Olympiad problems

India EGMO 2021 TST, 2

Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

2018 Thailand TSTST, 4

Tags: algebra , recurrence relation , Sequence , Inequality , IMO Shortlist

Define the numbers $a_0, a_1, \ldots, a_n$ in the following way: \[ a_0 = \frac{1}{2}, \quad a_{k+1} = a_k + \frac{a^2_k}{n} \quad (n > 1, k = 0,1, \ldots, n-1). \] Prove that \[ 1 - \frac{1}{n} < a_n < 1.\]

Russian TST 2016, P1

Tags: algebra , polynomial , Inequality

For which even natural numbers $d{}$ does there exists a constant $\lambda>0$ such that any reduced polynomial $f(x)$ of degree $d{}$ with integer coefficients that does not have real roots satisfies the inequality $f(x) > \lambda$ for all real numbers?

1988 IMO Longlists, 42

Tags: algebra , polynomial , Vieta , Inequality , roots , IMO , IMO 1988

Show that the solution set of the inequality \[ \sum^{70}_{k \equal{} 1} \frac {k}{x \minus{} k} \geq \frac {5}{4} \] is a union of disjoint intervals, the sum of whose length is 1988.

2019 India Regional Mathematical Olympiad, 3

Tags: india , RMO , algebra , Inequality , system of equations

Find all triples of non-negative real numbers $(a,b,c)$ which satisfy the following set of equations $$a^2+ab=c$$ $$b^2+bc=a$$ $$c^2+ca=b$$

2018 German National Olympiad, 4

Tags: geometry , Inequality , geometric inequality , inequalities , triangle inequality

a) Let $a,b$ and $c$ be side lengths of a triangle with perimeter $4$. Show that \[a^2+b^2+c^2+abc<8.\] b) Is there a real number $d<8$ such that for all triangles with perimeter $4$ we have \[a^2+b^2+c^2+abc<d \quad\] where $a,b$ and $c$ are the side lengths of the triangle?

2007 Korea Junior Math Olympiad, 5

Tags: algebra , inequalities , Inequality , three variable inequality

For all positive real numbers $a, b,c.$ Prove the folllowing inequality$$\frac{a}{c+5b}+\frac{b}{a+5c}+\frac{c}{b+5a}\geq\frac{1}{2}.$$

2011 Germany Team Selection Test, 1

Tags: algebra , Inequality , Sequence , IMO Shortlist , get the smallest

A sequence $x_1, x_2, \ldots$ is defined by $x_1 = 1$ and $x_{2k}=-x_k, x_{2k-1} = (-1)^{k+1}x_k$ for all $k \geq 1.$ Prove that $\forall n \geq 1$ $x_1 + x_2 + \ldots + x_n \geq 0.$ [i]Proposed by Gerhard Wöginger, Austria[/i]

2005 Uzbekistan National Olympiad, 1

Tags: inequalities unsolved , Inequality , algebra

Given a,b c are lenth of a triangle (If ABC is a triangle then AC=b, BC=a, AC=b) and $a+b+c=2$. Prove that $1+abc<ab+bc+ca\leq \frac{28}{27}+abc$

1992 IMO Longlists, 79

Tags: floor function , algebra , Sequence , Fibonacci , Fibonacci sequence , Inequality , IMO Shortlist

Let $ \lfloor x \rfloor$ denote the greatest integer less than or equal to $ x.$ Pick any $ x_1$ in $ [0, 1)$ and define the sequence $ x_1, x_2, x_3, \ldots$ by $ x_{n\plus{}1} \equal{} 0$ if $ x_n \equal{} 0$ and $ x_{n\plus{}1} \equal{} \frac{1}{x_n} \minus{} \left \lfloor \frac{1}{x_n} \right \rfloor$ otherwise. Prove that \[ x_1 \plus{} x_2 \plus{} \ldots \plus{} x_n < \frac{F_1}{F_2} \plus{} \frac{F_2}{F_3} \plus{} \ldots \plus{} \frac{F_n}{F_{n\plus{}1}},\] where $ F_1 \equal{} F_2 \equal{} 1$ and $ F_{n\plus{}2} \equal{} F_{n\plus{}1} \plus{} F_n$ for $ n \geq 1.$

2003 Bosnia and Herzegovina Team Selection Test, 3

Tags: Inequality , algebra , positive integer

Prove that for every positive integer $n$ holds: $(n-1)^n+2n^n \leq (n+1)^{n} \leq 2(n-1)^n+2n^{n}$

2001 Korea Junior Math Olympiad, 6

Tags: algebra , KJMO , Inequality

For real variables $0 \leq x, y, z, w \leq 1$, find the maximum value of $$x(1-y)+2y(1-z)+3z(1-w)+4w(1-x)$$

2021 IMC, 7

Tags: complex analysis , IMC 2021 , polynomial , Inequality

Let $D \subseteq \mathbb{C}$ be an open set containing the closed unit disk $\{z : |z| \leq 1\}$. Let $f : D \rightarrow \mathbb{C}$ be a holomorphic function, and let $p(z)$ be a monic polynomial. Prove that $$ |f(0)| \leq \max_{|z|=1} |f(z)p(z)| $$

2021 Junior Macedonian Mathematical Olympiad, Problem 4

Tags: algebra , TST , Inequality

Let $a$, $b$, $c$ be positive real numbers such that $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2} = \frac{27}{4}.$ Show that: $$\frac{a^3+b^2}{a^2+b^2} + \frac{b^3+c^2}{b^2+c^2} + \frac{c^3+a^2}{c^2+a^2} \geq \frac{5}{2}.$$ [i]Authored by Nikola Velov[/i]

