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

1997 USAMO, 6
Tags: algorithm , inequalities , floor function , induction , algebra
Suppose the sequence of nonnegative integers $a_1, a_2, \ldots, a_{1997}$ satisfies \[ a_i + a_j \leq a_{i+j} \leq a_i + a_j + 1 \] for all $i,j \geq 1$ with $i + j \leq 1997$. Show that there exists a real number $x$ such that $a_n = \lfloor nx \rfloor$ (the greatest integer $\leq nx$) for all $1 \leq n \leq 1997$.

1976 Vietnam National Olympiad, 6
Tags: inequality , inequalities , algebra
Show that $\frac{1}{x_1^n} + \frac{1}{x_2^n} +...+ \frac{1}{x_k^n} \ge k^{n+1}$ for positive real numbers $x_i $ with sum $1$.

2014 Taiwan TST Round 1, 1
Tags: absolute value , polynomial , inequalities , algebra
Let $f(x) = x^n + a_{n-2} x^{n-2} + a_{n-3}x^{n-3} + \dots + a_1x + a_0$ be a polynomial with real coefficients $(n \ge 2)$. Suppose all roots of $f$ are real. Prove that the absolute value of each root is at most $\sqrt{\frac{2(1-n)}n a_{n-2}}$.

1961 AMC 12/AHSME, 20
Tags: asymptote , inequalities
The set of points satisfying the pair of inequalities $y>2x$ and $y>4-x$ is contained entirely in quadrants: ${{ \textbf{(A)}\ \text{I and II} \qquad\textbf{(B)}\ \text{II and III} \qquad\textbf{(C)}\ \text{I and III} \qquad\textbf{(D)}\ \text{III and IV} }\qquad\textbf{(E)}\ \text{I and IV} } $

2010 Laurențiu Panaitopol, Tulcea, 4
Tags: trigonometry , trigonometric inequality , inequalities
Let be a natural number $ n, $ and $ n $ real numbers $ a_1,a_2,\ldots ,a_n . $ Then, $$ \sum_{1\le i<j\le n} \cos\left( a_i-a_j \right)\ge -n/2. $$

1992 Vietnam National Olympiad, 3
Tags: limit , calculus , calculus computations , inequalities
Let $a,b,c$ be positive reals and sequences $\{a_{n}\},\{b_{n}\},\{c_{n}\}$ defined by $a_{k+1}=a_{k}+\frac{2}{b_{k}+c_{k}},b_{k+1}=b_{k}+\frac{2}{c_{k}+a_{k}},c_{k+1}=c_{k}+\frac{2}{a_{k}+b_{k}}$ for all $k=0,1,2,...$. Prove that $\lim_{k\to+\infty}a_{k}=\lim_{k\to+\infty}b_{k}=\lim_{k\to+\infty}c_{k}=+\infty$.

1986 Vietnam National Olympiad, 2
Tags: pyramid , circumcircle , trigonometry , 3d geometry , geometry , inradius , inequalities
Let $ R$, $ r$ be respectively the circumradius and inradius of a regular $ 1986$-gonal pyramid. Prove that \[ \frac{R}{r}\ge 1\plus{}\frac{1}{\cos\frac{\pi}{1986}}\] and find the total area of the surface of the pyramid when the equality occurs.

2014 AMC 12/AHSME, 7
Tags: inequalities
For how many positive integers $n$ is $\frac{n}{30-n}$ also a positive integer? ${ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}}\ 7\qquad\textbf{(E)}\ 8 $

2010 Peru IMO TST, 4
Tags: inequalities
Let $ \displaystyle{a,b,c}$ be positive real numbers such that $\displaystyle{a+b+c=1.}$ Prove that $$ \displaystyle{\frac{1+ab}{a+b}+\frac{1+bc}{b+c}+\frac{1+ca}{c+a}\geq 5.}$$

2023-IMOC, A3
Tags: inequalities , algebra
Given positive reals $x,y,z$ satisfying $x+y+z=3$, prove that \[\sum_{cyc}\left( x^2+y^2+x^2y^2+\frac{y^2}{x^2}\right)\geq 4\sum_{cyc}\frac{y}{x}.\] [i]Proposed by chengbilly.[/i]

2008 Mathcenter Contest, 8
Tags: inequalities , algebra
Let $a,b,c,d \in R^+$ with $abcd=1$. Prove that $$\left(\frac{1+ab}{1+a}\right)^{2008}+\left(\frac{1+bc}{1+b}\right)^{2008}+\left(\frac{1+cd }{1+c}\right)^{2008}+\left(\frac{1+da}{1+d}\right)^{2008} \geq 4$$ [i](dektep)[/i]

2019 Jozsef Wildt International Math Competition, W. 14
Tags: function , trigonometry , inequalities
If $a$, $b$, $c > 0$; $ab + bc + ca = 3$ then: $$4\left(\tan^{-1} 2\right)\left(\tan^{-1}\left(\sqrt[3]{abc}\right)\right) \leq \pi \tan^{-1}\left(1 + \sqrt[3]{abc}\right)$$

