Olympiad problems

1978 IMO, 2

Let $f$ be an injective function from ${1,2,3,\ldots}$ in itself. Prove that for any $n$ we have: $\sum_{k=1}^{n} f(k)k^{-2} \geq \sum_{k=1}^{n} k^{-1}.$

1973 IMO Shortlist, 10

Tags: sequence , algebra , inequality , construction , geometric sequence

Let $a_1, \ldots, a_n$ be $n$ positive numbers and $0 < q < 1.$ Determine $n$ positive numbers $b_1, \ldots, b_n$ so that: [i]a.)[/i] $ a_{k} < b_{k}$ for all $k = 1, \ldots, n,$ [i]b.)[/i] $q < \frac{b_{k+1}}{b_{k}} < \frac{1}{q}$ for all $k = 1, \ldots, n-1,$ [i]c.)[/i] $\sum \limits^n_{k=1} b_k < \frac{1+q}{1-q} \cdot \sum \limits^n_{k=1} a_k.$

Azerbaijan Al-Khwarizmi IJMO TST 2025, 2

Tags: inequality , algebra , inequalities

For $a,b,c$ positive real numbers satisfying $a^2+b^2+c^2 \geq 3$,show that: $\sqrt[3]{\frac{a^3+b^3+c^3}{3}}+\frac{a+b+c}{9} \geq \frac{4}{3}$.

2024 Junior Macedonian Mathematical Olympiad, 1

Tags: algebra , inequality

Let $a, b$, and $c$ be positive real numbers. Prove that \[\frac{a^4 + 3}{b} + \frac{b^4 + 3}{c} + \frac{c^4 + 3}{a} \ge 12.\] When does equality hold? [i]Proposed by Petar Filipovski[/i]

2020 Centroamerican and Caribbean Math Olympiad, 5

Tags: algebra , polynomial , inequality , inequalities , n-variable inequality

Let $P(x)$ be a polynomial with real non-negative coefficients. Let $k$ be a positive integer and $x_1, x_2, \dots, x_k$ positive real numbers such that $x_1x_2\cdots x_k=1$. Prove that $$P(x_1)+P(x_2)+\cdots+P(x_k)\geq kP(1).$$

2022 Azerbaijan Junior National Olympiad, A3

Tags: inequality , algebra

Let $x,y,z \in \mathbb{R}^{+}$ and $x^2+y^2+z^2=x+y+z$. Prove that $$x+y+z+3 \ge 6 \sqrt[3]{\frac{xy+yz+zx}{3}}$$

1985 IMO Longlists, 22

Tags: prime number , number theory , sequence , inequality

The positive integers $x_1, \cdots , x_n$, $n \geq 3$, satisfy $x_1 < x_2 <\cdots< x_n < 2x_1$. Set $P = x_1x_2 \cdots x_n.$ Prove that if $p$ is a prime number, $k$ a positive integer, and $P$ is divisible by $pk$, then $\frac{P}{p^k} \geq n!.$

2016 EGMO, 1

Tags: inequality , algebra , inequalities , n-variable inequality , sequence

Let $n$ be an odd positive integer, and let $x_1,x_2,\cdots ,x_n$ be non-negative real numbers. Show that \[ \min_{i=1,\ldots,n} (x_i^2+x_{i+1}^2) \leq \max_{j=1,\ldots,n} (2x_jx_{j+1}) \]where $x_{n+1}=x_1$.

2023 China Western Mathematical Olympiad, 7

Tags: algebra , inequality , number theory

For positive integers $x, y, $ $r_x(y)$ to represent the smallest positive integer $ r $ such that $ r \equiv y(\text{mod x})$ .For any positive integers $a, b, n ,$ Prove that $$\sum_{i=1}^{n} r_b(a i)\leq \frac{n(a+b)}{2}$$

2020 DMO Stage 1, 1.

Tags: inequality , holder , inequalities

[b]Q.[/b] Find the minimum value of the expression for $x,y,z\in \mathbb{R}^{+}$ $$\sum_{\text{cyc}}\frac{(x+1)^{4}+2(y+1)^{6}-(y+1)^{4}}{(y+1)^{6}}$$ [i]Proposed by Aritra12[/i]

2014 India Regional Mathematical Olympiad, 2

Tags: inequalities , inequality , 3-variable inequality

Let $x, y, z$ be positive real numbers. Prove that $\frac{y^2 + z^2}{x}+\frac{z^2 + x^2}{y}+\frac{x^2 + y^2}{z}\ge 2(x + y + z)$.

