Olympiad problems

2010 Contests, 2

Let $a,b,c$ be positive real numbers for which $a+b+c=3$. Prove the inequality \[\frac{a^3+2}{b+2}+\frac{b^3+2}{c+2}+\frac{c^3+2}{a+2}\ge3\]

1983 IMO Shortlist, 9

Tags: algebra , triangle inequality , inequality , rearrangement inequality , polynomial

Let $ a$, $ b$ and $ c$ be the lengths of the sides of a triangle. Prove that \[ a^{2}b(a \minus{} b) \plus{} b^{2}c(b \minus{} c) \plus{} c^{2}a(c \minus{} a)\ge 0. \] Determine when equality occurs.

2008 China Team Selection Test, 3

Tags: rearrangement inequality , induction , inequalities

Let $ 0 < x_{1}\leq\frac {x_{2}}{2}\leq\cdots\leq\frac {x_{n}}{n}, 0 < y_{n}\leq y_{n \minus{} 1}\leq\cdots\leq y_{1},$ Prove that $ (\sum_{k \equal{} 1}^{n}x_{k}y_{k})^2\leq(\sum_{k \equal{} 1}^{n}y_{k})(\sum_{k \equal{} 1}^{n}(x_{k}^2 \minus{} \frac {1}{4}x_{k}x_{k \minus{} 1})y_{k}).$ where $ x_{0} \equal{} 0.$

2010 Contests, 2

Tags: rearrangement inequality , function , parameterization , induction , algebra , inequalities

Let $n$ be a positive integer number and let $a_1, a_2, \ldots, a_n$ be $n$ positive real numbers. Prove that $f : [0, \infty) \rightarrow \mathbb{R}$, defined by \[f(x) = \dfrac{a_1 + x}{a_2 + x} + \dfrac{a_2 + x}{a_3 + x} + \cdots + \dfrac{a_{n-1} + x}{a_n + x} + \dfrac{a_n + x}{a_1 + x}, \] is a decreasing function. [i]Dan Marinescu et al.[/i]

1987 Romania Team Selection Test, 5

Tags: algebra , inequalities , rearrangement inequality

Let $A$ be the set $\{1,2,\ldots,n\}$, $n\geq 2$. Find the least number $n$ for which there exist permutations $\alpha$, $\beta$, $\gamma$, $\delta$ of the set $A$ with the property: \[ \sum_{i=1}^n \alpha(i) \beta (i) = \dfrac {19}{10} \sum^n_{i=1} \gamma(i)\delta(i) . \] [i]Marcel Chirita[/i]

1988 IMO Longlists, 55

Tags: rearrangement inequality , trigonometry , induction , inequalities , algebra

Suppose $\alpha_i > 0, \beta_i > 0$ for $1 \leq i \leq n, n > 1$ and that \[ \sum^n_{i=1} \alpha_i = \sum^n_{i=1} \beta_i = \pi. \] Prove that \[ \sum^n_{i=1} \frac{\cos(\beta_i)}{\sin(\alpha_i)} \leq \sum^n_{i=1} \cot(\alpha_i). \]

1987 IMO Shortlist, 6

Tags: rearrangement inequality , three variable inequality , inequality , triangle inequality

Show that if $a, b, c$ are the lengths of the sides of a triangle and if $2S = a + b + c$, then \[\frac{a^n}{b+c} + \frac{b^n}{c+a} +\frac{c^n}{a+b} \geq \left(\dfrac 23 \right)^{n-2}S^{n-1} \quad \forall n \in \mathbb N \] [i]Proposed by Greece.[/i]

2011 China National Olympiad, 2

Tags: function , derivative , rearrangement inequality , inequalities , calculus

Let $a_i,b_i,i=1,\cdots,n$ are nonnegitive numbers,and $n\ge 4$,such that $a_1+a_2+\cdots+a_n=b_1+b_2+\cdots+b_n>0$. Find the maximum of $\frac{\sum_{i=1}^n a_i(a_i+b_i)}{\sum_{i=1}^n b_i(a_i+b_i)}$

