Olympiad problems

2005 Croatia National Olympiad, 3

Tags: inequalities

If $k, l, m$ are positive integers with $\frac{1}{k}+\frac{1}{l}+\frac{1}{m}<1$, ﬁnd the maximum possible value of $\frac{1}{k}+\frac{1}{l}+\frac{1}{m}$.

2012 ELMO Shortlist, 8

Tags: binomial coefficient , algebra , polynomial , ceiling function , inequalities , absolute value , geometric series

Fix two positive integers $a,k\ge2$, and let $f\in\mathbb{Z}[x]$ be a nonconstant polynomial. Suppose that for all sufficiently large positive integers $n$, there exists a rational number $x$ satisfying $f(x)=f(a^n)^k$. Prove that there exists a polynomial $g\in\mathbb{Q}[x]$ such that $f(g(x))=f(x)^k$ for all real $x$. [i]Victor Wang.[/i]

2013 Saint Petersburg Mathematical Olympiad, 2

Tags: inequalities

if $a^2+b^2+c^2+d^2=1$ prove that \[ (1-a)(1-b)\ge cd. \] A. Khrabrov

2019 JBMO Shortlist, A5

Tags: algebra , inequalities

Let $a, b, c, d$ be positive real numbers such that $abcd = 1$. Prove the inequality $\frac{1}{a^3 + b + c + d} +\frac{1}{a + b^3 + c + d}+\frac{1}{a + b + c^3 + d} +\frac{1}{a + b + c + d^3} \leq \frac{a+b+c+d}{4}$ [i]Proposed by Romania[/i]

2013 ELMO Shortlist, 4

Tags: inequalities , algorithm , combinatorics

Let $n$ be a positive integer. The numbers $\{1, 2, ..., n^2\}$ are placed in an $n \times n$ grid, each exactly once. The grid is said to be [i]Muirhead-able[/i] if the sum of the entries in each column is the same, but for every $1 \le i,k \le n-1$, the sum of the first $k$ entries in column $i$ is at least the sum of the first $k$ entries in column $i+1$. For which $n$ can one construct a Muirhead-able array such that the entries in each column are decreasing? [i]Proposed by Evan Chen[/i]

2018 Cyprus IMO TST, 3

Tags: inequalities

Find all triples $(\alpha, \beta, \gamma)$ of positive real numbers for which the expression $$K = \frac{\alpha+3 \gamma}{\alpha + 2\beta + \gamma} + \frac{4\beta}{\alpha+\beta+2\gamma} - \frac{8 \gamma}{\alpha+ \beta + 3\gamma}$$ obtains its minimum value.

2017 Taiwan TST Round 1, 1

Tags: inequalities , three variable inequality , ineqstd

Let $a$, $b$, $c$ be positive real numbers such that $\min(ab,bc,ca) \ge 1$. Prove that $$\sqrt[3]{(a^2+1)(b^2+1)(c^2+1)} \le \left(\frac{a+b+c}{3}\right)^2 + 1.$$ [i]Proposed by Tigran Margaryan, Armenia[/i]

2003 Tuymaada Olympiad, 1

Tags: trigonometry , inequalities

Prove that for every $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}$ in the interval $(0,\pi/2)$ \[\left({1\over \sin \alpha_{1}}+{1\over \sin \alpha_{2}}+\ldots+{1\over \sin \alpha_{n}}\right) \left({1\over \cos \alpha_{1}}+{1\over \cos \alpha_{2}}+\ldots+{1\over \cos \alpha_{n}}\right) \leq\] \[\leq 2 \left({1\over \sin 2\alpha_{1}}+{1\over \sin 2\alpha_{2}}+\ldots+{1\over \sin 2\alpha_{n}}\right)^{2}.\] [i]Proposed by A. Khrabrov[/i]

2017 Austria Beginners' Competition, 1

Tags: inequalities , algebra

The nonnegative real numbers $a$ and $b$ satisfy $a + b = 1$. Prove that: $$\frac{1}{2} \leq \frac{a^3+b^3}{a^2+b^2} \leq 1$$ When do we have equality in the right inequality and when in the left inequality? [i]Proposed by Walther Janous [/i]

1999 Romania National Olympiad, 2a

Tags: inequalities

let $x_i,y_i 1 \le i \le n$ be positive numbers such that : $\displaystyle \sum_{i=1}^n x_i \ge \sum_{i=1}^n x_iy_i$ Prove : $\displaystyle \sum_{i=1}^n x_i \le \sum _{i=1}^n \frac{x_i}{y_i}$

2010 Federal Competition For Advanced Students, P2, 1

Tags: algebra , inequalities

Show that $\frac{(x - y)^7 + (y - z)^7 + (z - x)^7 - (x - y)(y - z)(z - x) ((x - y)^4 + (y - z)^4 + (z - x)^4)} {(x - y)^5 + (y - z)^5 + (z - x)^5} \ge 3$ holds for all triples of distinct integers $x, y, z$. When does equality hold?

