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

The Golden Digits 2024, P1
Tags: inequalities , algebra
Let $n\geqslant 2$ be an integer. Prove that for any positive real numbers $a_1, a_2,\ldots, a_n$, \[\frac{1}{2\sqrt{2}}\sum_{i=1}^{n}2^{i}a_i^2 \geqslant\sum_{1 \leqslant i < j \leqslant n}a_i a_j.\][i]Proposed by Andrei Vila[/i]

2009 Kyiv Mathematical Festival, 6
Tags: maximum , set , inequalities
Let $\{a_1,...,a_n\}\subset \{-1,1\}$ and $a>0$ . Denote by $X$ and $Y$ the number of collections $\{\varepsilon_1,...,\varepsilon_n\}\subset \{-1,1\}$, such that $$max_{1\le k\le n}(\varepsilon_1a_1+...+\varepsilon_ka_k) >\alpha$$ and $$\varepsilon_1a_1+...+\varepsilon_na_n>a$$ respectively. Prove that $X\le 2Y$.

1904 Eotvos Mathematical Competition, 3
Tags: locus , rectangle , geometry , inequalities , geometric inequality
Let $A_1A_2$ and $B_1B_2$ be the diagonals of a rectangle, and let $O$ be its center. Find and construct the set of all points $P$ that satisfy simultaneously the four inequaliies: $$A_1P > OP , \\A_2P > OP, \ \ B_1P > OP , \ \ B_2P > OP.$$

1990 Romania Team Selection Test, 2
Tags: inequalities , geometry , circumcircle , function
Prove that in any triangle $ABC$ the following inequality holds: \[ \frac{a^{2}}{b+c-a}+\frac{b^{2}}{a+c-b}+\frac{c^{2}}{a+b-c}\geq 3\sqrt{3}R. \] [i]Laurentiu Panaitopol[/i]

2002 Bulgaria National Olympiad, 6
Tags: inequalities
Find the smallest number $k$, such that $ \frac{l_a+l_b}{a+b}<k$ for all triangles with sides $a$ and $b$ and bisectors $l_a$ and $l_b$ to them, respectively. [i]Proposed by Sava Grodzev, Svetlozar Doichev, Oleg Mushkarov and Nikolai Nikolov[/i]

2023 China Northern MO, 2
Tags: inequality , inequalities , algebra
Let $ a,b,c \in (0,1) $ and $ab+bc+ca=4abc .$ Prove that $$\sqrt{a+b+c}\geq \sqrt{1-a}+\sqrt{1-b}+\sqrt{1-c}$$

2012 German National Olympiad, 4
Tags: real number , square root , inequalities , algebra
Let $a,b$ be positive real numbers and $n\geq 2$ a positive integer. Prove that if $x^n \leq ax+b$ holds for a positive real number $x$, then it also satisfies the inequality $x < \sqrt[n-1]{2a} + \sqrt[n]{2b}.$

2023 Bulgaria National Olympiad, 3
Tags: abstract algebra , inequalities , algebra , polynomial
Let $f(x)$ be a polynomial with positive integer coefficients. For every $n\in\mathbb{N}$, let $a_{1}^{(n)}, a_{2}^{(n)}, \dots , a_{n}^{(n)}$ be fixed positive integers that give pairwise different residues modulo $n$ and let \[g(n) = \sum\limits_{i=1}^{n} f(a_{i}^{(n)}) = f(a_{1}^{(n)}) + f(a_{2}^{(n)}) + \dots + f(a_{n}^{(n)})\] Prove that there exists a constant $M$ such that for all integers $m>M$ we have $\gcd(m, g(m))>2023^{2023}$.

2023 South East Mathematical Olympiad, 6
Tags: inequalities , algebra
Let $a_1\geq a_2\geq \cdots \geq a_n >0 .$ Prove that$$ \left( \frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}\right)^2\geq \sum_{k=1}^{n} \frac{k(2k-1)}{a^2_1+a^2_2+\cdots+a^2_k}$$

2010 Poland - Second Round, 3
Tags: prime number , inequalities , number theory
Positive integer numbers $k$ and $n$ satisfy the inequality $k > n!$. Prove that there exist pairwisely different prime numbers $p_1, p_2, \ldots, p_n$ which are divisors of the numbers $k+1, k+2, \ldots, k+n$ respectively (i.e. $p_i|k+i$).

2022 Regional Competition For Advanced Students, 1
Tags: algebra , inequalities
Let $a$ and $b$ be positive real numbers with $a^2 + b^2 =\frac12$. Prove that $$\frac{1}{1 - a}+\frac{1}{1-b}\ge 4.$$ When does equality hold? [i](Walther Janous)[/i]

2014 Moldova Team Selection Test, 2
Tags: inequalities
Let $a,b,c$ be positive real numbers such that $abc=1$. Determine the minimum value of $E(a,b,c) = \sum \dfrac{a^3+5}{a^3(b+c)}$ .

