Olympiad problems

2013 Finnish National High School Mathematics Competition, 2

In a particular European city, there are only $7$ day tickets and $30$ day tickets to the public transport. The former costs $7.03$ euro and the latter costs $30$ euro. Aina the Algebraist decides to buy at once those tickets that she can travel by the public transport the whole three year (2014-2016, 1096 days) visiting in the city. What is the cheapest solution?

2018 IFYM, Sozopol, 3

Tags: algebra , inequalities

The number 1 is a solution of the equation $(x + a)(x + b)(x + c)(x + d) = 16$, where $a, b, c, d$ are positive real numbers. Find the largest value of $abcd$.

2014 Bosnia and Herzegovina Junior BMO TST, 3

Tags: algebra , inequalities

Let $a$, $b$ and $c$ be positive real numbers such that $a+b+c=1$. Prove the inequality: $\frac{1}{\sqrt{(a+2b)(b+2a)}}+\frac{1}{\sqrt{(b+2c)(c+2b)}}+\frac{1}{\sqrt{(c+2a)(a+2c)}} \geq 3$

1987 Bundeswettbewerb Mathematik, 4

Tags: equality , inequality , inequalities , algebra

Let $1<k\leq n$ be positive integers and $x_1 , x_2 , \ldots , x_k$ be positive real numbers such that $x_1 \cdot x_2 \cdot \ldots \cdot x_k = x_1 + x_2 + \ldots +x_k.$ a) Show that $x_{1}^{n-1} +x_{2}^{n-1} + \ldots +x_{k}^{n-1} \geq kn.$ b) Find all numbers $k,n$ and $x_1, x_2 ,\ldots , x_k$ for which equality holds.

2012 ELMO Shortlist, 9

Tags: function , polynomial , inequalities , algebra , induction

Let $a,b,c$ be distinct positive real numbers, and let $k$ be a positive integer greater than $3$. Show that \[\left\lvert\frac{a^{k+1}(b-c)+b^{k+1}(c-a)+c^{k+1}(a-b)}{a^k(b-c)+b^k(c-a)+c^k(a-b)}\right\rvert\ge \frac{k+1}{3(k-1)}(a+b+c)\] and \[\left\lvert\frac{a^{k+2}(b-c)+b^{k+2}(c-a)+c^{k+2}(a-b)}{a^k(b-c)+b^k(c-a)+c^k(a-b)}\right\rvert\ge \frac{(k+1)(k+2)}{3k(k-1)}(a^2+b^2+c^2).\] [i]Calvin Deng.[/i]

2006 Romania National Olympiad, 2

Tags: inequalities , trigonometry

Prove that for all $\displaystyle a,b \in \left( 0 ,\frac{\pi}{4} \right)$ and $\displaystyle n \in \mathbb N^\ast$ we have \[ \frac{\sin^n a + \sin^n b}{\left( \sin a + \sin b \right)^n} \geq \frac{\sin^n 2a + \sin^n 2b}{\left( \sin 2a + \sin 2b \right)^n} . \]

2014 Ukraine Team Selection Test, 2

Tags: inequalities

Let $x_1,x_2,\cdots,x_n$ be postive real numbers such that $x_1x_2\cdots x_n=1$ ,$S=x^3_1+x^3_2+\cdots+x^3_n$.Find the maximum of $\frac{x_1}{S-x^3_1+x^2_1}+\frac{x_2}{S-x^3_2+x^2_2}+\cdots+\frac{x_n}{S-x^3_n+x^2_n}$

2010 Contests, 2

Tags: function , cauchy inequality , inequalities , absolute value , algebra

For each positive integer $n$, find the largest real number $C_n$ with the following property. Given any $n$ real-valued functions $f_1(x), f_2(x), \cdots, f_n(x)$ defined on the closed interval $0 \le x \le 1$, one can find numbers $x_1, x_2, \cdots x_n$, such that $0 \le x_i \le 1$ satisfying \[|f_1(x_1)+f_2(x_2)+\cdots f_n(x_n)-x_1x_2\cdots x_n| \ge C_n\] [i]Marko Radovanović, Serbia[/i]

