Olympiad problems

2014 NIMO Problems, 6

Tags: inequalities

For all positive integers $k$, define $f(k)=k^2+k+1$. Compute the largest positive integer $n$ such that \[2015f(1^2)f(2^2)\cdots f(n^2)\geq \Big(f(1)f(2)\cdots f(n)\Big)^2.\][i]Proposed by David Altizio[/i]

2000 Brazil Team Selection Test, Problem 4

Tags: inequalities , relatively prime , number theory

[b]Problem:[/b]For a positive integer $ n$,let $ V(n; b)$ be the number of decompositions of $ n$ into a product of one or more positive integers greater than $ b$. For example,$ 36 \equal{} 6.6 \equal{}4.9 \equal{} 3.12 \equal{} 3 .3. 4$, so that $ V(36; 2) \equal{} 5$.Prove that for all positive integers $ n$; b it holds that $ V(n;b)<\frac{n}{b}$. :)

1979 IMO Longlists, 78

Tags: inequalities , number theory

Denote the number of different prime divisors of the number $n$ by $\omega (n)$, where $n$ is an integer greater than $1$. Prove that there exist infinitely many numbers $n$ for which $\omega (n)< \omega (n+1)<\omega (n+2)$ holds.

2008 IMO, 2

Tags: inequalities , algebra , vieta , calculus

[b](a)[/b] Prove that \[\frac {x^{2}}{\left(x \minus{} 1\right)^{2}} \plus{} \frac {y^{2}}{\left(y \minus{} 1\right)^{2}} \plus{} \frac {z^{2}}{\left(z \minus{} 1\right)^{2}} \geq 1\] for all real numbers $x$, $y$, $z$, each different from $1$, and satisfying $xyz=1$. [b](b)[/b] Prove that equality holds above for infinitely many triples of rational numbers $x$, $y$, $z$, each different from $1$, and satisfying $xyz=1$. [i]Author: Walther Janous, Austria[/i]

2002 Abels Math Contest (Norwegian MO), 2ab

Tags: algebra , inequalities

a) Let $x$ be a positive real number. Show that $x + 1 / x\ge 2$. b) Let $n\ge 2$ be a positive integer and let $x _1,y_1,x_2,y_2,...,x_n,y_n$ be positive real numbers such that $x _1+x _2+...+x _n \ge x _1y_1+x _2y_2+...+x _ny_n$. Show that $x _1+x _2+...+x _n \le \frac{x _1}{y_1}+\frac{x _2}{y_2}+...+\frac{x _n}{y_n}$

2016 Croatia Team Selection Test, Problem 1

Tags: inequalities , n-variable inequality , kantorovich

Let $n \ge 1$ and $x_1, \ldots, x_n \ge 0$. Prove that $$ (x_1 + \frac{x_2}{2} + \ldots + \frac{x_n}{n}) (x_1 + 2x_2 + \ldots + nx_n) \le \frac{(n+1)^2}{4n} (x_1 + x_2 + \ldots + x_n)^2 .$$

2009 Iran MO (2nd Round), 3

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

Let $ ABC $ be a triangle and the point $ D $ is on the segment $ BC $ such that $ AD $ is the interior bisector of $ \angle A $. We stretch $ AD $ such that it meets the circumcircle of $ \Delta ABC $ at $ M $. We draw a line from $ D $ such that it meets the lines $ MB,MC $ at $ P,Q $, respectively ($ M $ is not between $ B,P $ and also is not between $ C,Q $). Prove that $ \angle PAQ\geq\angle BAC $.

1999 Bosnia and Herzegovina Team Selection Test, 2

Tags: inequalities , circumcircle , geometry

Prove the inequality $$\frac{a^2}{b+c-a}+\frac{b^2}{a+c-b}+\frac{c^2}{a+b-c} \geq 3\sqrt{3}R$$ in triangle $ABC$ where $a$, $b$ and $c$ are sides of triangle and $R$ radius of circumcircle of $ABC$

2004 Alexandru Myller, 3

Tags: trigonometry , inequalities

Consider three real numbers $ x,y,z $ satisfying $ \cos x+\cos y+\cos z =\cos 3x +\cos 3y +\cos 3z=0. $ Show that $ \cos 2x\cdot \cos 2y\cdot\cos 2z\le 0. $ [i]Bogdan Enescu[/i]

1990 Nordic, 2

Tags: power , inequalities

Let $a_1, a_2, . . . , a_n$ be real numbers. Prove $\sqrt[3]{a_1^3+ a_2^3+ . . . + a_n^3} \le \sqrt{a_1^2+ a_2^2+ . . . + a_n^2} $ (1) When does equality hold in (1)?

2020 Switzerland - Final Round, 8

Tags: algebra , inequalities

Let $u_1, u_2, \dots, u_{2019}$ be real numbers satisfying \[u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { and } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1.\] Let $a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$ and $b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$. Prove that \[ a b \leqslant-\frac{1}{2019}. \]

2006 Federal Competition For Advanced Students, Part 2, 2

Tags: inequalities

Let $ a,b,c$ be positive real numbers. Show that $ 3(a \plus{} b \plus{} c) \ge 8 \sqrt [3]{abc} \plus{} \sqrt [3]{\frac {a^3 \plus{} b^3 \plus{} c^3}{3} }.$

2001 Greece Junior Math Olympiad, 1

Tags: inequalities

Let $a, b, x, y$ be positive real numbers such that $a+b=1$. Prove that $\frac{1}{\frac{a}{x}+\frac{b}{y}}\leq ax+by$ and find when equality holds.

