Olympiad problems

2024 Belarus Team Selection Test, 1.3

Prove that for any real numbers $a,b,c,d \geq \frac{1}{3}$ the following inequality holds: $$\sqrt{\frac{a^6}{b^4+c^3}+\frac{b^6}{c^4+d^3}+\frac{c^6}{d^4+a^3}+\frac{d^6}{a^4+b^3}}\geq \frac{a+b+c+d}{4}$$ [i]D. Zmiaikou[/i]

2015 Turkey Team Selection Test, 5

Tags: Turkey , TST , 2015 , combinatorics

We are going to colour the cells of a $2015 \times 2015$ board such that there are none of the following: $1)$ Three cells with the same colour where two of them are in the same column, and the third is in the same row and to the right of the upper cell, $2)$ Three cells with the same colour where two of them are in the same column, and the third is in the same row and to the left of the lower cell. What is the minimum number of colours $k$ required to make such a colouring possible?

2023 Chile TST Ibero., 2

Tags: TST , Chile , algebra

Consider a function $ n \mapsto f(n) $ that satisfies the following conditions: $ f(n) $ is an integer for each $ n $. $ f(0) = 1 $. $ f(n+1) > f(n) + f(n-1) + \cdots + f(0) $ for each $ n = 0, 1, 2, \dots $. Determine the smallest possible value of $ f(2023) $.

2019 Greece JBMO TST, 2

Tags: number theory , TST

Find all pairs of positive integers $(x,n) $ that are solutions of the equation $3 \cdot 2^x +4 =n^2$.

2020 Taiwan TST Round 1, 1

Tags: geometry , IMO Shortlist , TST

Let $ABC$ be an acute-angled triangle and let $D, E$, and $F$ be the feet of altitudes from $A, B$, and $C$ to sides $BC, CA$, and $AB$, respectively. Denote by $\omega_B$ and $\omega_C$ the incircles of triangles $BDF$ and $CDE$, and let these circles be tangent to segments $DF$ and $DE$ at $M$ and $N$, respectively. Let line $MN$ meet circles $\omega_B$ and $\omega_C$ again at $P \ne M$ and $Q \ne N$, respectively. Prove that $MP = NQ$. (Vietnam)

2017 Azerbaijan BMO TST, 3

Tags: algebra , functional equation , AZE BMO TST , TST

Find all funtions $f:\mathbb R\to\mathbb R$ such that: $$f(xy-1)+f(x)f(y)=2xy-1$$ for all $x,y\in \mathbb{R}$.

2023 Chile TST IMO, 5

Tags: geometry , TST , Chile

Let $ \triangle ABC $ be an acute-angled triangle. Let $ D $ and $ E $ be the feet of the altitudes from $ B $ and $ C $, respectively. Let $ E' $ be the reflection of point $ E $ with respect to line $ BD $, which is assumed to lie on the circumcircle of triangle $ \triangle ABC $. Let $ C' $ be the reflection of point $ C $ with respect to line $ BD $. Prove that triangle $ C'AE $ is isosceles and determine the ratio $ AD : DC $.

2020 USA TSTST, 8

Tags: USA , TST , number theory , Tstst

For every positive integer $N$, let $\sigma(N)$ denote the sum of the positive integer divisors of $N$. Find all integers $m\geq n\geq 2$ satisfying \[\frac{\sigma(m)-1}{m-1}=\frac{\sigma(n)-1}{n-1}=\frac{\sigma(mn)-1}{mn-1}.\] [i]Ankan Bhattacharya[/i]

2005 Italy TST, 1

Tags: combinatorics proposed , combinatorics , Italy TST , TST , Team Selection Test , double counting

A stage course is attended by $n \ge 4$ students. The day before the final exam, each group of three students conspire against another student to throw him/her out of the exam. Prove that there is a student against whom there are at least $\sqrt[3]{(n-1)(n- 2)} $conspirators.

2018 Peru Cono Sur TST, 2

Tags: algebra , TST

Let $ x $ be a positive real number such that the numbers $ x^{-1} $, $ x $, and $ x^{2018} $ have the same fractional part: $$ \{x^{-1}\} = \{x\} = \{x^{2018}\}. $$ Prove that $ x = 1 $. [b]Note:[/b] If $ x $ is a real number, its fractional part is $ \{x\} = x - \lfloor x \rfloor $, where $ \lfloor x \rfloor $ denotes the greatest integer less than or equal to $ x $.

1986 China Team Selection Test, 4

Tags: combinatorics unsolved , combinatorics , points , Chords , China , TST , Team Selection Test

Mark $4 \cdot k$ points in a circle and number them arbitrarily with numbers from $1$ to $4 \cdot k$. The chords cannot share common endpoints, also, the endpoints of these chords should be among the $4 \cdot k$ points. [b]i.[/b] Prove that $2 \cdot k$ pairwisely non-intersecting chords can be drawn for each of whom its endpoints differ in at most $3 \cdot k - 1$. [b]ii.[/b] Prove that the $3 \cdot k - 1$ cannot be improved.