2016 JBMO Shortlist, 4

Tags: JBMO Shortlist , Inequality

If the non-negative reals $x,y,z$ satisfy $x^2+y^2+z^2=x+y+z$. Prove that $$\displaystyle\frac{x+1}{\sqrt{x^5+x+1}}+\frac{y+1}{\sqrt{y^5+y+1}}+\frac{z+1}{\sqrt{z^5+z+1}}\geq 3.$$ When does the equality occur? [i]Proposed by Dorlir Ahmeti, Albania[/i]

2012 Greece Team Selection Test, 3

Tags: inequalities proposed , algebra , Inequality

Let $a,b,c$ be positive real numbers satisfying $a+b+c=3$.Prove that $\sum_{sym} \frac{a^{2}}{(b+c)^{3}}\geq \frac{3}{8}$

1967 IMO Shortlist, 5

Tags: geometry , function , algebra , Inequality , Linear Function , IMO Shortlist , IMO Longlist

A linear binomial $l(z) = Az + B$ with complex coefficients $A$ and $B$ is given. It is known that the maximal value of $|l(z)|$ on the segment $-1 \leq x \leq 1$ $(y = 0)$ of the real line in the complex plane $z = x + iy$ is equal to $M.$ Prove that for every $z$ \[|l(z)| \leq M \rho,\] where $\rho$ is the sum of distances from the point $P=z$ to the points $Q_1: z = 1$ and $Q_3: z = -1.$

2022 China Team Selection Test, 5

Tags: algebra , Inequality , combinatorics

Let $n$ be a positive integer, $x_1,x_2,\ldots,x_{2n}$ be non-negative real numbers with sum $4$. Prove that there exist integer $p$ and $q$, with $0 \le q \le n-1$, such that \[ \sum_{i=1}^q x_{p+2i-1} \le 1 \mbox{ and } \sum_{i=q+1}^{n-1} x_{p+2i} \le 1, \] where the indices are take modulo $2n$. [i]Note:[/i] If $q=0$, then $\sum_{i=1}^q x_{p+2i-1}=0$; if $q=n-1$, then $\sum_{i=q+1}^{n-1} x_{p+2i}=0$.

2002 Rioplatense Mathematical Olympiad, Level 3, 2

Tags: Inequality , Integers , positive integers , number theory , algebra

Let $\lambda$ be a real number such that the inequality $0 <\sqrt {2002} - \frac {a} {b} <\frac {\lambda} {ab}$ holds for an infinite number of pairs $ (a, b)$ of positive integers. Prove that $\lambda \geq 5 $.

2019 Balkan MO, 2

Tags: Inequality , BMO , BMO Shortlist

Let $a,b,c$ be real numbers such that $0 \leq a \leq b \leq c$ and $a+b+c=ab+bc+ca >0.$ Prove that $\sqrt{bc}(a+1) \geq 2$ and determine the equality cases. (Edit: Proposed by sir Leonard Giugiuc, Romania)

2001 Mongolian Mathematical Olympiad, Problem 2

Tags: geometric inequality , Inequality , geometry

In an acute-angled triangle $ABC$, $a,b,c$ are sides, $m_a,m_b,m_c$ the corresponding medians, $R$ the circumradius and $r$ the inradius. Prove the inequality $$\frac{a^2+b^2}{a+b}\cdot\frac{b^2+c^2}{b+c}\cdot\frac{a^2+c^2}{a+c}\ge16R^2r\frac{m_a}a\cdot\frac{m_b}b\cdot\frac{m_c}c.$$

2021 Saudi Arabia IMO TST, 11

Tags: algebra , Inequality , IMO Shortlist , IMO Shortlist 2020 , 4-variable inequality , inequalities

Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

2018 JBMO Shortlist, A1

Tags: algebra , Inequality , inequalities

Let $x,y,z$ be positive real numbers . Prove: $\frac{x}{\sqrt{\sqrt[4]{y}+\sqrt[4]{z}}}+\frac{y}{\sqrt{\sqrt[4]{z}+\sqrt[4]{x}}}+\frac{z}{\sqrt{\sqrt[4]{x}+\sqrt[4]{y}}}\geq \frac{\sqrt[4]{(\sqrt{x}+\sqrt{y}+\sqrt{z})^7}}{\sqrt{2\sqrt{27}}}$

1998 IMO Shortlist, 5

Tags: matrix , combinatorics , IMO , Inequality , IMO 1998 , IMO Shortlist , Set systems

In a contest, there are $m$ candidates and $n$ judges, where $n\geq 3$ is an odd integer. Each candidate is evaluated by each judge as either pass or fail. Suppose that each pair of judges agrees on at most $k$ candidates. Prove that \[{\frac{k}{m}} \geq {\frac{n-1}{2n}}. \]

1987 IMO Longlists, 33

Tags: rearrangement inequality , three variable inequality , triangle inequality , Inequality , IMO Shortlist

Show that if $a, b, c$ are the lengths of the sides of a triangle and if $2S = a + b + c$, then \[\frac{a^n}{b+c} + \frac{b^n}{c+a} +\frac{c^n}{a+b} \geq \left(\dfrac 23 \right)^{n-2}S^{n-1} \quad \forall n \in \mathbb N \] [i]Proposed by Greece.[/i]