Olympiad problems

2004 China Team Selection Test, 2

Find the largest positive real $ k$, such that for any positive reals $ a,b,c,d$, there is always: \[ (a\plus{}b\plus{}c) \left[ 3^4(a\plus{}b\plus{}c\plus{}d)^5 \plus{} 2^4(a\plus{}b\plus{}c\plus{}2d)^5 \right] \geq kabcd^3\]

1987 IMO Longlists, 44

Tags: inequalities , trigonometry , modular arithmetic , inequalities unsolved

Let $\theta_1,\theta_2,\cdots,\theta_n$ be $n$ real numbers such that $\sin \theta_1+\sin \theta_2+\cdots+\sin \theta_n=0$. Prove that \[|\sin \theta_1+2 \sin \theta_2+\cdots +n \sin \theta_n| \leq \left[ \frac{n^2}{4} \right]\]

2012 Postal Coaching, 1

Tags: inequalities , inequalities unsolved

Given a triangle $ABC$, the internal bisectors through $A$ and $B$ meet the opposite sides in $D$ and $E$, respectively. Prove that \[DE \le (3 - 2\sqrt2)(AB + BC + CA)\] and determine the cases of equality.

2009 Hong Kong TST, 1

Tags: inequalities , trigonometry , inequalities unsolved

Let $ \theta_1, \theta_2,\ldots , \theta_{2008}$ be real numbers. Find the maximum value of $ \sin\theta_1\cos\theta_2 \plus{} \sin\theta_2\cos\theta_3 \plus{} \ldots \plus{} \sin\theta_{2007}\cos\theta_{2008} \plus{} \sin\theta_{2008}\cos\theta_1$

2012 Turkmenistan National Math Olympiad, 1

Tags: trigonometry , inequalities unsolved , inequalities

Find the max and min value of $a\cos^2 x+b\sin x\cos x+c\sin^2 x$.

1984 IMO Longlists, 35

Tags: inequalities , ratio , inequalities unsolved

Prove that there exist distinct natural numbers $m_1,m_2, \cdots , m_k$ satisfying the conditions \[\pi^{-1984}<25-\left(\frac{1}{m_1}+\frac{1}{m_2}+\cdots+\frac{1}{m_k}\right)<\pi^{-1960}\] where $\pi$ is the ratio between a circle and its diameter.

2011 Morocco National Olympiad, 2

Tags: inequalities , inequalities unsolved

Let $a,b,c$ be three postive real numbers such that $a+b+c=1$. Prove that $9abc\leq ab+ac+bc < 1/4 +3abc$.

2011 Brazil National Olympiad, 6

Tags: inequalities , algorithm , modular arithmetic , inequalities unsolved

Let $a_{1}, a_{2}, a_{3}, ... a_{2011}$ be nonnegative reals with sum $\frac{2011}{2}$, prove : $|\prod_{cyc} (a_{n} - a_{n+1})| = |(a_{1} - a_{2})(a_{2} - a_{3})...(a_{2011}-a_{1})| \le \frac{3 \sqrt3}{16}.$

2005 South africa National Olympiad, 5

Tags: inequalities , induction , inequalities unsolved , n-variable inequality

Let $x_1,x_2,\dots,x_n$ be positive numbers with product equal to 1. Prove that there exists a $k\in\{1,2,\dots,n\}$ such that \[\frac{x_k}{k+x_1+x_2+\cdots+x_k}\ge 1-\frac{1}{\sqrt[n]{2}}.\]

2006 Romania Team Selection Test, 4

Tags: inequalities , function , integration , calculus , derivative , inequalities unsolved

Let $p$, $q$ be two integers, $q\geq p\geq 0$. Let $n \geq 2$ be an integer and $a_0=0, a_1 \geq 0, a_2, \ldots, a_{n-1},a_n = 1$ be real numbers such that \[ a_{k} \leq \frac{ a_{k-1} + a_{k+1} } 2 , \ \forall \ k=1,2,\ldots, n-1 . \] Prove that \[ (p+1) \sum_{k=1}^{n-1} a_k^p \geq (q+1) \sum_{k=1}^{n-1} a_k^q . \]

1979 USAMO, 3

Tags: probability , inequalities , inequalities unsolved

Given three identical $n$- faced dice whose corresponding faces are identically numbered with arbitrary integers. Prove that if they are tossed at random, the probability that the sum of the bottom three face numbers is divisible by three is greater than or equal to $\frac{1}{4}$.

2005 China National Olympiad, 1

Tags: inequalities , trigonometry , inequalities unsolved

Suppose $\theta_{i}\in(-\frac{\pi}{2},\frac{\pi}{2}), i = 1,2,3,4$. Prove that, there exist $x\in \mathbb{R}$, satisfying two inequalities \begin{eqnarray*} \cos^2\theta_1\cos^2\theta_2-(\sin\theta\sin\theta_2-x)^2 &\geq& 0, \\ \cos^2\theta_3\cos^2\theta_4-(\sin\theta_3\sin\theta_4-x)^2 & \geq & 0 \end{eqnarray*} if and only if \[ \sum^4_{i=1}\sin^2\theta_i\leq2(1+\prod^4_{i=1}\sin\theta_i + \prod^4_{i=1}\cos\theta_i). \]

