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

2005 Putnam, B1
Tags: floor function , algebra , function , polynomial , inequalities
Find a nonzero polynomial $P(x,y)$ such that $P(\lfloor a\rfloor,\lfloor 2a\rfloor)=0$ for all real numbers $a.$ (Note: $\lfloor v\rfloor$ is the greatest integer less than or equal to $v.$)

2015 Azerbaijan JBMO TST, 1
Tags: inequalities
Let $x,y$ and $z$ be non-negative real numbers satisfying the equation $x+y+z=xyz$. Prove that $2(x^2+y^2+z^2)\geq3(x+y+z)$.

2011 Romania Team Selection Test, 2
Tags: inequalities
Let $n$ be an integer number greater than $2$, let $x_{1},x_{2},\ldots ,x_{n}$ be $n$ positive real numbers such that \[\sum_{i=1}^{n}\frac{1}{x_{i}+1}=1\] and let $k$ be a real number greater than $1$. Show that: \[\sum_{i=1}^{n}\frac{1}{x_{i}^{k}+1}\ge\frac{n}{(n-1)^{k}+1}\] and determine the cases of equality.

PEN G Problems, 2
Tags: inequalities , irrational number
Prove that for any positive integers $ a$ and $ b$ \[ \left\vert a\sqrt{2}\minus{}b\right\vert >\frac{1}{2(a\plus{}b)}.\]

2008 Hanoi Open Mathematics Competitions, 5
Tags: algebra , inequalities
Suppose $x, y, z, t$ are real numbers such that $\begin{cases} |x + y + z -t |\le 1 \\ |y + z + t - x|\le 1 \\ |z + t + x - y|\le 1 \\ |t + x + y - z|\le 1 \end{cases}$ Prove that $x^2 + y^2 + z^2 + t^2 \le 1$.

2011 Federal Competition For Advanced Students, Part 1, 2
Tags: inequalities
For a positive integer $k$ and real numbers $x$ and $y$, let \[f_k(x,y)=(x+y)-\left(x^{2k+1}+y^{2k+1}\right)\mbox{.}\] If $x^2+y^2=1$, then determine the maximal possible value $c_k$ of $f_k(x,y)$.

2000 Moldova National Olympiad, Problem 6
Tags: inequalities , inequality
Let $(a_n)_{n\ge0}$ be a sequence of positive numbers that satisfy the relations $a_{i-1}a_{i+1}\le a_i^2$ for all $i\in\mathbb N$. For any integer $n>1$, prove the inequality $$\frac{a_0+\ldots+a_n}{n+1}\cdot\frac{a_1+\ldots+a_{n-1}}{n-1}\ge\frac{a_0+\ldots+a_{n-1}}n\cdot\frac{a_1+\ldots+a_n}n.$$

1994 Tournament Of Towns, (438) 4
Tags: algebra , inequalities
Prove that for all positive $a_1. a_2, ..., a_n$ the inequality $$\left( 1+\frac{a_1^2}{a_2}\right) \left( 1+\frac{a_2^2}{a_3}\right) ...\left( 1+\frac{a_n^2}{a_1}\right) \ge (1+a_1)(1+a_2)...(1+a_n)$$ holds. (LD Kurliandchik)

1989 USAMO, 3
Tags: algebra , inequalities , polynomial
Let $P(z)= z^n + c_1 z^{n-1} + c_2 z^{n-2} + \cdots + c_n$ be a polynomial in the complex variable $z$, with real coefficients $c_k$. Suppose that $|P(i)| < 1$. Prove that there exist real numbers $a$ and $b$ such that $P(a + bi) = 0$ and $(a^2 + b^2 + 1)^2 < 4 b^2 + 1$.

2013 China National Olympiad, 3
Tags: inequalities , ceiling function , geometry
Find all positive real numbers $t$ with the following property: there exists an infinite set $X$ of real numbers such that the inequality \[ \max\{|x-(a-d)|,|y-a|,|z-(a+d)|\}>td\] holds for all (not necessarily distinct) $x,y,z\in X$, all real numbers $a$ and all positive real numbers $d$.

1989 Tournament Of Towns, (230) 4
Tags: inequalities , combinatorics , number theory
Given the natural number N, consider triples of different positive integers $(a, b, c)$ such that $a + b + c = N$. Take the largest possible system of these triples such that no two triples of the system have any common elements. Denote the number of triples of this system by $K(N)$. Prove that: (a) $K(N) >\frac{N}{6}-1$ (b) $K(N) <\frac{2N}{9}$ (L.D. Kurliandchik, Leningrad)

2003 China Team Selection Test, 3
Tags: inequalities , geometry , trigonometry , ratio
(1) $D$ is an arbitary point in $\triangle{ABC}$. Prove that: \[ \frac{BC}{\min{AD,BD,CD}} \geq \{ \begin{array}{c} \displaystyle 2\sin{A}, \ \angle{A}< 90^o \\ \\ 2, \ \angle{A} \geq 90^o \end{array} \] (2)$E$ is an arbitary point in convex quadrilateral $ABCD$. Denote $k$ the ratio of the largest and least distances of any two points among $A$, $B$, $C$, $D$, $E$. Prove that $k \geq 2\sin{70^o}$. Can equality be achieved?

