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

2008 Junior Balkan Team Selection Tests - Romania, 2
Tags: inequalities
Let $ a,b,c$ be positive reals with $ ab \plus{} bc \plus{} ca \equal{} 3$. Prove that: \[ \frac {1}{1 \plus{} a^2(b \plus{} c)} \plus{} \frac {1}{1 \plus{} b^2(a \plus{} c)} \plus{} \frac {1}{1 \plus{} c^2(b \plus{} a)}\le \frac {1}{abc}. \]

2024 ISI Entrance UGB, P2
Tags: algebra , polynomial , inequalities
Suppose $n\ge 2$. Consider the polynomial \[Q_n(x) = 1-x^n - (1-x)^n .\] Show that the equation $Q_n(x) = 0$ has only two real roots, namely $0$ and $1$.

1994 Tournament Of Towns, (419) 7
Tags: inequalities , geometry
Consider an arbitrary “figure” $F$ (non convex polygon). A chord of $F$ is defined to be a segment which lies entirely within $ F$ and whose ends are on its boundary. (a) Does there always exist a chord of $F$ that divides its area in half? (b) Prove that for any $F$ there exists a chord such that the area of each of the two parts of $F$ is not less than $ 1/3$ of the area of $F$. (c) Can the number $1/3$ in (b) be changed to a greater one? (V Proizvolov)

VMEO II 2005, 1
Tags: algebra , inequalities
Let $a, b, c$ be three positive real numbers. a) Prove that there exists a unique positive real number $d$ that satisfies $$\frac{1}{a + d}+ \frac{1}{b + d}+\frac{1}{c + d}=\frac{2}{d} .$$ b) With $x, y, z$ being positive real numbers such that $ax + by + cz = xyz$, prove the inequality $$x + y + z \ge \frac{2}{d}\sqrt{(a + d)(b + d)(c + d)}.$$

1989 National High School Mathematics League, 7
Tags: inequalities
If $\log_{a}\sqrt2<1$, then the range value of $a$ is________.

2019 Polish Junior MO Finals, 4.
Tags: geometrical inequalities , geometry , geometric inequality , inequalities
The point $D$ lies on the side $AB$ of the triangle $ABC$. Assume that there exists such a point $E$ on the side $CD$, that $$ \sphericalangle EAD = \sphericalangle AED \quad \text{and} \quad \sphericalangle ECB = \sphericalangle CEB. $$ Show that $AC + BC > AB + CE$.

2014 Contests, 3
Tags: greatest common divisor , least common multiple , inequalities , number theory
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 $\frac{a+b}{4}$

2017 Romania Team Selection Test, P3
Tags: inequalities
Given an interger $n\geq 2$, determine the maximum value the sum $\frac{a_1}{a_2}+\frac{a_2}{a_3}+...+\frac{a_{n-1}}{a_n}$ may achieve, and the points at which the maximum is achieved, as $a_1,a_2,...a_n$ run over all positive real numers subject to $a_k\geq a_1+a_2...+a_{k-1}$, for $k=2,...n$

1991 Putnam, B6
Tags: inequalities
Let $a$ and $b$ be positive numbers. Find the largest number $c$, in terms of $a$ and $b$, such that for all $x$ with $0<|x|\le c$ and for all $\alpha$ with $0<\alpha<1$, we have: $$a^\alpha b^{1-\alpha}\le\frac{a\sinh\alpha x}{\sinh x}+\frac{b\sinh x(1-\alpha)}{\sinh x}.$$

2009 Baltic Way, 9
Tags: modular arithmetic , inequalities , number theory
Determine all positive integers $n$ for which $2^{n+1}-n^2$ is a prime number.

2013 Miklós Schweitzer, 1
Tags: inequalities , combinatorics
Let $q$ be a positive integer. Prove there exists a constant $C_q$ such that the following inequality holds for any finite set $A$ of integers: \[|A+qA|\ge (q+1)|A|-C_q.\] [i]Proposed by Antal Balog.[/i]

1998 Akdeniz University MO, 3
Tags: inequalities , easy inequality , easy
Let $x,y,z$ be real numbers such that, $x \geq y \geq z >0$. Prove that $$\frac{x^2-y^2}{z}+\frac{z^2-y^2}{x}+\frac{x^2-z^2}{y} \geq 3x-4y+z$$

2009 District Olympiad, 3
Tags: inequalities
[b]a)[/b] For $ a,b\ge 0 $ and $ x,y>0, $ show that: $$ \frac{a^3}{x^2} +\frac{b^3}{y^2}\ge \frac{(a+b)^3}{(x+y)^2} . $$ [b]b)[/b] For $ a,b,c\ge 0 $ and $ x,y,z>0 $ under the condition $ a+b+c=x+y+z, $ prove that: $$ \frac{a^3}{x^2} +\frac{b^3}{y^2} +\frac{c^3}{z^2} \ge a+b+c. $$

