Olympiad problems

2006 Junior Balkan Team Selection Tests - Romania, 3

Find all real numbers $ a$ and $ b$ such that \[ 2(a^2 \plus{} 1)(b^2 \plus{} 1) \equal{} (a \plus{} 1)(b \plus{} 1)(ab \plus{} 1). \] [i]Valentin Vornicu[/i]

1992 Taiwan National Olympiad, 3

Tags: induction , inequalities proposed , inequalities , BPSQ , algebra , High school olympiad

If $x_{1},x_{2},...,x_{n}(n>2)$ are positive real numbers with $x_{1}+x_{2}+...+x_{n}=1$. Prove that $x_{1}^{2}x_{2}+x_{2}^{2}x_{3}+...+x_{n}^{2}x_{1}\leq\frac{4}{27}$.

2018 Azerbaijan BMO TST, 1

Tags: inequalities , algebra , High school olympiad , BMO

Problem Shortlist BMO 2017 Let $ a $,$ b$,$ c$, be positive real numbers such that $abc= 1 $. Prove that $$\frac{1}{a^{5}+b^{5}+c^{2}}+\frac{1}{b^{5}+c^{5}+a^{2}}+\frac{1}{c^{5}+b^{5}+b^{2}}\leq 1 . $$

1999 All-Russian Olympiad, 7

Tags: Russia , High school olympiad , algebra , Inequality

Positive numbers $x,y$ satisfy $x^2+y^3 \ge x^3+y^4$. Prove that $x^3+y^3 \le 2$.

2017 CHKMO, Q1

Tags: High school olympiad , combinatorics

A, B and C are three persons among a set P of n (n[u]>[/u]3) persons. It is known that A, B and C are friends of one another, and that every one of the three persons has already made friends with more than half the total number of people in P. Given that every three persons who are friends of one another form a [i]friendly group[/i], what is the minimum number of friendly groups that may exist in P?

2015 IFYM, Sozopol, 4

Tags: algebra , Inequality , inequalities , High school olympiad , inequalities proposed

For all real numbers $a,b,c>0$ such that $abc=1$, prove that $\frac{a}{1+b^3}+\frac{b}{1+c^3}+\frac{c}{1+a^3}\geq \frac{3}{2}$.

2020 Caucasus Mathematical Olympiad, 8

Tags: inequalities , algebra , High school olympiad

Let real $a$, $b$, and $c$ satisfy $$abc+a+b+c=ab+bc+ca+5.$$ Find the least possible value of $a^2+b^2+c^2$.

2013 Hong kong National Olympiad, 1

Tags: inequalities , function , inequalities proposed , China , High school olympiad

Let $a,b,c$ be positive real numbers such that $ab+bc+ca=1$. Prove that \[\sqrt[4]{\frac{\sqrt{3}}{a}+6\sqrt{3}b}+\sqrt[4]{\frac{\sqrt{3}}{b}+6\sqrt{3}c}+\sqrt[4]{\frac{\sqrt{3}}{c}+6\sqrt{3}a}\le\frac{1}{abc}\] When does inequality hold?

2020 Turkey EGMO TST, 6

Tags: inequalities , algebra , Turkey , High school olympiad , inequalities proposed

$x,y,z$ are positive real numbers such that: $$xyz+x+y+z=6$$ $$xyz+2xy+yz+zx+z=10$$ Find the maximum value of: $$(xy+1)(yz+1)(zx+1)$$

2015 JBMO Shortlist, A5

Tags: inequalities , JBMO , China , BPSQ , JBMO Shortlist , High school olympiad

The positive real $x, y, z$ are such that $x^2+y^2+z^2 = 3$. Prove that$$\frac{x^2+yz}{x^2+yz +1}+\frac{y^2+zx}{y^2+zx+1}+\frac{z^2+xy}{z^2+xy+1}\leq 2$$

2003 Mediterranean Mathematics Olympiad, 3

Tags: inequalities , blogs , inequalities unsolved , algebra , highschoolmath , High school olympiad

Let $a, b, c$ be non-negative numbers with $a+b+c = 3$. Prove the inequality \[\frac{a}{b^2+1}+\frac{b}{c^2+1}+\frac{c}{a^2+1} \geq \frac 32.\]

2013 Dutch IMO TST, 5

Tags: algebra , inequalities , Dutch MO , High school olympiad

Let $a, b$, and $c$ be positive real numbers satisfying $abc = 1$. Show that $a + b + c \ge \sqrt{\frac13 (a + 2)(b + 2)(c + 2)}$

