Olympiad problems

2017 Israel National Olympiad, 3

A large collection of congruent right triangles is given, each with side length 3,4,5. Find the maximal number of such triangles you can place inside a 20x20 square, with no two triangles intersecting (in their interiors).

2021 Ecuador NMO (OMEC), 3

Tags: geometry , perimeter , Locus , national olympiad

Let $T_1$ and $T_2$ internally tangent circumferences at $P$, with radius $R$ and $2R$, respectively. Find the locus traced by $P$ as $T_1$ rolls tangentially along the entire perimeter of $T_2$

2014 Indonesia MO Shortlist, N5

Tags: number theory , pigeonhole principle , national olympiad

Prove that we can give a color to each of the numbers $1,2,3,...,2013$ with seven distinct colors (all colors are necessarily used) such that for any distinct numbers $a,b,c$ of the same color, then $2014\nmid abc$ and the remainder when $abc$ is divided by $2014$ is of the same color as $a,b,c$.

2020 Ecuador NMO (OMEC), 1

Tags: combinatorics , national olympiad

The country OMEC is divided in $5$ regions, each region is divided in $5$ districts, and, in each district, $1001$ people vote. Each person choose between $A$ or $B$. In a district, a candidate's letter wins if it's the letter with the most votes. In a region, a candidate's letter wins if it won in most districts. A candidate is the new president of OMEC if the candidate won in most regions. The candidate $A$ can rearrange the people of each district in each region (for example, A moves someone in District M to District N in region 1), but he can't change them to a different region. Find the minimum number of votes that the candidate $A$ needs to become the new president.

2024 Israel National Olympiad (Gillis), P1

Tags: algebra , system of equations , national olympiad

Solve the following system (over the real numbers): \[\begin{cases}5x+5y+5xy-2xy^2-2x^2y=20 &\\ 3x+3y+3xy+xy^2+x^2y=23&\end{cases}\]

2015 Bangladesh Mathematical Olympiad, 7

Tags: national olympiad , geometry , algebra , area , Cevas Theorem

In triangle $\triangle ABC$, the points $A', B', C'$ are on sides $BC, AC, AB$ respectively. Also, $AA', BB', CC'$ intersect at the point $O$(they are concurrent at $O$). Also, $\frac {AO}{OA'}+\frac {BO}{OB'}+\frac {CO}{OC'} = 92$. Find the value of $\frac {AO}{OA'}\times \frac {BO}{OB'}\times \frac {CO}{OC'}$.

2020 Kosovo National Mathematical Olympiad, 1

Tags: national olympiad , Olympiad , algebra , number theory , Compare , powers , POW

Compare the following two numbers: $2^{2^{2^{2^{2}}}}$ and $3^{3^{3^{3}}}$.

2022 Macedonian Mathematical Olympiad, Problem 1

Tags: Sequence , Inequality , national olympiad , algebra

Let $(x_n)_{n=1}^\infty$ be a sequence defined recursively with: $x_1=2$ and $x_{n+1}=\frac{x_n(x_n+n)}{n+1}$ for all $n \ge 1$. Prove that $$n(n+1) >\frac{(x_1+x_2+ \ldots +x_n)^2}{x_{n+1}}.$$ [i]Proposed by Nikola Velov[/i]

2020 Kosovo National Mathematical Olympiad, 4

Tags: geometry , circumcircle , national olympiad , Olympiad

Let $B'$ and $C'$ be points in the circumcircle of triangle $\triangle ABC$ such that $AB=AB'$ and $AC=AC'$. Let $E$ and $F$ be the foot of altitudes from $B$ and $C$ to $AC$ and $AB$, respectively. Show that $B'E$ and $C'F$ intersect on the circumcircle of triangle $\triangle ABC$.

1983 Czech and Slovak Olympiad III A, 3

Tags: combinatorics , combinatorial geometry , Chessboard , national olympiad , rectangle

An $8\times 8$ chessboard is made of unit squares. We put a rectangular piece of paper with sides of length 1 and 2. We say that the paper and a single square overlap if they share an inner point. Determine the maximum number of black squares that can overlap the paper.

2021 Ecuador NMO (OMEC), 1

Tags: number theory , national olympiad

Find all integers $n$ such that $\frac{4n}{n^2 +3 }$is an integer.

