Olympiad problems

2024 China National Olympiad, 4

Let $a_1, a_2, \ldots, a_{2023}$ be nonnegative real numbers such that $a_1 + a_2 + \ldots + a_{2023} = 100$. Let $A = \left \{ (i,j) \mid 1 \leqslant i \leqslant j \leqslant 2023, \, a_ia_j \geqslant 1 \right\}$. Prove that $|A| \leqslant 5050$ and determine when the equality holds. [i]Proposed by Yunhao Fu[/i]

1988 Federal Competition For Advanced Students, P2, 1

Tags: inequalities , inequalities proposed

If $ a_1,...,a_{1988}$ are positive numbers whose arithmetic mean is $ 1988$, show that: $ \sqrt[1988]{\displaystyle\prod_{i,j\equal{}1}^{1988} \left( 1\plus{}\frac{a_i}{a_j} \right)} \ge 2^{1988}$ and determine when equality holds.

2023 China Second Round, 11

Tags: inequalities proposed , algebra , China , inequalities

Find all real numbers $ t $ not less than $1 $ that satisfy the following requirements: for any $a,b\in [-1,t]$ , there always exists $c,d \in [-1,t ]$ such that $ (a+c)(b+d)=1.$

1996 Iran MO (3rd Round), 3

Tags: inequalities proposed , inequalities

Let $a_1 \geq a_2 \geq \cdots \geq a_n$ be $n$ real numbers such that $a_1^k +a_2^k + \cdots + a_n^k \geq 0$ for all positive integers $k$. Suppose that $p=\max\{|a_1|,|a_2|, \ldots,|a_n|\}$. Prove that $p=a_1$, and \[(x-a_1)(x-a_2)\cdots(x-a_n)\leq x^n-a_1^n \qquad \forall x>a_1.\]

2011 Turkey Team Selection Test, 2

Tags: inequalities , inequalities proposed

Let $a,b,c$ be positive real numbers satisfying $a^2+b^2+c^2 \geq 3.$ Prove that \[ \frac{(a+1)(b+2)}{(b+1)(b+5)} + \frac{(b+1)(c+2)}{(c+1)(c+5)}+\frac{(c+1)(a+2)}{(a+1)(a+5)} \geq \frac{3}{2} \]

2015 China Team Selection Test, 2

Tags: inequalities , inequalities proposed , China TST

Let $a_1,a_2,a_3, \cdots ,a_n$ be positive real numbers. For the integers $n\ge 2$, prove that\[ \left (\frac{\sum_{j=1}^{n} \left (\prod_{k=1}^{j}a_k \right )^{\frac{1}{j}}}{\sum_{j=1}^{n}a_j} \right )^{\frac{1}{n}}+\frac{\left (\prod_{i=1}^{n}a_i \right )^{\frac{1}{n}}}{\sum_{j=1}^{n} \left (\prod_{k=1}^{j}a_k \right )^{\frac{1}{j}}}\le \frac{n+1}{n}\]

2014 Contests, 2

Tags: inequalities proposed , inequalities , algebra , China

Define a positive number sequence sequence $\{a_n\}$ by \[a_{1}=1,(n^2+1)a^2_{n-1}=(n-1)^2a^2_{n}.\]Prove that\[\frac{1}{a^2_1}+\frac{1}{a^2_2}+\cdots +\frac{1}{a^2_n}\le 1+\sqrt{1-\frac{1}{a^2_n}} .\]

2024 239 Open Mathematical Olympiad, 5

Tags: algebra , inequalities proposed

Let $a, b, c$ be reals such that $$a^2(c^2-2b-1)+b^2(a^2-2c-1)+c^2(b^2-2a-1)=0.$$ Show that $$3(a^2+b^2+c^2)+4(a+b+c)+3 \geq 6abc.$$

2023 Irish Math Olympiad, P8

Tags: algebra , inequalities , inequalities proposed

Suppose that $a, b, c$ are positive real numbers and $a + b + c = 3$. Prove that $$\frac{a+b}{c+2} + \frac{b+c}{a+2} + \frac{c+a}{b+2} \geq 2$$ and determine when equality holds.

1992 Baltic Way, 13

Tags: inequalities , inequalities proposed

Prove that for any positive $ x_1,x_2,\ldots,x_n,y_1,y_2,\ldots,y_n$ the inequality \[ \sum_{i\equal{}1}^n\frac1{x_iy_i}\ge\frac{4n^2}{\sum_{i\equal{}1}^n(x_i\plus{}y_i)^2} \] holds.