2018 Israel National Olympiad, 3

Tags: absolute value , minimum value , maximum value , maximum and minimum , algebra , inequality

Determine the minimal and maximal values the expression $\frac{|a+b|+|b+c|+|c+a|}{|a|+|b|+|c|}$ can take, where $a,b,c$ are real numbers.

2020 Australian Maths Olympiad, 1

Tags: inequality , 2-variable inequality , algebra

Determine all pairs $(a,b)$ of non-negative integers such that $$ \frac{a+b}{2}-\sqrt{ab}=1.$$

2022 Taiwan TST Round 2, A

Tags: algebra , inequality , inequalities

Let $n\geqslant 1$ be an integer, and let $x_0,x_1,\ldots,x_{n+1}$ be $n+2$ non-negative real numbers that satisfy $x_ix_{i+1}-x_{i-1}^2\geqslant 1$ for all $i=1,2,\ldots,n.$ Show that \[x_0+x_1+\cdots+x_n+x_{n+1}>\bigg(\frac{2n}{3}\bigg)^{3/2}.\][i]Pakawut Jiradilok and Wijit Yangjit, Thailand[/i]

2018 Macedonia JBMO TST, 3

Tags: algebra , inequality , inequalities

Let $x$, $y$, and $z$ be positive real numbers such that $x + y + z = 1$. Prove that $\frac{(x+y)^3}{z} + \frac{(y+z)^3}{x} + \frac{(z+x)^3}{y} + 9xyz \ge 9(xy + yz + zx)$. When does equality hold?

1994 IMO, 1

Tags: number theory , inequality , algebra

Let $ m$ and $ n$ be two positive integers. Let $ a_1$, $ a_2$, $ \ldots$, $ a_m$ be $ m$ different numbers from the set $ \{1, 2,\ldots, n\}$ such that for any two indices $ i$ and $ j$ with $ 1\leq i \leq j \leq m$ and $ a_i \plus{} a_j \leq n$, there exists an index $ k$ such that $ a_i \plus{} a_j \equal{} a_k$. Show that \[ \frac {a_1 \plus{} a_2 \plus{} ... \plus{} a_m}{m} \geq \frac {n \plus{} 1}{2}. \]

1969 IMO Shortlist, 56

Tags: decimal representation , inequality , calculus

Let $a$ and $b$ be two natural numbers that have an equal number $n$ of digits in their decimal expansions. The first $m$ digits (from left to right) of the numbers $a$ and $b$ are equal. Prove that if $m >\frac{n}{2},$ then $a^{\frac{1}{n}} -b^{\frac{1}{n}} <\frac{1}{n}$

2021 IMC, 7

Tags: complex analysis , polynomial , inequality

Let $D \subseteq \mathbb{C}$ be an open set containing the closed unit disk $\{z : |z| \leq 1\}$. Let $f : D \rightarrow \mathbb{C}$ be a holomorphic function, and let $p(z)$ be a monic polynomial. Prove that $$ |f(0)| \leq \max_{|z|=1} |f(z)p(z)| $$

1983 IMO Longlists, 66

Tags: algebra , polynomial , rearrangement inequality , triangle inequality , inequality

Let $ a$, $ b$ and $ c$ be the lengths of the sides of a triangle. Prove that \[ a^{2}b(a \minus{} b) \plus{} b^{2}c(b \minus{} c) \plus{} c^{2}a(c \minus{} a)\ge 0. \] Determine when equality occurs.

1977 Germany Team Selection Test, 1

Tags: rearrangement inequality , inequality , permutation , algebra

We consider two sequences of real numbers $x_{1} \geq x_{2} \geq \ldots \geq x_{n}$ and $\ y_{1} \geq y_{2} \geq \ldots \geq y_{n}.$ Let $z_{1}, z_{2}, .\ldots, z_{n}$ be a permutation of the numbers $y_{1}, y_{2}, \ldots, y_{n}.$ Prove that $\sum \limits_{i=1}^{n} ( x_{i} -\ y_{i} )^{2} \leq \sum \limits_{i=1}^{n}$ $( x_{i} - z_{i})^{2}.$