Olympiad problems

1914 Eotvos Mathematical Competition, 2

Tags: algebra , inequalities

Suppose that $$-1 \le ax^2 + bx + c \le 1 \ \ for \ \ -1 \le x \le 1 , $$ where a, b, c are real numbers. Prove that $$-4 \le 2ax + b \le 4 \ \ for \ \ -1 \le x \le 1 , $$

2003 Federal Competition For Advanced Students, Part 1, 2

Tags: geometry , inequalities

Find the greatest and smallest value of $f(x, y) = y-2x$, if x, y are distinct non-negative real numbers with $\frac{x^2+y^2}{x+y}\leq 4$.

2018 India IMO Training Camp, 2

Tags: algebra , inequalities

For an integer $n\ge 2$ find all $a_1,a_2,\cdots ,a_n, b_1,b_2,\cdots , b_n$ so that (a) $0\le a_1\le a_2\le \cdots \le a_n\le 1\le b_1\le b_2\le \cdots \le b_n;$ (b) $\sum_{k=1}^n (a_k+b_k)=2n;$ (c) $\sum_{k=1}^n (a_k^2+b_k^2)=n^2+3n.$

2016 Singapore MO Open, 2

Tags: inequalities , algebra

Let $a, b, c$ be real numbers such that $0 < a, b, c < 1/2$ and $a + b + c= 1$. Prove that for all real numbers $x,y,z$, $$abc(x + y + z)^2 \ge ayz( 1- 2a) + bxz( 1 - 2b) + cxy( 1 - 2c)$$. When does equality hold?

2010 ELMO Shortlist, 6

Tags: inequalities

For all positive real numbers $a,b,c$, prove that \[\sqrt{\frac{a^4 + 2b^2c^2}{a^2+2bc}} + \sqrt{\frac{b^4+2c^2a^2}{b^2+2ca}} + \sqrt{\frac{c^4 + 2a^2b^2}{c^2 + 2ab}} \geq a + b + c.\] [i]In-Sung Na.[/i]

2022 IMO Shortlist, A4

Tags: inequalities , algebra

Let $n \geqslant 3$ be an integer, and let $x_1,x_2,\ldots,x_n$ be real numbers in the interval $[0,1]$. Let $s=x_1+x_2+\ldots+x_n$, and assume that $s \geqslant 3$. Prove that there exist integers $i$ and $j$ with $1 \leqslant i<j \leqslant n$ such that \[2^{j-i}x_ix_j>2^{s-3}.\]

2000 Moldova National Olympiad, Problem 6

Tags: inequality , inequalities

Assuming that real numbers $x$ and $y$ satisfy $y\left(1+x^2\right)=x\left(\sqrt{1-4y^2}-1\right)$, find the maximum value of $xy$.

2012 Macedonia National Olympiad, 2

Tags: inequalities

If $~$ $a,\, b,\, c,\, d$ $~$ are positive real numbers such that $~$ $abcd=1$ $~$ then prove that the following inequality holds \[ \frac{1}{bc+cd+da-1} + \frac{1}{ab+cd+da-1} + \frac{1}{ab+bc+da-1} + \frac{1}{ab+bc+cd-1}\; \le\; 2\, . \] When does inequality hold?

1983 Czech and Slovak Olympiad III A, 5

Tags: inequality , algebra , inequalities

Find all pair $(x,y)$ of positive integers satisfying $$\left|\frac{x}{y}-\sqrt2\right|<\frac{1}{y^3}.$$

2019 Thailand TSTST, 1

Tags: inequalities , sequence , algebra

Let $\{x_i\}^{\infty}_{i=1}$ and $\{y_i\}^{\infty}_{i=1}$ be sequences of real numbers such that $x_1=y_1=\sqrt{3}$, $$x_{n+1}=x_n+\sqrt{1+x_n^2}\quad\text{and}\quad y_{n+1}=\frac{y_n}{1+\sqrt{1+y_n^2}}$$ for all $n\geq 1$. Prove that $2<x_ny_n<3$ for all $n>1$.

1997 Czech And Slovak Olympiad IIIA, 5

Tags: trigonometry , max , function , algebra , inequalities

For a given integer $n \ge 2$, find the maximum possible value of $V_n = \sin x_1 \cos x_2 +\sin x_2 \cos x_3 +...+\sin x_n \cos x_1$, where $x_1,x_2,...,x_n$ are real numbers.

1961 IMO, 2

Tags: inequalities , trigonometry , geometry , area of a triangle , heron's formula

Let $ a$, $ b$, $ c$ be the sides of a triangle, and $ S$ its area. Prove: \[ a^{2} \plus{} b^{2} \plus{} c^{2}\geq 4S \sqrt {3} \] In what case does equality hold?

