Olympiad problems

2003 China Team Selection Test, 3

(1) $D$ is an arbitary point in $\triangle{ABC}$. Prove that: \[ \frac{BC}{\min{AD,BD,CD}} \geq \{ \begin{array}{c} \displaystyle 2\sin{A}, \ \angle{A}< 90^o \\ \\ 2, \ \angle{A} \geq 90^o \end{array} \] (2)$E$ is an arbitary point in convex quadrilateral $ABCD$. Denote $k$ the ratio of the largest and least distances of any two points among $A$, $B$, $C$, $D$, $E$. Prove that $k \geq 2\sin{70^o}$. Can equality be achieved?

2015 Azerbaijan JBMO TST, 1

Tags: inequalities

Let $x,y$ and $z$ be non-negative real numbers satisfying the equation $x+y+z=xyz$. Prove that $2(x^2+y^2+z^2)\geq3(x+y+z)$.

2024 Malaysia IMONST 2, 1

Tags: inequalities

Suppose $a, b, c, d$ are positive reals such that $a \geq b \geq c \geq d$ and $ab^2c^3d^4 = 1$. Help Janson prove that $a+b+c+d \geq 4$.

2009 China Team Selection Test, 1

Tags: inequalities , number theory

Let $ a > b > 1, b$ is an odd number, let $ n$ be a positive integer. If $ b^n|a^n\minus{}1,$ then $ a^b > \frac {3^n}{n}.$

2018 JBMO Shortlist, A5

Tags: algebra , inequalities

Let a$,b,c,d$ and $x,y,z,t$ be real numbers such that $0\le a,b,c,d \le 1$ , $x,y,z,t \ge 1$ and $a+b+c+d +x+y+z+t=8$. Prove that $a^2+b^2+c^2+d^2+x^2+y^2+z^2+t^2\le 28$

2001 Moldova National Olympiad, Problem 1

Tags: algebra , inequalities

Find all real solutions of the equation $$x^2+y^2+z^2+t^2=xy+yz+zt+t-\frac25.$$

2012 Korea National Olympiad, 2

Tags: inequalities , combinatorics

There are $n$ students $ A_1 , A_2 , \cdots , A_n $ and some of them shaked hands with each other. ($ A_i $ and $ A_j$ can shake hands more than one time.) Let the student $ A_i $ shaked hands $ d_i $ times. Suppose $ d_1 + d_2 + \cdots + d_n > 0 $. Prove that there exist $ 1 \le i < j \le n $ satisfying the following conditions: (a) Two students $ A_i $ and $ A_j $ shaked hands each other. (b) $ \frac{(d_1 + d_2 + \cdots + d_n ) ^2 }{n^2 } \le d_i d_j $

2010 ELMO Shortlist, 1

Tags: inequalities , function , number theory

For a positive integer $n$, let $\mu(n) = 0$ if $n$ is not squarefree and $(-1)^k$ if $n$ is a product of $k$ primes, and let $\sigma(n)$ be the sum of the divisors of $n$. Prove that for all $n$ we have \[\left|\sum_{d|n}\frac{\mu(d)\sigma(d)}{d}\right| \geq \frac{1}{n}, \] and determine when equality holds. [i]Wenyu Cao.[/i]

2023 Romania Team Selection Test, P5

Tags: geometry , inequalities

Let $ABCDEF$ be a convex hexagon. The diagonals $AC$ and $BD$ cross at $P,$ the diagonals $AE{}$ and $DF$ cross at $Q,$ and the line $PQ$ crosses the sides $BC$ and $EF$ at $X$ and $Y,{}$ respectively. Prove that the length of the segment $XY$ does not exceed the sum of the lengths of one of the diagonals through $P{}$ and one of the diagonals through $Q{}$. [i]The Problem Selection Committee[/i]

1984 All Soviet Union Mathematical Olympiad, 372

Tags: inequalities , algebra , radical

Prove that every positive $a$ and $b$ satisfy inequality $$\frac{(a+b)^2}{2} + \frac{a+b}{4} \ge a\sqrt b + b\sqrt a$$

1968 Vietnam National Olympiad, 1

Tags: root , analysis , algebra , inequalities , function

Let $a$ and $b$ satisfy $a \ge b >0, a + b = 1$. i) Prove that if $m$ and $n$ are positive integers with $m < n$, then $a^m - a^n \ge b^m- b^n > 0$. ii) For each positive integer $n$, consider a quadratic function $f_n(x) = x^2 - b^nx- a^n$. Show that $f(x)$ has two roots that are in between $-1$ and $1$.

2016 Postal Coaching, 1

Tags: set theory , inequalities , algebra

The set of all positive real numbers is partitioned into three mutually disjoint non-empty subsets: $\mathbb R^+ = A \cup B\cup C$ and $A \cap B = B \cap C = C \cap A = \emptyset$ whereas none of $A, B, C$ is empty. [list=a][*] Show that one can choose $a \in A, b \in B$ and $c \in C$ such that $a,b, c$ are the sides of a triangle. [*] Is it always possible to choose three numbers from three different sets $A,B,C$ such that these three numbers are the sides of a right-angled triangle?[/list]

2018 VJIMC, 3

Tags: inequalities , algebra

Let $n$ be a positive integer and let $x_1,\dotsc,x_n$ be positive real numbers satisfying $\vert x_i-x_j\vert \le 1$ for all pairs $(i,j)$ with $1 \le i<j \le n$. Prove that \[\frac{x_1}{x_2}+\frac{x_2}{x_3}+\dots+\frac{x_{n-1}}{x_n}+\frac{x_n}{x_1} \ge \frac{x_2+1}{x_1+1}+\frac{x_3+1}{x_2+1}+\dots+\frac{x_n+1}{x_{n-1}+1}+\frac{x_1+1}{x_n+1}.\]

