Olympiad problems

1993 Poland - Second Round, 1

Tags: inequalities

If $ x,y,u,v$ are positiv real numbers, prove the inequality : \[ \frac {xu \plus{} xv \plus{} yu \plus{} yv}{x \plus{} y \plus{} u \plus{} v} \geq \frac {xy}{x \plus{} y} \plus{} \frac {uv}{u \plus{} v} \]

2019 Belarus Team Selection Test, 1.2

Tags: geometry , inequalities

Points $M$ and $N$ are the midpoints of the sides $BC$ and $AD$, respectively, of a convex quadrilateral $ABCD$. Is it possible that $$ AB+CD>\max(AM+DM,BN+CN)? $$ [i](Folklore)[/i]

2015 Saint Petersburg Mathematical Olympiad, 4

Tags: algebra , inequalities

Positive numbers $x, y, z$ satisfy the condition $$xy + yz + zx + 2xyz = 1.$$ Prove that $4x + y + z \ge 2.$ [i]A. Khrabrov[/i]

2018 China Team Selection Test, 3

Tags: inequalities , algebra

Prove that there exists a constant $C>0$ such that $$H(a_1)+H(a_2)+\cdots+H(a_m)\leq C\sqrt{\sum_{i=1}^{m}i a_i}$$ holds for arbitrary positive integer $m$ and any $m$ positive integer $a_1,a_2,\cdots,a_m$, where $$H(n)=\sum_{k=1}^{n}\frac{1}{k}.$$

Kvant 2022, M2712

Tags: trigonometry , inequalities , geometry

Let $ABC$ be a triangle, with $\angle A=\alpha,\angle B=\beta$ and $\angle C=\gamma$. Prove that \[\sum_{\text{cyc}}\tan \frac{\alpha}{2}\tan\frac{\beta}{2}\cot\frac{\gamma}{2}\geqslant\sqrt{3}.\][i]Proposed by R. Regimov (Azerbaijan)[/i]

PEN A Problems, 25

Tags: modular arithmetic , logarithm , least common multiple , function , inequalities , floor function , number theory

Show that ${2n \choose n} \; \vert \; \text{lcm}(1,2, \cdots, 2n)$ for all positive integers $n$.

2010 Romanian Masters In Mathematics, 4

Tags: inequalities , function , algebra , ceiling function , polynomial

Determine whether there exists a polynomial $f(x_1, x_2)$ with two variables, with integer coefficients, and two points $A=(a_1, a_2)$ and $B=(b_1, b_2)$ in the plane, satisfying the following conditions: (i) $A$ is an integer point (i.e $a_1$ and $a_2$ are integers); (ii) $|a_1-b_1|+|a_2-b_2|=2010$; (iii) $f(n_1, n_2)>f(a_1, a_2)$ for all integer points $(n_1, n_2)$ in the plane other than $A$; (iv) $f(x_1, x_2)>f(b_1, b_2)$ for all integer points $(x_1, x_2)$ in the plane other than $B$. [i]Massimo Gobbino, Italy[/i]

2019 Jozsef Wildt International Math Competition, W. 55

Tags: product , summation , inequalities

Let $a_1,a_2,\cdots ,a_n$ be $n$ positive numbers such that $\sum \limits_{i=1}^n\sqrt{a_i}=\sqrt{n}$. Then$$\prod \limits_{i=1}^{n-1}\left(1+\frac{1}{a_i}\right)^{a_{i+1}}\left(1+\frac{1}{a_n}\right)^{a_1}\geq 1+\frac{n}{\sum \limits_{i=1}^na_i}$$

1995 Poland - First Round, 5

Tags: inequalities , trigonometry

Given triangle $ABC$ in the plane such that $\angle CAB = a > \pi/2$. Let $PQ$ be a segment whose midpoint is the point $A$. Prove that $(BP+CQ) \tan a/2 \geq BC$.

2009 USA Team Selection Test, 9

Tags: function , inequalities , derivative , search , calculus , polynomial , algebra

Prove that for positive real numbers $x$, $y$, $z$, \[ x^3(y^2+z^2)^2 + y^3(z^2+x^2)^2+z^3(x^2+y^2)^2 \geq xyz\left[xy(x+y)^2 + yz(y+z)^2 + zx(z+x)^2\right].\] [i]Zarathustra (Zeb) Brady.[/i]

2000 Moldova National Olympiad, Problem 2

Tags: inequality , inequalities

Show that if real numbers $x<1<y$ satisfy the inequality $$2\log x+\log(1-x)\ge3\log y+\log(y-1),$$then $x^3+y^3<2$.

2006 China Team Selection Test, 2

Tags: cauchy inequality , function , inequalities

$x_{1}, x_{2}, \cdots, x_{n}$ are positive numbers such that $\sum_{i=1}^{n}x_{i}= 1$. Prove that \[\left( \sum_{i=1}^{n}\sqrt{x_{i}}\right) \left( \sum_{i=1}^{n}\frac{1}{\sqrt{1+x_{i}}}\right) \leq \frac{n^{2}}{\sqrt{n+1}}\]

2022 Greece Team Selection Test, 3

Tags: inequalities , algebra , sequence

Find largest possible constant $M$ such that, for any sequence $a_n$, $n=0,1,2,...$ of real numbers, that satisfies the conditions : i) $a_0=1$, $a_1=3$ ii) $a_0+a_1+...+a_{n-1} \ge 3 a_n - a_{n+1}$ for any integer $n\ge 1$ to be true that $$\frac{a_{n+1}}{a_n} >M$$ for any integer $n\ge 0$.