1998 IMO Shortlist, 3

Tags: rearrangement inequality , inequality , algebra

Let $x,y$ and $z$ be positive real numbers such that $xyz=1$. Prove that \[ \frac{x^{3}}{(1 + y)(1 + z)}+\frac{y^{3}}{(1 + z)(1 + x)}+\frac{z^{3}}{(1 + x)(1 + y)} \geq \frac{3}{4}. \]

2007 Baltic Way, 1

Tags: logarithm , quadratic , algebra , rearrangement inequality , function , inequalities , induction

For a positive integer $n$ consider any partition of the set $\{ 1,2,\ldots ,2n \}$ into $n$ two-element subsets $P_1,P_2\ldots,P_n$. In each subset $P_i$, let $p_i$ be the product of the two numbers in $P_i$. Prove that \[\frac{1}{p_1}+\frac{1}{p_2}+\ldots + \frac{1}{p_n}<1 \]

1987 IMO Longlists, 33

Tags: inequality , three variable inequality , rearrangement inequality , triangle inequality

Show that if $a, b, c$ are the lengths of the sides of a triangle and if $2S = a + b + c$, then \[\frac{a^n}{b+c} + \frac{b^n}{c+a} +\frac{c^n}{a+b} \geq \left(\dfrac 23 \right)^{n-2}S^{n-1} \quad \forall n \in \mathbb N \] [i]Proposed by Greece.[/i]

2006 Singapore Team Selection Test, 2

Tags: inequality , rearrangement inequality

Let n be an integer greater than 1 and let $x_1, x_2, . . . , x_n$ be real numbers such that $|x_1| + |x_2| + ... + |x_n| = 1$ and $x_1 + x_2 + ... + x_n = 0$ Prove that $\left| \frac{x_1}{1}+\frac{x_2}{2}+\cdots+\frac{x_n}{n} \right| \leq \frac{1}{2} \left(1-\frac{1}{n}\right)$

2016 Spain Mathematical Olympiad, 6

Tags: jensen , esp , rearrangement inequality , inequalities

Let $n\geq 2$ an integer. Find the least value of $\gamma$ such that for any positive real numbers $x_1,x_2,...,x_n$ with $x_1+x_2+...+x_n=1$ and any real $y_1+y_2+...+y_n=1$ and $0\leq y_1,y_2,...,y_n\leq \frac{1}{2}$ the following inequality holds: $$x_1x_2...x_n\leq \gamma \left(x_1y_1+x_2y_2+...+x_ny_n\right)$$

2013 India IMO Training Camp, 1

Tags: rearrangement inequality , induction , inequalities

Let $a, b, c$ be positive real numbers such that $a + b + c = 1$. If $n$ is a positive integer then prove that \[ \frac{(3a)^n}{(b + 1)(c + 1)} + \frac{(3b)^n}{(c + 1)(a + 1)} + \frac{(3c)^n}{(a + 1)(b + 1)} \ge \frac{27}{16} \,. \]

1978 IMO Shortlist, 6

Tags: algebra , rearrangement inequality , inequality , summation

Let $f$ be an injective function from ${1,2,3,\ldots}$ in itself. Prove that for any $n$ we have: $\sum_{k=1}^{n} f(k)k^{-2} \geq \sum_{k=1}^{n} k^{-1}.$

2010 Contests, 1

Tags: rearrangement inequality , inequalities

Let $a_1,a_2,\cdots, a_n$ and $b_1,b_2,\cdots, b_n$ be two permutations of the numbers $1,2,\cdots, n$. Show that \[\sum_{i=1}^n i(n+1-i) \le \sum_{i=1}^n a_ib_i \le \sum_{i=1}^n i^2\]

1983 IMO, 3

Tags: triangle inequality , algebra , rearrangement inequality , inequality , polynomial

Let $ a$, $ b$ and $ c$ be the lengths of the sides of a triangle. Prove that \[ a^{2}b(a \minus{} b) \plus{} b^{2}c(b \minus{} c) \plus{} c^{2}a(c \minus{} a)\ge 0. \] Determine when equality occurs.

