Olympiad problems

2024 Spain Mathematical Olympiad, 2

Tags: algebra , inequalities

Let $n$ be a positive integer. Let $x_1, x_2, \dots, x_n > 1$ be real numbers whose product is $n+1$. Prove that \[\left(\frac{1}{1^2(x_1-1)}+1\right)\left(\frac{1}{2^2(x_2-1)}+1\right)\cdots\left(\frac{1}{n^2(x_n-1)}+1\right)\geq n+1\] and find for which values equality holds.

1981 Austrian-Polish Competition, 3

Tags: geometry , trigonometry , inequalities

Given is a triangle $ABC$, the inscribed circle $G$ of which has radius $r$. Let $r_a$ be the radius of the circle touching $AB$, $AC$ and $G$. [This circle lies inside triangle $ABC$.] Define $r_b$ and $r_c$ similarly. Prove that $r_a + r_b + r_c \geq r$ and find all cases in which equality occurs. [i]Bosnia - Herzegovina Mathematical Olympiad 2002[/i]

Russian TST 2018, P1

Tags: algebra , inequalities

Let $a,b,c{}$ be positive real numbers. Prove that \[108\cdot(ab+bc+ca)\leqslant(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a})^4.\]

2004 Romania National Olympiad, 1

Tags: function , inequalities , algebra

Find the strictly increasing functions $f : \{1,2,\ldots,10\} \to \{ 1,2,\ldots,100 \}$ such that $x+y$ divides $x f(x) + y f(y)$ for all $x,y \in \{ 1,2,\ldots,10 \}$. [i]Cristinel Mortici[/i]

2008 APMO, 5

Tags: modular arithmetic , inequalities , geometric series , number theory , residue

Let $ a, b, c$ be integers satisfying $ 0 < a < c \minus{} 1$ and $ 1 < b < c$. For each $ k$, $ 0\leq k \leq a$, Let $ r_k,0 \leq r_k < c$ be the remainder of $ kb$ when divided by $ c$. Prove that the two sets $ \{r_0, r_1, r_2, \cdots , r_a\}$ and $ \{0, 1, 2, \cdots , a\}$ are different.

1998 Swedish Mathematical Competition, 2

Tags: trigonometry , geometry , inequalities

$ABC$ is a triangle. Show that $c \ge (a+b) \sin \frac{C}{2}$

2013 India IMO Training Camp, 1

Tags: induction , rearrangement inequality , inequalities

Let $a, b, c$ be positive real numbers such that $a + b + c = 1$. If $n$ is a positive integer then prove that \[ \frac{(3a)^n}{(b + 1)(c + 1)} + \frac{(3b)^n}{(c + 1)(a + 1)} + \frac{(3c)^n}{(a + 1)(b + 1)} \ge \frac{27}{16} \,. \]

2021 IMO Shortlist, A5

Tags: real analysis , inequalities , algebra

Let $n\geq 2$ be an integer and let $a_1, a_2, \ldots, a_n$ be positive real numbers with sum $1$. Prove that $$\sum_{k=1}^n \frac{a_k}{1-a_k}(a_1+a_2+\cdots+a_{k-1})^2 < \frac{1}{3}.$$

2002 China Team Selection Test, 1

Tags: inequalities

Given $ n \geq 3$, $ n$ is a integer. Prove that: \[ (2^n \minus{} 2) \cdot \sqrt{2i\minus{}1} \geq \left( \sum_{j\equal{}0}^{i\minus{}1}C_n^j \plus{} C_{n\minus{}1}^{i\minus{}1} \right) \cdot \sqrt{n}\] where if $ n$ is even, then $ \displaystyle 1 \leq i \leq \frac{n}{2}$; if $ n$ is odd, then $ \displaystyle 1 \leq i \leq \frac{n\minus{}1}{2}$.

Russian TST 2018, P4

Tags: algebra , inequalities

Let $a_1,\ldots,a_{n+1}$ be positive real numbers satisfying $1/(a_1+1)+\cdots+1/(a_{n+1}+1)=n$. Prove that \[\sum_{i=1}^{n+1}\prod_{j\neq i}\sqrt[n]{a_j}\leqslant\frac{n+1}{n}.\]

2017 India IMO Training Camp, 1

Tags: inequalities

Let $a,b,c$ be distinct positive real numbers with $abc=1$. Prove that $$\sum_{\text{cyc}} \frac{a^6}{(a-b)(a-c)}>15.$$

2002 APMO, 4

Tags: inequalities

Let $x,y,z$ be positive numbers such that \[ {1\over x}+{1\over y}+{1\over z}=1. \] Show that \[ \sqrt{x+yz}+\sqrt{y+zx}+\sqrt{z+xy}\ge\sqrt{xyz}+\sqrt{x}+\sqrt{y}+\sqrt{z} \]

2024 Korea Summer Program Practice Test, 6

Tags: inequalities , sequence , algebra

Find all possible values of $C\in \mathbb R$ such that there exists a real sequence $\{a_n\}_{n=1}^\infty$ such that $$a_na_{n+1}^2\ge a_{n+2}^4 +C$$ for all $n\ge 1$.

