Olympiad problems

2010 Putnam, A6

Tags: function , limit , integration , calculus , inequalities , logarithm

Let $f:[0,\infty)\to\mathbb{R}$ be a strictly decreasing continuous function such that $\lim_{x\to\infty}f(x)=0.$ Prove that $\displaystyle\int_0^{\infty}\frac{f(x)-f(x+1)}{f(x)}\,dx$ diverges.

2018 Czech-Polish-Slovak Match, 4

Tags: geometric inequalities , inequalities , geometry

Let $ABC$ be an acute triangle with the perimeter of $2s$. We are given three pairwise disjoint circles with pairwise disjoint interiors with the centers $A, B$, and $C$, respectively. Prove that there exists a circle with the radius of $s$ which contains all the three circles. [i]Proposed by Josef Tkadlec, Czechia[/i]

1985 IMO Longlists, 43

Tags: geometry , 3d geometry , inequalities , combinatorics

Suppose that $1985$ points are given inside a unit cube. Show that one can always choose $32$ of them in such a way that every (possibly degenerate) closed polygon with these points as vertices has a total length of less than $8 \sqrt 3.$

2016 IMO Shortlist, A1

Tags: inequalities , three variable inequality , ineqstd

Let $a$, $b$, $c$ be positive real numbers such that $\min(ab,bc,ca) \ge 1$. Prove that $$\sqrt[3]{(a^2+1)(b^2+1)(c^2+1)} \le \left(\frac{a+b+c}{3}\right)^2 + 1.$$ [i]Proposed by Tigran Margaryan, Armenia[/i]

2017 Caucasus Mathematical Olympiad, 7

Tags: 3d geometry , geometric inequality , inequalities , geometry

$8$ ants are placed on the edges of the unit cube. Prove that there exists a pair of ants at a distance not exceeding $1$.

2024 Tuymaada Olympiad, 8

Tags: inequalities , graph theory , graph

A graph $G$ has $n$ vertices ($n>1$). For each edge $e$ let $c(e)$ be the number of vertices of the largest complete subgraph containing $e$. Prove that the inequality (the summation is over all edges of $G$): \[\sum_{e} \frac{c(e)}{c(e)-1}\le \frac{n^2}{2}.\]

1977 Poland - Second Round, 2

Tags: geometry , inequalities , geometric inequality

Let $X$ be the interior point of triangle $ABC$. prove that the product of the distances of point $ X $ from the vertices $ A, B, C $ is at least eight times greater than the product of the distances of this point from the lines $ AB, BC, CA $.

2004 Swedish Mathematical Competition, 4

Tags: trigonometry , inequalities , algebra

If $0 < v <\frac{\pi}{2}$ and $\tan v = 2v$, decide whether $sinv < \frac{20}{21}$.

2015 Balkan MO Shortlist, A5

Tags: inequalities , algebra , exponential inequality , exponential , exponential equation

Let $m, n$ be positive integers and $a, b$ positive real numbers different from $1$ such thath $m > n$ and $$\frac{a^{m+1}-1}{a^m-1} = \frac{b^{n+1}-1}{b^n-1} = c$$. Prove that $a^m c^n > b^n c^{m}$ (Turkey)

2017 Junior Balkan Team Selection Tests - Romania, 4

Tags: inequalities , algebra

Let $a, b, c, d$ be non-negative real numbers satisfying $a + b + c + d = 3$. Prove that $$\frac{a}{1 + 2b^3} + \frac{b}{1 + 2c^3} +\frac{c}{1 + 2d^3} +\frac{d}{1 + 2a^3} \ge \frac{a^2 + b^2 + c^2 + d^2}{3}$$ When does the equality hold?

2015 Stars Of Mathematics, 3

Tags: inequalities

Let $n$ be a positive integer and let $a_1,a_2,...,a_n$ be non-zero positive integers.Prove that $$\sum_{k=1}^n\frac{\sqrt{a_k}}{1+a_1+a_2+...+a_k}<\sum_{k=1}^{n^2}\frac{1}{k}.$$

2016 Saudi Arabia GMO TST, 1

Tags: sum , algebra , inequalities

Let $S = x + y +z$ where $x, y, z$ are three nonzero real numbers satisfying the following system of inequalities: $$xyz > 1$$ $$x + y + z >\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$$ Prove that $S$ can take on any real values when $x, y, z$ vary

2022 Bulgarian Spring Math Competition, Problem 11.3

Tags: combinatorics , table , inequalities

In every cell of a table with $n$ rows and $m$ columns is written one of the letters $a$, $b$, $c$. Every two rows of the table have the same letter in at most $k\geq 0$ positions and every two columns coincide at most $k$ positions. Find $m$, $n$, $k$ if \[\frac{2mn+6k}{3(m+n)}\geq k+1\]

2015 AMC 12/AHSME, 2

Tags: geometry , perimeter , inequalities , triangle inequality

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? $\textbf{(A) }52\qquad\textbf{(B) }57\qquad\textbf{(C) }62\qquad\textbf{(D) }67\qquad\textbf{(E) }72$