2018 Switzerland - Final Round, 2

Tags: number theory , gcd , greatest common divisor , inequalities

Let $a, b$ and $c$ be natural numbers. Determine the smallest value that the following expression can take: $$\frac{a}{gcd\,\,(a + b, a - c)} + \frac{b}{gcd\,\,(b + c, b - a)} + \frac{c}{gcd\,\,(c + a, c - b)}.$$ . Remark: $gcd \,\, (6, 0) = 6$ and $gcd\,\,(3, -6) = 3$.

2009 Germany Team Selection Test, 3

Tags: algebra , inequalities

Prove that for any four positive real numbers $ a$, $ b$, $ c$, $ d$ the inequality \[ \frac {(a \minus{} b)(a \minus{} c)}{a \plus{} b \plus{} c} \plus{} \frac {(b \minus{} c)(b \minus{} d)}{b \plus{} c \plus{} d} \plus{} \frac {(c \minus{} d)(c \minus{} a)}{c \plus{} d \plus{} a} \plus{} \frac {(d \minus{} a)(d \minus{} b)}{d \plus{} a \plus{} b}\ge 0\] holds. Determine all cases of equality. [i]Author: Darij Grinberg (Problem Proposal), Christian Reiher (Solution), Germany[/i]

1997 APMO, 3

Tags: ratio , geometric inequality , angle bisector , geometry , circumcircle , trigonometry , inequalities

Let $ABC$ be a triangle inscribed in a circle and let \[ l_a = \frac{m_a}{M_a} \ , \ \ l_b = \frac{m_b}{M_b} \ , \ \ l_c = \frac{m_c}{M_c} \ , \] where $m_a$,$m_b$, $m_c$ are the lengths of the angle bisectors (internal to the triangle) and $M_a$, $M_b$, $M_c$ are the lengths of the angle bisectors extended until they meet the circle. Prove that \[ \frac{l_a}{\sin^2 A} + \frac{l_b}{\sin^2 B} + \frac{l_c}{\sin^2 C} \geq 3 \] and that equality holds iff $ABC$ is an equilateral triangle.

2019 Spain Mathematical Olympiad, 5

Tags: minimum value , inequalities

We consider all pairs (x, y) of real numbers such that $0\leq x \leq y \leq 1$.Let $M (x,y)$ the maximum value of the set $$A=\{xy, 1-x-y+xy, x+y-2xy\}.$$ Find the minimum value that $M(x,y)$ can take for all these pairs $(x,y)$.

Russian TST 2015, P1

Tags: algebra , inequalities

Let $n>4$ be a natural number. Prove that \[\sum_{k=2}^n\sqrt[k]{\frac{k}{k-1}}<n.\]

2014 Middle European Mathematical Olympiad, 8

Tags: inequalities , quadratic , 3d geometry , modular arithmetic , geometry , number theory

Determine all quadruples $(x,y,z,t)$ of positive integers such that \[ 20^x + 14^{2y} = (x + 2y + z)^{zt}.\]

2016 239 Open Mathematical Olympiad, 4

Tags: algebra , inequalities

Positive real numbers $a,b,c$ are given such that $abc=1$. Prove that$$a+b+c+\frac{3}{ab+bc+ca}\geq4.$$

1987 IMO Longlists, 14

Tags: inequalities

Given $n$ real numbers $0 < t_1 \leq t_2 \leq \cdots \leq t_n < 1$, prove that \[(1-t_n^2) \left( \frac{t_1}{(1-t_1^2)^2}+\frac{t_2}{(1-t_2^3)^2}+\cdots +\frac{t_n}{(1-t_n^{n+1})^2} \right) < 1.\]

VMEO III 2006, 12.3

Tags: inequalities

Let $a,b,c,d$ be positive real numbers such that \[(a+b+c+d)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}\right)=20. \] Prove that \[\left(a^{2}+b^{2}+c^{2}+d^{2}\right)\left(\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}+\frac{1}{d^{2}}\right)\ge 36. \] There are two solutions, one by Phan Thanh Nam, one by me, which are very nice.

