Olympiad problems

2010 Gheorghe Vranceanu, 2

Let be a natural number $ n, $ a number $ t\in (0,1) $ and $ n+1 $ numbers $ a_0\ge a_1\ge a_2\ge\cdots\ge a_n\ge 0. $ Prove the following matrix inequality: $$ \begin{vmatrix}\frac{(1+t\sqrt{-1})^2}{1+t^2} & -1 & 0& 0 & \cdots & 0 & 0 \\ 0 & \frac{(1+t\sqrt{-1})^2}{1+t^2} & -1 & 0 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & \cdots & \frac{(1+t\sqrt{-1})^2}{1+t^2} & -1 \\ a_0 & a_1 & a_2 & a_3 & \cdots & a_{n-1} & a_n \end{vmatrix}^2\le a_0^2\left( 1+\frac{1}{t^2} \right) $$

2000 Austria Beginners' Competition, 2

Tags: algebra , inequalities

Let $a,b$ positive real numbers. Prove that $$\frac{(a+b)^3}{a^2b}\ge \frac{27}{4}.$$ When does equality occur?

1961 IMO Shortlist, 5

Tags: trigonometry , geometry , inequalities , perpendicular bisector

Construct a triangle $ABC$ if $AC=b$, $AB=c$ and $\angle AMB=w$, where $M$ is the midpoint of the segment $BC$ and $w<90$. Prove that a solution exists if and only if \[ b \tan{\dfrac{w}{2}} \leq c <b \] In what case does the equality hold?

1961 IMO, 2

Tags: trigonometry , area of a triangle , geometry , heron's formula , inequalities

Let $ a$, $ b$, $ c$ be the sides of a triangle, and $ S$ its area. Prove: \[ a^{2} \plus{} b^{2} \plus{} c^{2}\geq 4S \sqrt {3} \] In what case does equality hold?

2007 Moldova National Olympiad, 12.3

Tags: inequalities , logarithm , function , algebra

For $a,b \in [1;\infty)$ show that \[ab\leq e^{a-1}+b\ln b\]

2005 Taiwan TST Round 2, 2

Tags: inequalities

Find all positive integers $n \ge 3$ such that there exists a positive constant $M_n$ satisfying the following inequality for any $n$ positive reals $a_1, a_2,\dots\>,a_n$: \[\displaystyle \frac{a_1+a_2+\cdots\>+a_n}{\sqrt[n]{a_1a_2\cdots\>a_n}} \le M_n \biggl( \frac{a_2}{a_1} + \frac{a_3}{a_2} +\cdots\>+ \frac{a_n}{a_{n-1}} + \frac {a_1}{a_n} \biggr).\] Moreover, find the minimum value of $M_n$ for such $n$. The difficulty is finding $M_n$...

2014 ELMO Shortlist, 1

Tags: trigonometry , inequalities , topology , three variable inequality , calculus

In a non-obtuse triangle $ABC$, prove that \[ \frac{\sin A \sin B}{\sin C} + \frac{\sin B \sin C}{\sin A} + \frac{\sin C \sin A}{ \sin B} \ge \frac 52. \][i]Proposed by Ryan Alweiss[/i]

1996 All-Russian Olympiad Regional Round, 9.7

Tags: inequalities , algebra

Prove that if $0 < a, b < 1,$ then $$\frac{ab(1 - a)(1 - b)}{(1- ab)^2 }< \frac14.$$

2015 Balkan MO Shortlist, A3

Tags: geometric inequality , median , inequalities

Let a$,b,c$ be sidelengths of a triangle and $m_a,m_b,m_c$ the medians at the corresponding sides. Prove that $$m_a\left(\frac{b}{a}-1\right)\left(\frac{c}{a}-1\right)+ m_b\left(\frac{a}{b}-1\right)\left(\frac{c}{b}-1\right) +m_c\left(\frac{a}{c}-1\right)\left(\frac{b}{c}-1\right)\geq 0.$$ (FYROM)

2015 India IMO Training Camp, 3

Tags: trigonometry , inequalities

Prove that for any triangle $ABC$, the inequality $\displaystyle\sum_{\text{cyclic}}\cos A\le\sum_{\text{cyclic}}\sin (A/2)$ holds.

2008 Swedish Mathematical Competition, 6

Tags: algebra , inequalities , max , product

A [i]sum decomposition[/i] of the number 100 is given by a positive integer $n$ and $n$ positive integers $x_1<x_2<\cdots <x_n$ such that $x_1 + x_2 + \cdots + x_n = 100$. Determine the largest possible value of the product $x_1x_2\cdots x_n$, and $n$ , as $x_1, x_2,\dots, x_n$ vary among all sum decompositions of the number $100$.