2007 India Regional Mathematical Olympiad, 6

Tags: inequalities unsolved , inequalities

Prove that: [b](a)[/b] $ 5<\sqrt {5}\plus{}\sqrt [3]{5}\plus{}\sqrt [4]{5}$ [b](b)[/b] $ 8>\sqrt {8}\plus{}\sqrt [3]{8}\plus{}\sqrt [4]{8}$ [b](c)[/b] $ n>\sqrt {n}\plus{}\sqrt [3]{n}\plus{}\sqrt [4]{n}$ for all integers $ n\geq 9 .$ [b][Weightage 16/100][/b]

1993 Vietnam Team Selection Test, 3

Tags: inequalities unsolved , inequalities

Let's consider the real numbers $x_1, x_2, x_3, x_4$ satisfying the condition \[ \dfrac{1}{2}\le x_1^2+x_2^2+x_3^2+x_4^2\le 1 \] Find the maximal and the minimal values of expression: \[ A = (x_1 - 2 \cdot x_2 + x_3)^2 + (x_2 - 2 \cdot x_3 + x_4)^2 + (x_2 - 2 \cdot x_1)^2 + (x_3 - 2 \cdot x_4)^2 \]

2008 South East Mathematical Olympiad, 1

Tags: inequalities , inequalities unsolved

Let $\lambda$ be a positive real number. Inequality $|\lambda xy+yz|\le \dfrac{\sqrt5}{2}$ holds for arbitrary real numbers $x, y, z$ satisfying $x^2+y^2+z^2=1$. Find the maximal value of $\lambda$.

1976 IMO Longlists, 21

Tags: inequalities , inequalities unsolved

Find the largest positive real number $p$ (if it exists) such that the inequality \[x^2_1+ x_2^2+ \cdots + x^2_n\ge p(x_1x_2 + x_2x_3 + \cdots + x_{n-1}x_n)\] is satisfied for all real numbers $x_i$, and $(a) n = 2; (b) n = 5.$ Find the largest positive real number $p$ (if it exists) such that the inequality holds for all real numbers $x_i$ and all natural numbers $n, n \ge 2.$

1989 Vietnam National Olympiad, 1

Tags: inequalities , trigonometry , inequalities unsolved

Let $ n$ and $ N$ be natural number. Prove that for any $ \alpha$, $ 0\le\alpha\le N$, and any real $ x$, it holds that \[{ |\sum_{k=0}^n}\frac{\sin((\alpha+k)x)}{N+k}|\le\min\{(n+1)|x|, \frac{1}{N|\sin\frac{x}{2}|}\}\]

Oliforum Contest I 2008, 3

Tags: calculus , derivative , symmetry , inequalities unsolved , inequalities

Let $ a,b,c$ be three pairwise distinct real numbers such that $ a\plus{}b\plus{}c\equal{}6\equal{}ab\plus{}bc\plus{}ca\minus{}3$. Prove that $ 0<abc<4$.

2011 Kyrgyzstan National Olympiad, 3

Tags: inequalities , inequalities unsolved

Given positive numbers ${a_1},{a_2},...,{a_n}$ with ${a_1} + {a_2} + ... + {a_n} = 1$. Prove that $\left( {\frac{1}{{a_1^2}} - 1} \right)\left( {\frac{1}{{a_2^2}} - 1} \right)...\left( {\frac{1}{{a_n^2}} - 1} \right) \geqslant {({n^2} - 1)^n}$.

2005 China Team Selection Test, 1

Tags: inequalities , function , floor function , inequalities unsolved

Find all positive integers $m$ and $n$ such that the inequality: \[ [ (m+n) \alpha ] + [ (m+n) \beta ] \geq [ m \alpha ] + [n \beta] + [ n(\alpha+\beta)] \] is true for any real numbers $\alpha$ and $\beta$. Here $[x]$ denote the largest integer no larger than real number $x$.

2014 Romania Team Selection Test, 3

Tags: inequalities unsolved , inequalities , algebra

Let $n$ a positive integer and let $f\colon [0,1] \to \mathbb{R}$ an increasing function. Find the value of : \[ \max_{0\leq x_1\leq\cdots\leq x_n\leq 1}\sum_{k=1}^{n}f\left ( \left | x_k-\frac{2k-1}{2n} \right | \right )\]

1986 Vietnam National Olympiad, 2

Tags: inequalities , function , inequalities unsolved

Find all $ n > 1$ such that the inequality \[ \sum_{i\equal{}1}^nx_i^2\ge x_n\sum_{i\equal{}1}^{n\minus{}1}x_i\] holds for all real numbers $ x_1$, $ x_2$, $ \ldots$, $ x_n$.

2012 China Second Round Olympiad, 4

Tags: ceiling function , floor function , inequalities unsolved , inequalities

Let $S_n=1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}$, where $n$ is a positive integer. Prove that for any real numbers $a,b,0\le a\le b\le 1$, there exist infinite many $n\in\mathbb{N}$ such that \[a<S_n-[S_n]<b\] where $[x]$ represents the largest integer not exceeding $x$.

2010 Contests, 4

Tags: inequalities , LaTeX , inequalities unsolved

Let's consider the inequality $ a^3\plus{}b^3\plus{}c^3<k(a\plus{}b\plus{}c)(ab\plus{}bc\plus{}ca)$ where $ a,b,c$ are the sides of a triangle and $ k$ a real number. [b]a)[/b] Prove the inequality for $ k\equal{}1$. [b]b) [/b]Find the smallest value of $ k$ such that the inequality holds for all triangles.

1987 IMO Longlists, 77

Tags: inequalities , function , inequalities unsolved

Find the least positive integer $k$ such that for any $a \in [0, 1]$ and any positive integer $n,$ \[a^k(1 - a)^n < \frac{1}{(n+1)^3}.\]