2011 JBMO Shortlist, 6

Tags: rectangle , trigonometry , vector , triangle inequality , geometry , inequalities

Let $ABCD$ be a convex quadrilateral and points $E$ and $F$ on sides $AB,CD$ such that \[\tfrac{AB}{AE}=\tfrac{CD}{DF}=n\] If $S$ is the area of $AEFD$ show that ${S\leq\frac{AB\cdot CD+n(n-1)AD^2+n^2DA\cdot BC}{2n^2}}$

1995 Taiwan National Olympiad, 6

Tags: vector , geometry , parallelogram , inequalities , analytic geometry , combinatorics

Let $a,b,c,d$ are integers such that $(a,b)=(c,d)=1$ and $ad-bc=k>0$. Prove that there are exactly $k$ pairs $(x_{1},x_{2})$ of rational numbers with $0\leq x_{1},x_{2}<1$ for which both $ax_{1}+bx_{2},cx_{1}+dx_{2}$ are integers.

2018 Dutch IMO TST, 2

Tags: functional inequality , function , inequalities , algebra

Find all functions $f : R \to R$ such that $f(x^2)-f(y^2) \le (f(x)+y) (x-f(y))$ for all $x, y \in R$.

1995 Singapore MO Open, 5

Tags: sum of powers , inequalities , algebra

Let $a, b, c, d$ be four positive real numbers. Prove that $$a^{10} + b^{10}+c^{10} + d^{10} \ge (0.1a + 0.2b + 0.3c + 0.4d)^{10} + (0.4a + 0.3b + 0.2c + 0.ld)^{10} + (0.2a + 0.4b + 0.1c + 0.3d)^{10} + (0.3a + 0.1b + 0.4c + 0.2d)^{10}$$

2013 APMO, 2

Tags: modular arithmetic , inequalities , quadratic , algebra , floor function , function

Determine all positive integers $n$ for which $\dfrac{n^2+1}{[\sqrt{n}]^2+2}$ is an integer. Here $[r]$ denotes the greatest integer less than or equal to $r$.

2022 Bulgarian Autumn Math Competition, Problem 12.3

Tags: inequalities

The sequence $a_{n}$ is defined by $a_{1}\geq 2$ and the recurrence formula \[a_{n+1}=a_{n}\sqrt{\frac{a_{n}^3+2}{2(a_{n}^3+1)}}\] for $n\geq 1$. Prove that for every integer $n$, the inequality $a_{n}>\sqrt{\frac{3}{n}}$ holds.

2011 China Team Selection Test, 3

Tags: inequalities

Let $n$ be a positive integer. Find the largest real number $\lambda$ such that for all positive real numbers $x_1,x_2,\cdots,x_{2n}$ satisfying the inequality \[\frac{1}{2n}\sum_{i=1}^{2n}(x_i+2)^n\geq \prod_{i=1}^{2n} x_i,\] the following inequality also holds \[\frac{1}{2n}\sum_{i=1}^{2n}(x_i+1)^n\geq \lambda\prod_{i=1}^{2n} x_i.\]

2012 Singapore Senior Math Olympiad, 4

Tags: recurrence relation , inequalities , sequence , algebra

Let $a_1, a_2, ..., a_n, a_{n+1}$ be a finite sequence of real numbers satisfying $a_0 = a_{n+1} = 0$ and $|a_{k-1} - 2a_{k} + a_{k+1}| \leq 1$ for $k = 1, 2, ..., n$ Prove that for $k=0, 1, ..., n+1,$ $|a_k| \leq \frac{k(n+1-k)}{2}$

2016 Macedonia National Olympiad, Problem 5

Tags: inequalities

Let $n\ge3$ and $a_1,a_2,...,a_n \in \mathbb{R^{+}}$, such that $\frac{1}{1+a_1^4} + \frac{1}{1+a_2^4} + ... + \frac{1}{1+a_n^4} = 1$. Prove that: $$a_1a_2...a_n \ge (n-1)^{\frac n4}$$

2007 China National Olympiad, 1

Tags: complex number , inequalities

Given complex numbers $a, b, c$, let $|a+b|=m, |a-b|=n$. If $mn \neq 0$, Show that \[\max \{|ac+b|,|a+bc|\} \geq \frac{mn}{\sqrt{m^2+n^2}}\]

2021 Iran MO (2nd Round), 5

Tags: combinatorics , algebra , inequalities

1400 real numbers are given. Prove that one can choose three of them like $x,y,z$ such that : $$\left|\frac{(x-y)(y-z)(z-x)}{x^4+y^4+z^4+1}\right| < 0.009$$

2009 Junior Balkan MO, 3

Tags: symmetry , inequalities

Let $ x$, $ y$, $ z$ be real numbers such that $ 0 < x,y,z < 1$ and $ xyz \equal{} (1 \minus{} x)(1 \minus{} y)(1 \minus{} z)$. Show that at least one of the numbers $ (1 \minus{} x)y,(1 \minus{} y)z,(1 \minus{} z)x$ is greater than or equal to $ \frac {1}{4}$