2010 Balkan MO, 1

Tags: rearrangement inequality , inequalities

Let $a,b$ and $c$ be positive real numbers. Prove that \[ \frac{a^2b(b-c)}{a+b}+\frac{b^2c(c-a)}{b+c}+\frac{c^2a(a-b)}{c+a} \ge 0. \]

2002 Junior Balkan MO, 4

Tags: rearrangement inequality , 3-variable inequality , cyclic inequality , inequalities , function

Prove that for all positive real numbers $a,b,c$ the following inequality takes place \[ \frac{1}{b(a+b)}+ \frac{1}{c(b+c)}+ \frac{1}{a(c+a)} \geq \frac{27}{2(a+b+c)^2} . \] [i]Laurentiu Panaitopol, Romania[/i]

1977 Germany Team Selection Test, 1

Tags: rearrangement inequality , inequality , permutation , algebra

We consider two sequences of real numbers $x_{1} \geq x_{2} \geq \ldots \geq x_{n}$ and $\ y_{1} \geq y_{2} \geq \ldots \geq y_{n}.$ Let $z_{1}, z_{2}, .\ldots, z_{n}$ be a permutation of the numbers $y_{1}, y_{2}, \ldots, y_{n}.$ Prove that $\sum \limits_{i=1}^{n} ( x_{i} -\ y_{i} )^{2} \leq \sum \limits_{i=1}^{n}$ $( x_{i} - z_{i})^{2}.$

1998 Iran MO (2nd round), 1

Tags: rearrangement inequality , induction , inequalities

If $a_1<a_2<\cdots<a_n$ be real numbers, prove that: \[ a_1a_2^4+a_2a_3^4+\cdots+a_{n-1}a_n^4+a_na_1^4\geq a_2a_1^4+a_3a_2^4+\cdots+a_na_{n-1}^4+a_1a_n^4. \]

2013 Albania Team Selection Test, 2

Tags: rearrangement inequality , inequalities

Let $a,b,c,d$ be positive real numbers such that $abcd=1$.Find with proof that $x=3 $ is the minimal value for which the following inequality holds: \[a^x+b^x+c^x+d^x\ge\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{d}\]

1977 Germany Team Selection Test, 1

Tags: rearrangement inequality , inequality , permutation , algebra

We consider two sequences of real numbers $x_{1} \geq x_{2} \geq \ldots \geq x_{n}$ and $\ y_{1} \geq y_{2} \geq \ldots \geq y_{n}.$ Let $z_{1}, z_{2}, .\ldots, z_{n}$ be a permutation of the numbers $y_{1}, y_{2}, \ldots, y_{n}.$ Prove that $\sum \limits_{i=1}^{n} ( x_{i} -\ y_{i} )^{2} \leq \sum \limits_{i=1}^{n}$ $( x_{i} - z_{i})^{2}.$

1997 India Regional Mathematical Olympiad, 5

Tags: rearrangement inequality , inequalities , triangle inequality

Let $x,y,z$ be three distinct real positive numbers, Determine whether or not the three real numbers \[ \left| \frac{x}{y} - \frac{y}{x}\right| ,\left| \frac{y}{z} - \frac{z}{y}\right |, \left| \frac{z}{x} - \frac{x}{z}\right| \] can be the lengths of the sides of a triangle.

2005 National Olympiad First Round, 7

Tags: rearrangement inequality , inequalities , trigonometry

What is the greatest value of $\sin x \cos y + \sin y \cos z + \sin z \cos x$, where $x,y,z$ are real numbers? $ \textbf{(A)}\ \sqrt 2 \qquad\textbf{(B)}\ \dfrac 32 \qquad\textbf{(C)}\ \dfrac {\sqrt 3}2 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 3 $