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

1998 All-Russian Olympiad Regional Round, 11.8
Tags: number theory , combinatorics , inequalities , combinatorial number theory , algebra
A sequence $a_1,a_2,\cdots$ of positive integers contains each positive integer exactly once. Moreover for every pair of distinct positive integer $m$ and $n$, $\frac{1}{1998} < \frac{|a_n- a_m|}{|n-m|} < 1998$, show that $|a_n - n | <2000000$ for all $n$.

2016 JBMO Shortlist, 5
Tags: algebra , inequalities
Let $x,y,z$ be positive real numbers such that $x+y+z=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$ Prove that \[x+y+z\geq \sqrt{\frac{xy+1}{2}}+\sqrt{\frac{yz+1}{2}}+\sqrt{\frac{zx+1}{2}} \ .\] [i]Proposed by Azerbaijan[/i] [hide=Second Suggested Version]Let $x,y,z$ be positive real numbers such that $x+y+z=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$ Prove that \[x+y+z\geq \sqrt{\frac{x^2+1}{2}}+\sqrt{\frac{y^2+1}{2}}+\sqrt{\frac{z^2+1}{2}} \ .\][/hide]

2017 India Regional Mathematical Olympiad, 6
Tags: inequalities
Let \(x,y,z\) be real numbers, each greater than \(1\). Prove that \(\dfrac{x+1}{y+1}+\dfrac{y+1}{z+1}+\dfrac{z+1}{x+1} \leq \dfrac{x-1}{y-1}+\dfrac{y-1}{z-1}+\dfrac{z-1}{x-1}\).

2007 Germany Team Selection Test, 1
Tags: function , inequalities , algebra
Prove the inequality: \[\sum_{i < j}{\frac {a_{i}a_{j}}{a_{i} \plus{} a_{j}}}\leq \frac {n}{2(a_{1} \plus{} a_{2} \plus{}\cdots \plus{} a_{n})}\cdot \sum_{i < j}{a_{i}a_{j}}\] for positive reals $ a_{1},a_{2},\ldots,a_{n}$. [i]Proposed by Dusan Dukic, Serbia[/i]

2015 South East Mathematical Olympiad, 1
Tags: inequalities , sequence
Suppose that the sequence $\{a_n\}$ satisfy $a_1=1$ and $a_{2k}=a_{2k-1}+a_k, \quad a_{2k+1}=a_{2k}$ for $k=1,2, \ldots$ \\Prove that $a_{2^n}< 2^{\frac{n^2}{2}}$ for any integer $n \geq 3$.

2010 Contests, 3
Tags: inequalities
Prove that for all $n \in \mathbb{Z^+}$ and for all positive real numbers satisfying $a_1a_2...a_n=1$ \[ \displaystyle\sum_{i=1}^{n} \frac{a_i}{\sqrt{{a_i}^4+3}} \leq \frac{1}{2}\displaystyle\sum_{i=1}^{n} \frac{1}{a_i} \]

I Soros Olympiad 1994-95 (Rus + Ukr), 10.7
Tags: algebra , inequalities
Without using a calculator, prove that $$2^{1995} >5^{854},$$

2005 Germany Team Selection Test, 2
Tags: inequalities , n-variable inequality
Let $ n$ be a positive integer such that $ n\geq 3$. Let $ a_1$, $ a_2$, ..., $ a_n$ and $ b_1$, $ b_2$, ..., $ b_n$ be $ 2n$ positive real numbers satisfying the equations \[ a_1 \plus{} a_2 \plus{} ... \plus{} a_n \equal{} 1, \quad \text{and} \quad b_1^2 \plus{} b_2^2 \plus{} ... \plus{} b_n^2 \equal{} 1.\] Prove the inequality \[a_1\left(b_1 \plus{} a_2\right) \plus{} a_2\left(b_2 \plus{} a_3\right) \plus{} ... \plus{} a_{n \minus{} 1}\left(b_{n \minus{} 1} \plus{} a_n\right) \plus{} a_n\left(b_n \plus{} a_1\right) < 1.\]

1985 IMO Longlists, 7
Tags: inequalities , triangle inequality , geometry , perimeter , trigonometry
A convex quadrilateral is inscribed in a circle of radius $1$. Prove that the difference between its perimeter and the sum of the lengths of its diagonals is greater than zero and less than $2.$

1957 Putnam, A3
Tags: inequalities , cosine , trigonometry
Let $a,b$ be real numbers and $k$ a positive integer. Show that $$ \left| \frac{ \cos kb \cos a - \cos ka \cos b}{\cos b -\cos a} \right|<k^2 -1$$ whenever the left side is defined.

