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

2013 ISI Entrance Examination, 3
Tags: inequalities , function , real analysis
Let $f:\mathbb R\to\mathbb R$ satisfy \[|f(x+y)-f(x-y)-y|\leq y^2\] For all $(x,y)\in\mathbb R^2.$ Show that $f(x)=\frac x2+c$ where $c$ is a constant.

1976 IMO Longlists, 2
Tags: inequalities , 3d geometry , tetrahedron , geometry
Let $P$ be a set of $n$ points and $S$ a set of $l$ segments. It is known that: $(i)$ No four points of $P$ are coplanar. $(ii)$ Any segment from $S$ has its endpoints at $P$. $(iii)$ There is a point, say $g$, in $P$ that is the endpoint of a maximal number of segments from $S$ and that is not a vertex of a tetrahedron having all its edges in $S$. Prove that $l \leq \frac{n^2}{3}$

2000 Harvard-MIT Mathematics Tournament, 1
Tags: inequalities , absolute value
How many integers $x$ satisfy $|x|+5<7$ and $|x-3|>2$?

1971 Canada National Olympiad, 3
Tags: inequalities , trigonometry , function
$ABCD$ is a quadrilateral with $AD=BC$. If $\angle ADC$ is greater than $\angle BCD$, prove that $AC>BD$.

Fractal Edition 1, P3
Tags: inequalities
Let \( a \), \( b \), and \( c \) be three positive real numbers that satisfy \( ab + bc + ca = 1 \). Show that: \[ \frac{a}{a^2+1} + \frac{b}{b^2+1} + \frac{c}{c^2+1} \le \frac{1}{4abc} \]

2010 IFYM, Sozopol, 2
Tags: inequalities
If $a,b,c>0$ and $abc=3$,find the biggest value of: $\frac{a^2b^2}{a^7+a^3b^3c+b^7}+\frac{b^2c^2}{b^7+b^3c^3a+c^7}+\frac{c^2a^2}{c^7+c^3a^3b+a^7}$

2011 IMAC Arhimede, 6
Tags: inequalities
Let $a$, $b$ and $c$ be positive real numbers such that $a+b+c=1$. Prove that $\frac{a}{a^3+b^2c+c^2b} + \frac{b}{b^3+c^2a+a^2c} + \frac{c}{c^3+a^2b+b^2a} \le 1+\frac{8}{27abc}$

2021 Iran Team Selection Test, 3
Tags: algebra , polynomial , inequalities , number theory , relatively prime
Prove there exist two relatively prime polynomials $P(x),Q(x)$ having integer coefficients and a real number $u>0$ such that if for positive integers $a,b,c,d$ we have: $$|\frac{a}{c}-1|^{2021} \le \frac{u}{|d||c|^{1010}}$$ $$| (\frac{a}{c})^{2020}-\frac{b}{d}| \le \frac{u}{|d||c|^{1010}}$$ Then we have : $$bP(\frac{a}{c})=dQ(\frac{a}{c})$$ (Two polynomials are relatively prime if they don't have a common root) Proposed by [i]Navid Safaii[/i] and [i]Alireza Haghi[/i]

2022 China Girls Math Olympiad, 1
Tags: sequence , inequalities
Consider all the real sequences $x_0,x_1,\cdots,x_{100}$ satisfying the following two requirements: (1)$x_0=0$; (2)For any integer $i,1\leq i\leq 100$,we have $1\leq x_i-x_{i-1}\leq 2$. Find the greatest positive integer $k\leq 100$,so that for any sequence $x_0,x_1,\cdots,x_{100}$ like this,we have \[x_k+x_{k+1}+\cdots+x_{100}\geq x_0+x_1+\cdots+x_{k-1}.\]

2010 BAMO, 3
Tags: algebra , inequalities
Suppose $a,b,c$ are real numbers such that $a+b \ge 0, b+c \ge 0$, and $c+a \ge 0$. Prove that $a+b+c \ge \frac{|a|+|b|+|c|}{3}$ . (Note: $|x|$ is called the absolute value of $x$ and is defined as follows. If $x \ge 0$ then $|x|= x$, and if $x < 0$ then $|x| = -x$. For example, $|6|= 6, |0| = 0$ and $|-6| = 6$.)

2002 China National Olympiad, 3
Tags: inequalities
Suppose that $c\in\left(\frac{1}{2},1\right)$. Find the least $M$ such that for every integer $n\ge 2$ and real numbers $0<a_1\le a_2\le\ldots \le a_n$, if $\frac{1}{n}\sum_{k=1}^{n}ka_{k}=c\sum_{k=1}^{n}a_{k}$, then we always have that $\sum_{k=1}^{n}a_{k}\le M\sum_{k=1}^{m}a_{k}$ where $m=[cn]$

1999 USAMO, 2
Tags: inequalities , geometry
Let $ABCD$ be a cyclic quadrilateral. Prove that \[ |AB - CD| + |AD - BC| \geq 2|AC - BD|. \]