2006 Victor Vâlcovici, 2
Tags: inequalities , geometry
Let $ ABC $ be a triangle with $ AB=AC $ and chose such that $ \angle BAC <120^{\circ } . $ On the altitude of $ ABC $ from $ A, $ consider the point $ O $ so that $ \angle BOC =120^{\circ } , $ and an arbitrary point $ M\neq O $ in the interior of $ ABC. $ Show that $ MA+MB+MC>OA+OB+OC. $ [i]Gheorghe Bucur[/i]

2008 Tournament Of Towns, 5
Tags: inequalities
Let $a_1,a_2,\cdots,a_n$ be a sequence of positive numbers, so that $a_1 + a_2 +\cdots + a_n \leq \frac 12$. Prove that \[(1 + a_1)(1 + a_2) \cdots (1 + a_n) < 2.\] [hide="Remark"]Remark. I think this problem was posted before, but I can't find the link now.[/hide]

2013 Saudi Arabia IMO TST, 2
Tags: algebra , combinatorics , inequalities
Given an integer $n \ge 2$, determine the number of ordered $n$-tuples of integers $(a_1, a_2,...,a_n)$ such that (a) $a_1 + a_2 + .. + a_n \ge n^2$ and (b) $a_1^2 + a_2^2 + ... + a_n^2 \le n^3 + 1$

2013 Bulgaria National Olympiad, 4
Tags: trigonometry , inequalities
Suppose $\alpha,\beta,\gamma \in [0.\pi/2)$ and $\tan \alpha + \tan\beta + \tan \gamma \leq 3$. Prove that: \[\cos 2\alpha + \cos 2\beta + \cos 2\gamma \ge 0\] [i]Proposed by Nikolay Nikolov[/i]

2016 Macedonia National Olympiad, Problem 5
Tags: inequalities
Let $n\ge3$ and $a_1,a_2,...,a_n \in \mathbb{R^{+}}$, such that $\frac{1}{1+a_1^4} + \frac{1}{1+a_2^4} + ... + \frac{1}{1+a_n^4} = 1$. Prove that: $$a_1a_2...a_n \ge (n-1)^{\frac n4}$$

2012 Indonesia TST, 3
Tags: inequalities
Let $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$ be positive reals such that \[a_1 + b_1 = a_2 + b_2 = \ldots + a_n + b_n\] and \[\sqrt[n]{\dfrac{a_1a_2\ldots a_n}{b_1b_2\ldots b_n}} \ge n.\] Prove that \[\sqrt[n]{\dfrac{a_1a_2\ldots a_n}{b_1b_2\ldots b_n}} \ge \dfrac{a_1+a_2+\ldots+a_n}{b_1+b_2+\ldots+b_n}.\]

2013 ELMO Shortlist, 5
Tags: derivative , function , inequalities , logarithm , 3-variable inequality , calculus
Let $a,b,c$ be positive reals satisfying $a+b+c = \sqrt[7]{a} + \sqrt[7]{b} + \sqrt[7]{c}$. Prove that $a^a b^b c^c \ge 1$. [i]Proposed by Evan Chen[/i]

2009 District Round (Round II), 2
Tags: inequalities , trigonometry , geometry
in a right-angled triangle $ABC$ with $\angle C=90$,$a,b,c$ are the corresponding sides.Circles $K.L$ have their centers on $a,b$ and are tangent to $b,c$;$a,c$ respectively,with radii $r,t$.find the greatest real number $p$ such that the inequality $\frac{1}{r}+\frac{1}{t}\ge p(\frac{1}{a}+\frac{1}{b})$ always holds.

2012 ELMO Shortlist, 2
Tags: inequalities
Let $a,b,c$ be three positive real numbers such that $ a \le b \le c$ and $a+b+c=1$. Prove that \[\frac{a+c}{\sqrt{a^2+c^2}}+\frac{b+c}{\sqrt{b^2+c^2}}+\frac{a+b}{\sqrt{a^2+b^2}} \le \frac{3\sqrt{6}(b+c)^2}{\sqrt{(a^2+b^2)(b^2+c^2)(c^2+a^2)}}.\] [i]Owen Goff.[/i]

2012 Greece Team Selection Test, 3
Tags: algebra , inequality , inequalities
Let $a,b,c$ be positive real numbers satisfying $a+b+c=3$.Prove that $\sum_{sym} \frac{a^{2}}{(b+c)^{3}}\geq \frac{3}{8}$

2010 239 Open Mathematical Olympiad, 8
Tags: inequalities , algebra
For positive numbers $x$, $y$, and $z$, we know that $x + y^2 + z^3 = 1$. Prove that $$\frac{x}{2 + xy} + \frac{y}{2 + yz} + \frac{z}{2 + zx} > \frac{1}{2} .$$

1999 Nordic, 4
Tags: positive , inequalities
Let $a_1, a_2, . . . , a_n$ be positive real numbers and $n \ge 1$. Show that $n (\frac{1}{a_1}+...+\frac{1}{a_n}) \ge (\frac{1}{1+a_1}+...+\frac{1}{1+a_n})(n+\frac{1}{a_1}+...+\frac{1}{a_n})$ When does equality hold?

1969 Swedish Mathematical Competition, 3
Tags: algebra , inequalities
$a_1 \le a_2 \le ... \le a_n$ is a sequence of reals $b_1, _b2, b_3,..., b_n$ is any rearrangement of the sequence $B_1 \le B_2 \le ...\le B_n$. Show that $\sum a_ib_i \le \sum a_i B_i$.