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

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Found problems: 9

2016 Belarus Team Selection Test, 1
Tags: Inequality , inequalities , algebra , Belarus
Prove for positive $a,b,c$ that $$ (a^2+\frac{b^2}{c^2})(b^2+\frac{c^2}{a^2})(c^2+\frac{a^2}{b^2}) \geq abc (a+\frac{1}{a})(b+\frac{1}{b})(c+\frac{1}{c})$$

2018 Stars of Mathematics, 1
Tags: number theory , IMO Shortlist , Belarus , IMO 1999
Two natural numbers have the property that the product of their positive divisors are equal. Does this imply that they are equal? [i]Proposed by Belarus for the 1999th IMO[/i]

2015 Belarus Team Selection Test, 1
Tags: number theory , Diophantine equation , diophantine , Belarus , TST , 2015
Solve the equation in nonnegative integers $a,b,c$: $3^a+2^b+2015=3c!$ I.Gorodnin

1998 Belarus Team Selection Test, 2
Tags: inequalities , Ring Theory , inequalities proposed , 109 , Belarus , algebra
Let $a$, $b$, $c$ be real positive numbers. Show that \[\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq \frac{a+b}{b+c}+\frac{b+c}{a+b}+1\]

2011 Belarus Team Selection Test, 4
Tags: algebra , system of equations , Belarus
Given nonzero real numbers a,b,c with $a+b+c=a^2+b^2+c^2=a^3+b^3+c^3$. ($*$) a) Find $\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{b}\right)(a+b+c-2)$ b) Do there exist pairwise different nonzero $a,b,c$ satisfying ($*$)? D. Bazylev

2019 Belarus Team Selection Test, 2.2
Tags: geometry , circumcircle , perpendicular bisector , Belarus , geometry solved , Nine point center , Euler Line
Let $O$ be the circumcenter and $H$ be the orthocenter of an acute-angled triangle $ABC$. Point $T$ is the midpoint of the segment $AO$. The perpendicular bisector of $AO$ intersects the line $BC$ at point $S$. Prove that the circumcircle of the triangle $AST$ bisects the segment $OH$. [i](M. Berindeanu, RMC 2018 book)[/i]

2009 Belarus Team Selection Test, 1
Tags: inequalities , algebra , Belarus
Prove that any positive real numbers a,b,c satisfy the inequlaity $$\frac{1}{(a+b)b}+\frac{1}{(b+c)c}+\frac{1}{(c+a)a}\ge \frac{9}{2(ab+bc+ca)}$$ I.Voronovich

2020 Greece National Olympiad, 1
Tags: algebra , polynomial , functional equation , functional , polynomial equation , Belarus
Find all non constant polynomials $P(x),Q(x)$ with real coefficients such that: $P((Q(x))^3)=xP(x)(Q(x))^3$

2014 Belarus Team Selection Test, 2
Tags: Inequality , algebra , Belarus
Let $a,b,c$ be positive real numbers such that $a+b+c=1$. Prove that $$\frac{a^2}{(b+c)^3}+\frac{b^2}{(c+a)^3}+\frac{c^2}{(a+b)^3}\geq \frac98$$