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.

AND:
OR:
NO:

Found problems: 6530

2014 Abels Math Contest (Norwegian MO) Final, 1a
Tags: algebra , inequalities
Assume that $x, y \ge 0$. Show that $x^2 + y^2 + 1 \le \sqrt{(x^3 + y + 1)(y^3 + x + 1)}$.

2019 China Second Round Olympiad, 1
Tags: inequalities
Suppose that $a_1,a_2,\cdots,a_{100}\in\mathbb{R}^+$ such that $a_i\geq a_{101-i}\,(i=1,2,\cdots,50).$ Let $x_k=\frac{ka_{k+1}}{a_1+a_2+\cdots+a_k}\,(k=1,2,\cdots,99).$ Prove that $$x_1x_2^2\cdots x_{99}^{99}\leq 1.$$

2012 Indonesia TST, 1
Tags: inequalities
Let $a,b,c \in \mathbb{C}$ such that $a|bc| + b|ca| + c|ab| = 0$. Prove that $|(a-b)(b-c)(c-a)| \ge 3\sqrt{3}|abc|$.

2004 Moldova Team Selection Test, 9
Tags: inequalities
Let $a,b$ and $c$ be positive real numbers . Prove that\[\left | \frac{4(b^3-c^3)}{b+c}+ \frac{4(c^3-a^3)}{c+a}+ \frac{4(a^3-b^3)}{a+b} \right |\leq (b-c)^2+(c-a)^2+(a-b)^2.\]

1996 Baltic Way, 15
Tags: inequalities
For which positive real numbers $a,b$ does the inequality \[x_1x_2+x_2x_3+\ldots x_{n-1}x_n+x_nx_1\ge x_1^ax_2^bx_3^a+ x_2^ax_3^bx_4^a+\ldots +x_n^ax_1^bx_2^a\] hold for all integers $n>2$ and positive real numbers $x_1,\ldots ,x_n$?

2013 239 Open Mathematical Olympiad, 8
Tags: algebra , inequalities
The product of the positive numbers $a, b, c, d$ and $e$ is equal to $1$. Prove that $$ \frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{d^2}+\frac{d^2}{e^2}+\frac{e^2}{a^2} \geq a+b+c+d+e .$$

JOM 2015 Shortlist, A9
Tags: inequalities
Let \(2n\) positive reals \(a_1, a_2, \cdots, a_n, b_1, b_2, \cdots, b_n\) satisfy \(a_{i+1}\ge 2a_i\) and \(b_{i+1} \le b_i\) for \(1\le i\le n-1\). Find the least constant \(C\) that satisfy: \[\displaystyle \sum^{n}_{i=1}{\frac{a_i}{b_i}} \ge \displaystyle \frac{C(a_1+a_2+\cdots+a_n)}{b_1+b_2+\cdots+b_n}\] and determine all equality case with that constant \(C\).

1989 APMO, 4
Tags: floor function , inequalities , graph theory , combinatorics
Let $S$ be a set consisting of $m$ pairs $(a,b)$ of positive integers with the property that $1 \leq a < b \leq n$. Show that there are at least \[ 4m \cdot \dfrac{(m - \dfrac{n^2}{4})}{3n} \] triples $(a,b,c)$ such that $(a,b)$, $(a,c)$, and $(b,c)$ belong to $S$.

2005 Junior Balkan Team Selection Tests - Moldova, 6
Tags: algebra , inequalities
Let $n$ be a nonzero natural number, and $x_1, x_2,..., x_n$ positive real numbers that $ \frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}= n$. Find the minimum value of the expression $x_1 +\frac{x_2^2}{2}++\frac{x_3^3}{3}+...++\frac{x_n^n}{n}$.

2011 Iran MO (3rd Round), 1
Tags: inequalities , polynomial , algebra
We define the recursive polynomial $T_n(x)$ as follows: $T_0(x)=1$ $T_1(x)=x$ $T_{n+1}(x)=2xT_n(x)+T_{n-1}(x)$ $\forall n \in \mathbb N$. [b]a)[/b] find $T_2(x),T_3(x),T_4(x)$ and $T_5(x)$. [b]b)[/b] find all the roots of the polynomial $T_n(x)$ $\forall n \in \mathbb N$. [i]Proposed by Morteza Saghafian[/i]

2015 Turkmenistan National Math Olympiad, 4
Tags: algebra , inequality , inequalities
Find the max and minimum without using dervivate: $\sqrt{x} +4 \cdot \sqrt{\frac{1}{2} - x}$

2013 Putnam, 5
Tags: geometry , inequalities , symmetry , vector , integration , geometric transformation
For $m\ge 3,$ a list of $\binom m3$ real numbers $a_{ijk}$ $(1\le i<j<k\le m)$ is said to be [i]area definite[/i] for $\mathbb{R}^n$ if the inequality \[\sum_{1\le i<j<k\le m}a_{ijk}\cdot\text{Area}(\triangle A_iA_jA_k)\ge0\] holds for every choice of $m$ points $A_1,\dots,A_m$ in $\mathbb{R}^n.$ For example, the list of four numbers $a_{123}=a_{124}=a_{134}=1, a_{234}=-1$ is area definite for $\mathbb{R}^2.$ Prove that if a list of $\binom m3$ numbers is area definite for $\mathbb{R}^2,$ then it is area definite for $\mathbb{R}^3.$

2011 Brazil Team Selection Test, 1
Tags: inequalities , quadratic , number theory
Let $a,b,c$ be positive integers. Prove that it is impossible to have all of the three numbers $a^2+b+c,b^2+c+a,c^2+a+b$ to be perfect squares.

