Olympiad problems

2017 Israel National Olympiad, 5

A regular pentagon $ABCDE$ is given. The point $X$ is on his circumcircle, on the arc $\overarc{AE}$. Prove that $|AX|+|CX|+|EX|=|BX|+|DX|$. [u][b]Remark:[/b][/u] Here's a more general version of the problem: Prove that for any point $X$ in the plane, $|AX|+|CX|+|EX|\ge|BX|+|DX|$, with equality only on the arc $\overarc{AE}$.

1983 Czech and Slovak Olympiad III A, 1

Tags: maximization , algebra , national olympiad

Let $n$ be a positive integer and $k\in[0,n]$ be a fixed real constant. Find the maximum value of $$\left|\sum_{i=1}^n\sin(2x_i)\right|$$ where $x_1,\ldots,x_n$ are real numbers satisfying $$\sum_{i=1}^n\sin^2(x_i)=k.$$

2019 Czech and Slovak Olympiad III A, 4

Tags: geometry , excircle , Triangle , perpendicular lines , national olympiad

Let be $ABC$ an acute-angled triangle. Consider point $P$ lying on the opposite ray to the ray $BC$ such that $|AB|=|BP|$. Similarly, consider point $Q$ on the opposite ray to the ray $CB$ such that $|AC|=|CQ|$. Denote $J$ the excenter of $ABC$ with respect to $A$ and $D,E$ tangent points of this excircle with the lines $AB$ and $AC$, respectively. Suppose that the opposite rays to $DP$ and $EQ$ intersect in $F\neq J$. Prove that $AF\perp FJ$.

2018 Brazil Undergrad MO, 1

Tags: college contests , national olympiad , geometry , Brazilian Math Olympiad

An equilateral triangle is cut as shown in figure 1 and the parts are used to form figure 2. What is the shape of figure 2?

2025 Kosovo National Mathematical Olympiad`, P1

Tags: algebra , system of equations , Kosovo , national olympiad , nice

Find all real numbers $a$, $b$ and $c$ that satisfy the following system of equations: $$\begin{cases} ab-c = 3 \\ a+bc = 4 \\ a^2+c^2 = 5\end{cases}$$

2022 Israel National Olympiad, P1

Tags: combinatorics , national olympiad

In a room are several people, some of which always lie and all others always tell the truth. Their ages are pairwise distinct. Each person says one of the following phrases: "In this room, there is an equal number of truth-sayers older than me and of liars younger than me" or "In this room, there is an equal number of truth-sayers younger than me and of liars older than me" What is the maximum possible number of truth-sayers in the room? Find an example in which this maximum is achieved and prove a higher number is impossible.

2019 Czech and Slovak Olympiad III A, 5

Tags: number theory , national olympiad

Prove that there are infinitely many integers which cannot be expressed as $2^a+3^b-5^c$ for non-negative integers $a,b,c$.

2025 Kosovo National Mathematical Olympiad`, P3

Tags: Kosovo , national olympiad , number theory , Sets

A subset $S$ of the natural numbers is called [i]dense [/i] for every $7$ consecutive natural numbers, at least $5$ of them are in $S$. Show that there exists a dense subset for which the equation $a^2+b^2=c^2$ has no solution for $a,b,c \in S$.

2017 Spain Mathematical Olympiad, 5

Tags: algebra , inequalities , maximum , national olympiad , Spain

Let $a,b,c$ be positive real numbers so that $a+b+c = \frac{1}{\sqrt{3}}$. Find the maximum value of $$27abc+a\sqrt{a^2+2bc}+b\sqrt{b^2+2ca}+c\sqrt{c^2+2ab}.$$

2016 Spain Mathematical Olympiad, 3

Tags: Olympiad , national olympiad , geometry , circumcircle

In the circumscircle of a triangle $ABC$, let $A_1$ be the point diametrically opposed to the vertex $A$. Let $A'$ the intersection point of $AA'$ and $BC$. The perpendicular to the line $AA'$ from $A'$ meets the sides $AB$ and $AC$ at $M$ and $N$, respectively. Prove that the points $A,M,A_1$ and $N$ lie on a circle which has the center on the height from $A$ of the triangle $ABC$.

2025 Kosovo National Mathematical Olympiad`, P4

Tags: geometry , national olympiad , Kosovo

Let $D$ be a point on the side $AC$ of triangle $\triangle ABC$ such that $AB=AD=DC$ and let $E$ be a point on the side $BC$ such that $BE=2CE$. Prove that $\angle BDE = 90 ^{\circ}$.

2023 Ecuador NMO (OMEC), 5

Tags: number theory , national olympiad

Find all positive integers $n$ such that $4^n + 4n + 1$ is a perfect square.

2021 Israel National Olympiad, P2

Tags: number theory , national olympiad

Does there exist an infinite sequence of primes $p_1, p_2, p_3, \dots $ for which, \[p_{n+1}=2p_n+1\] for each $n$?

2020 Kosovo National Mathematical Olympiad, 4

Tags: national olympiad , Olympiad , Kosovo , algebra , number theory , Sequence

Let $a_0$ be a fixed positive integer. We define an infinite sequence of positive integers $\{a_n\}_{n\ge 1}$ in an inductive way as follows: if we are given the terms $a_0,a_1,...a_{n-1}$ , then $a_n$ is the smallest positive integer such that $\sqrt[n]{a_0\cdot a_1\cdot ...\cdot a_n}$ is a positive integer. Show that the sequence $\{a_n\}_{n\ge 1}$ is eventually constant. [b]Note:[/b] The sequence $\{a_n\}_{n\ge 1}$ is eventually constant if there exists a positive integer $k$ such that $a_n=c$, for every $n\ge k$.