2009 Tuymaada Olympiad, 3
Tags: cyclic quadrilateral , geometry , inequalities
On the side $ AB$ of a cyclic quadrilateral $ ABCD$ there is a point $ X$ such that diagonal $ BD$ bisects $ CX$ and diagonal $ AC$ bisects $ DX$. What is the minimum possible value of $ AB\over CD$? [i]Proposed by S. Berlov[/i]

2013 APMO, 2
Tags: floor function , algebra , function , modular arithmetic , quadratic , inequalities
Determine all positive integers $n$ for which $\dfrac{n^2+1}{[\sqrt{n}]^2+2}$ is an integer. Here $[r]$ denotes the greatest integer less than or equal to $r$.

OMMC POTM, 2022 9
Tags: least common multiple , inequalities , number theory
For positive integers $a_1 < a_2 < \dots < a_n$ prove that $$\frac{1}{\operatorname{lcm}(a_1, a_2)}+\frac{1}{\operatorname{lcm}(a_2, a_3)}+\dots+\frac{1}{\operatorname{lcm}(a_{n-1}, a_n)} \leq 1-\frac{1}{2^{n-1}}.$$ [i]Proposed by Evan Chang (squareman), USA[/i]

2017 Macedonia National Olympiad, Problem 3
Tags: inequalities
Let $x,y,z \in \mathbb{R}$ such that $xyz = 1$. Prove that: $$\left(x^4 + \frac{z^2}{y^2}\right)\left(y^4 + \frac{x^2}{z^2}\right)\left(z^4 + \frac{y^2}{x^2}\right) \ge \left(\frac{x^2}{y} + 1 \right)\left(\frac{y^2}{z} + 1 \right)\left(\frac{z^2}{x} + 1 \right).$$

2002 Baltic Way, 4
Tags: inequalities
Let $n$ be a positive integer. Prove that \[\sum_{i=1}^nx_i(1-x_i)^2\le\left(1-\frac{1}{n}\right)^2 \] for all nonnegative real numbers $x_1,x_2,\ldots ,x_n$ such that $x_1+x_2+\ldots x_n=1$.

2023 Peru MO (ONEM), 2
Tags: geometry , triangle inequality , algebra , inequalities
For each positive real number $x$, let $f(x)=\frac{x}{1+x}$ . Prove that if $a$, $b,$ $c$ are the sidelengths of a triangle, then $f(a)$, $f(b),$ $f(c)$ are sidelengths of a triangle.

2022 Korea National Olympiad, 7
Tags: algebra , sequence , inequalities
Suppose that the sequence $\{a_n\}$ of positive reals satisfies the following conditions: [list] [*]$a_i \leq a_j$ for every positive integers $i <j$. [*]For any positive integer $k \geq 3$, the following inequality holds: $$(a_1+a_2)(a_2+a_3)\cdots(a_{k-1}+a_k)(a_k+a_1)\leq (2^k+2022)a_1a_2\cdots a_k$$ [/list] Prove that $\{a_n\}$ is constant.

1979 IMO Longlists, 65
Tags: function , inequalities , algebra
Given a function $f$ such that $f(x)\le x\forall x\in\mathbb{R}$ and $f(x+y)\le f(x)+f(y)\forall \{x,y\}\in\mathbb{R}$, prove that $f(x)=x\forall x\in\mathbb{R}$.

2004 Poland - First Round, 4
Tags: inequalities
4.Given is $n \in \mathbb Z$ and positive reals a,b. Find possible maximal value of the sum: $x_1y_1 + x_2y_2 + ... + x_ny_n$ when $x_1,x_2,...,x_n$ and $y_1,y_2,...,y_n$ are in $<0;1>$ and satisfies: $x_1 + x_2 + ... + x_n \leq a$ and $y_1 + y_2 + ... + y_n \leq b$

1997 Taiwan National Olympiad, 3
Tags: induction , inequalities
Let $n>2$ be an integer. Suppose that $a_{1},a_{2},...,a_{n}$ are real numbers such that $k_{i}=\frac{a_{i-1}+a_{i+1}}{a_{i}}$ is a positive integer for all $i$(Here $a_{0}=a_{n},a_{n+1}=a_{1}$). Prove that $2n\leq a_{1}+a_{2}+...+a_{n}\leq 3n$.

2006 Junior Balkan Team Selection Tests - Romania, 1
Tags: algebra , inequalities
Prove that $\frac{a^3}{bc} + \frac{b^3}{ca} + \frac{c^3}{ba} \ge a + b + c$, for all positive real numbers $a, b$, and $c$.

1992 China National Olympiad, 1
Tags: inequalities , algebra , triangle inequality
Let equation $x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\dots +a_1x+a_0=0$ with real coefficients satisfy $0<a_0\le a_1\le a_2\le \dots \le a_{n-1}\le 1$. Suppose that $\lambda$ ($|\lambda|>1$) is a complex root of the equation, prove that $\lambda^{n+1}=1$.

2018 ELMO Shortlist, 3
Tags: inequalities
Let $a, b, c,x, y, z$ be positive reals such that $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1$. Prove that \[a^x+b^y+c^z\ge \frac{4abcxyz}{(x+y+z-3)^2}.\] [i]Proposed by Daniel Liu[/i]