Olympiad problems

1989 Swedish Mathematical Competition, 5

Tags: algebra , inequalities

Assume $x_1,x_2,..,x_5$ are positive numbers such that $x_1 < x_2$ and $x_3,x_4, x_5$ are all greater than $x_2$. Prove that if $a > 0$, then $$\frac{1}{(x_1 +x_3)^a}+ \frac{1}{(x_2 +x_4)^a}+ \frac{1}{(x_2 +x_5)^a} <\frac{1}{(x_1 +x_2)^a}+ \frac{1}{(x_2 +x_3)^a}+ \frac{1}{(x_4 +x_5)^a}$$

2023 ISL, A3

Tags: inequalities

Let $x_1,x_2,\dots,x_{2023}$ be pairwise different positive real numbers such that \[a_n=\sqrt{(x_1+x_2+\dots+x_n)\left(\frac{1}{x_1}+\frac{1}{x_2}+\dots+\frac{1}{x_n}\right)}\] is an integer for every $n=1,2,\dots,2023.$ Prove that $a_{2023} \geqslant 3034.$

1948 Putnam, A1

Tags: complex number , inequalities

What is the maximum of $|z^3 -z+2|$, where $z$ is a complex number with $|z|=1?$

2024-IMOC, A8

Tags: algebra , inequalities

$a$, $b$, $c$ are three distinct real numbers, given $\lambda >0$. Proof that \[\frac{1+ \lambda ^2a^2b^2}{(a-b)^2}+\frac{1+ \lambda ^2b^2c^2}{(b-c)^2}+\frac{1+ \lambda ^2c^2a^2}{(c-a)^2} \geq \frac 32 \lambda.\] [hide=Remark]Old problem, can be found [url=https://artofproblemsolving.com/community/c6h588854p3487434]here[/url]. Double post to have a cleaner thread for collection (as the original one contains a messy quote)[/hide]

2011 Saudi Arabia Pre-TST, 3.1

Tags: inequalities , trigonometry , algebra

Prove that $$\frac{\sin^3 a}{\sin b} +\frac{\cos^3 a}{\cos b} \ge \frac{1}{\cos(a - b)}$$ for all $a$ and $b$ in the interval $(0, \pi/2)$ .

2011 Irish Math Olympiad, 2

Tags: geometry , inequalities

Let $ABC$ be a triangle whose side lengths are, as usual, denoted by $a=|BC|,$ $b=|CA|,$ $c=|AB|.$ Denote by $m_a,m_b,m_c$, respectively, the lengths of the medians which connect $A,B,C$, respectively, with the centers of the corresponding opposite sides. (a) Prove that $2m_a<b+c$. Deduce that $m_a+m_b+m_c<a+b+c$. (b) Give an example of (i) a triangle in which $m_a>\sqrt{bc}$; (ii) a triangle in which $m_a\le \sqrt{bc}$.

2011 Romania National Olympiad, 4

Tags: number theory , floor , inequalities

Let be a natural number $ n. $ Prove that there exists a number $ k\in\{ 0,1,2,\ldots n \} $ such that the floor of $ 2^{n+k}\sqrt 2 $ is even.

2021 Macedonian Balkan MO TST, Problem 3

Tags: inequalities

Suppose that $a_1, a_2, \dots a_{2021}$ are non-negative numbers such that $\sum_{k=1}^{2021} a_k=1$. Prove that $$ \sum_{k=1}^{2021}\sqrt[k]{a_1 a_2\dots a_k} \leq 3. $$

2024 China Team Selection Test, 21

Tags: inequalities

Let integer $n\ge 3,$ $\tbinom n2$ nonnegative real numbers $a_{i,j}$ satisfy $ a_{i,j}+a_{j,k}\le a_{i,k}$ holds for all $1\le i <j<k\le n$. Proof $$\left\lfloor\frac{n^2}4\right\rfloor\sum_{1\le i<j\le n}a_{i,j}^4\ge \left(\sum_{1\le i<j\le n}a_{i,j}^2\right)^2.$$ [i]Proposed by Jingjun Han, Dongyi Wei[/i]

2004 India IMO Training Camp, 1

Tags: function , inequalities , logarithm

Let $x_1, x_2 , x_3, .... x_n$ be $n$ real numbers such that $0 < x_j < \frac{1}{2}$. Prove that \[ \frac{ \prod\limits_{j=1}^{n} x_j } { \left( \sum\limits_{j=1}^{n} x_j \right)^n} \leq \frac{ \prod\limits_{j=1}^{n} (1-x_j) } { \left( \sum\limits_{j=1}^{n} (1 - x_j) \right)^n} \]

1999 Akdeniz University MO, 3

Tags: algebra , inequalities

Let $a$,$b$,$c$ and $d$ positive reals. Prove that $$\frac{1}{a+b+c+d} \leq \frac{1}{64}(\frac{1}{a}+\frac{1}{b}+\frac{4}{c}+\frac{16}{d})$$

2010 Thailand Mathematical Olympiad, 7

Tags: algebra , inequalities

Let $a, b, c$ be positive reals. Show that $\frac{a^5}{bc^2} + \frac{b^5}{ca^2} + \frac{c^5}{ab^2} \ge a^2 + b^2 + c^2.$

