Olympiad problems

2020 South East Mathematical Olympiad, 4

Let $0\leq a_1\leq a_2\leq \cdots\leq a_{n-1}\leq a_n $ and $a_1+a_2+\cdots+a_n=1.$ Prove that: For any non-negative numbers $x_1,x_2,\cdots,x_n ; y_1, y_2,\cdots, y_n$ , have $$\left(\sum_{i=1}^n a_ix_i - \prod_{i=1}^n x_i^{a_i}\right) \left(\sum_{i=1}^n a_iy_i - \prod_{i=1}^n y_i^{a_i}\right) \leq a_n^2\left(n\sqrt{\sum_{i=1}^n x_i\sum_{i=1}^n y_i} - \sum_{i=1}^n\sqrt{x_i} \sum_{i=1}^n\sqrt{y_i}\right)^2.$$

2018 Tuymaada Olympiad, 7

Tags: inequalities , algebra

Prove the inequality $$(x^3+2y^2+3z)(4y^3+5z^2+6x)(7z^3+8x^2+9y)\geq720(xy+yz+xz)$$ for $x, y, z \geq 1$. [i]Proposed by K. Kokhas[/i]

1999 Mongolian Mathematical Olympiad, Problem 3

Tags: rectangle , trigonometry , geometry , analytic geometry , inequalities

I couldn't solve this problem and the only solution I was able to find was very unnatural (it was an official solution, I think) and I couldn't be satisfied with it, so I ask you if you can find some different solutions. The problem is really great one! If $M$ is the centroid of a triangle $ABC$, prove that the following inequality holds: \[\sin\angle CAM+\sin\angle CBM\leq\frac{2}{\sqrt3}.\] The equality occurs in a very strange case, I don't remember it.

1992 Tournament Of Towns, (326) 3

Tags: algebra , inequalities , compare , radical

Let $n, m, k$ be natural numbers, with $m > n$. Which of the numbers is greater: $$\sqrt{n+\sqrt{m+\sqrt{n+...}}}\,\,\, or \,\,\,\, \sqrt{m+\sqrt{n+\sqrt{m+...}}}\,\, ?$$ Note: Each of the expressions contains $k$ square root signs; $n, m$ alternate within each expression. (N. Kurlandchik)

KoMaL A Problems 2017/2018, A. 721

Tags: algebra , inequalities

Let $n\ge 2$ be a positive integer, and suppose $a_1,a_2,\cdots ,a_n$ are positive real numbers whose sum is $1$ and whose squares add up to $S$. Prove that if $b_i=\tfrac{a^2_i}{S} \;(i=1,\cdots ,n)$, then for every $r>0$, we have $$\sum_{i=1}^n \frac{a_i}{{(1-a_i)}^r}\le \sum_{i=1}^n \frac{b_i}{{(1-b_i)}^r}.$$

2010 All-Russian Olympiad Regional Round, 10.5

Tags: inequalities , algebra

Non-zero numbers $a, b, c$ are such that $ax^2+bx+c > cx$ for any $x$. Prove that $cx^2-bx + a > cx-b$ for any $x$.

2015 China Team Selection Test, 2

Tags: inequalities

Let $a_1,a_2,a_3, \cdots ,a_n$ be positive real numbers. For the integers $n\ge 2$, prove that\[ \left (\frac{\sum_{j=1}^{n} \left (\prod_{k=1}^{j}a_k \right )^{\frac{1}{j}}}{\sum_{j=1}^{n}a_j} \right )^{\frac{1}{n}}+\frac{\left (\prod_{i=1}^{n}a_i \right )^{\frac{1}{n}}}{\sum_{j=1}^{n} \left (\prod_{k=1}^{j}a_k \right )^{\frac{1}{j}}}\le \frac{n+1}{n}\]

1999 Romania National Olympiad, 2b

Tags: algebra , inequalities

Let $a, b, c$ be positive real numbers such that $ab +be + ba \le 3abc$. Prove that $$a^3+b^3+c^3 \ge a+b+c.$$

MathLinks Contest 5th, 4.3

Tags: inequalities , algebra

Let $a_1,..., a_n$ be positive reals and let $x_1, ... , x_n$ be real numbers such that $a_1x_1 +...+ a_nx_n = 0$. Prove that $$\sum_{1\le i<j \le n} x_ix_j |a_i - a_j | \le 0.$$ When does the equality take place?

2019 Dutch Mathematical Olympiad, 4

Tags: inequalities , fibonacci sequence , fibonacci , sum , algebra

The sequence of Fibonacci numbers $F_0, F_1, F_2, . . .$ is defined by $F_0 = F_1 = 1 $ and $F_{n+2} = F_n+F_{n+1}$ for all $n > 0$. For example, we have $F_2 = F_0 + F_1 = 2, F_3 = F_1 + F_2 = 3, F_4 = F_2 + F_3 = 5$, and $F_5 = F_3 + F_4 = 8$. The sequence $a_0, a_1, a_2, ...$ is defined by $a_n =\frac{1}{F_nF_{n+2}}$ for all $n \ge 0$. Prove that for all $m \ge 0$ we have: $a_0 + a_1 + a_2 + ... + a_m < 1$.

