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

1984 Austrian-Polish Competition, 3
Tags: inequalities
Show that for $n>1$ and any positive real numbers $k,x_{1},x_{2},...,x_{n}$ then \[\frac{f(x_{1}-x_{2})}{x_{1}+x_{2}}+\frac{f(x_{2}-x_{3})}{x_{2}+x_{3}}+...+\frac{f(x_{n}-x_{1})}{x_{n}+x_{1}}\geq \frac{n^2}{2(x_{1}+x_{2}+...+x_{n})}\] Where $f(x)=k^x$. When does equality hold.

2010 District Olympiad, 3
Tags: function , real analysis , inequalities
Let $ f: \mathbb{R}\rightarrow \mathbb{R}$ a strictly increasing function such that $ f\circ f$ is continuos. Prove that $ f$ is continuos.

2009 Indonesia MO, 2
Tags: function , inequalities
Find the lowest possible values from the function \[ f(x) \equal{} x^{2008} \minus{} 2x^{2007} \plus{} 3x^{2006} \minus{} 4x^{2005} \plus{} 5x^{2004} \minus{} \cdots \minus{} 2006x^3 \plus{} 2007x^2 \minus{} 2008x \plus{} 2009\] for any real numbers $ x$.

2017 Morocco TST-, 1
Tags: inequalities
Let $a,b,c$ be non-negative real numbers such that $a^2+b^2+c^2 \le 3$ then prove that; $$(a+b+c)(a+b+c-abc)\ge2(a^2b+b^2c+c^2a)$$

2013 Princeton University Math Competition, 3
Tags: inequalities , i lost the game
A graph consists of a set of vertices, some of which are connected by (undirected) edges. A [i]star[/i] of a graph is a set of edges with a common endpoint. A [i]matching[/i] of a graph is a set of edges such that no two have a common endpoint. Show that if the number of edges of a graph $G$ is larger than $2(k-1)^2$, then $G$ contains a matching of size $k$ or a star of size $k$.

2021 New Zealand MO, 7
Tags: inequalities , max , algebra
Let $a, b, c, d$ be integers such that $a > b > c > d \ge -2021$ and $$\frac{a + b}{b + c}=\frac{c + d}{d + a}$$ (and $b + c \ne 0 \ne d + a$). What is the maximum possible value of $ac$?

2004 Romania National Olympiad, 2
Tags: inequalities
The sidelengths of a triangle are $a,b,c$. (a) Prove that there is a triangle which has the sidelengths $\sqrt a,\sqrt b,\sqrt c$. (b) Prove that $\displaystyle \sqrt{ab}+\sqrt{bc}+\sqrt{ca} \leq a+b+c < 2 \sqrt{ab} + 2 \sqrt{bc} + 2 \sqrt{ca}$.

2013 Middle European Mathematical Olympiad, 1
Tags: inequalities
Let $ a, b, c$ be positive real numbers such that \[ a+b+c=\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2} . \] Prove that \[ 2(a+b+c) \ge \sqrt[3]{7 a^2 b +1 } + \sqrt[3]{7 b^2 c +1 } + \sqrt[3]{7 c^2 a +1 } . \] Find all triples $ (a,b,c) $ for which equality holds.

2000 Abels Math Contest (Norwegian MO), 2b
Tags: inequalities , algebra , sum
Let $a,b,c$ and $d$ be non-negative real numbers such that $a+b+c+d = 4$. Show that $\sqrt{a+b+c}+\sqrt{b+c+d}+\sqrt{c+d+a}+\sqrt{d+a+b}\ge 6$.

1953 AMC 12/AHSME, 47
Tags: inequalities , logarithm , calculus
If $ x$ is greater than zero, then the correct relationship is: $ \textbf{(A)}\ \log (1\plus{}x) \equal{} \frac{x}{1\plus{}x} \qquad\textbf{(B)}\ \log (1\plus{}x) < \frac{x}{1\plus{}x} \\ \textbf{(C)}\ \log(1\plus{}x) > x \qquad\textbf{(D)}\ \log (1\plus{}x) < x \qquad\textbf{(E)}\ \text{none of these}$

2016 Latvia National Olympiad, 3
Tags: inequalities , algebra
Assume that real numbers $x$, $y$ and $z$ satisfy $x + y + z = 3$. Prove that $xy + xz + yz \leq 3$.

