Olympiad problems

2012 AMC 12/AHSME, 23

Consider all polynomials of a complex variable, $P(z)=4z^4+az^3+bz^2+cz+d$, where $a, b, c$ and $d$ are integers, $0 \le d \le c \le b \le a \le 4$, and the polynomial has a zero $z_0$ with $|z_0|=1$. What is the sum of all values $P(1)$ over all the polynomials with these properties? $ \textbf{(A)}\ 84\qquad\textbf{(B)}\ 92\qquad\textbf{(C)}\ 100\qquad\textbf{(D)}\ 108 \qquad\textbf{(E)}\ 120 $

2008 Bulgaria National Olympiad, 3

Tags: induction , algebra , trigonometry , inequalities , modular arithmetic , ceiling function , pigeonhole principle

Let $n\in\mathbb{N}$ and $0\leq a_1\leq a_2\leq\ldots\leq a_n\leq\pi$ and $b_1,b_2,\ldots ,b_n$ are real numbers for which the following inequality is satisfied : \[\left|\sum_{i\equal{}1}^{n} b_i\cos(ka_i)\right|<\frac{1}{k}\] for all $ k\in\mathbb{N}$. Prove that $ b_1\equal{}b_2\equal{}\ldots \equal{}b_n\equal{}0$.

2009 Croatia Team Selection Test, 1

Tags: algebra , inequalities

Solve in the set of real numbers: \[ 3\left(x^2 \plus{} y^2 \plus{} z^2\right) \equal{} 1, \] \[ x^2y^2 \plus{} y^2z^2 \plus{} z^2x^2 \equal{} xyz\left(x \plus{} y \plus{} z\right)^3. \]

2024 Iran Team Selection Test, 3

Tags: inequalities , algebra

For any real numbers $x , y ,z$ prove that : $$(x+y+z)^2 + \sum_{cyc}{\frac{(x+y)(y+z)}{1+|x-z|}} \ge xy+yz+zx$$ [i]Proposed by Navid Safaei[/i]

1996 Putnam, 2

Tags: geometry , inequalities , induction , rectangle , logarithm , integration

Prove the inequality for all positive integer $n$ : \[ \left(\frac{2n-1}{e}\right)^{\frac{2n-1}{2}}<1\cdot 3\cdot 5\cdots (2n-1)<\left(\frac{2n+1}{e}\right)^{\frac{2n+1}{2}} \]

2019 Jozsef Wildt International Math Competition, W. 23

Tags: inequalities , triangle

If $b$, $c$ are the legs, and $a$ is the hypotenuse of a right triangle, prove that$$\left(a+b+c\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\geq 5+3\sqrt{2}$$

2000 Putnam, 1

Tags: inequalities , geometric series , putnam inequalities , function

Let $A$ be a positive real number. What are the possible values of $\displaystyle\sum_{j=0}^{\infty} x_j^2, $ given that $x_0, x_1, \cdots$ are positive numbers for which $\displaystyle\sum_{j=0}^{\infty} x_j = A$?

1989 Irish Math Olympiad, 3

Tags: geometry , trigonometry , inequalities

Suppose P is a point in the interior of a triangle ABC, that x; y; z are the distances from P to A; B; C, respectively, and that p; q; r are the per- pendicular distances from P to the sides BC; CA; AB, respectively. Prove that $xyz \geq 8pqr$; with equality implying that the triangle ABC is equilateral.

2010 China Girls Math Olympiad, 4

Tags: inequalities

Let $x_1,x_2,\cdots,x_n$ be real numbers with $x_1^2+x_2^2+\cdots+x_n^2=1$. Prove that \[\sum_{k=1}^{n}\left(1-\dfrac{k}{{\displaystyle \sum_{i=1}^{n} ix_i^2}}\right)^2 \cdot \dfrac{x_k^2}{k} \leq \left(\dfrac{n-1}{n+1}\right)^2 \sum_{k=1}^{n} \dfrac{x_k^2}{k}\] Determine when does the equality hold?

2013 ELMO Shortlist, 2

Tags: inequalities

Prove that for all positive reals $a,b,c$, \[\frac{1}{a+\frac{1}{b}+1}+\frac{1}{b+\frac{1}{c}+1}+\frac{1}{c+\frac{1}{a}+1}\ge \frac{3}{\sqrt[3]{abc}+\frac{1}{\sqrt[3]{abc}}+1}. \][i]Proposed by David Stoner[/i]

1970 IMO Longlists, 1

Tags: inequalities

Prove that $\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{ca}{c+a}\le \frac{a+b+c}{2}$, where $a,b,c\in\mathbb{R}^{+}$.

