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

2000 Switzerland Team Selection Test, 6
Tags: sum , inequalities , algebra
Positive real numbers $x,y,z$ have the sum $1$. Prove that $\sqrt{7x+3}+ \sqrt{7y+3}+\sqrt{7z+3} \le 7$. Can number $7$ on the right hand side be replaced with a smaller constant?

1995 Abels Math Contest (Norwegian MO), 4
Tags: inequalities , algebra
Let $x_i,y_i$ be positive real numbers, $i = 1,2,...,n$. Prove that $$\left( \sum_{i=1}^n (x_i +y_i)^2\right)\left( \sum_{i=1}^n\frac{1}{x_iy_i}\right)\ge 4n^2$$

2008 Bundeswettbewerb Mathematik, 1
Tags: inequalities , algebra
Determine all real $ x$ satisfying the equation \[ \sqrt[5]{x^3 \plus{} 2x} \equal{} \sqrt[3]{x^5\minus{}2x}.\] Odd roots for negative radicands shall be included in the discussion.

2011 JBMO Shortlist, 2
Tags: inequalities , algebra
Let $x, y, z$ be positive real numbers. Prove that: $\frac{x + 2y}{z + 2x + 3y}+\frac{y + 2z}{x + 2y + 3z}+\frac{z + 2x}{y + 2z + 3x} \le \frac{3}{2}$

2020 Jozsef Wildt International Math Competition, W41
Tags: summation , inequalities
If $m,n\in\mathbb N_{\ge2}$, find the best constant $k\in\mathbb R$ for which $$\sum_{j=2}^m\sum_{i=2}^n\frac1{i^j}<k$$ [i]Proposed by Dorin Mărghidanu[/i]

1999 Kazakhstan National Olympiad, 5
Tags: inequalities , n-variable inequality , algebra
For real numbers $ x_1, x_2, \dots, x_n $ and $ y_1, y_2, \dots, y_n $ , the inequalities hold $ x_1 \geq x_2 \geq \ldots \geq x_n> 0 $ and $$ y_1 \geq x_1, ~ y_1y_2 \geq x_1x_2, ~ \dots, ~ y_1y_2 \dots y_n \geq x_1x_2 \dots x_n. $$ Prove that $ ny_1 + (n-1) y_2 + \dots + y_n \geq x_1 + 2x_2 + \dots + nx_n $.

2011 Saudi Arabia IMO TST, 1
Tags: inequalities , algebra
Let $a, b, c$ be real numbers such that $ab + bc + ca = 1$. Prove that $$\frac{(a + b)^2 + 1}{c^2+2}+\frac{(b + c)^2 + 1}{a^2+2}+ \frac{(c + a)^2 + 1}{b^2+2} \ge 3$$

1997 India Regional Mathematical Olympiad, 5
Tags: rearrangement inequality , inequalities , triangle inequality
Let $x,y,z$ be three distinct real positive numbers, Determine whether or not the three real numbers \[ \left| \frac{x}{y} - \frac{y}{x}\right| ,\left| \frac{y}{z} - \frac{z}{y}\right |, \left| \frac{z}{x} - \frac{x}{z}\right| \] can be the lengths of the sides of a triangle.

1959 AMC 12/AHSME, 25
Tags: absolute value , inequalities , inequality
The symbol $|a|$ means $+a$ if $a$ is greater than or equal to zero, and $-a$ if $a$ is less than or equal to zero; the symbol $<$ means "less than"; the symbol $>$ means "greater than." The set of values $x$ satisfying the inequality $|3-x|<4$ consists of all $x$ such that: $ \textbf{(A)}\ x^2<49 \qquad\textbf{(B)}\ x^2>1 \qquad\textbf{(C)}\ 1<x^2<49\qquad\textbf{(D)}\ -1<x<7\qquad\textbf{(E)}\ -7<x<1 $

2011 Costa Rica - Final Round, 4
Tags: inequalities , algebra
Let $p_1, p_2, ..., p_n$ be positive real numbers, such that $p_1 + p_2 +... + p_n = 1$. Let $x \in [0,1]$ and let $y_1, y_2, ..., y_n$ be such that $y^2_1 + y^2_2 +...+ y^2_n= x$. Prove that $$\left( \sum_{nx\le k \le n }y_k \sqrt{p_k} \right)^2 \le \sum_{k=1}^{n}\frac{k}{n} p_k$$

1979 Romania Team Selection Tests, 3.
Tags: simultaneous equation , algebra , system of equations , inequalities
Let $a,b,c\in \mathbb{R}$ with $a^2+b^2+c^2=1$ and $\lambda\in \mathbb{R}_{>0}\setminus\{1\}$. Then for each solution $(x,y,z)$ of the system of equations: \[ \begin{cases} x-\lambda y=a,\\ y-\lambda z=b,\\ z-\lambda x=c. \end{cases} \] we have $\displaystyle x^2+y^2+z^2\leqslant \frac1{(\lambda-1)^2}$. [i]Radu Gologan[/i]

2019 Balkan MO Shortlist, A4
Tags: inequalities , n-variable inequality , inequality , algebra
Let $a_{ij}, i = 1, 2, \dots, m$ and $j = 1, 2, \dots, n$ be positive real numbers. Prove that \[ \sum_{i = 1}^m \left( \sum_{j = 1}^n \frac{1}{a_{ij}} \right)^{-1} \le \left( \sum_{j = 1}^n \left( \sum_{i = 1}^m a_{ij} \right)^{-1} \right)^{-1} \]

2012 ELMO Shortlist, 6
Tags: calculus , inequalities , induction
Let $a,b,c\ge0$. Show that $(a^2+2bc)^{2012}+(b^2+2ca)^{2012}+(c^2+2ab)^{2012}\le (a^2+b^2+c^2)^{2012}+2(ab+bc+ca)^{2012}$. [i]Calvin Deng.[/i]

2002 China Western Mathematical Olympiad, 1
Tags: inequalities , number theory
Find all positive integers $ n$ such that $ n^4\minus{}4n^3\plus{}22n^2\minus{}36n\plus{}18$ is a perfect square.

