Olympiad problems

2006 Kyiv Mathematical Festival, 3

Tags: inequalities

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] Let $x,y>0$ and $xy\ge1.$ Prove that $x^3+y^3+4xy\ge x^2+y^2+x+y+2.$ Let $x,y>0$ and $xy\ge1.$ Prove that $2(x^3+y^3+xy+x+y)\ge5(x^2+y^2).$

1997 Korea National Olympiad, 5

Tags: geometry , inequalities

Let $a,b,c$ be the side lengths of any triangle $\triangle ABC$ opposite to $A,B$ and $C,$ respectively. Let $x,y,z$ be the length of medians from $A,B$ and $C,$ respectively. If $T$ is the area of $\triangle ABC$, prove that $\frac{a^2}{x}+\frac{b^2}{y}+\frac{c^2}{z}\ge\sqrt{\sqrt{3}T}$

2011 Indonesia MO, 7

Tags: inequalities

Let $a,b,c \in \mathbb{R}^+$ and $abc = 1$ such that $a^{2011} + b^{2011} + c^{2011} < \dfrac{1}{a^{2011}} + \dfrac{1}{b^{2011}} + \dfrac{1}{c^{2011}}$. Prove that $a + b + c < \dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c}$.

PEN I Problems, 8

Tags: inequalities , floor function , calculus , derivative

Prove that $\lfloor \sqrt[3]{n}+\sqrt[3]{n+1}+\sqrt[3]{n+2}\rfloor =\lfloor \sqrt[3]{27n+26}\rfloor$ for all positive integers $n$.

2005 Switzerland - Final Round, 3

Tags: algebra , inequalities

Prove for all $a_1, ..., a_n > 0$ the following inequality and determine all cases in where the equaloty holds: $$\sum_{k=1}^{n}ka_k\le {n \choose 2}+\sum_{k=1}^{n}a_k^k.$$

1969 Spain Mathematical Olympiad, 6

Tags: algebra , inequalities , polynomial

Given a polynomial of real coefficients P(x) , can it be affirmed that for any real value of x is true of one of the following inequalities: $$P(x) \le P(x)^2; \,\,\, P(x) < 1 + P(x)^2; \,\,\,P(x) \le \frac12 +\frac12 P(x)^2.$$ Find a simple general procedure (among the many existing ones) that allows, provided we are given two polynomials $P(x)$ and $Q(x)$ , find another $M(x)$ such that for every value of $x$, at the same time $-M(x) < P(x)<M(x)$ and $-M(x)< Q(x)<M(x)$.

2008 International Zhautykov Olympiad, 1

Tags: number theory , inequalities , modular arithmetic

For each positive integer $ n$,denote by $ S(n)$ the sum of all digits in decimal representation of $ n$. Find all positive integers $ n$,such that $ n\equal{}2S(n)^3\plus{}8$.

2002 Tuymaada Olympiad, 5

Tags: algebra , inequalities

Prove that for all $ x, y \in \[0, 1\] $ the inequality $ 5 (x^2+ y^2) ^2 \leq 4 + (x +y) ^4$ holds.

1974 Czech and Slovak Olympiad III A, 5

Tags: geometry , inequalities , geometric inequalities , geometrical inequalities , cyclic , hexagon , area

Let $ABCDEF$ be a cyclic hexagon such that \[AB=BC,\quad CD=DE,\quad EF=FA.\] Show that \[[ACE]\le[BDF]\] and determine when the equality holds. ($[XYZ]$ denotes the area of the triangle $XYZ.$)

2011 N.N. Mihăileanu Individual, 2

Tags: inequalities

Let be three real numbers $ x,y,z>1 $ that satisfy $ xyz=8. $ Prove that: $$ \left( \sqrt{\log_2 x} +\sqrt{\log_2 y} \right)\cdot \left( \sqrt{\log_2 y} +\sqrt{\log_2 z} \right)\cdot \left( \sqrt{\log_2 z} +\sqrt{\log_2 x} \right)\le 8 $$ [i]Gabriela Constantinescu[/i]

2005 IMO Shortlist, 3

Tags: algebra , inequalities , calculus

Four real numbers $ p$, $ q$, $ r$, $ s$ satisfy $ p+q+r+s = 9$ and $ p^{2}+q^{2}+r^{2}+s^{2}= 21$. Prove that there exists a permutation $ \left(a,b,c,d\right)$ of $ \left(p,q,r,s\right)$ such that $ ab-cd \geq 2$.

2002 India National Olympiad, 3

Tags: algebra , calculus , polynomial , inequalities , inequality

If $x$, $y$ are positive reals such that $x + y = 2$ show that $x^3y^3(x^3+ y^3) \leq 2$.

1960 Czech and Slovak Olympiad III A, 1

Tags: inequalities , trigonometry

Determine all real $x$ satisfying $$\frac{1}{\sin^2 x} -\frac{1}{\cos^2x} \ge \frac83.$$

1992 All Soviet Union Mathematical Olympiad, 558

Tags: inequalities , algebra

Show that $x^4 + y^4 + z^2\ge xyz \sqrt8$ for all positive reals $x, y, z$.