2021-IMOC, A7
Tags: algebra , inequality , inequalities
For any positive reals $a,b,c,d$ that satisfy $a^2 + b^2 + c^2 + d^2 = 4,$ show that $$\frac{a^3}{a+b} + \frac{b^3}{b+c} + \frac{c^3}{c+d} + \frac{d^3}{d+a} + 4abcd \leq 6.$$

2021 Estonia Team Selection Test, 2
Tags: algebra , inequalities
Positive real numbers $a, b, c$ satisfy $abc = 1$. Prove that $$\frac{a}{1+b}+\frac{b}{1+c}+\frac{c}{1+a} \ge \frac32$$

2014 India National Olympiad, 3
Tags: number theory , inequalities , greatest common divisor , least common multiple
Let $a,b$ be natural numbers with $ab>2$. Suppose that the sum of their greatest common divisor and least common multiple is divisble by $a+b$. Prove that the quotient is at most $\frac{a+b}{4}$. When is this quotient exactly equal to $\frac{a+b}{4}$

2015 Cuba MO, 9
Tags: inequalities , algebra
Determine the largest possible value of$ M$ for which it holds that: $$\frac{x}{1 +\dfrac{yz}{x}}+ \frac{y}{1 + \dfrac{zx}{y}}+ \frac{z}{1 + \dfrac{xy}{z}} \ge M,$$ for all real numbers $x, y, z > 0$ that satisfy the equation $xy + yz + zx = 1$.

2023 Irish Math Olympiad, P8
Tags: algebra , inequalities
Suppose that $a, b, c$ are positive real numbers and $a + b + c = 3$. Prove that $$\frac{a+b}{c+2} + \frac{b+c}{a+2} + \frac{c+a}{b+2} \geq 2$$ and determine when equality holds.

1989 Polish MO Finals, 3
Tags: inequalities
Show that for positive reals $a, b, c, d$ we have \[ \left(\dfrac{ab + ac + ad + bc + bd + cd}{6} \right)^3 \geq \left(\dfrac{abc + abd + acd + bcd}{4}\right)^2 \]

2019 Regional Competition For Advanced Students, 1
Tags: algebra , inequalities
Let $x,y$ be real numbers such that $(x+1)(y+2)=8.$ Prove that $$(xy-10)^2\ge 64.$$

2018 China Western Mathematical Olympiad, 7
Tags: least common multiple , number theory , inequalities
Let $p$ and $c$ be an prime and a composite, respectively. Prove that there exist two integers $m,n,$ such that $$0<m-n<\frac{\textup{lcm}(n+1,n+2,\cdots,m)}{\textup{lcm}(n,n+1,\cdots,m-1)}=p^c.$$

2010 Contests, 2b
Tags: algebra , inequalities
Show that $abc \le (ab + bc + ca)(a^2 + b^2 + c^2)^2$ for all positive real numbers $a, b$ and $c$ such that $a + b + c = 1$.

Maryland University HSMC part II, 2023.5
Tags: am-gm , inequalities
Let $0 \le a_1 \le a_2 \le \dots \le a_n \le 1$ be $n$ real numbers with $n \ge 2$. Assume $a_1 + a_2 + \dots + a_n \ge n-1$. Prove that \[ a_2a_3\dots a_n \ge \left( 1 - \frac 1n \right)^{n-1} \]

2013 BMT Spring, P1
Tags: algebra , inequalities , binomial coefficient , summation
Prove that for all positive integers $m$ and $n$, $$\frac1m\cdot\binom{2n}0-\frac1{m+1}\cdot\binom{2n}1+\frac1{m+2}\cdot\binom{2n}2-\ldots+\frac1{m+2n}\cdot\binom{2n}{n2}>0$$

1998 Tuymaada Olympiad, 3
Tags: geometry , area , inradius , geometric inequality , inequalities , area of a triangle
The segment of length $\ell$ with the ends on the border of a triangle divides the area of that triangle in half. Prove that $\ell >r\sqrt2$, where $r$ is the radius of the inscribed circle of the triangle.

2010 Iran Team Selection Test, 9
Tags: concavity , induction , inequalities
Sequence of real numbers $a_0,a_1,\dots,a_{1389}$ are called concave if for each $0<i<1389$, $a_i\geq\frac{a_{i-1}+a_{i+1}}2$. Find the largest $c$ such that for every concave sequence of non-negative real numbers: \[\sum_{i=0}^{1389}ia_i^2\geq c\sum_{i=0}^{1389}a_i^2\]

2007 Regional Competition For Advanced Students, 1
Tags: inequalities
Let $ 0<x_0,x_1, \dots , x_{669}<1$ be pairwise distinct real numbers. Show that there exists a pair $ (x_i,x_j)$ with $ 0<x_ix_j(x_j\minus{}x_i)<\frac{1}{2007}$