This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 592

1983 IMO Shortlist, 21
Tags: algebra , summation , series summation , approximation , inequality
Find the greatest integer less than or equal to $\sum_{k=1}^{2^{1983}} k^{\frac{1}{1983} -1}.$

2025 Kosovo National Mathematical Olympiad`, P2
Tags: inequality , algebra , inequalities
Let $x$ and $y$ be real numbers where at least one of them is bigger than $2$ and $xy+4 > 2(x+y)$ holds. Show that $xy>x+y$.

2019 District Olympiad, 4
Tags: inequality , algebra
Find the smallest positive real number $\lambda$ such that for every numbers $a_1,a_2,a_3 \in \left[0, \frac{1}{2} \right]$ and $b_1,b_2,b_3 \in (0, \infty)$ with $\sum\limits_{i=1}^3a_i=\sum\limits_{i=1}^3b_i=1,$ we have $$b_1b_2b_3 \le \lambda (a_1b_1+a_2b_2+a_3b_3).$$

2017 Bosnia and Herzegovina EGMO TST, 4
Tags: triplets , equality , inequality , algebra
Let $a$, $b$, $c$, $d$ and $e$ be distinct positive real numbers such that $a^2+b^2+c^2+d^2+e^2=ab+ac+ad+ae+bc+bd+be+cd+ce+de$ $a)$ Prove that among these $5$ numbers there exists triplet such that they cannot be sides of a triangle $b)$ Prove that, for $a)$, there exists at least $6$ different triplets

2016 Bulgaria JBMO TST, 2
Tags: inequalities , inequality
a, b, c are positive real numbers and a+b+c=k. Find the minimum value of $ b^2/(ka+bc)^1/2+c^2/(kb+ac)^1/2+a^2/(kc+ab)^1/2 $

1971 IMO Longlists, 52
Tags: inequality , 4-variable inequality , four variables
Prove the inequality \[ \frac{a_1+ a_3}{a_1 + a_2} + \frac{a_2 + a_4}{a_2 + a_3} + \frac{a_3 + a_1}{a_3 + a_4} + \frac{a_4 + a_2}{a_4 + a_1} \geq 4, \] where $a_i > 0, i = 1, 2, 3, 4.$

2010 Saint Petersburg Mathematical Olympiad, 6
Tags: algebra , inequality
For positive numbers is true that $$ab+ac+bc=a+b+c$$ Prove $$a+b+c+1 \geq 4abc$$

1998 Bosnia and Herzegovina Team Selection Test, 2
Tags: algebra , inequality , inequalities
For positive real numbers $x$, $y$ and $z$ holds $x^2+y^2+z^2=1$. Prove that $$\frac{x}{1+x^2}+\frac{y}{1+y^2}+\frac{z}{1+z^2} \leq \frac{3\sqrt{3}}{4}$$

2013 Macedonian Team Selection Test, Problem 5
Tags: geometry , inequality , rectangle
Let $ABC$ be a triangle with given sides $a,b,c$. Determine the minimal possible length of the diagonal of an inscribed rectangle in this triangle. [i]Note: A rectangle is inscribed in the triangle if two of its consecutive vertices lie on one side of the triangle, while the other two vertices lie on the other two sides of the triangle. [/i]

2015 Turkmenistan National Math Olympiad, 4
Tags: algebra , inequality , inequalities
Find the max and minimum without using dervivate: $\sqrt{x} +4 \cdot \sqrt{\frac{1}{2} - x}$

1988 IMO Shortlist, 24
Tags: linear recurrence , inequality system , inequality , sequence
Let $ \{a_k\}^{\infty}_1$ be a sequence of non-negative real numbers such that: \[ a_k \minus{} 2 a_{k \plus{} 1} \plus{} a_{k \plus{} 2} \geq 0 \] and $ \sum^k_{j \equal{} 1} a_j \leq 1$ for all $ k \equal{} 1,2, \ldots$. Prove that: \[ 0 \leq a_{k} \minus{} a_{k \plus{} 1} < \frac {2}{k^2} \] for all $ k \equal{} 1,2, \ldots$.

