This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 6530

2020 Thailand Mathematical Olympiad, 7
Tags: functional equation , inequalities
Determine all functions $f:\mathbb{R}\to\mathbb{Z}$ satisfying the inequality $(f(x))^2+(f(y))^2 \leq 2f(xy)$ for all reals $x,y$.

2011 China National Olympiad, 3
Tags: ceiling function , inequalities , integration , calculus , combinatorics
Let $A$ be a set consist of finite real numbers,$A_1,A_2,\cdots,A_n$ be nonempty sets of $A$, such that [b](a)[/b] The sum of the elements of $A$ is $0,$ [b](b)[/b] For all $x_i \in A_i(i=1,2,\cdots,n)$,we have $x_1+x_2+\cdots+x_n>0$. Prove that there exist $1\le k\le n,$ and $1\le i_1<i_2<\cdots<i_k\le n$, such that \[|A_{i_1}\bigcup A_{i_2} \bigcup \cdots \bigcup A_{i_k}|<\frac{k}{n}|A|.\] Where $|X|$ denote the numbers of the elements in set $X$.

2021 Mediterranean Mathematics Olympiad, 4
Tags: inequalities , algebra
Let $x_1,x_2,x_3,x_4,x_5$ ve non-negative real numbers, so that $x_1\le4$ and $x_1+x_2\le13$ and $x_1+x_2+x_3\le29$ and $x_1+x_2+x_3+x_4\le54$ and $x_1+x_2+x_3+x_4+x_5\le90$. Prove that $\sqrt{x_1}+\sqrt{x_2}+\sqrt{x_3}+\sqrt{x_4}+\sqrt{x_5}\le20$.

2001 USAMO, 3
Tags: inequalities , symmetry , trigonometry , quadratic
Let $a, b, c \geq 0$ and satisfy \[ a^2+b^2+c^2 +abc = 4 . \] Show that \[ 0 \le ab + bc + ca - abc \leq 2. \]

2014 Belarus Team Selection Test, 2
Tags: algebra , inequalities
Given positive real numbers $a,b,c$ with $ab+bc+ca\ge a+b+c$ , prove that $$(a + b + c)(ab + bc+ca) + 3abc \ge 4(ab + bc + ca).$$ (I. Gorodnin)

V Soros Olympiad 1998 - 99 (Russia), 11.1
Tags: algebra , inequalities , trigonometry
Find all $x$ for which the inequality holds $$9 \sin x +40 \cos x \ge 41.$$

2007 Iran Team Selection Test, 2
Tags: analytic geometry , trigonometry , inequalities , circumcircle , geometry
Triangle $ABC$ is isosceles ($AB=AC$). From $A$, we draw a line $\ell$ parallel to $BC$. $P,Q$ are on perpendicular bisectors of $AB,AC$ such that $PQ\perp BC$. $M,N$ are points on $\ell$ such that angles $\angle APM$ and $\angle AQN$ are $\frac\pi2$. Prove that \[\frac{1}{AM}+\frac1{AN}\leq\frac2{AB}\]

2000 Mongolian Mathematical Olympiad, Problem 2
Tags: inequalities , geometry , vector
Let $n\ge2$. For any two $n$-vectors $\vec x=(x_1,\ldots,x_n)$ and $\vec y=(y_1,\ldots,y_n)$, we define $$f\left(\vec x,\vec y\right)=x_1\overline{y_1}-\sum_{i=2}^nx_i\overline{y_i}.$$Prove that if $f\left(\vec x,\vec x\right)\ge0$, and $f\left(\vec y,\vec y\right)\ge0$, then $\left|f\left(\vec x,\vec y\right)\right|^2\ge f\left(\vec x,\vec x\right)f\left(\vec y,\vec y\right)$.

2020 Dutch IMO TST, 1
Tags: algebra , polynomial , inequalities
Given are real numbers $a_1, a_2,..., a_{2020}$, not necessarily different. For every $n \ge 2020$, define $a_{n + 1}$ as the smallest real zero of the polynomial $$P_n (x) = x^{2n} + a_1x^{2n - 2} + a_2x^{2n - 4} +... + a_{n -1}x^2 + a_n$$, if it exists. Assume that $a_{n + 1}$ exists for all $n \ge 2020$. Prove that $a_{n + 1} \le a_n$ for all $n \ge 2021$.

2020 Turkey MO (2nd round), 3
Tags: inequalities
If $x, y, z$ are positive real numbers find the minimum value of $$2\sqrt{(x+y+z) \left( \frac{1}{x}+ \frac{1}{y} + \frac{1}{z} \right)} - \sqrt{ \left( 1+ \frac{x}{y} \right) \left( 1+ \frac{y}{z} \right)}$$

1991 Putnam, A5
Tags: integration , calculus , inequalities , function
A5) Find the maximum value of $\int_{0}^{y}\sqrt{x^{4}+(y-y^{2})^{2}}dx$ for $0\leq y\leq 1$. I don't have a solution for this yet. I figure this may be useful: Let the integral be denoted $f(y)$, then according to the [url=http://mathworld.wolfram.com/LeibnizIntegralRule.html]Leibniz Integral Rule[/url] we have $\frac{df}{dy}=\int_{0}^{y}\frac{y(1-y)(1-2y)}{\sqrt{x^{4}+(y-y^{2})^{2}}}dx+\sqrt{y^{4}+(y-y^{2})^{2}}$ Now what?