1984 Polish MO Finals, 3

Tags: geometry , circles , octahedron , 3d geometry , inequalities

Let $W$ be a regular octahedron and $O$ be its center. In a plane $P$ containing $O$ circles $k_1(O,r_1)$ and $k_2(O,r_2)$ are chosen so that $k_1 \subset P\cap W \subset k_2$. Prove that $\frac{r_1}{r_2}\le \frac{\sqrt3}{2}$

2014 Tajikistan Team Selection Test, 3

Tags: inequalities , geometry

Let $a$, $b$, $c$ be side length of a triangle. Prove the inequality \begin{align*} \sqrt{a^2 + ab + b^2} + \sqrt{b^2 + bc + c^2} + \sqrt{c^2 + ca + a^2} \leq \sqrt{5a^2 + 5b^2 + 5c^2 + 4ab + 4 bc + 4ca}.\end{align*}

2013 Math Prize for Girls Olympiad, 1

Tags: inequalities

Let $n$ be a positive integer. Let $a_1$, $a_2$, $\ldots\,$, $a_n$ be real numbers such that $-1 \le a_i \le 1$ (for all $1 \le i \le n$). Let $b_1$, $b_2$, $\ldots$, $b_n$ be real numbers such that $-1 \le b_i \le 1$ (for all $1 \le i \le n$). Prove that \[ \left| \prod_{i=1}^n a_i - \prod_{i=1}^n b_i \right| \le \sum_{i = 1}^n \left| a_i - b_i \right| \, . \]

2014 Cuba MO, 2

Tags: algebra , inequalities

Let $a$ and $b$ be real numbers with $0 \le a, b \le 1$. (a) Prove that $ \frac{a}{b+1} +\frac{b}{a+1} \le 1.$ (b) Find the case of equality.

2011 Today's Calculation Of Integral, 764

Tags: calculus , integration , trigonometry , function , inequalities , calculus computations

Let $f(x)$ be a continuous function defined on $0\leq x\leq \pi$ and satisfies $f(0)=1$ and \[\left\{\int_0^{\pi} (\sin x+\cos x)f(x)dx\right\}^2=\pi \int_0^{\pi}\{f(x)\}^2dx.\] Evaluate $\int_0^{\pi} \{f(x)\}^3dx.$

1995 IMO Shortlist, 4

Tags: polynomial , function , inequalities , algebra

Suppose that $ x_1, x_2, x_3, \ldots$ are positive real numbers for which \[ x^n_n \equal{} \sum^{n\minus{}1}_{j\equal{}0} x^j_n\] for $ n \equal{} 1, 2, 3, \ldots$ Prove that $ \forall n,$ \[ 2 \minus{} \frac{1}{2^{n\minus{}1}} \leq x_n < 2 \minus{} \frac{1}{2^n}.\]

2010 India IMO Training Camp, 9

Tags: rectangle , algebra , geometry , inequalities

Let $A=(a_{jk})$ be a $10\times 10$ array of positive real numbers such that the sum of numbers in row as well as in each column is $1$. Show that there exists $j<k$ and $l<m$ such that \[a_{jl}a_{km}+a_{jm}a_{kl}\ge \frac{1}{50}\]

1969 Swedish Mathematical Competition, 4

Tags: algebra , min , inequalities

Define $g(x)$ as the largest value of$ |y^2 - xy|$ for $y$ in $[0, 1]$. Find the minimum value of $g$ (for real $x$).

2011 Abels Math Contest (Norwegian MO), 3b

Tags: functional , algebra , function , functional inequalities , inequalities

Find all functions $f$ from the real numbers to the real numbers such that $f(xy) \le \frac12 \left(f(x) + f(y) \right)$ for all real numbers $x$ and $y$.

2018 Saudi Arabia JBMO TST, 3

Tags: inequalities

Prove that in every triangle there are two sides with lengths $x$ and $y$ such that $$\frac{\sqrt{5}-1}{2}\leq\frac{x}{y}\leq\frac{\sqrt{5}+1}{2}$$

2013 Korea - Final Round, 3

Tags: inequalities , floor function , logarithm

For a positive integer $n \ge 2 $, define set $ T = \{ (i,j) | 1 \le i < j \le n , i | j \} $. For nonnegative real numbers $ x_1 , x_2 , \cdots , x_n $ with $ x_1 + x_2 + \cdots + x_n = 1 $, find the maximum value of \[ \sum_{(i,j) \in T} x_i x_j \] in terms of $n$.

2013 Bosnia Herzegovina Team Selection Test, 3

Tags: combinatorics , inequalities , induction

Prove that in the set consisting of $\binom{2n}{n}$ people we can find a group of $n+1$ people in which everyone knows everyone or noone knows noone.

2013 ELMO Problems, 2

Tags: function , inequalities , calculus , derivative , 3-variable inequality , logarithm

Let $a,b,c$ be positive reals satisfying $a+b+c = \sqrt[7]{a} + \sqrt[7]{b} + \sqrt[7]{c}$. Prove that $a^a b^b c^c \ge 1$. [i]Proposed by Evan Chen[/i]