Olympiad problems

2011 Morocco National Olympiad, 3

Tags: inequalities

Let $a$ and $b$ be two real numbers and let$M(a,b)=\max\left \{ 3a^{2}+2b; 3b^{2}+2a\right \}$. Find the values of $a$ and $b$ for which $M(a,b)$ is minimal.

1981 All Soviet Union Mathematical Olympiad, 325

Tags: polynomial , algebra , minimum , inequalities

a) Find the minimal value of the polynomial $$P(x,y) = 4 + x^2y^4 + x^4y^2 - 3x^2y^2$$ b) Prove that it cannot be represented as a sum of the squares of some polynomials of $x,y$.

2005 Cuba MO, 2

Tags: trinomial , algebra , inequalities

Determine the quadratic functions $f(x) = ax^2 + bx + c$ for which there exists an interval $(h, k)$ such that for all $x \in (h, k)$ it holds that $f(x)f(x + 1) < 0$ and $f(x)f(x -1) < 0$.

1980 Swedish Mathematical Competition, 4

Tags: integration , inequalities , analysis

The functions $f$ and $g$ are positive and continuous. $f$ is increasing and $g$ is decreasing. Show that \[ \int\limits_0^1 f(x)g(x) dx \leq \int\limits_0^1 f(x)g(1-x) dx \]

2001 Moldova National Olympiad, Problem 6

Tags: inequalities , algebra , sequence , inequality

Set $a_n=\frac{2n}{n^4+3n^2+4},n\in\mathbb N$. Prove that $\frac14\le a_1+a_2+\ldots+a_n\le\frac12$ for all $n$.

2005 MOP Homework, 4

Tags: geometry , inradius , trigonometry , inequalities

Let $ABC$ be an obtuse triangle with $\angle A>90^{\circ}$, and let $r$ and $R$ denote its inradius and circumradius. Prove that \[\frac{r}{R} \le \frac{a\sin A}{a+b+c}.\]

2012 Middle European Mathematical Olympiad, 2

Tags: function , geometry , polynomial , algebra , inequalities

Let $ a,b$ and $ c $ be positive real numbers with $ abc = 1 $. Prove that \[ \sqrt{ 9 + 16a^2}+\sqrt{ 9 + 16b^2}+\sqrt{ 9 + 16c^2} \ge 3 +4(a+b+c)\]

2001 Italy TST, 2

Tags: algebra , inequalities

Let $0\le a\le b\le c$ be real numbers. Prove that \[(a+3b)(b+4c)(c+2a)\ge 60abc \]

2006 IMS, 1

Tags: combinatorics , inequalities

Prove that for each $m\geq1$: \[\sum_{|k|<\sqrt m}\binom{2m}{m+k}\geq 2^{2m-1}\] [hide="Hint"]Maybe probabilistic method works[/hide]

KoMaL A Problems 2017/2018, A. 709

Tags: inequalities , algebra

Let $a>0$ be a real number. Find the minimal constant $C_a$ for which the inequality$$\displaystyle C_a\sum_{k=1}^n \frac1{x_k-x_{k-1}} >\sum_{k=1}^n \frac{k+a}{x_k}$$holds for any positive integer $n$ and any sequence $0=x_0<x_1<\cdots <x_n$ of real numbers.

2024 Turkey Junior National Olympiad, 4

Tags: inequality , algebra , inequalities

Let $n\geq 2$ be an integer and $a_1,a_2,\cdots,a_n>1$ be real numbers. Prove that the inequality below holds. $$\prod_{i=1}^n\left(a_ia_{i+1}-\frac{1}{a_ia_{i+1}}\right)\geq 2^n\prod_{i=1}^n\left(a_i-\frac{1}{a_i}\right)$$

1997 Junior Balkan MO, 1

Tags: geometry , symmetry , combinatorics , area of a triangle , inequalities , pigeonhole principle

Show that given any 9 points inside a square of side 1 we can always find 3 which form a triangle with area less than $\frac 18$. [i]Bulgaria[/i]

2023 Swedish Mathematical Competition, 3

Tags: algebra , inequalities

Let $n$ be a positive integer and let $a_1$, $a_2$,..., $a_n$ be different real numbers, placed one after the other in any order. We say we have a [i]local minimum[/i] in one of the numbers if this is less than both of their neighbors. Which is the average number of local minima over all possible ways of ordering the numbers each other?

2001 Singapore Team Selection Test, 2

Tags: algebra , inequalities

Determine all the integers $n > 1$ such that $$\sum_{i=1}^{n}x_i^2 \ge x_n \sum_{i=1}^{n-1}x_i$$ for all real numbers $x_1, x_2, ... , x_n$.