2018 Regional Competition For Advanced Students, 2

Tags: geometry , Equilateral Triangle , perpendicular bisector , Chords , Austria , national olympiad , AUT

Let $k$ be a circle with radius $r$ and $AB$ a chord of $k$ such that $AB > r$. Furthermore, let $S$ be the point on the chord $AB$ satisfying $AS = r$. The perpendicular bisector of $BS$ intersects $k$ in the points $C$ and $D$. The line through $D$ and $S$ intersects $k$ for a second time in point $E$. Show that the triangle $CSE$ is equilateral. [i]Proposed by Stefan Leopoldseder[/i]

2020 Kosovo National Mathematical Olympiad, 1

Tags: Sequence , terms , national olympiad , Olympiad , Kosovo , algebra , combinatorics

Some positive integers, sum of which is $23$, are written in sequential form. Neither one of the terms nor the sum of some consecutive terms in the sequence is equal to $3$. [b]a) [/b]Is it possible that the sequence contains exactly $11$ terms? [b]b)[/b]Is it possible that the sequence contains exactly $12$ terms?

2016 Thailand Mathematical Olympiad, 7

Tags: polynomial , algebra , national olympiad , absolute value

Given $P(x)=a_{2016}x^{2016}+a_{2015}x^{2015}+...+a_1x+a_0$ be a polynomial with real coefficients and $a_{2016} \neq 0$ satisfies $|a_1+a_3+...+a_{2015}| > |a_0+a_2+...+a_{2016}|$ Prove that $P(x)$ has an odd number of complex roots with absolute value less than $1$ (count multiple roots also) edited: complex roots

2018 Peru MO (ONEM), 2

Tags: Peru , national olympiad , algebra , inequalities , High School Olympiads

2) Let $a, b, c$ be real numbers such that $$a+\frac{b}{c}=b+\frac{c}{a}=c+\frac{a}{b}=1$$a) Prove that $ab+bc+ca=0$ and $a+b+c=3$. b) Prove that $|a|+|b|+|c|< 5$

2018 Czech and Slovak Olympiad III A, 6

Tags: Combinatorial Number Theory , national olympiad , combinatorics , number theory

Determine the least positive integer $n$ with the following property – for every 3-coloring of numbers $1,2,\ldots,n$ there are two (different) numbers $a,b$ of the same color such that $|a-b|$ is a perfect square.

2024 Israel National Olympiad (Gillis), P7

Tags: combinatorics , grid , Rooks , paths , national olympiad

A rook stands in one cell of an infinite square grid. A different cell was colored blue and mines were placed in $n$ additional cells: the rook cannot stand on or pass through them. It is known that the rook can reach the blue cell in finitely many moves. Can it do so (for every $n$ and such a choice of mines, starting point, and blue cell) in at most [b](a)[/b] $1.99n+100$ moves? [b](b)[/b] $2n-2\sqrt{3n}+100$ moves? [b]Remark.[/b] In each move, the rook goes in a vertical or horizontal line.

2025 Kosovo National Mathematical Olympiad`, P4

Tags: remainder , number theory , Kosovo , national olympiad

When a number is divided by $2$ it has quotient $x$ and remainder $1$. Whereas, when the same number is divided by $3$ it has quotient $y$ and remainder $2$. What is the remainder when $x+y$ is divided by $5$?

2020 Macedonian Nationаl Olympiad, 4

Tags: combinatorics , set theory , national olympiad

Let $S$ be a nonempty finite set, and $\mathcal {F}$ be a collection of subsets of $S$ such that the following conditions are met: (i) $\mathcal {F}$ $\setminus$ {$S$} $\neq$ $\emptyset$ ; (ii) if $F_1, F_2 \in \mathcal {F}$, then $F_1 \cap F_2 \in \mathcal {F}$ and $F_1 \cup F_2 \in \mathcal {F}$. Prove that there exists $a \in S$ which belongs to at most half of the elements of $\mathcal {F}$.

2017 Spain Mathematical Olympiad, 3

Tags: algebra , USAMO , Spain , national olympiad

Let $p$ be an odd prime and $S_{q} = \frac{1}{2*3*4} + \frac{1}{5*6*7} + ... + \frac{1}{q(q+1)(q+2)}$, where $q = \frac{3p-5}{2}$. We write $\frac{1}{2}-2S_{q}$ in the form $\frac{m}{n}$, where $m$ and $n$ are integers. Prove that $m \equiv n (mod p)$

2018 Peru MO (ONEM), 4

Tags: Peru , national olympiad

4) A $100\times 200$ board has $k$ black cells. An operations consists of choosing a $2\times 3$ or $3\times 2$ sub-board having exactly $5$ black cells and painting of black the remaining cell. Find the least value of $k$ for which exists an initial distribution of the black cells such that after some operations the board is completely black.

2019 Czech and Slovak Olympiad III A, 3

Tags: number theory , Divisibility , Perfect Powers , national olympiad

Let $a,b,c,n$ be positive integers such that the following conditions hold (i) numbers $a,b,c,a+b+c$ are pairwise coprime, (ii) number $(a+b)(b+c)(c+a)(a+b+c)(ab+bc+ca)$ is a perfect $n$-th power. Prove, that the product $abc$ can be expressed as a difference of two perfect $n$-th powers.