2008 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: inequalities , logarithms , circumcircle , High school olympiad , Bosnia , algebra

If $ a$, $ b$ and $ c$ are positive reals prove inequality: \[ \left(1\plus{}\frac{4a}{b\plus{}c}\right)\left(1\plus{}\frac{4b}{a\plus{}c}\right)\left(1\plus{}\frac{4c}{a\plus{}b}\right) > 25.\]

2017 Brazil Undergrad MO, 4

Tags: real analysis , Sequences , High school olympiad

Let $(a_n)_{n\geq 1}$ be a sequence of positive real numbers in which $\lim_{n\to\infty} a_n = 0$ such that there is a constant $c >0$ so that for all $n \geq 1$, $|a_{n+1}-a_n| \leq c\cdot a_n^2$. Show that exists $d>0$ with $na_n \geq d, \forall n \geq 1$.

2013 Dutch IMO TST, 5

Tags: algebra , inequalities , Dutch MO , High school olympiad

Let $a, b$, and $c$ be positive real numbers satisfying $abc = 1$. Show that $a + b + c \ge \sqrt{\frac13 (a + 2)(b + 2)(c + 2)}$

2019 All-Russian Olympiad, 8

Tags: algebra , inequalities , Russia , High school olympiad

For $a,b,c$ be real numbers greater than $1$, prove that \[\frac{a+b+c}{4} \geq \frac{\sqrt{ab-1}}{b+c}+\frac{\sqrt{bc-1}}{c+a}+\frac{\sqrt{ca-1}}{a+b}.\]

2019 Federal Competition For Advanced Students, P2, 4

Tags: inequalities , BPSQ , High school olympiad , China , Austria

Let $a, b, c$ be the positive real numbers such that $a+b+c+2=abc .$ Prove that $$(a+1)(b+1)(c+1)\geq 27.$$

2011 JBMO Shortlist, 3

Tags: inequalities , algebra , JBMO Shortlist , High school olympiad

$\boxed{\text{A3}}$If $a,b$ be positive real numbers, show that:$$ \displaystyle{\sqrt{\dfrac{a^2+ab+b^2}{3}}+\sqrt{ab}\leq a+b}$$

2008 Vietnam National Olympiad, 6

Tags: inequalities , quadratics , inequalities unsolved , algebra , High school olympiad , Vietnam

Let $ x, y, z$ be distinct non-negative real numbers. Prove that \[ \frac{1}{(x\minus{}y)^2} \plus{} \frac{1}{(y\minus{}z)^2} \plus{} \frac{1}{(z\minus{}x)^2} \geq \frac{4}{xy \plus{} yz \plus{} zx}.\] When does the equality hold?

2007 China Northern MO, 2

Tags: inequalities , geometric inequality , inequalities proposed , China , High school olympiad

Let $ a,\, b,\, c$ be side lengths of a triangle and $ a+b+c = 3$. Find the minimum of \[ a^{2}+b^{2}+c^{2}+\frac{4abc}{3}\]

1987 Nordic, 4

Tags: inequalities , algebra , High school olympiad

Let $a, b$, and $c$ be positive real numbers. Prove: $\frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\le \frac{a^2}{b^2} + \frac{b^2}{c^2} + \frac{c^2}{a^2}$ .

2019 CHKMO, 1

Tags: inequalities , algebra , High school olympiad

Given that $a,b$, and $c$ are positive real numbers such that $ab + bc + ca \geq 1$, prove that \[ \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2} \geq \frac{\sqrt{3}}{abc} .\]

2018 Singapore Senior Math Olympiad, 4

Tags: algebra , inequalities , High school olympiad , singapore

Let $a,b,c,d$ be positive integers such that $a+c=20$ and $\frac{a}{b}+\frac{c}{d}<1$. Find the maximum possible value of $\frac{a}{b}+\frac{c}{d}$.

2019 Taiwan TST Round 2, 1

Tags: inequalities , algebra , High school olympiad

Prove that for any positive reals $ a,b,c,d $ with $ a+b+c+d = 4 $, we have $$ \sum\limits_{cyc}{\frac{3a^3}{a^2+ab+b^2}}+\sum\limits_{cyc}{\frac{2ab}{a+b}} \ge 8 $$

2020 Turkey Team Selection Test, 8

Tags: inequalities , algebra , Turkey , High school olympiad

Let $x,y,z$ be real numbers such that $0<x,y,z<1$. Find the minimum value of: $$\frac {xyz(x+y+z)+(xy+yz+zx)(1-xyz)}{xyz\sqrt {1-xyz}}$$