2008 Mathcenter Contest, 1

Tags: inequalities , algebra

Let $x,y,z$ be a positive real numbers. Prove that $$\frac {x}{\sqrt {x + y}} + \frac {y}{\sqrt {y + z}} + \frac { z}{\sqrt {z + x}}\geq\sqrt [4]{\frac {27(yz + zx + xy)}{4}}$$ [i](dektep)[/i]

2008 Junior Balkan MO, 1

Tags: function , polynomial , inequalities , algebra

Find all real numbers $ a,b,c,d$ such that \[ \left\{\begin{array}{cc}a \plus{} b \plus{} c \plus{} d \equal{} 20, \\ ab \plus{} ac \plus{} ad \plus{} bc \plus{} bd \plus{} cd \equal{} 150. \end{array} \right.\]

2014 Baltic Way, 15

Tags: triangle inequality , geometry , inequalities

The sum of the angles $A$ and $C$ of a convex quadrilateral $ABCD$ is less than $180^{\circ} .$ Prove that \[AB \cdot CD + AD \cdot BC < AC(AB + AD).\]

2023 India IMO Training Camp, 1

Tags: algebra , inequalities

Let $\mathbb{Z}_{\ge 0}$ be the set of non-negative integers and $\mathbb{R}^+$ be the set of positive real numbers. Let $f: \mathbb{Z}_{\ge 0}^2 \rightarrow \mathbb{R}^+$ be a function such that $f(0, k) = 2^k$ and $f(k, 0) = 1$ for all integers $k \ge 0$, and $$f(m, n) = \frac{2f(m-1, n) \cdot f(m, n-1)}{f(m-1, n)+f(m, n-1)}$$ for all integers $m, n \ge 1$. Prove that $f(99, 99)<1.99$. [i]Proposed by Navilarekallu Tejaswi[/i]

2015 Thailand TSTST, 1

Tags: inequalities

Let $a, b, c$ be positive real numbers. Prove that $$\frac{a}{a+\sqrt{(a+b)(a+c)}}+\frac{b}{b+\sqrt{(b+c)(b+a)}}+\frac{c}{c+\sqrt{(c+a)(c+b)}}\leq\frac{2a^2+ab}{(b+\sqrt{ca}+c)^2}+\frac{2b^2+bc}{(c+\sqrt{ab}+a)^2}+\frac{2c^2+ca}{(a+\sqrt{bc}+b)^2}.$$

1971 Bulgaria National Olympiad, Problem 4

Tags: circumcircle , inequalities , geometry , triangle , geometric inequalities

It is given a triangle $ABC$. Let $R$ be the radius of the circumcircle of the triangle and $O_1,O_2,O_3$ be the centers of excircles of the triangle $ABC$ and $q$ is the perimeter of the triangle $O_1O_2O_3$. Prove that $q\le6R\sqrt3$. When does equality hold?

1999 IMC, 2

Tags: inequalities , rearrangement inequality

Does there exist a bijective map $f:\mathbb{N} \rightarrow \mathbb{N}$ so that $\sum^{\infty}_{n=1}\frac{f(n)}{n^2}$ is finite?

2013 Junior Balkan Team Selection Tests - Moldova, 5

Tags: algebra , inequalities

The real numbers $a, b, c$ are positive, and the real numbers $p, q, r \in [0,1/2]$ satisfy equality $p + q + r = 1$. Prove the inequality $$pab + qbc + rca \le \frac18 (a + b + c)^2.$$

2011 Mathcenter Contest + Longlist, 11

Tags: algebra , inequalities

Let $a,b,c\in R^+$ with $a+b+c=3$. Prove that $$2(ab+bc+ca)\le 5+ abc$$ [i](Real Matrik)[/i]

2006 Taiwan TST Round 1, 1

Tags: inequalities , trigonometry

Let the three sides of $\triangle ABC$ be $a,b,c$. Prove that $\displaystyle \frac{\sin^2A}{a}+\frac{\sin^2B}{b}+\frac{\sin^2C}{c} \le \frac{S^2}{abc}$ where $\displaystyle S=\frac{a+b+c}{2}$. Find the case where equality holds.

2015 AMC 12/AHSME, 2

Tags: triangle inequality , geometry , inequalities , perimeter

Two of the three sides of a triangle are $20$ and $15$. Which of the following numbers is not a possible perimeter of the triangle? $\textbf{(A) }52\qquad\textbf{(B) }57\qquad\textbf{(C) }62\qquad\textbf{(D) }67\qquad\textbf{(E) }72$

2021 China Team Selection Test, 3

Tags: inequalities , algebra

Determine the greatest real number $ C $, such that for every positive integer $ n\ge 2 $, there exists $ x_1, x_2,..., x_n \in [-1,1]$, so that $$\prod_{1\le i<j\le n}(x_i-x_j) \ge C^{\frac{n(n-1)}{2}}$$.

2021 CIIM, 4

Tags: inequalities , function

Let $\mathbb{Z}^{+}$ be the set of positive integers. [b]a)[/b] Prove that there is only one function $f:\mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+}$, strictly increasing, such that $f(f(n))=2n+1$ for every $n\in \mathbb{Z}^{+}$. [b]b)[/b] For the function in [b]a[/b]. Prove that for every $n\in \mathbb{Z}^{+}$ $\frac{4n+1}{3}\leq f(n)\leq \frac{3n+1}{2}$ [b]c) [/b] Prove that in each inequality side of [b]b[/b] the equality can reach by infinite positive integers $n$.

2019 Greece JBMO TST, 3

Tags: inequalities , algebra

Let $a,b,c$ be positive real numbers . Prove that$$ \frac{1}{ab(b+1)(c+1)}+\frac{1}{bc(c+1)(a+1)}+\frac{1}{ca(a+1)(b+1)}\geq\frac{3}{(1+abc)^2}.$$