1995 Baltic Way, 6

Tags: search , inequalities

Prove that for positive $a,b,c,d$ \[\frac{a+c}{a+b}+\frac{b+d}{b+c}+\frac{c+a}{c+d}+\frac{d+b}{d+a}\ge 4\]

2000 USA Team Selection Test, 1

Tags: inequalities

Let $a, b, c$ be nonnegative real numbers. Prove that \[ \frac{a+b+c}{3} - \sqrt[3]{abc} \leq \max\{(\sqrt{a} - \sqrt{b})^2, (\sqrt{b} - \sqrt{c})^2, (\sqrt{c} - \sqrt{a})^2\}. \]

2000 Bosnia and Herzegovina Team Selection Test, 4

Tags: algebra , inequality , inequalities

Prove that for all positive real $a$, $b$ and $c$ holds: $$ \frac{bc}{a^2+2bc}+\frac{ac}{b^2+2ac}+\frac{ab}{c^2+2ab} \leq 1 \leq \frac{a^2}{a^2+2bc}+\frac{b^2}{b^2+2ac}+\frac{c^2}{c^2+2ab}$$

1987 AMC 12/AHSME, 23

Tags: quadratic , inequalities

If $p$ is a prime and both roots of $x^2+px-444p=0$ are integers, then $ \textbf{(A)}\ 1<p\le 11 \qquad\textbf{(B)}\ 11<p \le 21 \qquad\textbf{(C)}\ 21< p \le 31 \\ \qquad\textbf{(D)}\ 31< p \le 41 \qquad\textbf{(E)}\ 41< p \le 51 $

2021 Polish Junior MO Finals, 2

Tags: geometry , triangle geometry , inequalities

Point $M$ is the midpoint of the hypotenuse $AB$ of a right angled triangle $ABC$. Points $P$ and $Q$ lie on segments $AM$ and $MB$ respectively and $PQ=CQ$. Prove that $AP\leq 2\cdot MQ$.

2008 District Olympiad, 4

Tags: inequalities , algebra

Determine $ x,y,z>0$ for which $ x^3y\plus{}3<\equal{}4z, y^3z\plus{}3<\equal{}4x,z^3x\plus{}3<\equal{}4y.$

2020 Moldova Team Selection Test, 10

Tags: inequalities , maximum value , algebra

Let $n$ be a positive integer. Positive numbers $a$, $b$, $c$ satisfy $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$. Find the greatest possible value of $$E(a,b,c)=\frac{a^{n}}{a^{2n+1}+b^{2n} \cdot c + b \cdot c^{2n}}+\frac{b^{n}}{b^{2n+1}+c^{2n} \cdot a + c \cdot a^{2n}}+\frac{c^{n}}{c^{2n+1}+a^{2n} \cdot b + a \cdot b^{2n}}$$

2011 All-Russian Olympiad Regional Round, 11.8

Tags: inequalities

$b$ and $c$ are positive. Prove the inequality \[ \left(b-c\right)^{2011}\left(b+c\right)^{2011}\left(c-b\right)^{2011} \geq \left(b^{2011}-c^{2011}\right)\left(b^{2011}+c^{2011}\right)\left(c^{2011}-b^{2011}\right). \] (Author: V. Senderov)

2011 Vietnam Team Selection Test, 3

Tags: inequalities

Let $n$ be a positive integer $\geq 3.$ There are $n$ real numbers $x_1,x_2,\cdots x_n$ that satisfy: \[\left\{\begin{aligned}&\ x_1\ge x_2\ge\cdots \ge x_n;\\& \ x_1+x_2+\cdots+x_n=0;\\& \ x_1^2+x_2^2+\cdots+x_n^2=n(n-1).\end{aligned}\right.\] Find the maximum and minimum value of the sum $S=x_1+x_2.$

2002 China Team Selection Test, 2

Tags: induction , function , inequalities , algebra

For any two rational numbers $ p$ and $ q$ in the interval $ (0,1)$ and function $ f$, there is always $ \displaystyle f \left( \frac{p\plus{}q}{2} \right) \leq \frac{f(p) \plus{} f(q)}{2}$. Then prove that for any rational numbers $ \lambda, x_1, x_2 \in (0,1)$, there is always: \[ f( \lambda x_1 \plus{} (1\minus{}\lambda) x_2 ) \leq \lambda f(x_i) \plus{} (1\minus{}\lambda) f(x_2)\]

2016 Bosnia and Herzegovina Junior BMO TST, 4

Tags: algebra , inequalities

Let $x$, $y$ and $z$ be positive real numbers such that $\sqrt{xy} + \sqrt{yz} + \sqrt{zx} = 3$. Prove that $\sqrt{x^3+x} + \sqrt{y^3+y} + \sqrt{z^3+z} \geq \sqrt{6(x+y+z)}$

2020-IMOC, A3

Tags: inequalities , quadratic

$\definecolor{A}{RGB}{250,120,0}\color{A}\fbox{A3.}$ Assume that $a, b, c$ are positive reals such that $a + b + c = 3$. Prove that $$\definecolor{A}{RGB}{200,0,200}\color{A} \frac{1}{8a^2-18a+11}+\frac{1}{8b^2-18b+11}+\frac{1}{8c^2-18c+11}\le 3.$$ [i]Proposed by [/i][b][color=#419DAB]ltf0501[/color][/b]. [color=#3D9186]#1734[/color]

2023 Regional Olympiad of Mexico West, 3

Tags: algebra , inequalities , floor function

Let $x>1$ be a real number that is not an integer. Denote $\{x\}$ as its decimal part and $\lfloor x\rfloor$ the floor function. Prove that $$ \left(\frac{x+\{x\}}{\lfloor x\rfloor}-\frac{\lfloor x\rfloor}{x+\{x\}}\right)+\left(\frac{x+\lfloor x\rfloor}{\{x\}}-\frac{\{x\}}{x+\lfloor x\rfloor}\right)>\frac{16}{3}$$

2000 Finnish National High School Mathematics Competition, 3

Tags: logarithm , induction , inequalities , algebra

Determine the positive integers $n$ such that the inequality \[n! > \sqrt{n^n}\] holds.