2003 Germany Team Selection Test, 3
Tags: floor function , inequalities , least common multiple , ceiling function , number theory
Let $N$ be a natural number and $x_1, \ldots , x_n$ further natural numbers less than $N$ and such that the least common multiple of any two of these $n$ numbers is greater than $N$. Prove that the sum of the reciprocals of these $n$ numbers is always less than $2$: $\sum^n_{i=1} \frac{1}{x_i} < 2.$

2016 India National Olympiad, P2
Tags: algebra , inequalities
For positive real numbers $a,b,c$ which of the following statements necessarily implies $a=b=c$: (I) $a(b^3+c^3)=b(c^3+a^3)=c(a^3+b^3)$, (II) $a(a^3+b^3)=b(b^3+c^3)=c(c^3+a^3)$ ? Justify your answer.

1991 China Team Selection Test, 1
Tags: polynomial , inequalities , function , algebra
Let real coefficient polynomial $f(x) = x^n + a_1 \cdot x^{n-1} + \ldots + a_n$ has real roots $b_1, b_2, \ldots, b_n$, $n \geq 2,$ prove that $\forall x \geq max\{b_1, b_2, \ldots, b_n\}$, we have \[f(x+1) \geq \frac{2 \cdot n^2}{\frac{1}{x-b_1} + \frac{1}{x-b_2} + \ldots + \frac{1}{x-b_n}}.\]

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

2008 Germany Team Selection Test, 1
Tags: inequalities
Let $ a_1, a_2, \ldots, a_{100}$ be nonnegative real numbers such that $ a^2_1 \plus{} a^2_2 \plus{} \ldots \plus{} a^2_{100} \equal{} 1.$ Prove that \[ a^2_1 \cdot a_2 \plus{} a^2_2 \cdot a_3 \plus{} \ldots \plus{} a^2_{100} \cdot a_1 < \frac {12}{25}. \] [i]Author: Marcin Kuzma, Poland[/i]

1992 All Soviet Union Mathematical Olympiad, 558
Tags: algebra , inequalities
Show that $x^4 + y^4 + z^2\ge xyz \sqrt8$ for all positive reals $x, y, z$.

1982 IMO Longlists, 51
Tags: inequalities
Let n numbers $x_1, x_2, \ldots, x_n$ be chosen in such a way that $1 \geq x_1 \geq x_2 \geq \cdots \geq x_n \geq 0$. Prove that \[(1 + x_1 + x_2 + \cdots + x_n)^\alpha \leq 1 + x_1^\alpha+ 2^{\alpha-1}x_2^\alpha+ \cdots+ n^{\alpha-1}x_n^\alpha\] if $0 \leq \alpha \leq 1$.

2010 Contests, 2
Tags: inequalities , combinatorics
There are $100$ random, distinct real numbers corresponding to $100$ points on a circle. Prove that you can always choose $4$ consecutive points in such a way that the sum of the two numbers corresponding to the points on the outside is always greater than the sum of the two numbers corresponding to the two points on the inside.

2007 China Team Selection Test, 1
Tags: inequalities
Let $ a_{1},a_{2},\cdots,a_{n}$ be positive real numbers satisfying $ a_{1} \plus{} a_{2} \plus{} \cdots \plus{} a_{n} \equal{} 1$. Prove that \[\left(a_{1}a_{2} \plus{} a_{2}a_{3} \plus{} \cdots \plus{} a_{n}a_{1}\right)\left(\frac {a_{1}}{a_{2}^2 \plus{} a_{2}} \plus{} \frac {a_{2}}{a_{3}^2 \plus{} a_{3}} \plus{} \cdots \plus{} \frac {a_{n}}{a_{1}^2 \plus{} a_{1}}\right)\ge\frac {n}{n \plus{} 1}\]

2010 Contests, 4
Tags: function , inequalities
Let $a,b,c$ be positive real numbers such that $ab+bc+ca\le 3abc$. Prove that \[\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}+3\le \sqrt{2} (\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a})\]

2011 Purple Comet Problems, 21
Tags: inequalities , function
If a, b, and c are non-negative real numbers satisfying $a + b + c = 400$, fi nd the maximum possible value of $\sqrt{2a+b}+\sqrt{2b+c}+\sqrt{2c+a}$.

2015 BMT Spring, 5
Tags: inequalities , algebra
Let $x$ and $y$ be real numbers satisfying the equation $x^2-4x+y^2+3=0$. If the maximum and minimum values of $x^2+y^2$ are $M$ and $m$ respectively, compute the numerical value of $M-m$.