1990 Irish Math Olympiad, 1

Tags: inequalities proposed , inequalities

Let $n>3$ be a natural number . Prove that \[\frac{1}{3^3}+\frac{1}{4^3}+\cdots+\frac{1}{n^3}<\frac{1}{12}.\]

2010 Baltic Way, 2

Tags: inequalities , trigonometry , inequalities proposed

Let $x$ be a real number such that $0<x<\frac{\pi}{2}$. Prove that \[\cos^2(x)\cot (x)+\sin^2(x)\tan (x)\ge 1\]

2000 Macedonia National Olympiad, 2

Tags: inequalities , inequalities proposed

If $a_1,a_2,a_3\ldots a_n$ are positive numbers, find the maximum value of \[\frac{a_1a_2\ldots a_{n-1}a_n}{(1+a_1)(a_1+a_2)\ldots (a_{n-1}+a_n)(a_n+2^{n+1})} \]

2006 Baltic Way, 4

Tags: inequalities , inequalities proposed

Let $a,b,c,d,e,f$ be non-negative real numbers satisfying $a+b+c+d+e+f=6$. Find the maximal possible value of $\color{white}\ .\quad \ \color{black}\ \quad abc+bcd+cde+def+efa+fab$ and determine all $6$-tuples $(a,b,c,d,e,f)$ for which this maximal value is achieved.

2014 Switzerland - Final Round, 6

Tags: inequalities , inequalities proposed , algebra

Let $a,b,c\in \mathbb{R}_{\ge 0}$ satisfy $a+b+c=1$. Prove the inequality : \[ \frac{3-b}{a+1}+\frac{a+1}{b+1}+\frac{b+1}{c+1}\ge 4 \]

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: $$(xy+1)(yz+1)(zx+1)$$

2006 Moldova Team Selection Test, 3

Tags: inequalities , inequalities proposed

Positive real numbers $a,b,c$ satisfy the relation $abc=1$. Prove the inequality: $\frac{a+3}{(a+1)^{2}}+\frac{b+3}{(b+1)^{2}}+\frac{c+3}{(c+1)^{2}}\geq3$.

2010 Korea National Olympiad, 1

Tags: inequalities , trigonometry , AMC , USA(J)MO , USAMO , inequalities proposed

$ x, y, z $ are positive real numbers such that $ x+y+z=1 $. Prove that \[ \sqrt{ \frac{x}{1-x} } + \sqrt{ \frac{y}{1-y} } + \sqrt{ \frac{z}{1-z} } > 2 \]

2011 Laurențiu Duican, 4

Tags: inequalities , inequalities proposed

For $a, b, c>0,$ and $k\geq1,$ prove that \[\frac{a^{k+1}}{b^k+c^k}+\frac{b^{k+1}}{c^k+a^k}+\frac{c^{k+1}}{a^k+b^k}\geq\frac{3}{2}\sqrt{\frac{a^{k+1}+b^{k+1}+c^{k+1}}{{a^{k-1}+b^{k-1}+c^{k-1}}}}\] Author: MIHALY BENCZE

2013 Greece Team Selection Test, 3

Tags: function , inequalities proposed , inequalities

Find the largest possible value of $M$ for which $\frac{x}{1+\frac{yz}{x}}+\frac{y}{1+\frac{zx}{y}}+\frac{z}{1+\frac{xy}{z}}\geq M$ for all $x,y,z>0$ with $xy+yz+zx=1$

2014 Korea National Olympiad, 3

Tags: inequalities , inequalities proposed

Let $x, y, z$ be the real numbers that satisfies the following. $(x-y)^2+(y-z)^2+(z-x)^2=8, x^3+y^3+z^3=1$ Find the minimum value of $x^4+y^4+z^4$.

2010 Contests, 2

Tags: inequalities , rearrangement inequality , inequalities proposed

Let $a,b,c$ be positive real numbers for which $a+b+c=3$. Prove the inequality \[\frac{a^3+2}{b+2}+\frac{b^3+2}{c+2}+\frac{c^3+2}{a+2}\ge3\]

2006 Costa Rica - Final Round, 2

Tags: inequalities , inequalities proposed

If $ a$, $ b$, $ c$ are the sidelengths of a triangle, then prove that $ \frac {3\left(a^4 \plus{} b^4 \plus{} c^4\right)}{\left(a^2 \plus{} b^2 \plus{} c^2\right)^2} \plus{} \frac {bc \plus{} ca \plus{} ab}{a^2 \plus{} b^2 \plus{} c^2}\geq 2$.

2003 China National Olympiad, 3

Tags: inequalities , inequalities proposed

Suppose $a,b,c,d$ are positive reals such that $ab+cd=1$ and $x_i,y_i$ are real numbers such that $x_i^2+y_i^2=1$ for $i=1,2,3,4$. Prove that \[(ax_1+bx_2+cx_3+dx_4)^2+(ay_4+by_3+cy_2+dy_1)^2\le 2\left(\frac{a^2+b^2}{ab}+\frac{c^2+d^2}{cd}\right).\] [i]Li Shenghong[/i]

2013 China Team Selection Test, 1

Tags: inequalities proposed , inequalities , China TST

Let $n$ and $k$ be two integers which are greater than $1$. Let $a_1,a_2,\ldots,a_n,c_1,c_2,\ldots,c_m$ be non-negative real numbers such that i) $a_1\ge a_2\ge\ldots\ge a_n$ and $a_1+a_2+\ldots+a_n=1$; ii) For any integer $m\in\{1,2,\ldots,n\}$, we have that $c_1+c_2+\ldots+c_m\le m^k$. Find the maximum of $c_1a_1^k+c_2a_2^k+\ldots+c_na_n^k$.