1983 IMO Longlists, 32

Tags: function , inequalities , algebra , logarithm

Let $a, b, c$ be positive real numbers and let $[x]$ denote the greatest integer that does not exceed the real number $x$. Suppose that $f$ is a function defined on the set of non-negative integers $n$ and taking real values such that $f(0) = 0$ and \[f(n) \leq an + f([bn]) + f([cn]), \qquad \text{ for all } n \geq 1.\] Prove that if $b + c < 1$, there is a real number $k$ such that \[f(n) \leq kn \qquad \text{ for all } n \qquad (1)\] while if $b + c = 1$, there is a real number $K$ such that $f(n) \leq K n \log_2 n$ for all $n \geq 2$. Show that if $b + c = 1$, there may not be a real number $k$ that satisfies $(1).$

2016 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: algebra , inequalities , maximum value

Let $a$ and $b$ be real numbers bigger than $1$. Find maximal value of $c \in \mathbb{R}$ such that $$\frac{1}{3+\log _{a} b}+\frac{1}{3+\log _{b} a} \geq c$$

2022 APMO, 5

Tags: inequalities

Let $a,b,c,d$ be real numbers such that $a^2+b^2+c^2+d^2=1$. Determine the minimum value of $(a-b)(b-c)(c-d)(d-a)$ and determine all values of $(a,b,c,d)$ such that the minimum value is achived.

2000 Kazakhstan National Olympiad, 6

Tags: algebra , inequalities

For positive numbers $ a $, $ b $ and $ c $ satisfying the equality $ a + b + c = 1 $, prove the inequality $$ \frac {a ^ 7 + b ^ 7} {a ^ 5 + b ^ 5} + \frac {b ^ 7 + c ^ 7} {b ^ 5 + c ^ 5} + \frac {c ^ 7 + a ^ 7} {c ^ 5 + a ^ 5} \geq \frac {1} {3}. $$

MathLinks Contest 6th, 4.3

Tags: inequalities

Let $a, b, c$ be positive real numbers such that $abc = 1$. Prove that $$\sqrt{\frac{a+b}{b+1}}+\sqrt{\frac{b+c}{c+1}}+\sqrt{\frac{c+a}{a+1}} \ge 3$$

1991 APMO, 3

Tags: inequalities , n-variable inequality

Let $a_1$, $a_2$, $\cdots$, $a_n$, $b_1$, $b_2$, $\cdots$, $b_n$ be positive real numbers such that $a_1 + a_2 + \cdots + a_n = b_1 + b_2 + \cdots + b_n$. Show that \[ \frac{a_1^2}{a_1 + b_1} + \frac{a_2^2}{a_2 + b_2} + \cdots + \frac{a_n^2}{a_n + b_n} \geq \frac{a_1 + a_2 + \cdots + a_n}{2} \]

2019 Tuymaada Olympiad, 7

Tags: inequalities , grid

$N$ cells chosen on a rectangular grid. Let $a_i$ is number of chosen cells in $i$-th row, $b_j$ is number of chosen cells in $j$-th column. Prove that $$ \prod_{i} a_i! \cdot \prod_{j} b_j! \leq N! $$

II Soros Olympiad 1995 - 96 (Russia), 9.1

Tags: algebra , inequalities

Solve the inequality $$(x-1)(x^2-1)(x^3-1)\cdot ...\cdot (x^{100}-1)(x^{101}-1)\ge 0$$

2015 Indonesia MO Shortlist, A3

Tags: inequality , inequalities

Let $a,b,c$ positive reals such that $a^2+b^2+c^2=1$. Prove that $$\frac{a+b}{\sqrt{ab+1}}+\frac{b+c}{\sqrt{bc+1}}+\frac{c+a}{\sqrt{ac+1}}\le 3$$

2008 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: inequalities

For arbitrary reals $ x$, $ y$ and $ z$ prove the following inequality: $ x^{2} \plus{} y^{2} \plus{} z^{2} \minus{} xy \minus{} yz \minus{} zx \geq \max \{\frac {3(x \minus{} y)^{2}}{4} , \frac {3(y \minus{} z)^{2}}{4} , \frac {3(y \minus{} z)^{2}}{4} \}$

2003 China Team Selection Test, 1

Tags: real analysis , combinatorics , inequalities

Let $S$ be the set of points inside and on the boarder of a regular haxagon with side length 1. Find the least constant $r$, such that there exists one way to colour all the points in $S$ with three colous so that the distance between any two points with same colour is less than $r$.

2019 New Zealand MO, 3

Tags: algebra , inequalities

Let $a, b$ and $c$ be positive real numbers such that $a + b + c = 3$. Prove that $$a^a + b^b + c^c \ge 3$$

2023 NMTC Junior, P6

Tags: inequalities

The sum of squares of four reals $x,y,z,u$ is $1$. Find the minimum value of the expression $E=(x-y)(y-z)(z-u)(u-x)$. Find also the minimum values of $x$, $y$, $z$ and $u$ when this minimum occurs.