1957 Czech and Slovak Olympiad III A, 2

Tags: geometry , computational geometry , constructive geometry , 3D geometry , pyramid , national olympiad

Consider a (right) square pyramid $ABCDV$ with the apex $V$ and the base (square) $ABCD$. Denote $d=AB/2$ and $\varphi$ the dihedral angle between planes $VAD$ and $ABC$. (1) Consider a line $XY$ connecting the skew lines $VA$ and $BC$, where $X$ lies on line $VA$ and $Y$ lies on line $BC$. Describe a construction of line $XY$ such that the segment $XY$ is of the smallest possible length. Compute the length of segment $XY$ in terms of $d,\varphi$. (2) Compute the distance $v$ between points $V$ and $X$ in terms of $d,\varphi.$

2017 Spain Mathematical Olympiad, 1

Tags: algebra , national olympiad , Spain

Find the amount of different values given by the following expression: $\frac{n^2-2}{n^2-n+2}$ where $ n \in \{1,2,3,..,100\}$

2025 Kosovo National Mathematical Olympiad`, P2

Tags: algebra , Kosovo , national olympiad , get the smallest , beautiful , Holder

Find the smallest natural number $k$ such that the system of equations $$x+y+z=x^2+y^2+z^2=\dots=x^k+y^k+z^k $$ has only one solution for positive real numbers $x$, $y$ and $z$.

2022 Ecuador NMO (OMEC), 5

Tags: geometry , similar triangles , national olympiad

Let $ABC$ be a 90-degree triangle with hypotenuse $BC$. Let $D$ and $E$ distinct points on segment $BC$ and $P, Q$ be the foot of the perpendicular from $D$ to $AB$ and $E$ to $AC$, respectively. $DP$ and $EQ$ intersect at $R$. Lines $CR$ and $AB$ intersect at $M$ and lines $BR$ and $AC$ intersect at $N$. Prove that $MN \parallel BC$ if and only if $BD=CE$.

2018 Czech and Slovak Olympiad III A, 4

Tags: number theory , national olympiad

Let $a,b,c$ be integers which are lengths of sides of a triangle, $\gcd(a,b,c)=1$ and all the values $$\frac{a^2+b^2-c^2}{a+b-c},\quad\frac{b^2+c^2-a^2}{b+c-a},\quad\frac{c^2+a^2-b^2}{c+a-b}$$ are integers as well. Show that $(a+b-c)(b+c-a)(c+a-b)$ or $2(a+b-c)(b+c-a)(c+a-b)$ is a perfect square.

1987 Czech and Slovak Olympiad III A, 2

Tags: number theory , Diophantine equation , prime numbers , number of solutions , national olympiad

Given a prime $p>3$ and an odd integer $n>0$, show that the equation $$xyz=p^n(x+y+z)$$ has at least $3(n+1)$ different solutions up to symmetry. (That is, if $(x',y',z')$ is a solution and $(x'',y'',z'')$ is a permutation of the previous, they are considered to be the same solution.)

2023 Ecuador NMO (OMEC), 2

Tags: geometry , national olympiad , geometric transformation , reflection

Let $ABCD$ a cyclic convex quadrilateral. There is a line $l$ parallel to $DC$ containing $A$. Let $P$ a point on $l$ closer to $A$ than to $B$. Let $B'$ the reflection of $B$ over the midpoint of $AD$. Prove that $\angle B'AP = \angle BAC$

1957 Czech and Slovak Olympiad III A, 4

Tags: geometry , constructive geometry , trapezoid , national olympiad

Consider a non-zero convex angle $\angle POQ$ and its inner point $M$. Moreover, let $m>0$ be given. Construct a trapezoid $ABCD$ satisfying the following conditions: (1) vertices $A, D$ lie on ray $OP$ and vertices $B,C$ lie on ray $OQ$, (2) diagonals $AC$ and $BD$ intersect in $M$, (3) $AB=m$. Prove that your construction is correct and discuss conditions of solvability.

1989 Czech And Slovak Olympiad IIIA, 6

Tags: combinatorics , Combinatorial Number Theory , Sequence , Number sequence , national olympiad

Consider a finite sequence $a_1, a_2,...,a_n$ whose terms are natural numbers at most equal to $n$. Determine the maximum number of terms of such a sequence, if you know that every two of its neighboring terms are different and at the same time there is no quartet of terms in it such that $a_p = a_r \ne a_q = a_s$ for $p < q < r < s$.