1996 China National Olympiad, 2

Tags: inequalities , trigonometry , calculus , derivative , integration

Let $n$ be a natural number. Suppose that $x_0=0$ and that $x_i>0$ for all $i\in\{1,2,\ldots ,n\}$. If $\sum_{i=1}^nx_i=1$ , prove that \[1\leq\sum_{i=1}^{n} \frac{x_i}{\sqrt{1+x_0+x_1+\ldots +x_{i-1}}\sqrt{x_i+\ldots+x_n}} < \frac{\pi}{2} \]

1986 Polish MO Finals, 1

Tags: geometry , rectangle , perimeter , combinatorial geometry , combinatorics , inequalities

A square of side $1$ is covered with $m^2$ rectangles. Show that there is a rectangle with perimeter at least $\frac{4}{m}$.

2015 Korea National Olympiad, 4

Tags: inequalities , number theory , relatively prime , prime number , coprime integers

For a positive integer $n$, $a_1, a_2, \cdots a_k$ are all positive integers without repetition that are not greater than $n$ and relatively prime to $n$. If $k>8$, prove the following. $$\sum_{i=1}^k |a_i-\frac{n}{2}|<\frac{n(k-4)}{2}$$

2006 Korea National Olympiad, 6

Tags: inequalities , algebra

Prove that for any positive real numbers $x,y$ and $z,$ $xyz(x+2)(y+2)(z+2)\le(1+\frac{2(xy+yz+zx)}{3})^3$

1999 Brazil National Olympiad, 3

Tags: geometry , rectangle , inequalities , graph theory , combinatorics

How many coins can be placed on a $10 \times 10$ board (each at the center of its square, at most one per square) so that no four coins form a rectangle with sides parallel to the sides of the board?

1993 Miklós Schweitzer, 10

Tags: probability and stats , inequalities

Let $U_1 , U_2 , U_3$ be iid random variables on [0,1], which in order of magnitude, $U_1^{\ast} \le U_2^{\ast} \leq U_3 ^ {\ast}$. Let $\alpha, p_1 , p_2 , p_3 \in [0,1]$ such that $P(U_j ^ {\ast} \ge p_j)= \alpha$ ( j = 1,2,3). Prove that $$P \left( p_1 + (p_2-p_1) U_3^{\ast} + (p_3- p_2) U_2^{\ast} + (1-p_3) U_1^{\ast} \geq \frac{1}{2} \right) \geq 1-\alpha$$

2019 Kosovo Team Selection Test, 5

Tags: inequalities , algebra

$a,b,c,d$ are fixed positive real numbers. Find the maximum value of the function $f: \mathbb{R^{+}}_{0} \rightarrow \mathbb{R}$ $f(x)=\frac{a+bx}{b+cx}+\frac{b+cx}{c+dx}+\frac{c+dx}{d+ax}+\frac{d+ax}{a+bx}, x \geq 0$

2007 USAMO, 4

Tags: search , inequalities , induction , graph theory , combinatorics

An [i]animal[/i] with $n$ [i]cells[/i] is a connected figure consisting of $n$ equal-sized cells[1]. A [i]dinosaur[/i] is an animal with at least $2007$ cells. It is said to be [i]primitive[/i] it its cells cannot be partitioned into two or more dinosaurs. Find with proof the maximum number of cells in a primitive dinosaur. (1) Animals are also called [i]polyominoes[/i]. They can be defined inductively. Two cells are [i]adjacent[/i] if they share a complete edge. A single cell is an animal, and given an animal with $n$ cells, one with $n+1$ cells is obtained by adjoining a new cell by making it adjacent to one or more existing cells.

2019 Jozsef Wildt International Math Competition, W. 16

Tags: integration , trigonometry , inequalities , calculus

If $f : [a, b] \to (0,\infty)$; $0 < a \leq b$; $f$ derivable; $f'$ continuous then:$$\int \limits_{a}^{b}\frac{f'(x)\sqrt{f(x)}}{f^3(x) + 1}\leq \tan^{-1}\left(\frac{f(b)-f(a)}{1 + f(a)f(b)}\right)$$

2004 Croatia National Olympiad, Problem 3

Tags: absolute value , inequalities

Prove that for any three real numbers $x,y,z$ the following inequality holds: $$|x|+|y|+|z|-|x+y|-|y+z|-|z+x|+|x+y+z|\ge0.$$

2012 USA TSTST, 6

Tags: inequalities , algebra , polynomial , vieta , quadratic

Positive real numbers $x, y, z$ satisfy $xyz+xy+yz+zx = x+y+z+1$. Prove that \[ \frac{1}{3} \left( \sqrt{\frac{1+x^2}{1+x}} + \sqrt{\frac{1+y^2}{1+y}} + \sqrt{\frac{1+z^2}{1+z}} \right) \le \left( \frac{x+y+z}{3} \right)^{5/8} . \]

2005 Romania National Olympiad, 4

Tags: inequalities , algebra

a) Prove that for all positive reals $u,v,x,y$ the following inequality takes place: \[ \frac ux + \frac vy \geq \frac {4(uy+vx)}{(x+y)^2} . \] b) Let $a,b,c,d>0$. Prove that \[ \frac a{b+2c+d} + \frac b{c+2d+a} + \frac c{d+2a+b} + \frac d{a+2b+c} \geq 1.\] [i]Traian Tămâian[/i]