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

JOM 2013, 1.
Tags: inequality , inequalities , algebra
Determine the minimum value of $\dfrac{m^m}{1\cdot 3\cdot 5\cdot \ldots \cdot(2m-1)}$ for positive integers $m$.

2004 China Team Selection Test, 2
Tags: inequalities
Find the largest positive real $ k$, such that for any positive reals $ a,b,c,d$, there is always: \[ (a\plus{}b\plus{}c) \left[ 3^4(a\plus{}b\plus{}c\plus{}d)^5 \plus{} 2^4(a\plus{}b\plus{}c\plus{}2d)^5 \right] \geq kabcd^3\]

2023 Bulgaria National Olympiad, 5
Tags: algebra , n-variable inequality , inequalities
For every positive integer $n$ determine the least possible value of the expression \[|x_{1}|+|x_{1}-x_{2}|+|x_{1}+x_{2}-x_{3}|+\dots +|x_{1}+x_{2}+\dots +x_{n-1}-x_{n}|\] given that $x_{1}, x_{2}, \dots , x_{n}$ are real numbers satisfying $|x_{1}|+|x_{2}|+\dots+|x_{n}| = 1$.

2011 Korea National Olympiad, 4
Tags: function , inequalities
Let $ x_1, x_2, \cdots, x_{25} $ real numbers such that $ 0 \le x_i \le i (i=1, 2, \cdots, 25) $. Find the maximum value of \[x_{1}^{3}+x_{2}^{3}+\cdots +x_{25}^{3} - ( x_1x_2x_3 + x_2x_3x_4 + \cdots x_{25}x_1x_2 ) \]

1994 India Regional Mathematical Olympiad, 8
Tags: inequalities
If $a,b,c$ are positive real numbers such that $a+b+c = 1$, prove that \[ (1+a)(1+b)(1+c) \geq 8 (1-a)(1-b)(1-c) . \]

2020 Kazakhstan National Olympiad, 2
Tags: algebra , inequalities
Let $x_1, x_2, ... , x_n$ be a real numbers such that\\ 1) $1 \le x_1, x_2, ... , x_n \le 160$ 2) $x^{2}_{i} + x^{2}_{j} + x^{2}_{k} \ge 2(x_ix_j + x_jx_k + x_kx_i)$ for all $1\le i < j < k \le n$ Find the largest possible $n$.

2021 Israel TST, 1
Tags: algebra , inequalities
Which is greater: \[\frac{1^{-3}-2^{-3}}{1^{-2}-2^{-2}}-\frac{2^{-3}-3^{-3}}{2^{-2}-3^{-2}}+\frac{3^{-3}-4^{-3}}{3^{-2}-4^{-2}}-\cdots +\frac{2019^{-3}-2020^{-3}}{2019^{-2}-2020^{-2}}\] or \[1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\cdots +\frac{1}{5781}?\]

2014 Singapore Senior Math Olympiad, 32
Tags: inequalities , trigonometry
Determine the maximum value of $\frac{8(x+y)(x^3+y^3)}{(x^2+y^2)^2}$ for all $(x,y)\neq (0,0)$

2007 India National Olympiad, 6
Tags: geometry , inequalities , trigonometry , inradius , circumcircle , incenter
If $ x$, $ y$, $ z$ are positive real numbers, prove that \[ \left(x \plus{} y \plus{} z\right)^2 \left(yz \plus{} zx \plus{} xy\right)^2 \leq 3\left(y^2 \plus{} yz \plus{} z^2\right)\left(z^2 \plus{} zx \plus{} x^2\right)\left(x^2 \plus{} xy \plus{} y^2\right) .\]

2017 Korea Junior Math Olympiad, 4
Tags: inequalities
4. Let $a \geq b \geq c \geq d>0$. Show that \[ \frac{b^3}{a} + \frac{c^3}{b} + \frac{d^3}{c} + \frac{a^3}{d} + 3 \left( ab+bc+cd+da \right) \geq 4 {\left( a^2 + b^2 + c^2 +d^2 \right)}. \] Other problems (in Korean) are also available at https://www.facebook.com/KoreanMathOlympiad

1995 AMC 12/AHSME, 23
Tags: inequalities , calculus , integration , trigonometry , triangle inequality , pythagorean theorem , geometry
The sides of a triangle have lengths $11$,$15$, and $k$, where $k$ is an integer. For how many values of $k$ is the triangle obtuse? $\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 13 \qquad \textbf{(E)}\ 14$

1990 Swedish Mathematical Competition, 3
Tags: inequalities , algebra , trigonometry
Find all $a, b$ such that $\sin x + \sin a\ge b \cos x$ for all $x$.

