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: 6530

1997 India Regional Mathematical Olympiad, 5
Tags: inequalities , triangle inequality , rearrangement inequality
Let $x,y,z$ be three distinct real positive numbers, Determine whether or not the three real numbers \[ \left| \frac{x}{y} - \frac{y}{x}\right| ,\left| \frac{y}{z} - \frac{z}{y}\right |, \left| \frac{z}{x} - \frac{x}{z}\right| \] can be the lengths of the sides of a triangle.

2024 Junior Balkan Team Selection Tests - Moldova, 7
Tags: inequalities , algebra
Find all the real numbers $x,y,z$ which satisfy the following conditions: $$ \begin{cases} 3(x^2+y^2+z^2)=1\\ x^2y^2+y^2z^2+z^2x^2=xyz(x+y+z)^3\\ \end{cases} $$

2005 Croatia National Olympiad, 3
Tags: logarithm , inequalities
If $a, b, c$ are real numbers greater than $1$, prove that for any real number $r$ \[(\log_{a}bc)^{r}+(\log_{b}ca)^{r}+(\log_{c}ab)^{r}\geq 3 \cdot 2^{r}. \]

2007 ISI B.Stat Entrance Exam, 5
Tags: trigonometry , inequalities
Show that \[-2 \le \cos \theta\left(\sin \theta + \sqrt{\sin ^2 \theta +3}\right) \le 2\] for all values of $\theta$.

2010 Saudi Arabia Pre-TST, 4.3
Tags: algebra , inequalities
Let $a, b, c$ be positive real numbers such that $abc = 8$. Prove that $$\frac{a-2}{a+1}+\frac{b-2}{b+1}+\frac{c-2}{c+1} \le 0$$

1990 India Regional Mathematical Olympiad, 5
Tags: inequalities , geometry , triangle inequality
$P$ is any point inside a triangle $ABC$. The perimeter of the triangle $AB + BC + Ca = 2s$. Prove that $s < AP +BP +CP < 2s$.

2005 International Zhautykov Olympiad, 1
Tags: inequalities
For the positive real numbers $ a,b,c$ prove the inequality \[ \frac {c}{a \plus{} 2b} \plus{} \frac {a}{b \plus{} 2c} \plus{} \frac {b}{c \plus{} 2a}\ge1. \]

2005 Germany Team Selection Test, 3
Tags: inequalities , modular arithmetic , algorithm , induction , number theory
We have $2p-1$ integer numbers, where $p$ is a prime number. Prove that we can choose exactly $p$ numbers (from these $2p-1$ numbers) so that their sum is divisible by $p$.

2010 Ukraine Team Selection Test, 4
Tags: inequalities , algebra
For the nonnegative numbers $a, b, c$ prove the inequality: $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}+\sqrt{\frac{ab+bc+ca}{a^2+b^2+c^2}}\ge \frac52$$

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 \[ \frac{x_1}{1+x_1^2} + \frac{x_2}{1+x_1^2 + x_2^2} + \cdots + \frac{x_n}{1 + x_1^2 + \cdots + x_n^2} < \sqrt{n}. \]

2010 CentroAmerican, 5
Tags: polynomial , inequalities , triangle inequality , algebra
If $p$, $q$ and $r$ are nonzero rational numbers such that $\sqrt[3]{pq^2}+\sqrt[3]{qr^2}+\sqrt[3]{rp^2}$ is a nonzero rational number, prove that $\frac{1}{\sqrt[3]{pq^2}}+\frac{1}{\sqrt[3]{qr^2}}+\frac{1}{\sqrt[3]{rp^2}}$ is also a rational number.

1980 AMC 12/AHSME, 6
Tags: inequalities
A positive number $x$ satisfies the inequality $\sqrt{x} < 2x$ if and only if $\text{(A)} \ x > \frac{1}{4} \qquad \text{(B)} \ x > 2 \qquad \text{(C)} x > 4 \qquad \text{(D)} \ x < \frac{1}{4}\qquad \text{(E)} x < 4$

2020 Austrian Junior Regional Competition, 1
Tags: inequalities , algebra
Let $a$ be a real number and $b$ a real number with $b\neq-1$ and $b\neq0. $ Find all pairs $ (a, b)$ such that $$\frac{(1 + a)^2 }{1 + b}\leq 1 + \frac{a^2}{b}.$$ For which pairs (a, b) does equality apply? (Walther Janous)

2014 Estonia Team Selection Test, 3
Tags: max , area , inequalities , algebra
Three line segments, all of length $1$, form a connected figure in the plane. Any two different line segments can intersect only at their endpoints. Find the maximum area of the convex hull of the figure.