2014 Putnam, 5
Tags: algebra , polynomial , inequalities
Let $P_n(x)=1+2x+3x^2+\cdots+nx^{n-1}.$ Prove that the polynomials $P_j(x)$ and $P_k(x)$ are relatively prime for all positive integers $j$ and $k$ with $j\ne k.$

2005 International Zhautykov Olympiad, 1
Tags: inequalities
For the positive real numbers $ a,b,c$ prove that \[ \frac c{a \plus{} 2b} \plus{} \frac d{b \plus{} 2c} \plus{} \frac a{c \plus{} 2d} \plus{} \frac b{d \plus{} 2a} \geq \frac 43.\]

2008 Harvard-MIT Mathematics Tournament, 6
Tags: quadratic , inequalities , function , parabola , analytic geometry , absolute value , conic
Determine all real numbers $ a$ such that the inequality $ |x^2 \plus{} 2ax \plus{} 3a|\le2$ has exactly one solution in $ x$.

2008 Romania Team Selection Test, 3
Tags: inequalities , vector , trigonometry , analytic geometry , geometry
Show that each convex pentagon has a vertex from which the distance to the opposite side of the pentagon is strictly less than the sum of the distances from the two adjacent vertices to the same side. [i]Note[/i]. If the pentagon is labeled $ ABCDE$, the adjacent vertices of $ A$ are $ B$ and $ E$, the ones of $ B$ are $ A$ and $ C$ etc.

2003 China Team Selection Test, 2
Tags: ratio , inequalities , analytic geometry , linear algebra , matrix , geometry
In triangle $ABC$, the medians and bisectors corresponding to sides $BC$, $CA$, $AB$ are $m_a$, $m_b$, $m_c$ and $w_a$, $w_b$, $w_c$ respectively. $P=w_a \cap m_b$, $Q=w_b \cap m_c$, $R=w_c \cap m_a$. Denote the areas of triangle $ABC$ and $PQR$ by $F_1$ and $F_2$ respectively. Find the least positive constant $m$ such that $\frac{F_1}{F_2}<m$ holds for any $\triangle{ABC}$.

2014 Contests, 3
Tags: limit , induction , inequalities , algebra , difference of squares
Let $a_0=5/2$ and $a_k=a_{k-1}^2-2$ for $k\ge 1.$ Compute \[\prod_{k=0}^{\infty}\left(1-\frac1{a_k}\right)\] in closed form.

2009 Romania National Olympiad, 3
Tags: inequalities , permutation , arrangement
Let be a natural number $ n, $ a permutation $ \sigma $ of order $ n, $ and $ n $ nonnegative real numbers $ a_1,a_2,\ldots , a_n. $ Prove the following inequality. $$ \left( a_1^2+a_{\sigma (1)} \right)\left( a_2^2+a_{\sigma (2)} \right)\cdots \left( a_n^2+a_{\sigma (n)} \right)\ge \left( a_1^2+a_1 \right)\left( a_2^2+a_{2} \right)\cdots \left( a_n^2+a_n \right) $$

2017 Saint Petersburg Mathematical Olympiad, 5
Tags: inequalities
Let $x,y,z>0 $ and $\sqrt{xyz}=xy+yz+zx$. Prove that$$x+y+z\leq \frac{1}{3}.$$

2009 JBMO Shortlist, 4
Tags: symmetry , inequalities
Let $ x$, $ y$, $ z$ be real numbers such that $ 0 < x,y,z < 1$ and $ xyz \equal{} (1 \minus{} x)(1 \minus{} y)(1 \minus{} z)$. Show that at least one of the numbers $ (1 \minus{} x)y,(1 \minus{} y)z,(1 \minus{} z)x$ is greater than or equal to $ \frac {1}{4}$

2016 Junior Balkan Team Selection Tests - Moldova, 2
Tags: inequalities
This is a really easy one for Junior level :p $a^2+b^2+c^2+ab+bc+ac=6$ a,b,c>0 Find max{a+b+c}

Kvant 2025, M2836
Tags: inequalities , geometry
The heights $BD$ and $CE$ of the acute-angled triangle $ABC$ intersect at point $H$, the heights of the triangle $ADE$ intersect at point $F$, point $M$ is the midpoint of side $BC$. Prove that $BH + CH \geqslant 2 FM$. [i]A. Kuznetsov[/i]

2009 Croatia Team Selection Test, 1
Tags: inequalities
Prove for all positive reals a,b,c,d: $ \frac{a\minus{}b}{b\plus{}c}\plus{}\frac{b\minus{}c}{c\plus{}d}\plus{}\frac{c\minus{}d}{d\plus{}a}\plus{}\frac{d\minus{}a}{a\plus{}b} \geq 0$

2020 Jozsef Wildt International Math Competition, W39
Tags: inequalities
Prove that: i) $$\sum_{k=1}^{n-1}(1+\ln k)\le n^2-n+1$$ ii) $$\sum_{k=1}^{n-1}\sqrt{\ln k}\le\frac{n^2-n+1}2$$ [i]Proposed by Laurențiu Modan[/i]