2011 Iran Team Selection Test, 10
Tags: calculus , inequalities
Find the least value of $k$ such that for all $a,b,c,d \in \mathbb{R}$ the inequality \[ \begin{array} c \sqrt{(a^2+1)(b^2+1)(c^2+1)} +\sqrt{(b^2+1)(c^2+1)(d^2+1)} +\sqrt{(c^2+1)(d^2+1)(a^2+1)} +\sqrt{(d^2+1)(a^2+1)(b^2+1)} \\ \ \\ \ge 2( ab+bc+cd+da+ac+bd)-k \end{array}\] holds.

1998 Romania Team Selection Test, 3
Tags: inequalities , divisibility theory , induction
Find all positive integers $(x, n)$ such that $x^{n}+2^{n}+1$ divides $x^{n+1}+2^{n+1}+1$.

2024 Romania Team Selection Tests, P2
Tags: algebra , inequalities
Let $n\geqslant 2$ be a fixed integer. Consider $n$ real numbers $a_1,a_2,\ldots,a_n$ not all equal and let\[d:=\max_{1\leqslant i<j\leqslant n}|a_i-a_j|;\qquad s=\sum_{1\leqslant i<j\leqslant n}|a_i-a_j|.\]Determine in terms of $n{}$ the smalest and largest values the quotient $s/d$ may achieve. [i]Selected from the Kvant Magazine[/i]

2017 Moscow Mathematical Olympiad, 3
Tags: inequalities , algebra , polynomial
Let $x_0$ - is positive root of $x^{2017}-x-1=0$ and $y_0$ - is positive root of $y^{4034}-y=3x_0$ a) Compare $x_0$ and $y_0$ b) Find tenth digit after decimal mark in decimal representation of $|x_0-y_0|$

1986 French Mathematical Olympiad, Problem 2
Tags: geometry , geometric inequalities , inequalities
Points $A,B,C$, and $M$ are given in the plane. (a) Let $D$ be the point in the plane such that $DA\le CA$ and $DB\le CB$. Prove that there exists point $N$ satisfying $NA\le MA,NB\le MB$, and $ND\le MC$. (b) Let $A',B',C'$ be the points in the plane such that $A'B'\le AB,A'C'\le AC,B'C'\le BC$. Does there exist a point $M'$ which satisfies the inequalities $M'A'\le MA,M'B'\le MB,M'C'\le MC$?

VI Soros Olympiad 1999 - 2000 (Russia), 10.4
Tags: geometry , geometric inequalities , exradius , geometric inequality , inequalities
Prove that the inequality $ r^2+r_a^2+r_b^2+ r_c^2 \ge 2S$ holds for an arbitrary triangle, where $r$ is the radius of the circle inscribed in the triangle, $r_a$, $r_b$, $r_c$ are the radii of its three excribed circles, $S$ is the area of the triangle.

2020 Jozsef Wildt International Math Competition, W49
Tags: inequalities
Let $a,b,c>0$ so that $a+b+c=1$. Then prove that $$(a+2ab+2ac+bc)^a(b+2bc+2ba+ca)^b(c+2ca+2cb+ab)^c\le1.$$ [i]Proposed by Marius Drăgan[/i]

2020 Jozsef Wildt International Math Competition, W11
Tags: inequalities
If $a,b,c\in\mathbb N\setminus\{0,1,2,3\}$ then prove: $$b^2\cdot\sqrt[a]a+c^2\cdot\sqrt[b]b+a^2\cdot\sqrt[c]c\ge48\sqrt2$$ [i]Proposed by Daniel Sitaru[/i]

2006 IMO Shortlist, 6
Tags: algebra , inequalities , function
Determine the least real number $M$ such that the inequality \[|ab(a^{2}-b^{2})+bc(b^{2}-c^{2})+ca(c^{2}-a^{2})| \leq M(a^{2}+b^{2}+c^{2})^{2}\] holds for all real numbers $a$, $b$ and $c$.

2005 Romania National Olympiad, 3
Tags: calculus , limit , real analysis , function , integration , inequalities
Let $f:[0,\infty)\to(0,\infty)$ a continous function such that $\lim_{n\to\infty} \int^x_0 f(t)dt$ exists and it is finite. Prove that \[ \lim_{x\to\infty} \frac 1{\sqrt x} \int^x_0 \sqrt {f(t)} dt = 0. \] [i]Radu Miculescu[/i]

Durer Math Competition CD 1st Round - geometry, 2008.D1
Tags: geometry , geometric inequality , algebra , inequalities
Prove the following inequality if we know that $a$ and $b$ are the legs of a right triangle , and $c$ is the length of the hypotenuse of this triangle: $$3a + 4b \le 5c.$$ When does equality holds?