Olympiad problems

2014 China Second Round Olympiad, 1

Let $a,b,c$ be real numbers such that $a+b+c=1$ and $abc>0$ . Prove that\[bc+ca+ab<\frac{\sqrt{abc}}{2}+\frac{1}{4}.\]

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 $x_{1}^{2}x_{2}+x_{2}^{2}x_{3}+...+x_{n}^{2}x_{1}\leq\frac{4}{27}$.

2014 Contests, 1

Tags: inequalities proposed , inequalities , China , BPSQ

Let $a,b,c$ be real numbers such that $a+b+c=1$ and $abc>0$ . Prove that\[bc+ca+ab<\frac{\sqrt{abc}}{2}+\frac{1}{4}.\]

2019 China National Olympiad, 1

Tags: inequalities , China , High School Olympiads , BPSQ , Hi

Let $a,b,c,d,e\geq -1$ and $a+b+c+d+e=5.$ Find the maximum and minimum value of $S=(a+b)(b+c)(c+d)(d+e)(e+a).$

2015 Junior Balkan Team Selection Tests - Romania, 3

Tags: inequalities , BPSQ

Let $x$,$y$,$z>0$ . Show that : $$\frac{x^3}{z^3+x^2y}+\frac{y^3}{x^3+y^2z}+\frac{z^3}{y^3+z^2x} \geq \frac{3}{2}$$

2013 China Second Round Olympiad, 1

Tags: modular arithmetic , number theory proposed , number theory , China , BPSQ

For any positive integer $n$ , Prove that there is not exist three odd integer $x,y,z$ satisfing the equation $(x+y)^n+(y+z)^n=(x+z)^n$.

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$$\frac{x^2+yz}{x^2+yz +1}+\frac{y^2+zx}{y^2+zx+1}+\frac{z^2+xy}{z^2+xy+1}\leq 2$$

2014 Czech and Slovak Olympiad III A, 6

Tags: inequalities , BPSQ

For arbitrary non-negative numbers $a$ and $b$ prove inequality $\frac{a}{\sqrt{b^2+1}}+\frac{b}{\sqrt{a^2+1}}\ge\frac{a+b}{\sqrt{ab+1}}$, and find, where equality occurs. (Day 2, 6th problem authors: Tomáš Jurík, Jaromír Šimša)

2019 Kyiv Mathematical Festival, 2

Tags: Kyiv mathematical festival , inequalities , BPSQ

Let $a,b,c>0$ and $abc\ge1.$ Prove that $a^4+b^3+c^2\ge a^3+b^2+c.$

2017 China National Olympiad, 6

Tags: BPSQ , China , PRC , inequalities , Inequality proposed , convex-concave inequalities , sqing orz

Given an integer $n \geq2$ and real numbers $a,b$ such that $0<a<b$. Let $x_1,x_2,\ldots, x_n\in [a,b]$ be real numbers. Find the maximum value of $$\frac{\frac{x^2_1}{x_2}+\frac{x^2_2}{x_3}+\cdots+\frac{x^2_{n-1}}{x_n}+\frac{x^2_n}{x_1}}{x_1+x_2+\cdots +x_{n-1}+x_n}.$$

2014 Contests, 1

Tags: inequalities proposed , algebra , China , BPSQ , Inequality

Let $x,y$ be positive real numbers .Find the minimum of $x+y+\frac{|x-1|}{y}+\frac{|y-1|}{x}$.

2019 China Girls Math Olympiad, 6

Tags: inequalities , algebra , China , BPSQ

Let $0\leq x_1\leq x_2\leq \cdots \leq x_n\leq 1 $ $(n\geq 2).$ Prove that $$\sqrt[n]{x_1x_2 \cdots x_n}+ \sqrt[n]{(1-x_1)(1-x_2)\cdots (1-x_n)}\leq \sqrt[n]{1-(x_1- x_n)^2}.$$

2019 Regional Competition For Advanced Students, 1

Tags: inequalities , BPSQ , algebra , inequalities proposed

Let $x,y$ be real numbers such that $(x+1)(y+2)=8.$ Prove that $$(xy-10)^2\ge 64.$$

2002 Tournament Of Towns, 1

Tags: inequalities , inequalities proposed , BPSQ

Let $a,b,c$ be sides of a triangle. Show that $a^3+b^3+3abc>c^3$.

2014 China Western Mathematical Olympiad, 1

Tags: inequalities proposed , algebra , China , BPSQ , Inequality

Let $x,y$ be positive real numbers .Find the minimum of $x+y+\frac{|x-1|}{y}+\frac{|y-1|}{x}$.

2017 South East Mathematical Olympiad, 3

Tags: inequalities , algebra , China , BPSQ

Let $a_1,a_2,\cdots,a_{n+1}>0$. Prove that$$\sum_{i-1}^{n}a_i\sum_{i=1}^{n}a_{i+1}\geq \sum_{i=1}^{n}\frac{a_i a_{i+1}}{a_i+a_{i+1}}\cdot \sum_{i=1}^{n}(a_i+a_{i+1})$$

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 $$(a+1)(b+1)(c+1)\geq 27.$$

2016 China Second Round Olympiad, 1

Tags: inequalities , algebra , China , BPSQ

Let $a_1, a_2, \ldots, a_{2016}$ be real numbers such that $9a_i\ge 11a^2_{i+1}$ $(i=,2,\cdots,2015)$. Find the maximum value of $(a_1-a^2_2)(a_2-a^2_3)\cdots (a_{2015}-a^2_{2016})(a_{2016}-a^2_{1}).$

1988 India National Olympiad, 4

Tags: inequalities , function , inequalities unsolved , india , BPSQ

If $ a$ and $ b$ are positive and $ a \plus{} b \equal{} 1$, prove that \[ \left(a\plus{}\frac{1}{a}\right)^2\plus{}\left(b\plus{}\frac{1}{b}\right)^2 \geq \frac{25}{2}\]

2008 China Northern MO, 6

Tags: trigonometry , inequalities , China , BPSQ

Let $a, b, c$ be side lengths of a right triangle and $c$ be the length of the hypotenuse .Find the minimum value of $\frac{a^3+b^3+c^3}{abc}$.

2017 China Second Round Olympiad, 10

Tags: inequalities , China , BPSQ , inequalities proposed

Let $x_1,x_2,x_3\geq 0$ and $x_1+x_2+x_3=1$. Find the minimum value and the maximum value of $(x_1+3x_2+5x_3)\left(x_1+\frac{x_2}{3}+\frac{x_3}{5}\right).$

2019 Kyiv Mathematical Festival, 3

Tags: Kyiv mathematical festival , inequalities , BPSQ

Let $a,b,c\ge0$ and $a+b+c\ge3.$ Prove that $a^4+b^3+c^2\ge a^3+b^2+c.$