Olympiad problems

2004 China Team Selection Test, 3

Tags: inequalities

Let $k \geq 2, 1 < n_1 < n_2 < \ldots < n_k$ are positive integers, $a,b \in \mathbb{Z}^+$ satisfy \[ \prod^k_{i=1} \left( 1 - \frac{1}{n_i} \right) \leq \frac{a}{b} < \prod^{k-1}_{i=1} \left( 1 - \frac{1}{n_i} \right) \] Prove that: \[ \prod^k_{i=1} n_i \geq (4 \cdot a)^{2^k - 1}. \]

1976 AMC 12/AHSME, 23

Tags: inequalities , binomial coefficient

For integers $k$ and $n$ such that $1\le k<n$, let $C^n_k=\frac{n!}{k!(n-k)!}$. Then $\left(\frac{n-2k-1}{k+1}\right)C^n_k$ is an integer $\textbf{(A) }\text{for all }k\text{ and }n\qquad$ $\textbf{(B) }\text{for all even values of }k\text{ and }n,\text{ but not for all }k\text{ and }n\qquad$ $\textbf{(C) }\text{for all odd values of }k\text{ and }n,\text{ but not for all }k\text{ and }n\qquad$ $\textbf{(D) }\text{if }k=1\text{ or }n-1,\text{ but not for all odd values }k\text{ and }n\qquad$ $\textbf{(E) }\text{if }n\text{ is divisible by }k,\text{ but not for all even values }k\text{ and }n$

2018 Balkan MO Shortlist, A6

Tags: inequalities

Let $ x_1, x_2, \cdots, x_n$ be positive real numbers . Prove that: $$\sum_ {i = 1}^n x_i ^2\geq \frac {1} {n + 1} \left (\sum_ {i = 1}^n x_i \right)^2+\frac{12(\sum_ {i = 1}^n i x_i)^2}{n (n + 1) (n + 2) (3n + 1)}. $$

2017 ELMO Shortlist, 2

Tags: inequalities , algebra

Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for all real numbers $a,b,$ and $c$: (i) If $a+b+c\ge 0$ then $f(a^3)+f(b^3)+f(c^3)\ge 3f(abc).$ (ii) If $a+b+c\le 0$ then $f(a^3)+f(b^3)+f(c^3)\le 3f(abc).$ [i]Proposed by Ashwin Sah[/i]

2009 Today's Calculation Of Integral, 461

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

Let $ I_n\equal{}\int_0^{\sqrt{3}} \frac{1}{1\plus{}x^{n}}\ dx\ (n\equal{}1,\ 2,\ \cdots)$. (1) Find $ I_1,\ I_2$. (2) Find $ \lim_{n\to\infty} I_n$.

2006 Estonia Team Selection Test, 5

Tags: inequalities , mean , algebra

Let $a_1, a_2, a_3, ...$ be a sequence of positive real numbers. Prove that for any positive integer $n$ the inequality holds $\sum_{i=1}^n b_i^2 \le 4 \sum_{i=1}^n a_i^2$ where $b_i$ is the arithmetic mean of the numbers $a_1, a_2, ..., a_n$

2012 Puerto Rico Team Selection Test, 1

Tags: inequalities , algebra

Let $x, y$ and $z$ be consecutive integers such that \[\frac 1x+\frac 1y+\frac 1z >\frac{1}{45}.\] Find the maximum value of $x + y + z$.

VI Soros Olympiad 1999 - 2000 (Russia), 9.4

Tags: algebra , inequalities

For real numbers $x \ge 0$ and $y \ge 0$, prove the inequality $$x^4+y^3+x^2+y+1 >\frac92 xy.$$

1969 Vietnam National Olympiad, 3

Tags: inequalities , algebra

Consider $x_1 > 0, y_1 > 0, x_2 < 0, y_2 > 0, x_3 < 0, y_3 < 0, x_4 > 0, y_4 < 0.$ Suppose that for each $i = 1, ... ,4$ we have $ (x_i -a)^2 +(y_i -b)^2 \le c^2$. Prove that $a^2 + b^2 < c^2$. Restate this fact in the form of geometric result in plane geometry.

2013 VTRMC, Problem 5

Tags: inequalities

Prove that $$\frac{x}{\sqrt{1+x^2}}+\frac{y}{\sqrt{1+y^2}}+\frac{z}{\sqrt{1+z^2}} \leq\frac{3\sqrt{3}}{2}$$ for any positive real numbers $x, y,z$ such that $x+y+z = xyz.$ [url=https://artofproblemsolving.com/community/c7h236610p10925499]2008 VTRMC #1[/url] [url=http://www.math.vt.edu/people/plinnell/Vtregional/solutions.pdf]here[/url]

2006 Romania Team Selection Test, 4

Tags: inequalities , calculus , algebra , three variable inequality

Let $a,b,c$ be positive real numbers such that $a+b+c=3$. Prove that: \[ \frac 1{a^2}+\frac 1{b^2}+\frac 1{c^2} \geq a^2+b^2+c^2. \]

2010 Greece National Olympiad, 2

Tags: inequalities

If $ x,y$ are positive real numbers with sum $ 2a$, prove that : $ x^3y^3(x^2\plus{}y^2)^2 \leq 4a^{10}$ When does equality hold ? Babis

2011 CentroAmerican, 5

Tags: inequalities , algebra

If $x$, $y$, $z$ are positive numbers satisfying \[x+\frac{y}{z}=y+\frac{z}{x}=z+\frac{x}{y}=2.\] Find all the possible values of $x+y+z$.

