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

2009 JBMO Shortlist, 5
Tags: inequality , algebra
$\boxed{\text{A5}}$ Let $x,y,z$ be positive reals. Prove that $(x^2+y+1)(x^2+z+1)(y^2+x+1)(y^2+z+1)(z^2+x+1)(z^2+y+1)\geq (x+y+z)^6$

2021 Azerbaijan IZhO TST, 1
Tags: algebra , inequality
Let $a, b, c$ be real numbers with the property as $ab + bc + ca = 1$. Show that: $$\frac {(a + b) ^ 2 + 1} {c ^ 2 + 2} + \frac {(b + c) ^ 2 + 1} {a ^ 2 + 2} + \frac {(c + a) ^ 2 + 1} {b ^ 2 + 2} \ge 3 $$.

2002 India National Olympiad, 3
Tags: calculus , algebra , polynomial , inequality , inequalities
If $x$, $y$ are positive reals such that $x + y = 2$ show that $x^3y^3(x^3+ y^3) \leq 2$.

2022 JBMO Shortlist, A3
Tags: inequality , algebra
Let $a, b,$ and $c$ be positive real numbers such that $a + b + c = 1$. Prove the following inequality $$a \sqrt[3]{\frac{b}{a}} + b \sqrt[3]{\frac{c}{b}} + c \sqrt[3]{\frac{a}{c}} \le ab + bc + ca + \frac{2}{3}.$$ Proposed by [i]Anastasija Trajanova, Macedonia[/i]

1983 IMO, 3
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.

1975 IMO Shortlist, 2
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 $\sum \limits_{i=1}^{n} ( x_{i} -\ y_{i} )^{2} \leq \sum \limits_{i=1}^{n}$ $( x_{i} - z_{i})^{2}.$

1996 IMO Shortlist, 2
Tags: algebra , sequence , inequality
Let $ a_1 \geq a_2 \geq \ldots \geq a_n$ be real numbers such that for all integers $ k > 0,$ \[ a^k_1 \plus{} a^k_2 \plus{} \ldots \plus{} a^k_n \geq 0.\] Let $ p \equal{}\max\{|a_1|, \ldots, |a_n|\}.$ Prove that $ p \equal{} a_1$ and that \[ (x \minus{} a_1) \cdot (x \minus{} a_2) \cdots (x \minus{} a_n) \leq x^n \minus{} a^n_1\] for all $ x > a_1.$

2015 Irish Math Olympiad, 10
Tags: binomial , inequalities , binomial summation , inequality
Prove that, for all pairs of nonnegative integers, $j,n$, $$\sum_{K=0}^{n}k^j\binom n k \ge 2^{n-j} n^j$$

1970 IMO Shortlist, 9
Tags: inequality , n-variable inequality
Let $u_1, u_2, \ldots, u_n, v_1, v_2, \ldots, v_n$ be real numbers. Prove that \[1+ \sum_{i=1}^n (u_i+v_i)^2 \leq \frac 43 \Biggr( 1+ \sum_{i=1}^n u_i^2 \Biggl) \Biggr( 1+ \sum_{i=1}^n v_i^2 \Biggl) .\]

Kvant 2024, M2797
Tags: inequality , algebra , inequalities
For real numbers $0 \leq a_1 \leq a_2 \leq ... \leq a_n$ and $0 \leq b_1 \leq b_2 \leq ... \leq b_n$ prove that \[ \left( \frac{a_1}{1 \cdot 2}+\frac{a_2}{2 \cdot 3}+...+\frac{a_n}{n(n+1)} \right) \times \left( \frac{b_1}{1 \cdot 2}+\frac{b_2}{2 \cdot 3}+...+\frac{b_n}{n(n+1)} \right) \leq \frac{a_1b_1}{1 \cdot 2}+\frac{a_2b_2}{2 \cdot 3}+...+\frac{a_nb_n}{n(n+1)}.\] [i]Proposed by A. Antropov[/i]

1983 Czech and Slovak Olympiad III A, 2
Tags: geometric inequality , geometry , inequality , inequalities
Given a triangle $ABC$, prove that for every inner point $P$ of the side $AB$ the inequality $$PC\cdot AB<PA\cdot BC+PB\cdot AC$$ holds.