1979 IMO Longlists, 49
Tags: inequalities , number theory
Let there be given two sequences of integers $f_i(1), f_i(2), \cdots (i = 1, 2)$ satisfying: $(i) f_i(nm) = f_i(n)f_i(m)$ if $\gcd(n,m) = 1$; $(ii)$ for every prime $P$ and all $k = 2, 3, 4, \cdots$, $f_i(P^k) = f_i(P)f_i(P^{k-1}) - P^2f(P^{k-2}).$ Moreover, for every prime $P$: $(iii) f_1(P) = 2P,$ $(iv) f_2(P) < 2P.$ Prove that $|f_2(n)| < f_1(n)$ for all $n$.

VMEO III 2006, 10.4
Tags: geometry , perimeter , trigonometry , inequalities , combinatorics
Given a convex polygon $ G$, show that there are three vertices of $ G$ which form a triangle so that it's perimeter is not less than 70% of the polygon's perimeter.

2017 Mediterranean Mathematics Olympiad, Problem 4
Tags: inequalities
Let $x,y,z$ and $a,b,c$ be positive real numbers with $a+b+c=1$. Prove that $$\left(x^2+y^2+z^2\right) \left( \frac{a^3}{x^2+2y^2} + \frac{b^3}{y^2+2z^2} + \frac{c^3}{z^2+2x^2} \right) \ge\frac19.$$

2022 Moldova Team Selection Test, 10
Tags: inequalities
Let $P(X)$ be a polynomial with positive coefficients. Show that for every integer $n \geq 2$ and every $n$ positive numbers $x_1, x_2,..., x_n$ the following inequality is true: $$P\left(\frac{x_1}{x_2} \right)^2+P\left(\frac{x_2}{x_3} \right)^2+ ... +P\left(\frac{x_n}{x_1} \right)^2 \geq n \cdot P(1)^2.$$ When does the equality take place?

2017 IMEO, 4
Tags: inequalities , algebra
Let $a,b,c$ be positive real numbers such that $abc=1$. Prove that $$\sqrt{\frac{a^3}{1+bc}}+\sqrt{\frac{b^3}{1+ac}}+\sqrt{\frac{c^3}{1+ab}}\geq 2$$ Are there any triples $(a,b,c)$, for which the equality holds? [i]Proposed by Konstantinos Metaxas.[/i]

1970 Czech and Slovak Olympiad III A, 6
Tags: inequalities , trigonometry , algebra , logarithm
Determine all real $x$ such that \[\sqrt{\tan(x)-1}\,\Bigl(\log_{\tan(x)}\bigl(2+4\cos^2(x)-2\bigr)\Bigr)\ge0.\]

1983 Vietnam National Olympiad, 2
Tags: inequalities
Decide whether $S_n$ or $T_n$ is larger, where \[S_n =\displaystyle\sum_{k=1}^n \frac{k}{(2n - 2k + 1)(2n - k + 1)}, T_n =\displaystyle\sum_{k=1}^n\frac{1}{k}\]

2020 China Northern MO, BP1
Tags: inequalities , algebra
For all positive real numbers $a,b,c$, prove that $$\frac{a^3+b^3}{ \sqrt{a^2-ab+b^2} } + \frac{b^3+c^3}{ \sqrt{b^2-bc+c^2} } + \frac{c^3+a^3}{ \sqrt{c^2-ca+a^2} } \geq 2(a^2+b^2+c^2)$$

2014 National Olympiad First Round, 7
Tags: inequalities
If $ (x^2+1)(y^2+1)+9=6(x+y)$ where $x,y$ are real numbers, what is $x^2+y^2$? $ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 3 $

2002 Greece National Olympiad, 1
Tags: quadratic , analytic geometry , inequalities
The real numbers $a,b,c$ with $bc\neq0$ satisfy $\frac{1-c^2}{bc}\geq0.$ Prove that $10(a^2+b^2+c^2-bc^3)\geq2ab+5ac.$

2018 Stars of Mathematics, 4
Tags: inequalities
Let be a natural number $ n\ge 4 $ and $ n $ nonnegative numbers $ a,b,\ldots ,c. $ Prove that $$ \prod_{\text{cyc} } (a+b+c)^2 \ge 2^n\prod_{\text{cyc} } (a+b)^2, $$ and tell in which circumstances equality happens.

2015 Thailand TSTST, 1
Tags: inequalities
Let $a, b, c$ be positive real numbers. Prove that $$\frac{a}{a+\sqrt{(a+b)(a+c)}}+\frac{b}{b+\sqrt{(b+c)(b+a)}}+\frac{c}{c+\sqrt{(c+a)(c+b)}}\leq\frac{2a^2+ab}{(b+\sqrt{ca}+c)^2}+\frac{2b^2+bc}{(c+\sqrt{ab}+a)^2}+\frac{2c^2+ca}{(a+\sqrt{bc}+b)^2}.$$