2010 Serbia National Math Olympiad, 1

Tags: inequalities , combinatorics , floor function

Some of $n$ towns are connected by two-way airlines. There are $m$ airlines in total. For $i = 1, 2, \cdots, n$, let $d_i$ be the number of airlines going from town $i$. If $1\le d_i \le 2010$ for each $i = 1, 2,\cdots, 2010$, prove that \[\displaystyle\sum_{i=1}^n d_i^2\le 4022m- 2010n\] Find all $n$ for which equality can be attained. [i]Proposed by Aleksandar Ilic[/i]

2005 Iran MO (3rd Round), 3

Tags: trigonometry , inequalities , inradius , geometry , circumcircle

Prove that in acute-angled traingle ABC if $r$ is inradius and $R$ is radius of circumcircle then: \[a^2+b^2+c^2\geq 4(R+r)^2\]

2020 Argentina National Olympiad, 5

Tags: algebra , inequalities , sum , permutation

Determine the highest possible value of: $$S = a_1a_2a_3 + a_4a_5a_6 +... + a_{2017}a_{2018}a_{2019} + a_{2020}$$ where $(a_1, a_2, a_3,..., a_{2020})$ is a permutation of $(1,2,3,..., 2020)$. Clarification: In $S$, each term, except the last one, is the multiplication of three numbers.

2009 Hanoi Open Mathematics Competitions, 10

Tags: geometric inequality , inequalities , geometry

Prove that $d^2+(a-b)^2<c^2$ ,where $d$ is diameter of the inscribed circle of $\vartriangle ABC$

2008 Romania National Olympiad, 3

Tags: prime number , number theory , inequalities

Let $ p,q,r$ be 3 prime numbers such that $ 5\leq p <q<r$. Knowing that $ 2p^2\minus{}r^2 \geq 49$ and $ 2q^2\minus{}r^2\leq 193$, find $ p,q,r$.

2007 Romania National Olympiad, 1

Tags: calculus , inequalities , derivative , logarithm , integration , trigonometry , function

Let $\mathcal{F}$ be the set of functions $f: [0,1]\to\mathbb{R}$ that are differentiable, with continuous derivative, and $f(0)=0$, $f(1)=1$. Find the minimum of $\int_{0}^{1}\sqrt{1+x^{2}}\cdot \big(f'(x)\big)^{2}\ dx$ (where $f\in\mathcal{F}$) and find all functions $f\in\mathcal{F}$ for which this minimum is attained. [hide="Comment"] In the contest, this was the b) point of the problem. The a) point was simply ``Prove the Cauchy inequality in integral form''. [/hide]

the 15th XMO, 2

Tags: inequalities

$n$ is a integer and $a_1, a_2, \ldots, a_n\in[-1,1]$ are real numbers with $ \sum_{i=1}^{n}a_{i}=0$ ,try to find the maximum value of $$ \sum_{1\leq i , j \leq n , i\ne j}|a_{i}-a^2_j|$$

1984 Swedish Mathematical Competition, 3

Tags: algebra , inequalities

Prove that if $a,b$ are positive numbers, then $$\left( \frac{a+1}{b+1}\right)^{b+1} \ge \left( \frac{a}{b}\right)^{b}$$

2022 Durer Math Competition Finals, 3

Tags: inequalities , algebra

Let $x, y, z$ denote positive real numbers for which $x+y+z = 1$ and $x > yz$, $y > zx$, $z > xy$. Prove that $$\left(\frac{x - yz}{x + yz}\right)^2+ \left(\frac{y - zx}{y + zx}\right)^2+\left(\frac{z - xy}{z + xy}\right)^2< 1.$$

VMEO I 2004, 1

Tags: algebra , inequalities

Let $x, y, z$ be non-negative numbers, so that $x + y + z = 1$. Prove that $$\sqrt{x+\frac{(y-z)^2}{12}}+\sqrt{y+\frac{(x-z)^2}{12}}+\sqrt{z+\frac{(x-y)^2}{12}}\le \sqrt{3}$$

2009 Today's Calculation Of Integral, 514

Tags: domain , algebra , trigonometry , integration , calculus , inequalities , function

Prove the following inequalities: (1) $ x\minus{}\sin x\leq \tan x\minus{}x\ \ \left(0\leq x<\frac{\pi}{2}\right)$ (2) $ \int_0^x \cos (\tan t\minus{}t)\ dt\leq \sin (\sin x)\plus{}\frac 12 \left(x\minus{}\frac{\sin 2x}{2}\right)\ \left(0\leq x\leq \frac{\pi}{3}\right)$

2014 Romania National Olympiad, 1

Tags: inequalities

Let $a,b,c\in \left( 0,\infty \right)$.Prove the inequality $\frac{a-\sqrt{bc}}{a+2\left( b+c \right)}+\frac{b-\sqrt{ca}}{b+2\left( c+a \right)}+\frac{c-\sqrt{ab}}{c+2\left( a+b \right)}\ge 0.$