1998 China National Olympiad, 2
Tags: inequalities
Given a positive integer $n>1$, determine with proof if there exist $2n$ pairwise different positive integers $a_1,\ldots ,a_n,b_1,\ldots b_n$ such that $a_1+\ldots +a_n=b_1+\ldots +b_n$ and \[n-1>\sum_{i=1}^{n}\frac{a_i-b_i}{a_i+b_i}>n-1-\frac{1}{1998}.\]

2002 Czech-Polish-Slovak Match, 1
Tags: inequalities , ceiling function , floor function , algebra
Let $a, b$ be distinct real numbers and $k,m$ be positive integers $k + m = n \ge 3, k \le 2m, m \le 2k$. Consider sequences $x_1,\dots , x_n$ with the following properties: (i) $k$ terms $x_i$, including $x_1$, are equal to $a$; (ii) $m$ terms $x_i$, including $x_n$, are equal to $b$; (iii) no three consecutive terms are equal. Find all possible values of $x_nx_1x_2 + x_1x_2x_3 + \cdots + x_{n-1}x_nx_1$.

2020 German National Olympiad, 3
Tags: polynomial , root , algebra , inequalities
Show that the equation \[x(x+1)(x+2)\dots (x+2020)-1=0\] has exactly one positive solution $x_0$, and prove that this solution $x_0$ satisfies \[\frac{1}{2020!+10}<x_0<\frac{1}{2020!+6}.\]

1971 Bulgaria National Olympiad, Problem 6
Tags: geometry , 3d geometry , pyramid , inequalities , geometric inequalities
In a triangular pyramid $SABC$ one of the plane angles with vertex $S$ is a right angle and the orthogonal projection of $S$ on the base plane $ABC$ coincides with the orthocenter of the triangle $ABC$. Let $SA=m$, $SB=n$, $SC=p$, $r$ is the inradius of $ABC$. $H$ is the height of the pyramid and $r_1,r_2,r_3$ are radii of the incircles of the intersections of the pyramid with the plane passing through $SA,SB,SC$ and the height of the pyramid. Prove that (a) $m^2+n^2+p^2\ge18r^2$; (b) $\frac{r_1}H,\frac{r_2}H,\frac{r_3}H$ are in the range $(0.4,0.5)$.

2008 Mathcenter Contest, 3
Tags: geometry , trigonometry , trigonometric inequality , inequalities , geometric inequality
Let $ABC$ be a triangle whose side lengths are opposite the angle $A,B,C$ are $a,b,c$ respectively. Prove that $$\frac{ab\sin{2C}+bc\sin{ 2A}+ca\sin{2B}}{ab+bc+ca}\leq\frac{\sqrt{3}}{2}$$. [i](nooonuii)[/i]

1988 All Soviet Union Mathematical Olympiad, 480
Tags: algebra , inequalities , minimum
Find the minimum value of $\frac{xy}{z} + \frac{yz}{x} +\frac{ zx}{y}$ for positive reals $x, y, z$ with $x^2 + y^2 + z^2 = 1$.

1998 Switzerland Team Selection Test, 9
Tags: inequalities , algebra
If $x$ and $y$ are positive numbers, prove the inequality $\frac{x}{x^4 +y^2 }+\frac{y}{x^2 +y^4} \le \frac{1}{xy}$ .

2005 China National Olympiad, 2
Tags: geometry , inequalities , trigonometry , conic
A circle meets the three sides $BC,CA,AB$ of a triangle $ABC$ at points $D_1,D_2;E_1,E_2; F_1,F_2$ respectively. Furthermore, line segments $D_1E_1$ and $D_2F_2$ intersect at point $L$, line segments $E_1F_1$ and $E_2D_2$ intersect at point $M$, line segments $F_1D_1$ and $F_2E_2$ intersect at point $N$. Prove that the lines $AL,BM,CN$ are concurrent.

1993 All-Russian Olympiad Regional Round, 10.3
Tags: inequalities
Solve in positive numbers the system $ x_1\plus{}\frac{1}{x_2}\equal{}4, x_2\plus{}\frac{1}{x_3}\equal{}1, x_3\plus{}\frac{1}{x_4}\equal{}4, ..., x_{99}\plus{}\frac{1}{x_{100}}\equal{}4, x_{100}\plus{}\frac{1}{x_1}\equal{}1$

2013 IMAC Arhimede, 6
Tags: inequalities , algebra , sums and products , product , sum
Let $p$ be an odd positive integer. Find all values of the natural numbers $n\ge 2$ for which holds $$\sum_{i=1}^{n} \prod_{j\ne i} (x_i-x_j)^p\ge 0$$ where $x_1,x_2,..,x_n$ are any real numbers.

the 9th XMO, 1
Tags: algebra , inequalities
For any $n$ consecutive integers $a_1, \cdots, a_n$, prove that $$(a_1+\cdots+a_n)\cdot\left(\frac{1}{a_1}+\cdots+\frac{1}{a_n}\right)\leqslant \frac{n(n+1)\ln(\text{e}n)}{2}.$$

2006 Romania Team Selection Test, 2
Tags: inequalities , geometry
Let $ABC$ be a triangle with $\angle B = 30^{\circ }$. We consider the closed disks of radius $\frac{AC}3$, centered in $A$, $B$, $C$. Does there exist an equilateral triangle with one vertex in each of the 3 disks? [i]Radu Gologan, Dan Schwarz[/i]

1957 Kurschak Competition, 3
Tags: algebra , inequalities , permutation , combinatorics
What is the largest possible value of $|a_1 - 1| + |a_2-2|+...+ |a_n- n|$ where $a_1, a_2,..., a_n$ is a permutation of $1,2,..., n$?