Olympiad problems

1979 IMO Longlists, 54

Tags: number theory , Sequences , recurrence relation , Inequality , IMO Shortlist , IMO Longlist

Consider the sequences $(a_n), (b_n)$ defined by \[a_1=3, \quad b_1=100 , \quad a_{n+1}=3^{a_n} , \quad b_{n+1}=100^{b_n} \] Find the smallest integer $m$ for which $b_m > a_{100}.$

2011 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: algebra , Inequality , inequalities proposed

If for real numbers $x$ and $y$ holds $\left(x+\sqrt{1+y^2}\right)\left(y+\sqrt{1+x^2}\right)=1$ prove that $$\left(x+\sqrt{1+x^2}\right)\left(y+\sqrt{1+y^2}\right)=1$$

2016 EGMO, 1

Tags: Inequality , algebra , inequalities , EGMO , n-variable inequality , Sequence

Let $n$ be an odd positive integer, and let $x_1,x_2,\cdots ,x_n$ be non-negative real numbers. Show that \[ \min_{i=1,\ldots,n} (x_i^2+x_{i+1}^2) \leq \max_{j=1,\ldots,n} (2x_jx_{j+1}) \]where $x_{n+1}=x_1$.

2021 Polish MO Finals, 4

Tags: algebra , Inequality

Prove that for every pair of positive real numbers $a, b$ and for every positive integer $n$, $$(a+b)^n-a^n-b^n \ge \frac{2^n-2}{2^{n-2}} \cdot ab(a+b)^{n-2}.$$

1967 IMO Longlists, 55

Tags: trigonometry , Trigonometric inequality , Inequality , algebra , IMO Shortlist , IMO Longlist

Find all $x$ for which, for all $n,$ \[\sum^n_{k=1} \sin {k x} \leq \frac{\sqrt{3}}{2}.\]

2020 International Zhautykov Olympiad, 3

Tags: geometry , hexagon , Inequality , IZHO 2020 , Inversion

Given convex hexagon $ABCDEF$, inscribed in the circle. Prove that $AC*BD*DE*CE*EA*FB \geq 27 AB * BC * CD * DE * EF * FA$

2014 Balkan MO, 1

Tags: algebra , Inequality , BMO , BMO Shortlist

Let $x,y$ and $z$ be positive real numbers such that $xy+yz+xz=3xyz$. Prove that \[ x^2y+y^2z+z^2x \ge 2(x+y+z)-3 \] and determine when equality holds. [i]UK - David Monk[/i]

2017 Azerbaijan Senior National Olympiad, A5

Tags: Inequality , algebra , inequalities

$a,b,c \in (0,1)$ and $x,y,z \in ( 0, \infty)$ reals satisfies the condition $a^x=bc,b^y=ca,c^z=ab$. Prove that \[ \dfrac{1}{2+x}+\dfrac{1}{2+y}+\dfrac{1}{2+z} \leq \dfrac{3}{4} \] \\

2006 Federal Math Competition of S&M, Problem 1

Tags: quadratics , Inequality , algebra , inequalities

2022 Greece National Olympiad, 3

Tags: Inequality , inequalities

The positive real numbers $a,b,c,d$ satisfy the equality $$a+bc+cd+db+\frac{1}{ab^2c^2d^2}=18.$$ Find the maximum possible value of $a$.

2013 Kyiv Mathematical Festival, 1

Tags: Inequality , 4-variable inequality , inequalities , algebra

For every positive $a, b, c, d$ such that $a + c\le ac$ and $b + d \le bd$ prove that $ab + cd \ge 8$.

2023 All-Russian Olympiad Regional Round, 9.9

Tags: algebra , Russia , inequalities proposed , Inequality

Find the largest real $m$, such that for all positive real $a, b, c$ with sum $1$, the inequality $\sqrt{\frac{ab} {ab+c}}+\sqrt{\frac{bc} {bc+a}}+\sqrt{\frac{ca} {ca+b}} \geq m$ is satisfied.

2005 IMAR Test, 1

Tags: AMC , USAMO , Inequality , three variable inequality

Let $a,b,c$ be positive real numbers such that $abc\geq 1$. Prove that \[ \frac{1}{1+b+c}+\frac{1}{1+c+a}+\frac{1}{1+a+b}\leq 1. \] [hide="Remark"]This problem derives from the well known inequality given in [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=185470#p185470]USAMO 1997, Problem 5[/url]. [/hide]

2019 Balkan MO Shortlist, A3

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)

2016 Azerbaijan Junior Mathematical Olympiad, 6

Tags: Inequality , algebra , inequalities , Azerbaijan

For all reals $x,y,z$ prove that $$\sqrt {x^2+\frac {1}{y^2}}+ \sqrt {y^2+\frac {1}{z^2}}+ \sqrt {z^2+\frac {1}{x^2}}\geq 3\sqrt {2}. $$

2019 District Olympiad, 2

Tags: characteristic polynomial , Inequality , Determinants , linear algebra

Let $n \in \mathbb{N},n \ge 2,$ and $A,B \in \mathcal{M}_n(\mathbb{R}).$ Prove that there exists a complex number $z,$ such that $|z|=1$ and $$\Re \left( {\det(A+zB)} \right) \ge \det(A)+\det(B),$$ where $\Re(w)$ is the real part of the complex number $w.$

1983 Czech and Slovak Olympiad III A, 2

Tags: geometric inequality , geometry , Inequality , national olympiad , inequalities

Given a triangle $ABC$, prove that for every inner point $P$ of the side $AB$ the inequality $$PC\cdot AB<PA\cdot BC+PB\cdot AC$$ holds.

2011 Akdeniz University MO, 3

Tags: inequalities , Inequality

For all $x \geq 2$, $y \geq 2$ real numbers, prove that $$x(\frac{4x}{y-1}+\frac{1}{2y+x})+y(\frac{y}{6x-9}+\frac{1}{2x+y}) > \frac{26}{3}$$

1988 IMO Shortlist, 16

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.

2015 IMC, 10

Tags: college contests , Polynomials , Inequality , IMC2015

Let $n$ be a positive integer, and let $p(x)$ be a polynomial of degree $n$ with integer coefficients. Prove that $$ \max_{0\le x\le1} \big|p(x)\big| > \frac1{e^n}. $$ Proposed by Géza Kós, Eötvös University, Budapest

2024 HMIC, 2

Tags: Inequality , algebra , inequalities proposed

Suppose that $a$, $b$, $c$, and $d$ are real numbers such that $a+b+c+d=8$. Compute the minimum possible value of \[20(a^2+b^2+c^2+d^2)-\sum_{\text{sym}}a^3b,\] where the sum is over all $12$ symmetric terms. [i]Derek Liu[/i]

1982 IMO Longlists, 53

Tags: limit , geometric sequence , geometric series , Inequality , minimization , IMO , IMO 1982

Consider infinite sequences $\{x_n\}$ of positive reals such that $x_0=1$ and $x_0\ge x_1\ge x_2\ge\ldots$. [b]a)[/b] Prove that for every such sequence there is an $n\ge1$ such that: \[ {x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}\ge3.999. \] [b]b)[/b] Find such a sequence such that for all $n$: \[ {x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}<4. \]