Olympiad problems

2010 Contests, 3

Tags: inequalities

let $n>2$ be a fixed integer.positive reals $a_i\le 1$(for all $1\le i\le n$).for all $k=1,2,...,n$,let $A_k=\frac{\sum_{i=1}^{k}a_i}{k}$ prove that $|\sum_{k=1}^{n}a_k-\sum_{k=1}^{n}A_k|<\frac{n-1}{2}$.

2009 Singapore Senior Math Olympiad, 4

Tags: inequalities , weighted am-gm , logarithm

Given that $ a,b,c, x_1, x_2, ... , x_5 $ are real positives such that $ a+b+c=1 $ and $ x_1.x_2.x_3.x_4.x_5 = 1 $. Prove that \[ (ax_1^2+bx_1+c)(ax_2^2+bx_2+c)...(ax_5^2+bx_5+c)\ge 1\]

2009 IMO Shortlist, 7

Tags: geometry , incenter , reflection , inequalities

Let $ABC$ be a triangle with incenter $I$ and let $X$, $Y$ and $Z$ be the incenters of the triangles $BIC$, $CIA$ and $AIB$, respectively. Let the triangle $XYZ$ be equilateral. Prove that $ABC$ is equilateral too. [i]Proposed by Mirsaleh Bahavarnia, Iran[/i]

2009 Hanoi Open Mathematics Competitions, 10

Tags: inequalities , geometric inequality , geometry

Prove that $d^2+(a-b)^2<c^2$ ,where $d$ is diameter of the inscribed circle of $\vartriangle ABC$

2010 Today's Calculation Of Integral, 558

Tags: calculus , integration , quadratic , function , inequalities , algebra , calculus computations

For a positive constant $ t$, let $ \alpha ,\ \beta$ be the roots of the quadratic equation $ x^2 \plus{} t^2x \minus{} 2t \equal{} 0$. Find the minimum value of $ \int_{ \minus{} 1}^2 \left\{\left(x \plus{} \frac {1}{\alpha ^ 2}\right)\left(x \plus{} \frac {1}{\beta ^ 2}\right) \plus{} \frac {1}{\alpha \beta}\right\}dx.$

2016 Korea - Final Round, 4

Tags: inequality , algebra , inequalities

If $x,y,z$ satisfies $x^2+y^2+z^2=1$, find the maximum possible value of $$(x^2-yz)(y^2-zx)(z^2-xy)$$

2008 Junior Balkan Team Selection Tests - Moldova, 2

Tags: algebra , inequalities

[b]BJ2. [/b] Positive real numbers $ a,b,c$ satisfy inequality $ \frac {3}{2}\geq a \plus{} b \plus{} c$. Find the smallest possible value for $ S \equal{} abc \plus{} \frac {1}{abc}$

2008 AMC 12/AHSME, 19

Tags: function , inequalities , complex number , triangle inequality

A function $ f$ is defined by $ f(z) \equal{} (4 \plus{} i) z^2 \plus{} \alpha z \plus{} \gamma$ for all complex numbers $ z$, where $ \alpha$ and $ \gamma$ are complex numbers and $ i^2 \equal{} \minus{} 1$. Suppose that $ f(1)$ and $ f(i)$ are both real. What is the smallest possible value of $ | \alpha | \plus{} |\gamma |$? $ \textbf{(A)} \; 1 \qquad \textbf{(B)} \; \sqrt {2} \qquad \textbf{(C)} \; 2 \qquad \textbf{(D)} \; 2 \sqrt {2} \qquad \textbf{(E)} \; 4 \qquad$

2018 India IMO Training Camp, 3

Tags: sequence , inequalities , algebra

Let $a_n, b_n$ be sequences of positive reals such that,$$a_{n+1}= a_n + \frac{1}{2b_n}$$ $$b_{n+1}= b_n + \frac{1}{2a_n}$$ for all $n\in\mathbb N$. Prove that, $\text{max}\left(a_{2018}, b_{2018}\right) >44$.

2012 Balkan MO Shortlist, C1

Tags: induction , inequalities , floor function , combinatorics , logarithm

Let $n$ be a positive integer. Let $P_n=\{2^n,2^{n-1}\cdot 3, 2^{n-2}\cdot 3^2, \dots, 3^n \}.$ For each subset $X$ of $P_n$, we write $S_X$ for the sum of all elements of $X$, with the convention that $S_{\emptyset}=0$ where $\emptyset$ is the empty set. Suppose that $y$ is a real number with $0 \leq y \leq 3^{n+1}-2^{n+1}.$ Prove that there is a subset $Y$ of $P_n$ such that $0 \leq y-S_Y < 2^n$

2005 France Pre-TST, 2

Tags: inequalities

Let $\omega (n)$ denote the number of prime divisors of the integer $n>1$. Find the least integer $k$ such that the inequality $2^{\omega (n) } \leq k \cdot n^{\frac 1 4}$ holds for all $n > 1.$ Pierre.