1972 Dutch Mathematical Olympiad, 2

Tags: functional inequality , functional equation , functional , algebra , inequalities

Prove that there exists exactly one function $ƒ$ which is defined for all $x \in R$, and for which holds: $\bullet$ $x \le y \Rightarrow f(x) \le f(y)$, for all $x, y \in R$, and $\bullet$ $f(f(x)) = x$, for all $x \in R$.

2018 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: logarithm , algebra , inequalities

If numbers $x_1$, $x_2$,...,$x_n$ are from interval $\left( \frac{1}{4},1 \right)$ prove the inequality: $\log _{x_1} {\left(x_2-\frac{1}{4} \right)} + \log _{x_2} {\left(x_3-\frac{1}{4} \right)}+ ... + \log _{x_{n-1}} {\left(x_n-\frac{1}{4} \right)} + \log _{x_n} {\left(x_1-\frac{1}{4} \right)} \geq 2n$

2006 Mediterranean Mathematics Olympiad, 4

Tags: inequalities

Let $0\le x_{i,j} \le 1$, where $i=1,2, \ldots m$ and $j=1,2, \ldots n$. Prove the inequality \[ \prod_{j=1}^n\left(1-\prod_{i=1}^mx_{i,j} \right)+ \prod_{i=1}^m\left(1-\prod_{j=1}^n(1-x_{i,j}) \right) \ge 1 \]

1991 Tournament Of Towns, (285) 1

Tags: algebra , inequalities

Prove that the product of the $99$ fractions $$\frac{k^3-1}{k^3+1} \,\, , \,\,\,\,\,\, k=2,3,...,100$$ is greater than $2/3$. (D. Fomin, Leningrad)

2012 German National Olympiad, 5

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

Let $a,b$ be the lengths of two nonadjacent edges of a tetrahedron with inradius $r$. Prove that \[r<\frac{ab}{2(a+b)}.\]

2024 South Africa National Olympiad, 5

Tags: geometric inequality , geometry , inequalities

Consider three circles $\Gamma_1$, $\Gamma_2$, and $\Gamma_3$, with centres $O_1$, $O_2$ and $O_3$, respectively, such that each pair of circles is externally tangent. Suppose we have another circle $\Gamma$ with centre $O$ on the line segment $O_1O_3$ such that $\Gamma_1$, $\Gamma_2$ and $\Gamma_3$ are each internally tangent to $\Gamma$. Show that $\angle O_1O_2O_3$ measures less than $90^\circ$.

1983 Vietnam National Olympiad, 2

Tags: trigonometry , inequalities

$(a)$ Prove that $\sqrt{2}(\sin t + \cos t) \ge 2\sqrt[4]{\sin 2t}$ for $0 \le t \le\frac{\pi}{2}.$ $(b)$ Find all $y, 0 < y < \pi$, such that $1 +\frac{2 \cot 2y}{\cot y} \ge \frac{\tan 2y}{\tan y}$. .

2010 Romania Team Selection Test, 1

Tags: strong induction , induction , inequalities

Given a positive integer number $n$, determine the minimum of \[\max \left\{\dfrac{x_1}{1 + x_1},\, \dfrac{x_2}{1 + x_1 + x_2},\, \cdots,\, \dfrac{x_n}{1 + x_1 + x_2 + \cdots + x_n}\right\},\] as $x_1, x_2, \ldots, x_n$ run through all non-negative real numbers which add up to $1$. [i]Kvant Magazine[/i]