This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 38

2023 Azerbaijan IZhO TST, 2
Tags: algebra , polynomial , TST , AZE IzHO TST
P(x) is polynomial such that, polynomial P(P(x)) is strictly monotone in all real number line. Prove that polynomial P(x) is also strictly monotone in all real number line.

2020 Azerbaijan IZHO TST, 4
Tags: number theory , TST , AZE IzHO TST
Consider an odd prime number $p$ and $p$ consecutive positive integers $m_1,m_2,…,m_p$. Choose a permutation $\sigma$ of $1,2,…,p$ . Show that there exist two different numbers $k,l\in{(1,2,…,p)}$ such that $p\mid{m_k.m_{\sigma(k)}-m_l.m_{\sigma(l)}}$

2018 Iranian Geometry Olympiad, 2
Tags: geometry , IGO , TST , AZE IzHO TST
In acute triangle $ABC, \angle A = 45^o$. Points $O,H$ are the circumcenter and the orthocenter of $ABC$, respectively. $D$ is the foot of altitude from $B$. Point $X$ is the midpoint of arc $AH$ of the circumcircle of triangle $ADH$ that contains $D$. Prove that $DX = DO$. Proposed by Fatemeh Sajadi

2018 Azerbaijan IZhO TST, 4
Tags: combinatorics , TST , AZE IzHO TST
There are $10$ cities in each of the three countries. Each road connects two cities from two different countries (there is at most one road between any two cities.) There are more than $200$ roads between these three countries. Prove that three cities, one city from each country, can be chosen such that there is a road between any two of these cities.

2012 Serbia National Math Olympiad, 2
Tags: number theory , TST , AZE IzHO TST
Find all natural numbers $a$ and $b$ such that \[a|b^2, \quad b|a^2 \mbox{ and } a+1|b^2+1.\]

2021 Azerbaijan IZhO TST, 1
Tags: algebra , Inequality , TST , AZE IzHO TST
Let $a, b, c$ be real numbers with the property as $ab + bc + ca = 1$. Show that: $$\frac {(a + b) ^ 2 + 1} {c ^ 2 + 2} + \frac {(b + c) ^ 2 + 1} {a ^ 2 + 2} + \frac {(c + a) ^ 2 + 1} {b ^ 2 + 2} \ge 3 $$.

2023 Austrian MO National Competition, 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 $x, y \in \mathbb{R}$.

2022 JBMO Shortlist, C5
Tags: combinatorics , JBMO Shortlist , TST , AZE IzHO TST
Let $S$ be a finite set of points in the plane, such that for each $2$ points $A$ and $B$ in $S$, the segment $AB$ is a side of a regular polygon all of whose vertices are contained in $S$. Find all possible values for the number of elements of $S$. Proposed by [i]Viktor Simjanoski, Macedonia[/i]

2021 Azerbaijan IZhO TST, 3
Tags: number theory , TST , AZE IzHO TST
For each $n \in N$ let $S(n)$ be the sum of all numbers in the set {1,2,3,…,n} which are relatively prime to $n$. a. Show that $2S(n) $ is not aperfect square for any $n$. b. Given positive integers $m,n$ with odd n, show that the equation $2S(x)=y^n$ has at least one solution $(x,y)$ among positive integers such that $m|x$.

2018 Tajikistan Team Selection Test, 3
Tags: algebra , TST , AZE IzHO TST
Problem 3. Suppose that the equation x^3-ax^2+bx-a=0 has three positive real roots (b>0). Find the minimum value of the expression: (b-a)(b^3+3a^3)

2020 Azerbaijan IZHO TST, 6
Tags: number theory , TST , AZE IzHO TST
Define a sequence ${{a_n}}_{n\ge1}$ such that $a_1=1$ , $a_2=2$ and $a_{n+1}$ is the smallest positive integer $m$ such that $m$ hasn't yet occurred in the sequence and also $gcd(m,a_n)\neq{1}$. Show that all positive integers occur in the sequence.

2020 Azerbaijan IZHO TST, 3
Tags: algebra , functional equation , TST , AZE IzHO TST
Find all functions $u:R\rightarrow{R}$ for which there exists a strictly monotonic function $f:R\rightarrow{R}$ such that $f(x+y)=f(x)u(y)+f(y)$ for all $x,y\in{\mathbb{R}}$

2023 Azerbaijan IZhO TST, 1
Tags: geometry , IGO , TST , AZE IzHO TST
In acute triangle $ABC, \angle A = 45^o$. Points $O,H$ are the circumcenter and the orthocenter of $ABC$, respectively. $D$ is the foot of altitude from $B$. Point $X$ is the midpoint of arc $AH$ of the circumcircle of triangle $ADH$ that contains $D$. Prove that $DX = DO$. Proposed by Fatemeh Sajadi

2010 Junior Balkan Team Selection Tests - Romania, 3
Tags: algebra , Inequality , TST , AZE IzHO TST
Let $a, b, c$ be real numbers with the property as $ab + bc + ca = 1$. Show that: $$\frac {(a + b) ^ 2 + 1} {c ^ 2 + 2} + \frac {(b + c) ^ 2 + 1} {a ^ 2 + 2} + \frac {(c + a) ^ 2 + 1} {b ^ 2 + 2} \ge 3 $$.