2021 Germany Team Selection Test, 3
Tags: algebra , inequality , 4-variable inequality , inequalities
Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

2025 NEPALTST, 1
Tags: algebra , recurrence , inequality
Let the sequence $\{a_n\}_{n \geq 1}$ be defined by \[ a_1 = 1, \quad a_{n+1} = a_n + \frac{1}{\sqrt[2024]{a_n}} \quad \text{for } n \geq 1, \, n \in \mathbb{N} \] Prove that \[ a_n^{2025} >n^{2024} \] for all positive integers $n \geq 2$. $\textbf{Proposed by Prajit Adhikari, Nepal.}$

2021 Silk Road, 2
Tags: algebra , inequality
For every positive integer $m$ prove the inquality $|\{\sqrt{m}\} - \frac{1}{2}| \geq \frac{1}{8(\sqrt m+1)} $ (The integer part $[x]$ of the number $x$ is the largest integer not exceeding $x$. The fractional part of the number $x$ is a number $\{x\}$ such that $[x]+\{x\}=x$.) A. Golovanov

2024 Korea Junior Math Olympiad (First Round), 19.
Tags: algebra , maximum value , inequality
For all integers $ {a}_{0},{a}_{1}, \cdot\cdot\cdot {a}_{100}$, find the maximum of ${a}_{5}-2{a}_{40}+3{a}_{60}-4{a}_{95} $ $\bigstar$ 1) ${a}_{0}={a}_{100}=0$ 2) for all $i=0,1,\cdot \cdot \cdot 99, $ $|{a}_{i+1}-{a}_{i}|\le1$ 3) $ {a}_{10}={a}_{90} $

2020 Balkan MO Shortlist, A2
Tags: algebra , inequality
Given are positive reals $a, b, c$, such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=3$. Prove that $\frac{\sqrt{a+\frac{b}{c}}+\sqrt{b+\frac{c}{a}}+\sqrt{c+\frac{a}{b}}}{3}\leq \frac{a+b+c-1}{\sqrt{2}}$. [i]Albania[/i]

2021 IMO Shortlist, A7
Tags: inequality , inequalities , algebra
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]

1981 IMO Shortlist, 3
Tags: maximization , optimization , algebra , inequality
Find the minimum value of \[\max(a + b + c, b + c + d, c + d + e, d + e + f, e + f + g)\] subject to the constraints (i) $a, b, c, d, e, f, g \geq 0,$ (ii)$ a + b + c + d + e + f + g = 1.$

2014 Contests, 1
Tags: algebra , inequality , inequalities
Prove that for $\forall$ $a,b,c\in [\frac{1}{3},3]$ the following inequality is true: $\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}\geq \frac{7}{5}$.

2018 Regional Competition For Advanced Students, 1
Tags: inequality
If $a, b$ are positive reals such that $a+b<2$. Prove that $$\frac{1}{1+a^2}+\frac{1}{1+b^2} \le \frac{2}{1+ab}$$ and determine all $a, b$ yielding equality. [i]Proposed by Gottfried Perz[/i]

2022 Azerbaijan EGMO/CMO TST, A2
Tags: algebra , inequality
Let $a, b$ and $c$ be pairwise different natural numbers. Prove $\frac{a^3 + b^3 + c^3}{3} \ge abc + a + b + c$. When does equality holds? (Karl Czakler)

2012 Balkan MO Shortlist, A2
Tags: inequality , algebra
Let $a,b,c\ge 0$ and $a+b+c=\sqrt2$. Show that \[\frac1{\sqrt{1+a^2}}+\frac1{\sqrt{1+b^2}}+\frac1{\sqrt{1+c^2}} \ge 2+\frac1{\sqrt3}\] [hide] In general if $a_1, a_2, \cdots , a_n \ge 0$ and $\sum_{i=1}^n a_i=\sqrt2$ we have \[\sum_{i=1}^n \frac1{\sqrt{1+a_i^2}} \ge (n-1)+\frac1{\sqrt3}\] [/hide]

1969 IMO Shortlist, 14
Tags: quadratic , algebra , inequality , root , polynomial
$(CZS 3)$ Let $a$ and $b$ be two positive real numbers. If $x$ is a real solution of the equation $x^2 + px + q = 0$ with real coefficients $p$ and $q$ such that $|p| \le a, |q| \le b,$ prove that $|x| \le \frac{1}{2}(a +\sqrt{a^2 + 4b})$ Conversely, if $x$ satisfies the above inequality, prove that there exist real numbers $p$ and $q$ with $|p|\le a, |q|\le b$ such that $x$ is one of the roots of the equation $x^2+px+ q = 0.$

JOM 2015 Shortlist, A5
Tags: inequality , inequalities
Let $ a, b, c $ be the side length of a triangle, with $ ab + bc + ca = 18 $ and $ a, b, c > 1 $. Prove that $$ \sum_{cyc}\frac{1}{(a - 1)^3} > \frac{1}{a + b + c - 3} $$

2012 Middle European Mathematical Olympiad, 1
Tags: function , functional equation , inequality , algebra
Let $ \mathbb{R} ^{+} $ denote the set of all positive real numbers. Find all functions $ \mathbb{R} ^{+} \to \mathbb{R} ^{+} $ such that \[ f(x+f(y)) = yf(xy+1)\] holds for all $ x, y \in \mathbb{R} ^{+} $.