1998 IberoAmerican Olympiad For University Students, 1

Tags: real analysis , inequalities , calculus , function , integration

The definite integrals between $0$ and $1$ of the squares of the continuous real functions $f(x)$ and $g(x)$ are both equal to $1$. Prove that there is a real number $c$ such that \[f(c)+g(c)\leq 2\]

2006 France Team Selection Test, 2

Tags: algebra , inequalities

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?

2008 International Zhautykov Olympiad, 3

Tags: invariant , three variable inequality , inequalities

Let $ a, b, c$ be positive integers for which $ abc \equal{} 1$. Prove that $ \sum \frac{1}{b(a\plus{}b)} \ge \frac{3}{2}$.

2014 Tuymaada Olympiad, 4

Tags: inequalities , function

Positive numbers $a,\ b,\ c$ satisfy $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=3$. Prove the inequality \[\dfrac{1}{\sqrt{a^3+1}}+\dfrac{1}{\sqrt{b^3+1}}+\dfrac{1}{\sqrt{c^3+1}}\le \dfrac{3}{\sqrt{2}}. \] [i](N. Alexandrov)[/i]

2014 Turkey Junior National Olympiad, 1

Tags: inequalities , algebra

Prove that for positive reals $a$,$b$,$c$ so that $a+b+c+abc=4$, \[\left (1+\dfrac{a}{b}+ca \right )\left (1+\dfrac{b}{c}+ab \right)\left (1+\dfrac{c}{a}+bc \right) \ge 27\] holds.

2008 Brazil Team Selection Test, 3

Tags: inequalities , algebra , function , calculus

Let $ n$ be a positive integer, and let $ x$ and $ y$ be a positive real number such that $ x^n \plus{} y^n \equal{} 1.$ Prove that \[ \left(\sum^n_{k \equal{} 1} \frac {1 \plus{} x^{2k}}{1 \plus{} x^{4k}} \right) \cdot \left( \sum^n_{k \equal{} 1} \frac {1 \plus{} y^{2k}}{1 \plus{} y^{4k}} \right) < \frac {1}{(1 \minus{} x) \cdot (1 \minus{} y)}. \] [i]Author: Juhan Aru, Estonia[/i]

2016 Federal Competition For Advanced Students, P1, 1

Tags: algebra , inequalities

Determine the largest constant $C$ such that $$(x_1 + x_2 + \cdots + x_6)^2 \ge C \cdot (x_1(x_2 + x_3) + x_2(x_3 + x_4) + \cdots + x_6(x_1 + x_2))$$ holds for all real numbers $x_1, x_2, \cdots , x_6$. For this $C$, determine all $x_1, x_2, \cdots x_6$ such that equality holds. (Walther Janous)

1989 Balkan MO, 3

Tags: inequalities , geometry

Let $G$ be the centroid of a triangle $ABC$ and let $d$ be a line that intersects $AB$ and $AC$ at $B_{1}$ and $C_{1}$, respectively, such that the points $A$ and $G$ are not separated by $d$. Prove that: $[BB_{1}GC_{1}]+[CC_{1}GB_{1}] \geq \frac{4}{9}[ABC]$.

2000 Tuymaada Olympiad, 4

Tags: inequalities , three variable inequality , algebra

Prove that if the product of positive numbers $a,b$ and $c$ equals one, then $\frac{1}{a(a+1)}+\frac{1}{b(b+1)}+\frac{1}{c(c+1)}\ge \frac{3}{2}$

2024 pOMA, 3

Tags: inequalities , geometric inequality , circumcircle , geometry

Let $ABC$ be a triangle with circumcircle $\Omega$, and let $P$ be a point on the arc $BC$ of $\Omega$ not containing $A$. Let $\omega_B$ and $\omega_C$ be circles respectively passing through $B$ and $C$ and such that both of them are tangent to line $AP$ at point $P$. Let $R$, $R_B$, $R_C$ be the radii of $\Omega$, $\omega_B$, and $\omega_C$, respectively. Prove that if $h$ is the distance from $A$ to line $BC$, then \[ \frac{R_B+R_C}{R} \le \frac{BC}{h}. \]

2020 Jozsef Wildt International Math Competition, W18

Tags: inequalities

Let $D:=\{(x, y)\mid x,y\in\mathbb R_+,x \ne y,x^y=y^x\}$. (Obvious that $x\ne1$ and $y\ne1$). And let $\alpha\le\beta$ be positive real numbers. Find $$\inf_{(x,y)\in D}x^\alpha y^\beta.$$ [i]Proposed by Arkady Alt[/i]