2020 Azerbaijan IZHO TST, 2
Tags: geometry , TST , AZE IzHO TST
Consider two circles $k_1,k_2$ touching at point $T$. A line touches $k_2$ at point $X$ and intersects $k_1$ at points $A,B$ where $B$ lies between $A$ and $X$.Let $S$ be the second intersection point of $k_1$ with $XT$. On the arc $\overarc{TS}$ not containing $A$ and $B$ , a point $C$ is choosen. Let $CY$ be the tangent line to $k_2$ with $Y\in{k_2}$ , such that the segment $CY$ doesn't intersect the segment $ST$ .If $I=XY\cap{SC}$ , prove that : $(a)$ the points $C,T,Y,I$ are concyclic. $(b)$ $I$ is the $A-excenter$ of $\triangle ABC$

2023 Azerbaijan IZhO TST, 3
Tags: combinatorics , JBMO Shortlist , TST , AZE IzHO TST
Let $S$ be a finite set of points in the plane, such that for each $2$ points $A$ and $B$ in $S$, the segment $AB$ is a side of a regular polygon all of whose vertices are contained in $S$. Find all possible values for the number of elements of $S$. Proposed by [i]Viktor Simjanoski, Macedonia[/i]

2024 Azerbaijan IZhO TST, 3
Tags: geometry , TST , AZE IzHO TST
In a triangle $ABC$, $I$ is the incenter. Line $CI$ intersects circumcircle of $ABC$ at $L$, and it is given that $CI=2IL$. $M;N$ are points chosen on $AB$ such that $\angle AIM=\angle BIN=90$. Prove that $AB=2MN$

2024 Azerbaijan IZhO TST, 2
Tags: combinatorics , TST , AZE IzHO TST
Find all positive integers $n$ such that one can place checkers on a $n\times n$ checkerboard such that any square chosen from the checkerboard has exactly $2$ adjacent squares with checkers on it. Two squares are considered adjacent if they both share a common side

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 $x, y \in \mathbb{R}$.

2025 Azerbaijan IZhO TST, 3
Tags: number theory , TST , AZE IzHO TST
Find all natural numbers $a$ and $b$ such that \[a|b^2, \quad b|a^2 \mbox{ and } a+1|b^2+1.\]

2023 Azerbaijan IZhO TST, 4
Tags: number theory , TST , AZE IzHO TST
A positive integer $t$ is called a Jane's integer if $t = x^3+y^2$ for some positive integers $x$ and $y$. Prove that for every integer $n \ge 2$ there exist infinitely many positive integers $m$ such that the set of $n^2$ consecutive integers $\{m+1,m+2,\dotsc,m+n^2\}$ contains exactly $n + 1$ Jane's integers.

2018 Tajikistan Team Selection Test, 5
Tags: TST , AZE IzHO TST
Problem 5. Consider the integer number n>2. Let a_1,a_2,…,a_n and b_1,b_2,…,b_n be two permutations of 0,1,2,…,n-1. Prove that there exist some i≠j such that: n|a_i b_i-a_j b_j [color=#00f]Moved to HSO. ~ oVlad[/color]

2020 Azerbaijan IZHO TST, 5
Tags: algebra , TST , AZE IzHO TST
Let $x,y,z$ be positive real numbers such that $x^4+y^4+z^4=1$ . Determine with proof the minimum value of $\frac{x^3}{1-x^8}+\frac{y^3}{1-y^8}+\frac{z^3}{1-z^8}$

2018 Azerbaijan IZhO TST, 3
Tags: TST , AZE IzHO TST
Problem 5. Consider the integer number n>2. Let a_1,a_2,…,a_n and b_1,b_2,…,b_n be two permutations of 0,1,2,…,n-1. Prove that there exist some i≠j such that: n|a_i b_i-a_j b_j [color=#00f]Moved to HSO. ~ oVlad[/color]

2020 Azerbaijan IZHO TST, 1
Tags: combinatorics , TST , AZE IzHO TST
Let $F$ be the set of all $n-tuples$ $(A_1,A_2,…,A_n)$ such that each $A_i$ is a subset of ${1,2,…,2019}$. Let $\mid{A}\mid$ denote the number of elements o the set $A$ . Find $\sum_{(A_1,…,A_n)\in{F}}^{}\mid{A_1\cup{A_2}\cup...\cup{A_n}}\mid$