2016 District Olympiad, 3
Tags: inequalities
Let be nonnegative real numbers $ a,b,c, $ holding the inequality: $ \sum_{\text{cyc}} \frac{a}{b+c+1} \le 1. $ Prove that $ \sum_{\text{cyc}} \frac{1}{b+c+1} \ge 1. $

2023 Korea Summer Program Practice Test, P7
Tags: inequalities
Determine the smallest value of $M$ for which for any choice of positive integer $n$ and positive real numbers $x_1<x_2<\ldots<x_n \le 2023$ the inequality $$\sum_{1\le i < j \le n , x_j-x_i \ge 1} 2^{i-j}\le M$$ holds.

1997 Brazil Team Selection Test, Problem 5
Tags: geometry , geometric inequality , inequalities , geometric inequalities , triangle , area
Let $ABC$ be an acute-angled triangle with incenter $I$. Consider the point $A_1$ on $AI$ different from $A$, such that the midpoint of $AA_1$ lies on the circumscribed circle of $ABC$. Points $B_1$ and $C_1$ are defined similarly. (a) Prove that $S_{A_1B_1C_1}=(4R+r)p$, where $p$ is the semi-perimeter, $R$ is the circumradius and $r$ is the inradius of $ABC$. (b) Prove that $S_{A_1B_1C_1}\ge9S_{ABC}$.

2014 Korea - Final Round, 1
Tags: inequalities
Suppose $x$, $y$, $z$ are positive numbers such that $x+y+z=1$. Prove that \[ \frac{(1+xy+yz+zx)(1+3x^3 + 3y^3 + 3z^3)}{9(x+y)(y+z)(z+x)} \ge \left( \frac{x \sqrt{1+x} }{\sqrt[4]{3+9x^2}} + \frac{y \sqrt{1+y} }{\sqrt[4]{3+9y^2}} + \frac{z \sqrt{1+z}}{\sqrt[4]{3+9z^2}} \right)^2. \]

2014 Harvard-MIT Mathematics Tournament, 8
Tags: symmetry , polynomial , inequalities , algebra
Find all real numbers $k$ such that $r^4+kr^3+r^2+4kr+16=0$ is true for exactly one real number $r$.

III Soros Olympiad 1996 - 97 (Russia), 9.9
Tags: radical , algebra , inequalities
What is the smallest value that the expression $$\sqrt{3x-2y-1}+\sqrt{2x+y+2}+\sqrt{3y-x}$$ can take?

2020 Jozsef Wildt International Math Competition, W36
Tags: trigonometry , inequalities
For all $x\in\left(0,\frac\pi4\right)$ prove $$\frac{(\sin^2x)^{\sin^2x}+(\tan^2x)^{\tan^2x}}{(\sin^2x)^{\tan^2x}+(\tan^2x)^{\sin^2x}}<\frac{\sin x}{4\sin x-3x}$$ [i]Proposed by Pirkulyiev Rovsen[/i]

1997 APMO, 1
Tags: inequalities
Given: \[ S = 1 + \frac{1}{1 + \frac{1}{3}} + \frac{1}{1 + \frac{1}{3} + \frac{1} {6}} + \cdots + \frac{1}{1 + \frac{1}{3} + \frac{1}{6} + \cdots + \frac{1} {1993006}} \] where the denominators contain partial sums of the sequence of reciprocals of triangular numbers (i.e. $k=\frac{n(n+1)}{2}$ for $n = 1$, $2$, $\ldots$,$1996$). Prove that $S>1001$.

2010 Costa Rica - Final Round, 4
Tags: number theory , inequalities
Find all integer solutions $(a,b)$ of the equation \[ (a+b+3)^2 + 2ab = 3ab(a+2)(b+2)\]

1975 Chisinau City MO, 103
Tags: inequalities , algebra
Prove the inequality: $$\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{1974}-\frac{1}{1975}<\frac{2}{5}$$

1997 India Regional Mathematical Olympiad, 4
Tags: geometry , inequalities
In a quadrilateral $ABCD$, it is given that $AB$ is parallel to $CD$ and the diagonals $AC$ and $BD$ are perpendicular to each other. Show that (a) $AD \cdot BC \geq AB \cdot CD$ (b) $AD + BC \geq AB + CD.$

2006 Balkan MO, 1
Tags: rearrangement inequality , inequalities theorems , inequalities
Let $ a$, $ b$, $ c$ be positive real numbers. Prove the inequality \[ \frac{1}{a\left(b+1\right)}+\frac{1}{b\left(c+1\right)}+\frac{1}{c\left(a+1\right)}\geq \frac{3}{1+abc}. \]

2019 Brazil Team Selection Test, 3
Tags: inequalities , algebra , symmetric inequality
Let $n \geq 2$ be an integer and $x_1, x_2, \ldots, x_n$ be positive real numbers such that $\sum_{i=1}^nx_i=1$. Show that $$\bigg(\sum_{i=1}^n\frac{1}{1-x_i}\bigg)\bigg(\sum_{1 \leq i < j \leq n}x_ix_j\bigg) \leq \frac{n}{2}.$$

2017 Macedonia JBMO TST, 3
Tags: inequalities
Let $x,y,z$ be positive reals such that $xyz=1$. Show that $$\frac{x^2+y^2+z}{x^2+2} + \frac{y^2+z^2+x}{y^2+2} + \frac{z^2+x^2+y}{z^2+2} \geq 3.$$ When does equality happen?