2013 India IMO Training Camp, 1
Tags: rearrangement inequality , inequalities , induction
Let $a, b, c$ be positive real numbers such that $a + b + c = 1$. If $n$ is a positive integer then prove that \[ \frac{(3a)^n}{(b + 1)(c + 1)} + \frac{(3b)^n}{(c + 1)(a + 1)} + \frac{(3c)^n}{(a + 1)(b + 1)} \ge \frac{27}{16} \,. \]

1999 Putnam, 5
Tags: integration , inequalities , polynomial , real analysis , calculus
Prove that there is a constant $C$ such that, if $p(x)$ is a polynomial of degree $1999$, then \[|p(0)|\leq C\int_{-1}^1|p(x)|\,dx.\]

2019 Bulgaria National Olympiad, 3
Tags: integer sequence , inequalities
Find all real numbers $a,$ which satisfy the following condition: For every sequence $a_1,a_2,a_3,\ldots$ of pairwise different positive integers, for which the inequality $a_n\leq an$ holds for every positive integer $n,$ there exist infinitely many numbers in the sequence with sum of their digits in base $4038,$ which is not divisible by $2019.$

1979 IMO Longlists, 78
Tags: inequalities , number theory
Denote the number of different prime divisors of the number $n$ by $\omega (n)$, where $n$ is an integer greater than $1$. Prove that there exist infinitely many numbers $n$ for which $\omega (n)< \omega (n+1)<\omega (n+2)$ holds.

VMEO III 2006, 10.4
Tags: trigonometry , inequalities , perimeter , combinatorics , geometry
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.

2018 Bulgaria JBMO TST, 2
Tags: inequalities
For all positive reals $a$ and $b$, show that $$\frac{a^2+b^2}{2a^5b^5} + \frac{81a^2b^2}{4} + 9ab > 18.$$

II Soros Olympiad 1995 - 96 (Russia), 9.1
Tags: inequalities , algebra
Solve the inequality $$(x-1)(x^2-1)(x^3-1)\cdot ...\cdot (x^{100}-1)(x^{101}-1)\ge 0$$

2011 Hanoi Open Mathematics Competitions, 8
Tags: inequalities , absolute value , minimum , algebra
Find the minimum value of $S = |x + 1| + |x + 5|+ |x + 14| + |x + 97| + |x + 1920|$.

VMEO II 2005, 1
Tags: inequalities , algebra
Let $a, b, c$ be three positive real numbers. a) Prove that there exists a unique positive real number $d$ that satisfies $$\frac{1}{a + d}+ \frac{1}{b + d}+\frac{1}{c + d}=\frac{2}{d} .$$ b) With $x, y, z$ being positive real numbers such that $ax + by + cz = xyz$, prove the inequality $$x + y + z \ge \frac{2}{d}\sqrt{(a + d)(b + d)(c + d)}.$$

2008 China National Olympiad, 3
Tags: inequalities , algebra
Given a positive integer $n$ and $x_1 \leq x_2 \leq \ldots \leq x_n, y_1 \geq y_2 \geq \ldots \geq y_n$, satisfying \[\displaystyle\sum_{i = 1}^{n} ix_i = \displaystyle\sum_{i = 1}^{n} iy_i\] Show that for any real number $\alpha$, we have \[\displaystyle\sum_{i =1}^{n} x_i[i\alpha] \geq \displaystyle\sum_{i =1}^{n} y_i[i\alpha]\] Here $[\beta]$ denotes the greastest integer not larger than $\beta$.

1996 German National Olympiad, 2
Tags: inequalities , algebra
Let $a$ and $b$ be positive real numbers smaller than $1$. Prove that the following two statements are equivalent: (i) $a+b = 1$, (ii) Whenever $x,y$ are positive real numbers such that $x < 1, y < 1, ax+by < 1$, the following inequlity holds: $$\frac{1}{1-ax-by} \le \frac{a}{1-x} + \frac{b}{1-y}$$

1951 AMC 12/AHSME, 21
Tags: inequalities
Given: $ x > 0, y > 0, x > y$ and $ z\not \equal{} 0$. The inequality which is not always correct is: $ \textbf{(A)}\ x \plus{} z > y \plus{} z \qquad\textbf{(B)}\ x \minus{} z > y \minus{} z \qquad\textbf{(C)}\ xz > yz$ $ \textbf{(D)}\ \frac {x}{z^2} > \frac {y}{z^2} \qquad\textbf{(E)}\ xz^2 > yz^2$

1988 All Soviet Union Mathematical Olympiad, 480
Tags: inequalities , algebra , minimum
Find the minimum value of $\frac{xy}{z} + \frac{yz}{x} +\frac{ zx}{y}$ for positive reals $x, y, z$ with $x^2 + y^2 + z^2 = 1$.

2014 IFYM, Sozopol, 4
Tags: inequalities , algebra , inequality
Prove that for $\forall$ $x,y,z\in \mathbb{R}^+$ the following inequality is true: $\frac{x}{y+z}+\frac{25y}{z+x}+\frac{4z}{x+y}>2$.

2010 Puerto Rico Team Selection Test, 3
Tags: inequalities , algebra
Prove that the inequality $x^2+y^2+1\ge 2(xy-x+y)$ is satisfied by any $x$, $y$ real numbers. Indicate when the equality is satisfied.