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

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Found problems: 583

1984 IMO, 1
Tags: Inequality , three variable inequality , calculus , maximization , IMO , IMO 1984 , Hi
Prove that $0\le yz+zx+xy-2xyz\le{7\over27}$, where $x,y$ and $z$ are non-negative real numbers satisfying $x+y+z=1$.

1983 IMO Longlists, 60
Tags: algebra , Summation , series summation , approximation , Inequality , IMO Shortlist
Find the greatest integer less than or equal to $\sum_{k=1}^{2^{1983}} k^{\frac{1}{1983} -1}.$

2022 IFYM, Sozopol, 5
Tags: algebra , Inequality , Sum
Prove that $\sum_{n=1}^{2022^{2022}} \frac{1}{\sqrt{n^3+2n^2+n}}<\frac{19}{10}$.

2021 China Team Selection Test, 4
Tags: Inequality , algebra , TST
Suppose $x_1,x_2,...,x_{60}\in [-1,1]$ , find the maximum of $$ \sum_{i=1}^{60}x_i^2(x_{i+1}-x_{i-1}),$$ where $x_{i+60}=x_i$.

1967 IMO Longlists, 58
Tags: geometry , function , algebra , Inequality , Linear Function , IMO Shortlist , IMO Longlist
A linear binomial $l(z) = Az + B$ with complex coefficients $A$ and $B$ is given. It is known that the maximal value of $|l(z)|$ on the segment $-1 \leq x \leq 1$ $(y = 0)$ of the real line in the complex plane $z = x + iy$ is equal to $M.$ Prove that for every $z$ \[|l(z)| \leq M \rho,\] where $\rho$ is the sum of distances from the point $P=z$ to the points $Q_1: z = 1$ and $Q_3: z = -1.$

2013 Bosnia And Herzegovina - Regional Olympiad, 1
Tags: algebra , Inequality
If $a$, $b$ and $c$ are nonnegative real numbers such that $a^2+b^2+c^2=1$, prove that $$\frac{1}{2} \leq \frac{a}{1+a^4}+\frac{b}{1+b^4}+\frac{c}{1+c^4} \leq \frac{9\sqrt{3}}{10}$$

2014 Belarus Team Selection Test, 2
Tags: Inequality , algebra , Belarus
Let $a,b,c$ be positive real numbers such that $a+b+c=1$. Prove that $$\frac{a^2}{(b+c)^3}+\frac{b^2}{(c+a)^3}+\frac{c^2}{(a+b)^3}\geq \frac98$$

2004 Bosnia and Herzegovina Team Selection Test, 5
Tags: Inequality , cosine , sine , algebra
For $0 \leq x < \frac{\pi}{2} $ prove the inequality: $a^2\tan(x)\cdot(\cos(x))^{\frac{1}{3}}+b^2\sin{x}\geq 2xab$ where $a$ and $b$ are real numbers.

2021-IMOC, A8
Tags: functional equation , algebra , Inequality , function , inequalities
Find all functions $f : \mathbb{N} \to \mathbb{N}$ with $$f(x) + yf(f(x)) < x(1 + f(y)) + 2021$$ holds for all positive integers $x,y.$

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 $$\frac{x}{1+x^2}+\frac{y}{1+y^2}+\frac{z}{1+z^2} \leq \frac{3\sqrt{3}}{4}$$

2025 239 Open Mathematical Olympiad, 4
Tags: algebra , Inequality
Positive numbers $a$, $b$ and $c$ are such that $a^2+b^2+c^2+abc=4$. Prove that \[\sqrt{2-a}+\sqrt{2-b}+\sqrt{2-c}\geqslant 2+\sqrt{(2-a)(2-b)(2-c)}.\]

2013 Balkan MO Shortlist, A1
Tags: algebra , Inequality , BMO Shortlist
Positive real numbers $a, b,c$ satisfy $ab + bc+ ca = 3$. Prove the inequality $$\frac{1}{4+(a+b)^2}+\frac{1}{4+(b+c)^2}+\frac{1}{4+(c+a)^2}\le \frac{3}{8}$$

2012 Kyiv Mathematical Festival, 2
Tags: inequalities , Inequality , algebra
Positive numbers $x, y, z$ satisfy $x^2+y^2+z^2+xy+yz+zy \le 1$. Prove that $\big( \frac{1}{x}-1\big) \big( \frac{1}{y}-1\big)\big( \frac{1}{z}-1\big) \ge 9 \sqrt6 -19$.