2001 Italy TST, 2

Tags: algebra , inequalities

Let $0\le a\le b\le c$ be real numbers. Prove that \[(a+3b)(b+4c)(c+2a)\ge 60abc \]

2024 China Team Selection Test, 21

Tags: inequalities

Let integer $n\ge 3,$ $\tbinom n2$ nonnegative real numbers $a_{i,j}$ satisfy $ a_{i,j}+a_{j,k}\le a_{i,k}$ holds for all $1\le i <j<k\le n$. Proof $$\left\lfloor\frac{n^2}4\right\rfloor\sum_{1\le i<j\le n}a_{i,j}^4\ge \left(\sum_{1\le i<j\le n}a_{i,j}^2\right)^2.$$ [i]Proposed by Jingjun Han, Dongyi Wei[/i]

2016 Latvia Baltic Way TST, 2

Tags: number theory , inequalities , algebra

Given natural numbers $m, n$ and $X$ such that $X \ge m$ and $X \ge n$. Prove that one can find two integers $u$ and $v$ such that $|u| + |v| > 0$, $|u| \le \sqrt{X}$, $|v| \le \sqrt{X}$ and $$0 \le mu + nv \le 2 \sqrt{X}.$$

MIPT Undergraduate Contest 2019, 2.2

Tags: inequalities , probability and stats , expectation , calculus , integration , probability , function

Petya and Vasya are playing the following game. Petya chooses a non-negative random value $\xi$ with expectation $\mathbb{E} [\xi ] = 1$, after which Vasya chooses his own value $\eta$ with expectation $\mathbb{E} [\eta ] = 1$ without reference to the value of $\xi$. For which maximal value $p$ can Petya choose a value $\xi$ in such a way that for any choice of Vasya's $\eta$, the inequality $\mathbb{P}[\eta \geq \xi ] \leq p$ holds?

1957 AMC 12/AHSME, 29

Tags: inequalities

The relation $ x^2(x^2 \minus{} 1)\ge 0$ is true only for: $ \textbf{(A)}\ x \ge 1\qquad \textbf{(B)}\ \minus{} 1 \le x \le 1\qquad \textbf{(C)}\ x \equal{} 0,\, x \equal{} 1,\, x \equal{} \minus{} 1\qquad \\\textbf{(D)}\ x \equal{} 0,\, x \le \minus{} 1,\, x \ge 1\qquad \textbf{(E)}\ x \ge 0$

2010 Contests, 2

Tags: search , inequalities

$a,b,c$ are positive real numbers. prove the following inequality: $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{(a+b+c)^2}\ge \frac{7}{25}(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{a+b+c})^2$ (20 points)

2022 South East Mathematical Olympiad, 5

Tags: inequalities , algebra , number theory

Let $a,b,c,d$ be non-negative integers. $(1)$ If $a^2+b^2-cd^2=2022 ,$ find the minimum of $a+b+c+d;$ $(1)$ If $a^2-b^2+cd^2=2022 ,$ find the minimum of $a+b+c+d .$

2010 China Second Round Olympiad, 3

Tags: inequalities

let $n>2$ be a fixed integer.positive reals $a_i\le 1$(for all $1\le i\le n$).for all $k=1,2,...,n$,let $A_k=\frac{\sum_{i=1}^{k}a_i}{k}$ prove that $|\sum_{k=1}^{n}a_k-\sum_{k=1}^{n}A_k|<\frac{n-1}{2}$.

2014 India IMO Training Camp, 2

Tags: inequalities

Let $a,b$ be positive real numbers.Prove that $(1+a)^{8}+(1+b)^{8}\geq 128ab(a+b)^{2}$.

2012 China Western Mathematical Olympiad, 4

Tags: inequalities , circumcircle , geometry

$P$ is a point in the $\Delta ABC$, $\omega $ is the circumcircle of $\Delta ABC $. $BP \cap \omega = \left\{ {B,{B_1}} \right\}$,$CP \cap \omega = \left\{ {C,{C_1}} \right\}$, $PE \bot AC$，$PF \bot AB$. The radius of the inscribed circle and circumcircle of $\Delta ABC $ is $r,R$. Prove $\frac{{EF}}{{{B_1}{C_1}}} \geqslant \frac{r}{R}$.

2024 Moldova EGMO TST, 5

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

$AD$ Is the angle bisector Of $\angle BAC$ Where $D$ lies on the The circumcircle of $\triangle ABC$. Show that $2AD>AB+AC$

2021 Switzerland - Final Round, 4

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

Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

2024 China National Olympiad, 2

Tags: inequalities , algebra

Find the largest real number $c$ such that $$\sum_{i=1}^{n}\sum_{j=1}^{n}(n-|i-j|)x_ix_j \geq c\sum_{j=1}^{n}x^2_i$$ for any positive integer $n $ and any real numbers $x_1,x_2,\dots,x_n.$

2012 China Team Selection Test, 1

Tags: inequalities , complex number , triangle inequality , algebra

Complex numbers ${x_i},{y_i}$ satisfy $\left| {{x_i}} \right| = \left| {{y_i}} \right| = 1$ for $i=1,2,\ldots ,n$. Let $x=\frac{1}{n}\sum\limits_{i=1}^n{{x_i}}$, $y=\frac{1}{n}\sum\limits_{i=1}^n{{y_i}}$ and $z_i=x{y_i}+y{x_i}-{x_i}{y_i}$. Prove that $\sum\limits_{i=1}^n{\left| {{z_i}}\right|}\leqslant n$.