$ .

LMT Team Rounds 2021+, 11

Tags: algebra , polynomial

Find the number of degree $8$ polynomials $f (x)$ with nonnegative integer coefficients satisfying both $f (1) = 16$ and $f (-1) = 8$.

2025 Sharygin Geometry Olympiad, 10

Tags: geometry

An acute-angled triangle with one side equal to the altitude from the opposite vertex is cut from paper. Construct a point inside this triangle such that the square of the distance from it to one of the vertices equals the sum of the squares of distances to to the remaining two vertices. No instruments are available, it is allowed only to fold the paper and to mark the common points of folding lines. Proposed by: M.Evdokimov

2016 Ukraine Team Selection Test, 7

Tags: number theory , divisibility

Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.

2021 Dutch BxMO TST, 5

Tags: ratio , geometry

Given is a triangle $ABC$ with the property that $|AB| + |AC| = 3|BC|$. Let $T$ be the point on segment $AC$ such that $|AC| = 4|AT|$. Let $K$ and $L$ be points on the interior of line segments $AB$ and $AC$ respectively such that $KL \parallel BC$ and $KL$ is tangent to the inscribed circle of $\vartriangle ABC$. Let $S$ be the intersection of $BT$ and $KL$. Determine the ratio $\frac{|SL|}{|KL|}$

2025 Sharygin Geometry Olympiad, 21

Tags: geometry

Let $P$ be a point inside a quadrilateral $ABCD$ such that $\angle APB+\angle CPD=180^{\circ}$. Points $P_{a}$, $P_{b}$, $P_{c},$ $P_{d}$ are isogonally conjugated to $P$ with respect to the triangles $BCD$, $CDA$, $DAB$, $ABC$ respectively. Prove that the diagonals of the quadrilaterals $ABCD$ and $P_{a}P_{b}P_{c}P_{d}$ concur. Proposed by: G.Galyapin

2005 Finnish National High School Mathematics Competition, 1

Tags: geometry , pythagorean theorem

In the figure below, the centres of four squares have been connected by two line segments. Prove that these line segments are perpendicular.

2021 Iberoamerican, 4

Tags: algebra

Let $a,b,c,x,y,z$ be real numbers such that \[ a^2+x^2=b^2+y^2=c^2+z^2=(a+b)^2+(x+y)^2=(b+c)^2+(y+z)^2=(c+a)^2+(z+x)^2 \] Show that $a^2+b^2+c^2=x^2+y^2+z^2$.

2010 Flanders Math Olympiad, 3

Tags: angle bisector , perpendicular , parallel , angle , geometry

In a triangle $ABC$, $\angle B= 2\angle A \ne 90^o$ . The inner bisector of $B$ intersects the perpendicular bisector of $[AC]$ at a point $D$. Prove that $AB \parallel CD$.

2014 PUMaC Combinatorics A, 3

Tags: pyramid , 3d geometry , octahedron , rotation , geometry

You have three colors $\{\text{red}, \text{blue}, \text{green}\}$ with which you can color the faces of a regular octahedron (8 triangle sided polyhedron, which is two square based pyramids stuck together at their base), but you must do so in a way that avoids coloring adjacent pieces with the same color. How many different coloring schemes are possible? (Two coloring schemes are considered equivalent if one can be rotated to fit the other.)

2015 Geolympiad Spring, 1

Tags:

Let $ABC$ be a triangle. Suppose $P,Q$ are on lines $AB, AC$ (on the same side of A) with $AP=AC$ and $AB=AQ$. Now suppose points $X,Y$ move along the sides $AB, AC$ of $ABC$ so that $XY || PQ$. Determine the locus of the circumcenters of the variable triangle $AXY$.