Kvant 2021, M2642

Tags: algebra , inequalities

The nonzero numbers $x{}$ and $y{}$ satisfy the inequalities $x^{2n}-y^{2n}>x$ and $y^{2n}-x^{2n}>y$ for some natural number $n{}$. Can the product $xy$ be a negative number? [i]Proposed by N. Agakhanov[/i]

2012 Turkey Junior National Olympiad, 4

Tags: induction , pigeonhole principle , geometry , geometric transformation , combinatorics , inequalities

We want to place $2012$ pockets, including variously colored balls, into $k$ boxes such that [b]i)[/b] For any box, all pockets in this box must include a ball with the same color or [b]ii)[/b] For any box, all pockets in this box must include a ball having a color which is not included in any other pocket in this box Find the smallest value of $k$ for which we can always do this placement whatever the number of balls in the pockets and whatever the colors of balls.

2015 China Western Mathematical Olympiad, 1

Tags: inequalities , algebra

Let the integer $n \ge 2$ , and $x_1,x_2,\cdots,x_n $ be real numbers such that $\sum_{k=1}^nx_k$ be integer . $d_k=\underset{m\in {Z}}{\min}\left|x_k-m\right| $, $1\leq k\leq n$ .Find the maximum value of $\sum_{k=1}^nd_k$.

2014 France Team Selection Test, 6

Tags: inequalities

Let $n$ be a positive integer and $x_1,x_2,\ldots,x_n$ be positive reals. Show that there are numbers $a_1,a_2,\ldots, a_n \in \{-1,1\}$ such that the following holds: \[a_1x_1^2+a_2x_2^2+\cdots+a_nx_n^2 \ge (a_1x_1+a_2x_2 +\cdots+a_nx_n)^2\]

2009 Dutch IMO TST, 3

Tags: inequalities , 3-variable inequality , three variable inequality

Let $a, b$ and $c$ be positive reals such that $a + b + c \ge abc$. Prove that $a^2 + b^2 + c^2 \ge \sqrt3 abc$.

1997 Israel National Olympiad, 4

Tags: function , inequalities , algebra

Let $f : [0,1] \to [0,1]$ be a continuous, strictly increasing function such that $f(0) = 0$ and $f(1) = 1$. Prove that $$f\left(\frac{1}{10}\right) + f\left(\frac{2}{10}\right) +...+f\left(\frac{9}{10}\right) +f^{-1}\left(\frac{1}{10}\right) +...+f^{-1}\left(\frac{9}{10}\right) \le \frac{99}{10}$$

1998 USAMTS Problems, 4

Tags: trigonometry , inequalities

Prove that if $0<x<\pi/2$, then $\sec^6 x+\csc^6 x+(\sec^6 x)(\csc^6 x)\geq 80$.

1965 Spain Mathematical Olympiad, 4

Tags: inequalities

Find all the intervals $I$ where any element of the interval $x \in I$ satisfies $$\cos x +\sin x >1.$$ Do the same computation when $x$ satisfies $$\cos x +\vert \sin x \vert>1.$$

2008 Tournament Of Towns, 3

Tags: algebra , polynomial , inequalities , sum , root

A polynomial $x^n + a_1x^{n-1} + a_2x^{n-2} +... + a_{n-2}x^2 + a_{n-1}x + a_n$ has $n$ distinct real roots $x_1, x_2,...,x_n$, where $n > 1$. The polynomial $nx^{n-1}+ (n - 1)a_1x^{n-2} + (n - 2)a_2x^{n-3} + ...+ 2a_{n-2}x + a_{n-1}$ has roots $y_1, y_2,..., y_{n_1}$. Prove that $\frac{x^2_1+ x^2_2+ ...+ x^2_n}{n}>\frac{y^2_1 + y^2_2 + ...+ y^2_{n-1}}{n - 1}$

2021 Vietnam TST, 1

Tags: number theory , inequalities

Define the sequence $(a_n)$ as $a_1 = 1$, $a_{2n} = a_n$ and $a_{2n+1} = a_n + 1$ for all $n\geq 1$. a) Find all positive integers $n$ such that $a_{kn} = a_n$ for all integers $1 \leq k \leq n$. b) Prove that there exist infinitely many positive integers $m$ such that $a_{km} \geq a_m$ for all positive integers $k$.

1999 Czech and Slovak Match, 1

Tags: search , inequalities

Leta,b,c are postive real numbers,proof that $ \frac{a}{b\plus{}2c}\plus{}\frac{b}{c\plus{}2a}\plus{}\frac{c}{a\plus{}2b}\geq1$

2006 Estonia Math Open Senior Contests, 10

Tags: inequalities , algebra

Let $ n \ge 2$ be a fixed integer and let $ a_{i,j} (1 \le i < j \le n)$ be some positive integers. For a sequence $ x_1, ... , x_n$ of reals, let $ K(x_1, .... , x_n)$ be the product of all expressions $ (x_i \minus{} x_j)^{a_{i,j}}$ where $ 1 \le i < j \le n$. Prove that if the inequality $ K(x_1, .... , x_n) \ge 0$ holds independently of the choice of the sequence $ x_1, ... , x_n$ then all integers $ a_{i,j}$ are even.