1999 Romania National Olympiad, 1
Tags: function , limit , inequalities , integration , logarithm , real analysis
„œ‚Find all continuous functions $ f: \mathbb{R}\to [1,\infty)$ for wich there exists $ a\in\mathbb{R}$ and a positive integer $ k$ such that \[ f(x)f(2x)\cdot...\cdot f(nx)\leq an^k\] for all real $ x$ and all positive integers $ n$. [i]author :Radu Gologan[/i]

2020 SJMO, 5
Tags: algebra , inequalities
A nondegenerate triangle with perimeter $1$ has side lengths $a, b,$ and $c$. Prove that \[\left|\frac{a - b}{c + ab}\right| + \left|\frac{b - c}{a + bc}\right| + \left|\frac{c - a}{b + ac}\right| < 2.\] [i]Proposed by Andrew Wen[/i]

2015 European Mathematical Cup, 2
Tags: inequalities
Let $m, n, p$ be fixed positive real numbers which satisfy $mnp = 8$. Depending on these constants, find the minimum of $$x^2+y^2+z^2+ mxy + nxz + pyz,$$ where $x, y, z$ are arbitrary positive real numbers satisfying $xyz = 8$. When is the equality attained? Solve the problem for: [list=a][*]$m = n = p = 2,$ [*] arbitrary (but fixed) positive real numbers $m, n, p.$[/list] [i]Stijn Cambie[/i]

1953 Moscow Mathematical Olympiad, 238
Tags: inequalities , radical , nested radical , algebra
Prove that if in the following fraction we have $n$ radicals in the numerator and $n - 1$ in the denominator, then $$\frac{2-\sqrt{2+\sqrt{2+...+\sqrt{2}}}}{2-\sqrt{2+\sqrt{2+...+\sqrt{2}}}}>\frac14$$

2011 Mongolia Team Selection Test, 3
Tags: inequalities , pigeonhole principle , combinatorics
Let $G$ be a graph, not containing $K_4$ as a subgraph and $|V(G)|=3k$ (I interpret this to be the number of vertices is divisible by 3). What is the maximum number of triangles in $G$?

2003 Hungary-Israel Binational, 1
Tags: inequalities
If $x_{1}, x_{2}, . . . , x_{n}$ are positive numbers, prove the inequality $\frac{x_{1}^{3}}{x_{1}^{2}+x_{1}x_{2}+x_{2}^{2}}+\frac{x_{2}^{3}}{x_{2}^{2}+x_{2}x_{3}+x_{3}^{2}}+...+\frac{x_{n}^{3}}{x_{n}^{2}+x_{n}x_{1}+x_{1}^{2}}\geq\frac{x_{1}+x_{2}+...+x_{n}}{3}$.

1998 North Macedonia National Olympiad, 5
Tags: sequence , recurrence relation , inequalities , algebra
The sequence $(a_n)$ is defined by $a_1 =\sqrt2$ and $a_{n+1} =\sqrt{2-\sqrt{4-a_n^2}}$. Let $b_n =2^{n+1}a_n$. Prove that $b_n \le 7$ and $b_n < b_{n+1}$ for all $n$.

2022-2023 OMMC, 21
Tags: inequalities
Define the minimum real $C$ where for any reals $0 = a_0 < a_{1} < \dots < a_{1000}$ then $$\min_{0 \le k \le 1000} (a_{k}^2 + (1000-k)^2) \le C(a_1+ \dots + a_{1000})$$ holds. Find $\lfloor 100C \rfloor.$

2007 Today's Calculation Of Integral, 202
Tags: calculus , integration , trigonometry , quadratic , inequalities , calculus computations
Let $a,\ b$ are real numbers such that $a+b=1$. Find the minimum value of the following integral. \[\int_{0}^{\pi}(a\sin x+b\sin 2x)^{2}\ dx \]

2000 India National Olympiad, 6
Tags: geometry , perimeter , inequalities , triangle inequality , combinatorics
For any natural numbers $n$, ( $n \geq 3$), let $f(n)$ denote the number of congruent integer-sided triangles with perimeter $n$. Show that (i) $f(1999) > f (1996)$; (ii) $f(2000) = f(1997)$.

PEN I Problems, 7
Tags: floor function , inequalities
Prove that for all positive integers $n$, \[\lfloor \sqrt[3]{n}+\sqrt[3]{n+1}\rfloor =\lfloor \sqrt[3]{8n+3}\rfloor.\]