2011 JBMO Shortlist, 1

Tags: inequalities

Let $a,b,c$ be positive real numbers such that $abc = 1$. Prove that: $\displaystyle\prod(a^5+a^4+a^3+a^2+a+1)\geq 8(a^2+a+1)(b^2+b+1)(c^2+c+1)$

1990 Federal Competition For Advanced Students, P2, 2

Tags: algebra , inequalities

Show that for all integers $ n \ge 2$, $ \sqrt { 2\sqrt[3]{3 \sqrt[4]{4...\sqrt[n]{n}}}}<2$

2010 Ukraine Team Selection Test, 8

Tags: algebra , inequalities , sequence

Consider an infinite sequence of positive integers in which each positive integer occurs exactly once. Let $\{a_n\}, n\ge 1$ be such a sequence. We call it [i]consistent [/i] if, for an arbitrary natural $k$ and every natural $n ,m$ such that $a_n <a_m$, the inequality $a_{kn} <a _{km}$ also holds. For example, the sequence $a_n = n$ is consistent . a) Prove that there are consistent sequences other than $a_n = n$. b) Are there consistent sequences for which $a_n \ne n, n\ge 2$ ? c) Are there consistent sequences for which $a n \ne n, n\ge 1$ ?

2016 Azerbaijan BMO TST, 1

Tags: algebra , inequalities

Let $a,b,c$ be nonnegative real numbers.Prove that $3(a^2+b^2+c^2)\ge (a+b+c)(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})+(a-b)^2+(b-c)^2+(c-a)^2\ge (a+b+c)^2$.

2008 Vietnam National Olympiad, 6

Tags: inequalities , quadratic , algebra

Let $ x, y, z$ be distinct non-negative real numbers. Prove that \[ \frac{1}{(x\minus{}y)^2} \plus{} \frac{1}{(y\minus{}z)^2} \plus{} \frac{1}{(z\minus{}x)^2} \geq \frac{4}{xy \plus{} yz \plus{} zx}.\] When does the equality hold?

2008 Gheorghe Vranceanu, 1

Tags: algebra , inequalities , function

Determine all functions $ f: \mathbb{R} \to \mathbb{R}$ satisfying the condition $ f(xy) \le xf(y)$ for all real numbers $ x$ and $ y$.

2011 Saudi Arabia Pre-TST, 1.1

Tags: inequalities , algebra

Let $a, b, c$ be positive real numbers. Prove that $$8(a+b+c) \left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \right) \le 9 \left(1+\frac{a}{b} \right)\left(1+\frac{b}{c} \right)\left(1+\frac{c}{a} \right)$$

2003 Junior Balkan Team Selection Tests - Moldova, 2

Tags: inequalities

Let $a, b, c>0$ such that $a^{2}+b^{2}+c^{2}=3abc.$ Prove the following inequality: \[ \frac{a}{b^{2}c^{2}}+\frac{b}{c^{2}a^{2}}+\frac{c}{a^{2}b^{2}}\geq\frac{9}{a+b+c} \]

1993 India Regional Mathematical Olympiad, 6

Tags: algebra , inequalities

If $a,b,c,d$ are four positive reals such that $abcd= 1$ , prove that $(1+a) (1+b) (1 +c ) (1 +d ) \geq 16.$

2002 Iran Team Selection Test, 9

Tags: function , number theory , integration , calculus , inequalities

$\pi(n)$ is the number of primes that are not bigger than $n$. For $n=2,3,4,6,8,33,\dots$ we have $\pi(n)|n$. Does exist infinitely many integers $n$ that $\pi(n)|n$?

1977 Poland - Second Round, 2

Tags: geometry , inequalities , geometric inequality

Let $X$ be the interior point of triangle $ABC$. prove that the product of the distances of point $ X $ from the vertices $ A, B, C $ is at least eight times greater than the product of the distances of this point from the lines $ AB, BC, CA $.

2008 Harvard-MIT Mathematics Tournament, 6

Tags: inequalities , conic , quadratic , parabola , analytic geometry , absolute value , function

Determine all real numbers $ a$ such that the inequality $ |x^2 \plus{} 2ax \plus{} 3a|\le2$ has exactly one solution in $ x$.

2005 China Team Selection Test, 2

Tags: inequalities

Let $a$, $b$, $c$ be nonnegative reals such that $ab+bc+ca = \frac{1}{3}$. Prove that \[\frac{1}{a^{2}-bc+1}+\frac{1}{b^{2}-ca+1}+\frac{1}{c^{2}-ab+1}\leq 3 \]

2004 Croatia National Olympiad, Problem 1

Tags: inequality , complex inequality , complex number , inequalities

Let $z_1,\ldots,z_n$ and $w_1,\ldots,w_n$ $(n\in\mathbb N)$ be complex numbers such that $$|\epsilon_1z_1+\ldots+\epsilon_nz_n|\le|\epsilon_1w_1+\ldots+\epsilon_nw_n|$$holds for every choice of $\epsilon_1,\ldots,\epsilon_n\in\{-1,1\}$. Prove that $$|z_1|^2+\ldots+|z_n|^2\le|w_1|^2+\ldots+|w_n|^2.$$

2022 Turkey EGMO TST, 6

Tags: inequalities , ratio , minimum value

Let $x,y,z$ be positive real numbers satisfying the equations $$xyz=1\text{ and }\frac yz(y-x^2)+\frac zx(z-y^2)+\frac xy(x-z^2)=0$$ What is the minimum value of the ratio of the sum of the largest and smallest numbers among $x,y,z$ to the median of them.