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

2009 Korea Junior Math Olympiad, 3
Tags: inequality , algebra , inequalities
For two arbitrary reals $x, y$ which are larger than $0$ and less than $1.$ Prove that$$\frac{x^2}{x+y}+\frac{y^2}{1-x}+\frac{(1-x-y)^2}{1-y}\geq\frac{1}{2}.$$

1990 IMO Longlists, 8
Tags: inequalities , area of a triangle , geometry
Let $a, b, c$ be the side lengths and $P$ be area of a triangle, respectively. Prove that \[(a^2+b^2+c^2-4\sqrt 3 P) (a^2+b^2+c^2) \geq 2 \left(a^2(b - c)^2 + b^2(c - a)^2 + c^2(a - b)^2\right).\]

2011 Iran MO (3rd Round), 4
Tags: cauchy inequality , inequalities , function
For positive real numbers $a,b$ and $c$ we have $a+b+c=3$. Prove $\frac{a}{1+(b+c)^2}+\frac{b}{1+(a+c)^2}+\frac{c}{1+(a+b)^2}\le \frac{3(a^2+b^2+c^2)}{a^2+b^2+c^2+12abc}$. [i]proposed by Mohammad Ahmadi[/i]

2014 Tajikistan Team Selection Test, 2
Tags: reflection , circumcircle , geometry , geometric transformation , inequalities
Let $M$be an interior point of triangle $ABC$. Let the line $AM$ intersect the circumcircle of the triangle $MBC$ for the second time at point $D$, the line $BM$ intersect the circumcircle of the triangle $MCA$ for the second time at point $E$, and the line $CM$ intersect the circumcircle of the triangle $MAB$ for the second time at point $F$. Prove that $\frac{AD}{MD} + \frac{BE}{ME} + \frac{CF}{MF} \geq \frac{9}{2}$. [i]Proposed by Nairy Sedrakyan[/i]

2022 Indonesia MO, 8
Tags: inequalities
Determine the smallest positive real $K$ such that the inequality \[ K + \frac{a + b + c}{3} \ge (K + 1) \sqrt{\frac{a^2 + b^2 + c^2}{3}} \]holds for any real numbers $0 \le a,b,c \le 1$. [i]Proposed by Fajar Yuliawan, Indonesia[/i]

2000 Pan African, 1
Tags: inequalities
Let $a$, $b$ and $c$ be real numbers such that $a^2+b^2=c^2$, solve the system: \[ z^2=x^2+y^2 \] \[ (z+c)^2=(x+a)^2+(y+b)^2 \] in real numbers $x, y$ and $z$.

2006 Turkey MO (2nd round), 1
Tags: inequalities
$x_{1},...,x_{n}$ are positive reals such that their sum and their squares' sum are equal to $t$. Prove that $\sum_{i\neq{j}}\frac{x_{i}}{x_{j}}\ge\frac{(n-1)^{2}\cdot{t}}{t-1}$

2005 Putnam, B2
Tags: inequalities
Find all positive integers $n,k_1,\dots,k_n$ such that $k_1+\cdots+k_n=5n-4$ and \[ \frac1{k_1}+\cdots+\frac1{k_n}=1. \]

1994 Flanders Math Olympiad, 1
Tags: function , inequalities
Let $a,b,c>0$ the sides of a right triangle. Find all real $x$ for which $a^x>b^x+c^x$, with $a$ is the longest side.

2014 ELMO Shortlist, 9
Tags: inequalities
Let $a$, $b$, $c$ be positive reals. Prove that \[ \sqrt{\frac{a^2(bc+a^2)}{b^2+c^2}}+\sqrt{\frac{b^2(ca+b^2)}{c^2+a^2}}+\sqrt{\frac{c^2(ab+c^2)}{a^2+b^2}}\ge a+b+c. \][i]Proposed by Robin Park[/i]

1997 USAMO, 5
Tags: symmetry , three variable inequality , inequalities
Prove that, for all positive real numbers $ a$, $ b$, $ c$, the inequality \[ \frac {1}{a^3 \plus{} b^3 \plus{} abc} \plus{} \frac {1}{b^3 \plus{} c^3 \plus{} abc} \plus{} \frac {1}{c^3 \plus{} a^3 \plus{} abc} \leq \frac {1}{abc} \] holds.

2023 Polish MO Finals, 3
Tags: combinatorics , algebra , inequalities
Given a positive integer $n \geq 2$ and real numbers $a_1, a_2, \ldots, a_n \in [0,1]$. Prove that there exist real numbers $b_1, b_2, \ldots, b_n \in \{0,1\}$, such that for all $1\leq k\leq l \leq n$ we have $$\left| \sum_{i=k}^l (a_i-b_i)\right| \leq \frac{n}{n+1}.$$

2008 Abels Math Contest (Norwegian MO) Final, 3
Tags: algebra , inequalities
a) Let $x$ and $y$ be positive numbers such that $x + y = 2$. Show that $\frac{1}{x}+\frac{1}{y} \le \frac{1}{x^2}+\frac{1}{y^2}$ b) Let $x,y$ and $z$ be positive numbers such that $x + y +z= 2$. Show that $\frac{1}{x}+\frac{1}{y} +\frac{1}{z} +\frac{9}{4} \le \frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}$ .