2023 Federal Competition For Advanced Students, P2, 1

Tags: algebra , functional equation , TST , AZE IzHO TST

Given is a nonzero real number $\alpha$. Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $$f(f(x+y))=f(x+y)+f(x)f(y)+\alpha xy$$ for all $x, y \in \mathbb{R}$.

2018 Azerbaijan JBMO TST, 2

Tags: geometry , Balkan MO Shortlist , TST , AZE JBMO TST

Let $ABC$ be an acute triangle and let $M$ be the midpoint of side $BC$. Let $D,E$ be the excircles of triangles $AMB,AMC$ respectively, towards $M$. Circumcirscribed circle of triangle $ABD$ intersects line $BC$ at points $B$ and $F$. Circumcirscribed circles of triangle $ACE$ intersects line $BC$ at points $C$ and $G$. Prove that $BF=CG$. by Petru Braica, Romania

2023 Junior Balkan Team Selection Tests - Moldova, 12

Tags: algebra , TST , Inequality

Let $a,b,c$ be positive real numbers such that $a^2+b^2+c^2=3. $ Prove that $$\frac{a^4+3ab^3}{a^3+2b^3}+\frac{b^4+3bc^3}{b^3+2c^3}+\frac{c^4+3ca^3}{c^3+2a^3}\leq4.$$

2021 Greece JBMO TST, 3

Tags: number theory , TST

Determine whether exists positive integer $n$ such that the number $A=8^n+47$ is prime.

2021 Bolivian Cono Sur TST, 1

Tags: geometry , rhombus , parallelogram , Equilateral Triangle , Bolivia , Cono Sur TST , TST

Inside a rhombus $ABCD$ with $\angle BAD=60$, points $F,H,G$ are choosen on lines $AD,DC,AC$ respectivily such that $DFGH$ is a paralelogram. Show that $BFH$ is a equilateral triangle.

2022 Macedonian Team Selection Test, Problem 1

Tags: combinatorics , TST

Let $n$ be a fixed positive integer. There are $n \geq 1$ lamps in a row, some of them are on and some are off. In a single move, we choose a positive integer $i$ ($1 \leq i \leq n$) and switch the state of the first $i$ lamps from the left. Determine the smallest number $k$ with the property that we can make all of the lamps be switched on using at most $k$ moves, no matter what the initial configuration was. [i]Proposed by Viktor Simjanoski and Nikola Velov[/i]

2024 Chile TST IMO, 1

Tags: TST , Chile , combinatorics

Consider a set of $ n \geq 3 $ points in the plane where no three are collinear. Prove that the points can be labeled as $ P_1, P_2, \dots, P_n $ so that the angles $ \angle P_i P_{i+1} P_{i+2} $ are less than $ 90^\circ $ for all $ i $.

2016 Kosovo Team Selection Test, 2

Tags: number theory , TST , Kosovo , 2016

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

2019 Bosnia and Herzegovina Junior BMO TST, 4

Tags: number theory , TST

$4.$ Let there be a variable positive integer whose last two digits are $3's$. Prove that this number is divisible by a prime greater than $7$.

2020 JBMO Shortlist, 2

Tags: number theory , JBMO Shortlist , TST

Find all positive integers $a$, $b$, $c$, and $p$, where $p$ is a prime number, such that $73p^2 + 6 = 9a^2 + 17b^2 + 17c^2$.

2017 Vietnam Team Selection Test, 1

Tags: geometry , Vietnam , TST

Triangle $ABC$ is inscribed in circle $(O)$. $A$ varies on $(O)$ such that $AB>BC$. $M$ is the midpoint of $AC$. The circle with diameter $BM$ intersects $(O)$ at $R$. $RM$ intersects $(O)$ at $Q$ and intersects $BC$ at $P$. The circle with diameter $BP$ intersects $AB, BO$ at $K,S$ in this order. a. Prove that $SR$ passes through the midpoint of $KP$. b. Let $N$ be the midpoint of $BC$. The radical axis of circles with diameters $AN, BM$ intersects $SR$ at $E$. Prove that $ME$ always passes through a fixed point.

2017 Azerbaijan Team Selection Test, 1

Tags: combinatorics , IMO Shortlist , AZE IMO TST , TST

Find all positive integers $n$ for which all positive divisors of $n$ can be put into the cells of a rectangular table under the following constraints: [list] [*]each cell contains a distinct divisor; [*]the sums of all rows are equal; and [*]the sums of all columns are equal. [/list]

2016 Junior Balkan Team Selection Tests - Romania, 1

Tags: geometry , TST

The altitudes $AA_1$,$BB_1$,$CC_1$ of $\triangle{ABC}$ intersect at $H$.$O$ is the circumcenter of $\triangle{ABC}$.Let $A_2$ be the reflection of $A$ wrt $B_1C_1$.Prove that: a)$O$,$A_2$,$B_1$,$C$ are all on a circle b)$O$,$H$,$A_1$,$A_2$ are all on a circle

2021 Bolivian Cono Sur TST, 2

Tags: combinatorics , monovariant , Cono Sur TST , Pre TST , TST , Bolivia

The numbers $1,2,...,100$ are written in a board. We are allowed to choose any two numbers from the board $a,b$ to delete them and replace on the board the number $a+b-1$. What are the possible numbers u can get after $99$ consecutive operations of these?