Olympiad problems

2009 Indonesia TST, 2

For every positive integer $ n$, let $ \phi(n)$ denotes the number of positive integers less than $ n$ that is relatively prime to $ n$ and $ \tau(n)$ denote the sum of all positive divisors of $ n$. Let $ n$ be a positive integer such that $ \phi(n)|n\minus{}1$ and that $ n$ is not a prime number. Prove that $ \tau(n)>2009$.

2004 Unirea, 2

Tags: inequalities , inequality system , algebra

Find the maximum value of the expression $ x+y+z, $ where $ x,y,z $ are real numbers satisfying $$ \left\{ \begin{matrix} x^2+yz\le 2 \\y^2+zx\le 2\\ z^2+xy\le 2 \end{matrix} \right. . $$

2019 Centroamerican and Caribbean Math Olympiad, 5

Tags: algebra , inequalities

Let $a,\ b$ and $c$ be positive real numbers so that $a+b+c=1$. Show that $$a\sqrt{a^2+6bc}+b\sqrt{b^2+6ac}+c\sqrt{c^2+6ab}\leq\frac{3\sqrt{2}}{4}$$

2005 Grigore Moisil Urziceni, 2

Tags: geometric sequence , number theory , inequalities

Find all triples $ (x,y,z) $ of natural numbers that are in geometric progression and verify the inequalities $$ 4016016\le x<y<z\le 4020025. $$

2007 Romania Team Selection Test, 1

Tags: function , inequalities , floor function , factorial , number theory

Prove that the function $f : \mathbb{N}\longrightarrow \mathbb{Z}$ defined by $f(n) = n^{2007}-n!$, is injective.

2002 Baltic Way, 12

Tags: inequalities , absolute value , triangle inequality , combinatorics

A set $S$ of four distinct points is given in the plane. It is known that for any point $X\in S$ the remaining points can be denoted by $Y,Z$ and $W$ so that $|XY|=|XZ|+|XW|$ Prove that all four points lie on a line.

2011 Balkan MO Shortlist, A1

Tags: inequalities , algebra , three variable inequality

Given real numbers $x,y,z$ such that $x+y+z=0$, show that \[\dfrac{x(x+2)}{2x^2+1}+\dfrac{y(y+2)}{2y^2+1}+\dfrac{z(z+2)}{2z^2+1}\ge 0\] When does equality hold?

2011 Mongolia Team Selection Test, 1

Tags: inequalities

Let $t,k,m$ be positive integers and $t>\sqrt{km}$. Prove that $\dbinom{2m}{0}+\dbinom{2m}{1}+\cdots+\dbinom{2m}{m-t-1}<\dfrac{2^{2m}}{2k}$ (proposed by B. Amarsanaa, folklore)

1997 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 10

Tags: geometry , geometric transformation , reflection , calculus , derivative , function , inequalities

The minimal value of $ f(x) \equal{} \sqrt{a^2 \plus{} x^2} \plus{} \sqrt{(x\minus{}b)^2 \plus{} c^2}$ is A. $ a\plus{}b\plus{}c$ B. $ \sqrt{a^2 \plus{} (b \plus{} c)^2}$ C. $ \sqrt{b^2 \plus{} (a\plus{}c)^2}$ D. $ \sqrt{(a\plus{}b)^2 \plus{} c^2}$ E. None of these

2007 South East Mathematical Olympiad, 4

Tags: inequalities

Let $a$,$b$,$c$ be positive real numbers satisfying $abc=1$. Prove that inequality $\dfrac{a^k}{a+b}+ \dfrac{b^k}{b+c}+\dfrac{c^k}{c+a}\ge \dfrac{3}{2}$ holds for all integer $k$ ($k \ge 2$).

2009 Romania Team Selection Test, 3

Tags: function , complex analysis , trigonometry , inequalities , floor function , complex number , algebra

Given an integer $n\geq 2$ and a closed unit disc, evaluate the maximum of the product of the lengths of all $\frac{n(n-1)}{2}$ segments determined by $n$ points in that disc.

2010 Polish MO Finals, 1

Tags: inequalities , perimeter , parallelogram , geometry

On the side $BC$ of the triangle $ABC$ there are two points $D$ and $E$ such that $BD < BE$. Denote by $p_1$ and $p_2$ the perimeters of triangles $ABC$ and $ADE$ respectively. Prove that \[p_1 > p_2 + 2\cdot \min\{BD, EC\}.\]

2018 Romania National Olympiad, 2

Tags: inequalities , number theory , algebra

Let $a, b, c, d$ be natural numbers such that $a + b + c + d = 2018$. Find the minimum value of the expression: $$E = (a-b)^2 + 2(a-c)^2 + 3(a-d)^2+4(b-c)^2 + 5(b-d)^2 + 6(c-d)^2.$$

2008 Balkan MO Shortlist, A1

Tags: inequality , inequalities

For all $\alpha_1, \alpha_2,\alpha_3 \in \mathbb{R}^+$, Prove \begin{align*} \sum \frac{1}{2\nu \alpha_1 +\alpha_2+\alpha_3} > \frac{2\nu}{2\nu +1} \left( \sum \frac{1}{\nu \alpha_1 + \nu \alpha_2 + \alpha_3} \right) \end{align*} for every positive real number $\nu$

2009 Austria Beginners' Competition, 2

Tags: inequalities , algebra

Let $x$ and $y$ be nonnegative real numbers. Prove that $(x +y^3) (x^3 +y) \ge 4x^2y^2$. When does equality holds? (Task committee)

2015 Romania Team Selection Tests, 2

Tags: circles , triangle inequality , inradius , inequalities , geometry

Let $ABC$ be a triangle, and let $r$ denote its inradius. Let $R_A$ denote the radius of the circle internally tangent at $A$ to the circle $ABC$ and tangent to the line $BC$; the radii $R_B$ and $R_C$ are defined similarly. Show that $\frac{1}{R_A} + \frac{1}{R_B} + \frac{1}{R_C}\leq\frac{2}{r}$.

2018 Belarus Team Selection Test, 1.4

Tags: inequalities , geometry

Let $A_1H_1,A_2H_2,A_3H_3$ be altitudes and $A_1L_1,A_2L_2,A_3L_3$ be bisectors of acute-angles triangle $A_1A_2A_3$. Prove the inequality $S(L_1L_2L_3)\ge S(H_1H_2H_3)$ where $S$ stands for the area of a triangle. [i](B. Bazylev)[/i]

2010 Today's Calculation Of Integral, 645

Tags: calculus , integration , function , inequalities , calculus computations

Prove the following inequality. \[\int_{-1}^1 \frac{e^x+e^{-x}}{e^{e^{e^x}}}dx<e-\frac{1}{e}\] Own

PEN A Problems, 107

Tags: inequalities , rearrangement inequality , divisibility theory , logarithm

Find four positive integers, each not exceeding $70000$ and each having more than $100$ divisors.

1997 Vietnam Team Selection Test, 3

Tags: induction , inequalities , algebra , polynomial , number theory

Find the greatest real number $ \alpha$ for which there exists a sequence of infinitive integers $ (a_n)$, ($ n \equal{} 1, 2, 3, \ldots$) satisfying the following conditions: 1) $ a_n > 1997n$ for every $ n \in\mathbb{N}^{*}$; 2) For every $ n\ge 2$, $ U_n\ge a^{\alpha}_n$, where $ U_n \equal{} \gcd\{a_i \plus{} a_k | i \plus{} k \equal{} n\}$.

1975 Polish MO Finals, 1