1968 Swedish Mathematical Competition, 5
Tags: min , inequalities , trigonometry , algebra
Let $a, b$ be non-zero integers. Let $m(a, b)$ be the smallest value of $\cos ax + \cos bx$ (for real $x$). Show that for some $r$, $m(a, b) \le r < 0$ for all $a, b$.

2001 Irish Math Olympiad, 5
Tags: inequalities
Prove that for all real numbers $ a,b$ with $ ab>0$ we have: $ \sqrt[3]{\frac{a^2 b^2 (a\plus{}b)^2}{4}} \le \frac{a^2\plus{}10ab\plus{}b^2}{12}$ and find the cases of equality. Hence, or otherwise, prove that for all real numbers $ a,b$ $ \sqrt[3]{\frac{a^2 b^2 (a\plus{}b)^2}{4}} \le \frac{a^2\plus{}ab\plus{}b^2}{3}$ and find the cases of equality.

1991 China Team Selection Test, 1
Tags: function , inequalities , algebra , polynomial
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 \[f(x+1) \geq \frac{2 \cdot n^2}{\frac{1}{x-b_1} + \frac{1}{x-b_2} + \ldots + \frac{1}{x-b_n}}.\]

2014 China Team Selection Test, 5
Tags: complex number , inequalities
Let $n$ be a given integer which is greater than $1$ . Find the greatest constant $\lambda(n)$ such that for any non-zero complex $z_1,z_2,\cdots,z_n$ ,have that \[\sum_{k\equal{}1}^n |z_k|^2\geq \lambda(n)\min\limits_{1\le k\le n}\{|z_{k+1}-z_k|^2\},\] where $z_{n+1}=z_1$.

2004 District Olympiad, 3
Tags: algebra , inequalities
One considers the set $$A = \left\{ n \in N^* \big | 1 < \sqrt{1 + \sqrt{n}} < 2 \right\}$$ a) Find the set $A$. b) Find the set of numbers $n \in A$ such that $$\sqrt{n} \cdot \left| 1-\sqrt{1 + \sqrt{n}}\right| <1 ?$$

III Soros Olympiad 1996 - 97 (Russia), 11.4
Tags: trigonometry , algebra , inequalities
Find the smallest value of a function $$y = \cos 8x + 3\cos 4x +3\cos2x + 2\cos x.$$

1997 Belarusian National Olympiad, 3
Tags: inequalities
Let $\ a,x,y,z>0$. Prove that: $\frac{a+y}{a+z}x+\frac{a+z}{a+x}y+\frac{a+x}{a+y}z\geq{x+y+z}\geq\frac{a+z}{a+x}x+\frac{a+x}{a+y}y+\frac{a+y}{a+z}z$

2020 Bulgaria National Olympiad, P2
Tags: n-variable inequality , sequence , algebra , inequalities
Let $b_1$, $\dots$ , $b_n$ be nonnegative integers with sum $2$ and $a_0$, $a_1$, $\dots$ , $a_n$ be real numbers such that $a_0=a_n=0$ and $|a_i-a_{i-1}|\leq b_i$ for each $i=1$, $\dots$ , $n$. Prove that $$\sum_{i=1}^n(a_i+a_{i-1})b_i\leq 2$$ [hide]I believe that the original problem was for nonnegative real numbers and it was a typo on the version of the exam paper we had but I'm not sure the inequality would hold[/hide]

1999 Korea - Final Round, 3
Tags: inequalities
Let $a_1, a_2, ..., a_{1999}$ be nonnegative real numbers satisfying the following conditions: a. $a_1+a_2+...+a_{1999}=2$ b. $a_1a_2+a_2a_3+...+a_{1999}a_1=1$. Let $S=a_1^ 2+a_2 ^ 2+...+a_{1999}^2$. Find the maximum and minimum values of $S$.

1996 Israel National Olympiad, 6
Tags: inequalities , min , algebra
Let $x,y,z$ be real numbers with $|x|,|y|,|z| > 2$. What is the smallest possible value of $|xyz+2(x+y+z)|$ ?

VI Soros Olympiad 1999 - 2000 (Russia), 8.7
Tags: inequalities , algebra
Prove that for any positive real $x$ and $y$, holds the inequality $$\frac{1}{(x+y)^2}+\frac{1}{x^2}+\frac{1}{y^2} \ge \frac{9}{4xy}$$

1915 Eotvos Mathematical Competition, 1
Tags: algebra , number theory , inequalities
Let $A, B, C$ be any three real numbers. Prove that there exists a number $\nu$ such that $$An^2 + Bn+ < n!$$ for every natural number $n > \nu.$

1994 Polish MO Finals, 3
Tags: inequalities
The distinct reals $x_1, x_2, ... , x_n$ ,($n > 3$) satisfy $\sum_{i=1}^n x_i = 0$, $\sum_{i=1}^n x_i ^2 = 1$. Show that four of the numbers $a, b, c, d$ must satisfy: \[ a + b + c + nabc \leq \sum_{i=1}^n x_i ^3 \leq a + b + d + nabd \].