2024 HMIC, 2
Tags: inequality , inequalities , algebra
Suppose that $a$, $b$, $c$, and $d$ are real numbers such that $a+b+c+d=8$. Compute the minimum possible value of \[20(a^2+b^2+c^2+d^2)-\sum_{\text{sym}}a^3b,\] where the sum is over all $12$ symmetric terms. [i]Derek Liu[/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)

2005 Bosnia and Herzegovina Team Selection Test, 2
Tags: algebra , inequality , three variable inequality
If $a_1$, $a_2$ and $a_3$ are nonnegative real numbers for which $a_1+a_2+a_3=1$, then prove the inequality $a_1\sqrt{a_2}+a_2\sqrt{a_3}+a_3\sqrt{a_1}\leq \frac{1}{\sqrt{3}}$

2021 Saudi Arabia IMO TST, 11
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]

1992 Irish Math Olympiad, 5
Tags: inequality , inequalities
If, for $k=1,2,\dots ,n$, $a_k$ and $b_k$ are positive real numbers, prove that $$\sqrt[n]{a_1a_2\cdots a_n}+\sqrt[n]{b_1b_2\cdots b_n}\le \sqrt[n]{(a_1+b_1)(a_2+b_2)\cdots (a_n+b_n)};$$ and that equality holds if, and only if, $$\frac{a_1}{b_1}=\frac{a_2}{b_2}=\cdots =\frac{a_n}{b_n}.$$

2012 Bosnia and Herzegovina Junior BMO TST, 4
Tags: triangle , perimeter , inequality , geometry , inequalities
If $a$, $b$ and $c$ are sides of triangle which perimeter equals $1$, prove that: $a^2+b^2+c^2+4abc<\frac{1}{2}$

1970 IMO, 3
Tags: limit , inequality , calculus , integration , area
The real numbers $a_0,a_1,a_2,\ldots$ satisfy $1=a_0\le a_1\le a_2\le\ldots. b_1,b_2,b_3,\ldots$ are defined by $b_n=\sum_{k=1}^n{1-{a_{k-1}\over a_k}\over\sqrt a_k}$. [b]a.)[/b] Prove that $0\le b_n<2$. [b]b.)[/b] Given $c$ satisfying $0\le c<2$, prove that we can find $a_n$ so that $b_n>c$ for all sufficiently large $n$.

2012 JBMO ShortLists, 1
Tags: algebra , inequality
Let $a,b,c$ be positive real numbers such that $a+b+c=1$. Prove that \[\frac {a}{b} + \frac {a}{c} + \frac {c}{b} + \frac {c}{a} + \frac {b}{c} + \frac {b}{a} + 6 \geq 2\sqrt{2}\left (\sqrt{\frac{1-a}{a}} + \sqrt{\frac{1-b}{b}} + \sqrt{\frac{1-c}{c}}\right ).\] When does equality hold?

2022 Brazil National Olympiad, 5
Tags: algebra , inequality
Let $n$ be a positive integer number. Define $S(n)$ to be the least positive integer such that $S(n) \equiv n \pmod{2}$, $S(n) \geq n$, and such that there are [b]not[/b] positive integers numbers $k,x_1,x_2,...,x_k$ such that $n=x_1+x_2+...+x_k$ and $S(n)=x_1^2+x_2^2+...+x_k^2$. Prove that there exists a real constant $c>0$ and a positive integer $n_0$ such that, for all $n \geq n_0$, $S(n) \geq cn^{\frac{3}{2}}$.

2021 Austrian MO National Competition, 1
Tags: inequality , algebra
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)

2021 Science ON all problems, 2
Tags: inequality , algebra
$a,b,c$ are nonnegative integers that satisfy $a^2+b^2+c^2=3$. Find the minimum and maximum value the sum $$\frac{1}{1+a+b}+\frac{1}{1+b+c}+\frac{1}{1+c+a}$$ may achieve and find all $a,b,c$ for which equality occurs.\\ \\ [i](Andrei Bâra)[/i]

2021 Indonesia TST, A
Tags: inequality , inequalities , algebra
A positive real $M$ is $strong$ if for any positive reals $a$, $b$, $c$ satisfying $$ \text{max}\left\{ \frac{a}{b+c} , \frac{b}{c+a} , \frac{c}{a+b} \right\} \geqslant M $$ then the following inequality holds: $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} > 20.$$ (a) Prove that $M=20-\frac{1}{20}$ is not $strong$. (b) Prove that $M=20-\frac{1}{21}$ is $strong$.

2023 Junior Balkan Team Selection Tests - Moldova, 12
Tags: algebra , inequality
Let $a,b,c$ be positive real numbers such that $a^2+b^2+c^2=3. $ Prove that $$\frac{a^4+3ab^3}{a^3+2b^3}+\frac{b^4+3bc^3}{b^3+2c^3}+\frac{c^4+3ca^3}{c^3+2a^3}\leq4.$$

1998 IMO Shortlist, 3
Tags: rearrangement inequality , algebra , inequality
Let $x,y$ and $z$ be positive real numbers such that $xyz=1$. Prove that \[ \frac{x^{3}}{(1 + y)(1 + z)}+\frac{y^{3}}{(1 + z)(1 + x)}+\frac{z^{3}}{(1 + x)(1 + y)} \geq \frac{3}{4}. \]