2004 France Team Selection Test, 1
Tags: derivative , calculus , function , inequalities
Let $n$ be a positive integer, and $a_1,...,a_n, b_1,..., b_n$ be $2n$ positive real numbers such that $a_1 + ... + a_n = b_1 + ... + b_n = 1$. Find the minimal value of $ \frac {a_1^2} {a_1 + b_1} + \frac {a_2^2} {a_2 + b_2} + ...+ \frac {a_n^2} {a_n + b_n}$.

2014 USAMTS Problems, 4:
Tags: ceiling function , pigeonhole principle , geometry , geometric transformation , rotation , inequalities
Nine distinct positive integers are arranged in a circle such that the product of any two non-adjacent numbers in the circle is a multiple of $n$ and the product of any two adjacent numbers in the circle is not a multiple of $n$, where $n$ is a fixed positive integer. Find the smallest possible value for $n$.

2011 Today's Calculation Of Integral, 766
Tags: calculus , trigonometry , calculus computations , integration , inequalities , function
Let $f(x)$ be a continuous function defined on $0\leq x\leq \pi$ and satisfies $f(0)=1$ and \[\left\{\int_0^{\pi} (\sin x+\cos x)f(x)dx\right\}^2=\pi \int_0^{\pi}\{f(x)\}^2dx.\] Evaluate $\int_0^{\pi} \{f(x)\}^3dx.$

2008 China Team Selection Test, 3
Tags: rearrangement inequality , induction , inequalities
Let $ 0 < x_{1}\leq\frac {x_{2}}{2}\leq\cdots\leq\frac {x_{n}}{n}, 0 < y_{n}\leq y_{n \minus{} 1}\leq\cdots\leq y_{1},$ Prove that $ (\sum_{k \equal{} 1}^{n}x_{k}y_{k})^2\leq(\sum_{k \equal{} 1}^{n}y_{k})(\sum_{k \equal{} 1}^{n}(x_{k}^2 \minus{} \frac {1}{4}x_{k}x_{k \minus{} 1})y_{k}).$ where $ x_{0} \equal{} 0.$

2021 Estonia Team Selection Test, 3
Tags: geometric inequality , disks , inequalities , geometry
In the plane, there are $n \geqslant 6$ pairwise disjoint disks $D_{1}, D_{2}, \ldots, D_{n}$ with radii $R_{1} \geqslant R_{2} \geqslant \ldots \geqslant R_{n}$. For every $i=1,2, \ldots, n$, a point $P_{i}$ is chosen in disk $D_{i}$. Let $O$ be an arbitrary point in the plane. Prove that \[O P_{1}+O P_{2}+\ldots+O P_{n} \geqslant R_{6}+R_{7}+\ldots+R_{n}.\] (A disk is assumed to contain its boundary.)

2019 PUMaC Individual Finals A, B, A3
Tags: geometry , area , inequalities
Let $ABCDEF$ be a convex hexagon with area $S$ such that $AB \parallel DE$, $BC \parallel EF$, $CD \parallel FA$ holds, and whose all angles are obtuse and opposite sides are not the same length. Prove that the following inequality holds: $$A_{ABC} + A_{BCD} + A_{CDE} + A_{DEF} + A_{EFA} + A_{FAB} < S$$ , where $A_{XYZ}$ is the area of triangle $XYZ$

2011 NZMOC Camp Selection Problems, 5
Tags: algebra , inequalities
Prove that for any three distinct positive real numbers $a, b$ and $c$: $$\frac{(a^2 - b^2)^3 + (b^2 - c^2)^3 + (c^2 - a^2)^3}{(a - b)^3 + (b - c)^3 + (c - a)^3}> 8abc.$$

1924 Eotvos Mathematical Competition, 1
Tags: number theory , geometric inequalities , triangle inequality , geometry , inequalities
Let $a, b, c$ be fìxed natural numbers. Suppose that, for every positive integer n, there is a triangle whose sides have lengths $a^n$, $b^n$, and $c^n$ respectively. Prove that these triangles are isosceles.

2019 239 Open Mathematical Olympiad, 7
Tags: algebra , inequalities
Given positive numbers $a_1, \ldots , a_n$, $b_1, \ldots , b_n$, $c_1, \ldots , c_n$. Let $m_k$ be the maximum of the products $a_ib_jc_l$ over the sets $(i, j, l)$ for which $max(i, j, l) = k$. Prove that $$(a_1 + \ldots + a_n) (b_1 +\ldots + b_n) (c_1 +\ldots + c_n) \leq n^2 (m_1 + \ldots + m_n).$$

2016 Turkey EGMO TST, 1
Tags: inequalities
Prove that \[ x^4y+y^4z+z^4x+xyz(x^3+y^3+z^3) \geq (x+y+z)(3xyz-1) \] for all positive real numbers $x, y, z$.

2005 Uzbekistan National Olympiad, 1
Tags: inequalities , inequality , algebra
Given a,b c are lenth of a triangle (If ABC is a triangle then AC=b, BC=a, AC=b) and $a+b+c=2$. Prove that $1+abc<ab+bc+ca\leq \frac{28}{27}+abc$

2005 Germany Team Selection Test, 3
Tags: algorithm , number theory , induction , inequalities , modular arithmetic
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 $p$.