2018 Azerbaijan Junior NMO, 3
Tags: Inequality , AZE JUNIOR NMO , algebra
$a;b\in\mathbb{R^+}$. Prove the following inequality: $$\sqrt[3]{\frac{a}{b}}+\sqrt[3]{\frac{b}{a}}\leq\sqrt[3]{2(a+b)(\frac1{a}+\frac1{b})}$$

2020 China Girls Math Olympiad, 2
Tags: maximum value , Inequality , algebra , CGMO
Let $n$ be an integer and $n \geq 2$, $x_1, x_2, \cdots , x_n$ are arbitrary real number, find the maximum value of $$2\sum_{1\leq i<j \leq n}\left \lfloor x_ix_j \right \rfloor-\left ( n-1 \right )\sum_{i=1}^{n}\left \lfloor x_i^2 \right \rfloor $$

2014 Dutch Mathematical Olympiad, 4
Tags: number theory , prime , divisible , Inequality , diophantine
A quadruple $(p, a, b, c)$ of positive integers is called a Leiden quadruple if - $p$ is an odd prime number, - $a, b$, and $c$ are distinct and - $ab + 1, bc + 1$ and $ca + 1$ are divisible by $p$. a) Prove that for every Leiden quadruple $(p, a, b, c)$ we have $p + 2 \le \frac{a+b+c}{3}$ . b) Determine all numbers $p$ for which a Leiden quadruple $(p, a, b, c)$ exists with $p + 2 = \frac{a+b+c}{3} $

2024 Greece Junior Math Olympiad, 1
Tags: algebra , Inequality
a) Prove that for all real numbers $k,l,m$ holds : $$(k+l+m)^2 \ge 3 (kl+lm+mk)$$ When does equality holds? b) If $x,y,z$ are positive real numbers and $a,b$ real numbers such that $$a(x+y+z)=b(xy+yz+zx)=xyz,$$ prove that $a \ge 3b^2$. When does equality holds?

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

1987 IMO Shortlist, 6
Tags: rearrangement inequality , three variable inequality , triangle inequality , Inequality , IMO Shortlist
Show that if $a, b, c$ are the lengths of the sides of a triangle and if $2S = a + b + c$, then \[\frac{a^n}{b+c} + \frac{b^n}{c+a} +\frac{c^n}{a+b} \geq \left(\dfrac 23 \right)^{n-2}S^{n-1} \quad \forall n \in \mathbb N \] [i]Proposed by Greece.[/i]

2023 Thailand Online MO, 7
Tags: Inequality , algebra
Let $a_0,a_1,\dots$ be a sequence of positive reals such that $$ a_{n+2} \leq \frac{2023a_n}{a_na_{n+1}+2023}$$ for all integers $n\geq 0$. Prove that either $a_{2023}<1$ or $a_{2024}<1$.

2021 Federal Competition For Advanced Students, P2, 1
Tags: algebra , AZE CMO TST , AZE EGMO TST , Inequality , TST
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)

2014 Contests, A2
Tags: algebra , Inequality , BMO , BMO Shortlist
Let $x,y$ and $z$ be positive real numbers such that $xy+yz+xz=3xyz$. Prove that \[ x^2y+y^2z+z^2x \ge 2(x+y+z)-3 \] and determine when equality holds. [i]UK - David Monk[/i]

2024 Azerbaijan JBMO TST, 4
Tags: JBMO Shortlist , algebra , AZE JBMO TST , Inequality , TST
Let $a \geq b \geq 1 \geq c \geq 0$ be real numbers such that $a+b+c=3$. Show that $$3 \left( \frac{a}{b}+\frac{b}{a} \right ) \geq 4c^2+\frac{a^2}{b}+\frac{b^2}{a}$$

2017 Pan-African Shortlist, I4
Tags: combinatorics , Inequality , maximum value , minimum value , permutations
Find the maximum and minimum of the expression \[ \max(a_1, a_2) + \max(a_2, a_3), + \dots + \max(a_{n-1}, a_n) + \max(a_n, a_1), \] where $(a_1, a_2, \dots, a_n)$ runs over the set of permutations of $(1, 2, \dots, n)$.

2025 6th Memorial "Aleksandar Blazhevski-Cane", P4
Tags: Inequality , algebra , inequalities
Prove that for all real numbers $a, b, c > 1$ the inequality \[a(b^2 + c) + b(c^2 + a) + c(a^2 + b) \ge a^2 + b^2 + c^2 + 3abc\] holds. When does equality hold? Proposed by [i]Ilija Jovcevski[/i]