2000 Croatia National Olympiad, Problem 1

Tags: inequalities

Let $a>0$ and $x_1,x_2,x_3$ be real numbers with $x_1+x_2+x_3=0$. Prove that $$\log_2\left(1+a^{x_1}\right)+\log_2\left(1+a^{x_2}\right)+\log_2\left(1+a^{x_3}\right)\ge3.$$

2013 Vietnam Team Selection Test, 1

Tags: geometric inequality , geometry , inequalities

The $ABCD$ is a cyclic quadrilateral with no parallel sides inscribed in circle $(O, R)$. Let $E$ be the intersection of two diagonals and the angle bisector of $AEB$ cut the lines $AB, BC, CD, DA$ at $M, N, P, Q$ respectively . a) Prove that the circles $(AQM), (BMN), (CNP), (DPQ)$ are passing through a point. Call that point $K$. b) Denote $min \,\{AC, BD\} = m$. Prove that $OK \le \dfrac{2R^2}{\sqrt{4R^2-m^2}}$.

2012 VJIMC, Problem 4

Tags: inequalities

Let $a,b,c,x,y,z,t$ be positive real numbers with $1\le x,y,z\le4$. Prove that $$\frac x{(2a)^t}+\frac y{(2b)^t}+\frac z{(2c)^t}\ge\frac{y+z-x}{(b+c)^t}+\frac{z+x-y}{(c+a)^t}+\frac{x+y-z}{(a+b)^t}.$$

1979 IMO Longlists, 61

Tags: inequalities , function

There are two non-decreasing sequences $\{a_i\}$ and $\{b_i\}$ of $n$ real numbers each, such that $a_i\le a_{i+1}$ for each $1\le i\le n-1$, and $b_i\le b_{i+1}$ for each $1\le i\le n-1$, and $\sum_{k=1}^{m}{a_k}\ge \sum_{k=1}^{m}{b_k}$ where $m\le n$ with equality for $m=n$. For a convex function $f$ defined on the real numbers, prove that $\sum_{k=1}^{n}{f(a_k)}\le \sum_{k=1}^{n}{f(b_k)}$.

2005 Morocco TST, 2

Tags: inequalities

Let $a,b,c$ be positive real numbers. Prove the inequality \[\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\geq a+b+c+\frac{4(a-b)^2}{a+b+c}.\] When does equality occur?

VMEO II 2005, 8

Tags: inequalities

If a,b,c>0, prove that: \[ \frac{1}{a\sqrt{(a+b)}}+\frac{1}{b\sqrt{(b+c)}}+\frac{1}{c\sqrt{(c+a)}} \geq \frac{3}{\sqrt{2abc}} \] thank u for ur help :oops:

2024 District Olympiad, P4

Tags: inequalities , real analysis , integral

Let $f:[0,\infty)\to\mathbb{R}$ be a differentiable function, with a continous derivative. Given that $f(0)=0$ and $0\leqslant f'(x)\leqslant 1$ for every $x>0$ prove that\[\frac{1}{n+1}\int_0^af(t)^{2n+1}\mathrm{d}t\leqslant\left(\int_0^af(t)^n\mathrm{d}t\right)^2,\]for any positive integer $n{}$ and real number $a>0.$

2013 Miklós Schweitzer, 9

Tags: inequalities , function , real analysis

Prove that there is a function ${f: (0,\infty) \rightarrow (0,\infty)}$ which is nowhere continuous and for all ${x,y \in (0,\infty)}$ and any rational ${\alpha}$ we have \[ \displaystyle f\left( \left(\frac{x^\alpha+y^\alpha}{2}\right)^{\frac{1}{\alpha}}\right)\leq \left(\frac{f(x)^\alpha +f(y)^\alpha }{2}\right)^{\frac{1}{\alpha}}. \] Is there such a function if instead the above relation holds for every ${x,y \in (0,\infty)}$ and for every irrational ${\alpha}?$ [i]Proposed by Maksa Gyula and Zsolt Páles[/i]

1995 Poland - Second Round, 4

Tags: algebra , inequalities , n-variable inequality , sum

Positive real numbers $x_1,x_2,...,x_n$ satisfy the condition $\sum_{i=1}^n x_i \le \sum_{i=1}^n x_i ^2$ . Prove the inequality $\sum_{i=1}^n x_i^t \le \sum_{i=1}^n x_i ^{t+1}$ for all real numbers $t > 1$.

2012 Vietnam Team Selection Test, 2

Tags: inequalities

Prove that $c=10\sqrt{24}$ is the largest constant such that if there exist positive numbers $a_1,a_2,\ldots ,a_{17}$ satisfying: \[\sum_{i=1}^{17}a_i^2=24,\ \sum_{i=1}^{17}a_i^3+\sum_{i=1}^{17}a_i<c \] then for every $i,j,k$ such that $1\le 1<j<k\le 17$, we have that $x_i,x_j,x_k$ are sides of a triangle.

2006 France Team Selection Test, 2

Tags: inequalities , algebra

Let $a,b,c$ be three positive real numbers such that $abc=1$. Show that: \[ \displaystyle \frac{a}{(a+1)(b+1)}+\frac{b}{(b+1)(c+1)}+ \frac{c}{(c+1)(a+1)} \geq \frac{3}{4}. \] When is there equality?