2007 District Olympiad, 1

Tags: inequalities , logarithm , algebra

Let be three real numbers $ a,b,c, $ all in the interval $ (0,\infty ) $ or all in the interval $ (0,1). $ Prove the following inequality: $$ \sum_{\text{cyc}}\log_a bc\ge 4\cdot\sum_{\text{cyc}} \log_{ab} c . $$

2006 QEDMO 2nd, 3

Tags: inequalities

Prove the inequality $\frac{b^2+c^2-a^2}{a\left(b+c\right)}+\frac{c^2+a^2-b^2}{b\left(c+a\right)}+\frac{a^2+b^2-c^2}{c\left(a+b\right)}\geq\frac32$ for any three positive reals $a$, $b$, $c$. [i]Comment.[/i] This was an attempt of creating a contrast to the (rather hard) inequality at the QEDMO before. However, it turned out to be more difficult than I expected (a wrong solution was presented during the competition). Darij

OMMC POTM, 2024 2

Tags: inequalities , algebra

Let $a,b,c$, and $d$ be real numbers such that $$a+b = c +d+ 12$$ and $$ab + cd - 28 = bc + ad.$$ Find the minimum possible value of $a^4+b^4+c^4+d^4$.

2020 Regional Olympiad of Mexico Center Zone, 2

Tags: algebra , inequalities

Let $a$, $b$ and $c$ be positive real numbers, prove that \[\frac{2a^2 b^2}{a^5+b^5}+\frac{2b^2 c^2}{b^5+c^5}+\frac{2c^2 a^2}{c^5+a^5}\le\frac{a+b}{2ab}+\frac{b+c}{2bc}+\frac{c+a}{2ca}\]

Russian TST 2016, P2

Tags: algebra , inequalities

Prove that \[1+\frac{2^1}{1-2^1}+\frac{2^2}{(1-2^1)(1-2^2)}+\cdots+\frac{2^{2016}}{(1-2^1)\cdots(1-2^{2016})}>0.\]

2015 China Western Mathematical Olympiad, 5

Tags: inequalities , geometry

Let $a,b,c,d$ are lengths of the sides of a convex quadrangle with the area equal to $S$, set $S =\{x_1, x_2,x_3,x_4\}$ consists of permutations $x_i$ of $(a, b, c, d)$. Prove that \[S \leq \frac{1}{2}(x_1x_2+x_3x_4).\]

1994 Dutch Mathematical Olympiad, 5

Tags: inequalities , polynomial , algebra

Three real numbers $ a,b,c$ satisfy the inequality $ |ax^2\plus{}bx\plus{}c| \le 1$ for all $ x \in [\minus{}1,1]$. Prove that $ |cx^2\plus{}bx\plus{}a| \le 2$ for all $ x \in [\minus{}1,1]$.

2012 Olympic Revenge, 6

Tags: incenter , circumcircle , inequalities , power of a point , radical axis , geometry

Let $ABC$ be an scalene triangle and $I$ and $H$ its incenter, ortocenter respectively. The incircle touchs $BC$, $CA$ and $AB$ at $D,E$ an $F$. $DF$ and $AC$ intersects at $K$ while $EF$ and $BC$ intersets at $M$. Shows that $KM$ cannot be paralel to $IH$. PS1: The original problem without the adaptation apeared at the Brazilian Olympic Revenge 2011 but it was incorrect. PS2:The Brazilian Olympic Revenge is a competition for teachers, and the problems are created by the students. Sorry if I had some English mistakes here.

2007 Middle European Mathematical Olympiad, 1

Tags: inequalities

Let $ a,b,c,d$ be real numbers which satisfy $ \frac{1}{2}\leq a,b,c,d\leq 2$ and $ abcd\equal{}1$. Find the maximum value of \[ \left(a\plus{}\frac{1}{b}\right)\left(b\plus{}\frac{1}{c}\right)\left(c\plus{}\frac{1}{d}\right)\left(d\plus{}\frac{1}{a}\right).\]

2013 District Olympiad, 1

Tags: inequalities , algebra

a) Prove that, whatever the real number x would be, the following inequality takes place ${{x}^{4}}-{{x}^{3}}-x+1\ge 0.$ b) Solve the following system in the set of real numbers: ${{x}_{1}}+{{x}_{2}}+{{x}_{3}}=3,x_{1}^{3}+x_{2}^{3}+x_{3}^{3}=x_{1}^{4}+x_{2}^{4}+x_{3}^{4}$. The Mathematical Gazette

1968 IMO Shortlist, 5

Tags: inequalities , geometry , polygon , geometric inequality

Let $h_n$ be the apothem (distance from the center to one of the sides) of a regular $n$-gon ($n \geq 3$) inscribed in a circle of radius $r$. Prove the inequality \[(n + 1)h_n+1 - nh_n > r.\] Also prove that if $r$ on the right side is replaced with a greater number, the inequality will not remain true for all $n \geq 3.$