2002 India IMO Training Camp, 15

Tags: inequalities , trigonometry , algebra

Let $x_1,x_2,\ldots,x_n$ be arbitrary real numbers. Prove the inequality \[ \frac{x_1}{1+x_1^2} + \frac{x_2}{1+x_1^2 + x_2^2} + \cdots + \frac{x_n}{1 + x_1^2 + \cdots + x_n^2} < \sqrt{n}. \]

the 15th XMO, 2

Tags: inequalities

$n$ is a integer and $a_1, a_2, \ldots, a_n\in[-1,1]$ are real numbers with $ \sum_{i=1}^{n}a_{i}=0$ ,try to find the maximum value of $$ \sum_{1\leq i , j \leq n , i\ne j}|a_{i}-a^2_j|$$

1995 Belarus Team Selection Test, 3

Tags: inequalities , algebra

If $0<a,b<1$ and $p,q\geq 0 ,\ p+q=1$ are real numbers , then prove that: \[a^pb^q+(1-a)^p(1-b)^q\le 1\]

1985 Canada National Olympiad, 4

Tags: floor function , inequalities , number theory , prime factorization , logarithm

Prove that $2^{n - 1}$ divides $n!$ if and only if $n = 2^{k - 1}$ for some positive integer $k$.

1998 Singapore MO Open, 4

Tags: inequalities , algebra

Let $n$ be a fixed positive integer. Find all the positive integers $m$ such that $$\frac{m^2+4m}{a_1}+\frac{m^2+8m}{a_1+a_2}+\frac{m^2+12m}{a_1+a_2+a_3}+...+\frac{m^2+4nm}{a_1+a_2+...+a_n}<2500 \left(\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}\right)$$ for any positive numbers $a_1,a_2,...,a_n$. Justify your answer.

1983 Dutch Mathematical Olympiad, 2

Tags: modular arithmetic , inequalities

Prove that if $ n$ is an odd positive integer, then the last two digits of $ 2^{2n}(2^{2n\plus{}1}\minus{}1)$ in base $ 10$ are $ 28$.

2021 Regional Competition For Advanced Students, 1

Tags: inequalities , algebra

Let $a$ and $b$ be positive integers and $c$ be a positive real number satisfying $$\frac{a + 1}{b + c}=\frac{b}{a}.$$ Prove that $c \ge 1$ holds. (Karl Czakler)

2010 All-Russian Olympiad Regional Round, 10.5

Tags: algebra , inequalities

Non-zero numbers $a, b, c$ are such that $ax^2+bx+c > cx$ for any $x$. Prove that $cx^2-bx + a > cx-b$ for any $x$.

2007 Germany Team Selection Test, 3

Tags: trigonometry , inequalities , search , geometry

Let $ ABC$ be a triangle and $ P$ an arbitrary point in the plane. Let $ \alpha, \beta, \gamma$ be interior angles of the triangle and its area is denoted by $ F.$ Prove: \[ \ov{AP}^2 \cdot \sin 2\alpha + \ov{BP}^2 \cdot \sin 2\beta + \ov{CP}^2 \cdot \sin 2\gamma \geq 2F \] When does equality occur?

2002 Moldova Team Selection Test, 1

Tags: inequalities , number theory

Consider the triangular numbers $T_n = \frac{n(n+1)}{2} , n \in \mathbb N$. [list][b](a)[/b] If $a_n$ is the last digit of $T_n$, show that the sequence $(a_n)$ is periodic and find its basic period. [b](b)[/b] If $s_n$ is the sum of the first $n$ terms of the sequence $(T_n)$, prove that for every $n \geq 3$ there is at least one perfect square between $s_{n-1} and $s_n$.[/list]

1966 IMO Shortlist, 13

Tags: inequalities , n-variable inequality

Let $a_1, a_2, \ldots, a_n$ be positive real numbers. Prove the inequality \[\binom n2 \sum_{i<j} \frac{1}{a_ia_j} \geq 4 \left( \sum_{i<j} \frac{1}{a_i+a_j} \right)^2\]

2014 China Team Selection Test, 1

Tags: floor function , inequalities , number theory

Prove that for any positive integers $k$ and $N$, \[\left(\frac{1}{N}\sum\limits_{n=1}^{N}(\omega (n))^k\right)^{\frac{1}{k}}\leq k+\sum\limits_{q\leq N}\frac{1}{q},\] where $\sum\limits_{q\leq N}\frac{1}{q}$ is the summation over of prime powers $q\leq N$ (including $q=1$). Note: For integer $n>1$, $\omega (n)$ denotes number of distinct prime factors of $n$, and $\omega (1)=0$.

2012 USAJMO, 3

Tags: inequalities , polynomial , ratio , algebra

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

2015 Junior Balkan MO, 2

Tags: inequalities

Let $a,b,c$ be positive real numbers such that $a+b+c = 3$. Find the minimum value of the expression \[A=\dfrac{2-a^3}a+\dfrac{2-b^3}b+\dfrac{2-c^3}c.\]