2007 Iran MO (3rd Round), 2

Tags: inequalities

$ a,b,c$ are three different positive real numbers. Prove that:\[ \left|\frac{a\plus{}b}{a\minus{}b}\plus{}\frac{b\plus{}c}{b\minus{}c}\plus{}\frac{c\plus{}a}{c\minus{}a}\right|>1\]

1959 AMC 12/AHSME, 25

Tags: inequalities , absolute value , inequality

The symbol $|a|$ means $+a$ if $a$ is greater than or equal to zero, and $-a$ if $a$ is less than or equal to zero; the symbol $<$ means "less than"; the symbol $>$ means "greater than." The set of values $x$ satisfying the inequality $|3-x|<4$ consists of all $x$ such that: $ \textbf{(A)}\ x^2<49 \qquad\textbf{(B)}\ x^2>1 \qquad\textbf{(C)}\ 1<x^2<49\qquad\textbf{(D)}\ -1<x<7\qquad\textbf{(E)}\ -7<x<1 $

2011 China Northern MO, 7

Tags: trigonometry , inequalities

In $\triangle ABC$ , then \[\frac{1}{1+\cos^2 A+\cos^2 B}+\frac{1}{1+\cos^2 B+\cos^2 C}+\frac{1}{1+\cos^2 C+\cos^2 A}\le 2\]

2004 Federal Math Competition of S&M, 3

Tags: inequalities

Let $M, N, P$ be arbitrary points on the sides $BC, CA, AB$ respectively of an acute-angled triangle $ABC$. Prove that at least one of the following inequalities is satisfied: $NP \geq \frac{1}{2}BC; PM \geq \frac{1}{2}CA; MN \geq \frac{1}{2}AB$

2010 Indonesia TST, 3

Tags: inequalities , induction , algebra

Let $ a_1,a_2,\dots$ be sequence of real numbers such that $ a_1\equal{}1$, $ a_2\equal{}\dfrac{4}{3}$, and \[ a_{n\plus{}1}\equal{}\sqrt{1\plus{}a_na_{n\minus{}1}}, \quad \forall n \ge 2.\] Prove that for all $ n \ge 2$, \[ a_n^2>a_{n\minus{}1}^2\plus{}\dfrac{1}{2}\] and \[ 1\plus{}\dfrac{1}{a_1}\plus{}\dfrac{1}{a_2}\plus{}\dots\plus{}\dfrac{1}{a_n}>2a_n.\] [i]Fajar Yuliawan, Bandung[/i]

2014 Chile National Olympiad, 1

Tags: inequalities , algebra

Let $a, b,c$ real numbers that are greater than $ 0$ and less than $1$. Show that there is at least one of these three values $ab(1-c)^2$, $bc(1-a)^2$ , $ca(1- b)^2$ which is less than or equal to $\frac{1}{16}$ .

2016 Stars of Mathematics, 3

Tags: inequalities

Let $ n $ be a natural number, and $ 2n $ nonnegative real numbers $ a_1,a_2,\ldots ,a_{2n} $ such that $ a_1a_2\cdots a_{2n}=1. $ Show that $$ 2^{n+1} +\left( a_1^2+a_2^2 \right)\left( a_3^2+a_4^2 \right)\cdots\left( a_{2n-1}^2+a_{2n}^2 \right) \ge 3\left( a_1+a_2 \right)\left( a_3+a_4 \right)\cdots\left( a_{2n-1}+a_{2n} \right) , $$ and specify in which circumstances equality happens.

2005 China Team Selection Test, 3

Tags: inequalities , algebra , polynomial , computer solutions , 4-variable inequality , symmetric inequality

Let $a,b,c,d >0$ and $abcd=1$. Prove that: \[ \frac{1}{(1+a)^2}+\frac{1}{(1+b)^2}+\frac{1}{(1+c)^2}+\frac{1}{(1+d)^2} \geq 1 \]

2015 Thailand Mathematical Olympiad, 2

Tags: algebra , inequalities

Let $a, b, c$ be positive reals with $abc = 1$. Prove the inequality $$\frac{a^5}{a^3 + 1}+\frac{b^5}{b^3 + 1}+\frac{c^5}{c^3 + 1} \ge \frac32$$ and determine all values of a, b, c for which equality is attained

2008 JBMO Shortlist, 8

Tags: inequalities , algebra

Show that $(x + y + z) \big(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\big) \ge 4 \big(\frac{x}{xy+1}+\frac{y}{yz+1}+\frac{z}{zx+1}\big)^2$ , for all real positive numbers $x, y $ and $z$.

2009 Indonesia TST, 2

Tags: inequalities , relatively prime , number theory , logarithm

For every positive integer $ n$, let $ \phi(n)$ denotes the number of positive integers less than $ n$ that is relatively prime to $ n$ and $ \tau(n)$ denote the sum of all positive divisors of $ n$. Let $ n$ be a positive integer such that $ \phi(n)|n\minus{}1$ and that $ n$ is not a prime number. Prove that $ \tau(n)>2009$.

2014 Contests, 1

Tags: algebra , inequalities , sequence , unique

Let $a_0 < a_1 < a_2 < \dots$ be an infinite sequence of positive integers. Prove that there exists a unique integer $n\geq 1$ such that \[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\] [i]Proposed by Gerhard Wöginger, Austria.[/i]