Olympiad problems

2005 All-Russian Olympiad Regional Round, 8.8

8.8, 9.8, 11.8 a) 99 boxes contain apples and oranges. Prove that we can choose 50 boxes in such a way that they contain at least half of all apples and half of all oranges. b) 100 boxes contain apples and oranges. Prove that we can choose 34 boxes in such a way that they contain at least a third of all apples and a third of all oranges. c) 100 boxes contain apples, oranges and bananas. Prove that we can choose 51 boxes in such a way that they contain at least half of all apples, and half of all oranges and half of all bananas. ([i]I. Bogdanov, G. Chelnokov, E. Kulikov[/i])

2013 Junior Balkan Team Selection Tests - Romania, 2

Tags: inequalities , algebra

Let $a, b, c$ be positive real numbers such that $a + b + c = 1$. Show that $$\frac{1 - a^2}{a + bc} + \frac{1 - b^2}{b + ca} + \frac{1 - c^2}{c + ab} \ge 6$$

2006 Lithuania National Olympiad, 3

Tags: inequalities , algebra

Show that if $a+b+c=0$ then $(\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b})(\frac{b-c}{a}+\frac{c-a}{b}+\frac{a-b}{c})=9$.

1983 IMO Longlists, 48

Tags: geometry , parallelogram , inequalities

Prove that in any parallelepiped the sum of the lengths of the edges is less than or equal to twice the sum of the lengths of the four diagonals.

1970 Bulgaria National Olympiad, Problem 6

Tags: geometry , inequalities , geometric inequalities

In space, we are given the points $A,B,C$ and a sphere with center $O$ and radius $1$. Find the point $X$ from the sphere for which the sum $f(X)=|XA|^2+|XB|^2+|XC|^2$ attains its maximal and minimal value. Prove that if the segments $OA,OB,OC$ are pairwise perpendicular and $d$ is the distance from the center $O$ to the centroid of the triangle $ABC$ then: (a) the maximum of $f(X)$ is equal to $9d^2+3+6d$; (b) the minimum of $f(X)$ is equal to $9d^2+3-6d$. [i]K. Dochev and I. Dimovski[/i]

2006 Petru Moroșan-Trident, 2

Tags: induction , inequalities , inequality

Let be an increasing, infinite sequence of natural numbers $ \left( a_n \right)_{n\ge 1} . $ [b]a)[/b] Prove that if $ a_n=n, $ for any natural numbers $ n, $ then $$ -2+2\sqrt{1+n} <\frac{1}{\sqrt{a_1}} +\frac{1}{\sqrt{a_2}} +\cdots +\frac{1}{\sqrt{a_n}} <2\sqrt n , $$ for any natural numbers $ n. $ [b]b)[/b] Disprove the converse of [b]a).[/b] [i]Vasile Radu[/i]

2013 Hanoi Open Mathematics Competitions, 6

Tags: inequalities , combination

Let be given $a\in\{0,1,2, 3,..., 100\}.$ Find all $n \in\{1,2, 3,..., 2013\}$ such that $C_n^{2013} > C_a^{2013}$ , where $C_k^m=\frac{m!}{k!(m -k)!}$.

2020 IMO Shortlist, A4

Tags: inequalities , algebra , mount inquality

The real numbers $a, b, c, d$ are such that $a\geq b\geq c\geq d>0$ and $a+b+c+d=1$. Prove that \[(a+2b+3c+4d)a^ab^bc^cd^d<1\] [i]Proposed by Stijn Cambie, Belgium[/i]

2000 Tuymaada Olympiad, 4

Tags: algebra , three variable inequality , inequalities

Prove that if the product of positive numbers $a,b$ and $c$ equals one, then $\frac{1}{a(a+1)}+\frac{1}{b(b+1)}+\frac{1}{c(c+1)}\ge \frac{3}{2}$

2008 Brazil National Olympiad, 3

Tags: calculus , trigonometry , inequalities

Let $ x,y,z$ real numbers such that $ x \plus{} y \plus{} z \equal{} xy \plus{} yz \plus{} zx$. Find the minimum value of \[ {x \over x^2 \plus{} 1} \plus{} {y\over y^2 \plus{} 1} \plus{} {z\over z^2 \plus{} 1}\]

2012 Israel National Olympiad, 3

Tags: algebra , inequality , symmetric inequality , inequalities

Let $a,b,c$ be real numbers such that $a^3(b+c)+b^3(a+c)+c^3(a+b)=0$. Prove that $ab+bc+ca\leq0$.

1932 Eotvos Mathematical Competition, 3

Tags: trigonometry , geometry , inequalities

Let $\alpha$, $\beta$ and $\gamma$ be the interior angles of an acute triangle. Prove that if $\alpha < \beta < \gamma$ then $$\sin 2\alpha >\ sin 2 \beta > \sin 2\gamma.$$

2013 Saudi Arabia GMO TST, 2

Tags: algebra , inequalities

For positive real numbers $a, b$ and $c$, prove that $$\frac{a^3}{a^2 + ab + b^2} +\frac{b^3}{b^2 + bc + c^2} +\frac{c^3}{ c^2 + ca + a^2} \ge\frac{ a + b + c}{3}$$

2015 IMO Shortlist, N8

Tags: inequalities , number theory , prime factorization , function

For every positive integer $n$ with prime factorization $n = \prod_{i = 1}^{k} p_i^{\alpha_i}$, define \[\mho(n) = \sum_{i: \; p_i > 10^{100}} \alpha_i.\] That is, $\mho(n)$ is the number of prime factors of $n$ greater than $10^{100}$, counted with multiplicity. Find all strictly increasing functions $f: \mathbb{Z} \to \mathbb{Z}$ such that \[\mho(f(a) - f(b)) \le \mho(a - b) \quad \text{for all integers } a \text{ and } b \text{ with } a > b.\] [i]Proposed by Rodrigo Sanches Angelo, Brazil[/i]

1966 Vietnam National Olympiad, 1

Tags: algebra , inequalities , maximum value , minimum value

Let $x, y$ and $z$ be nonnegative real numbers satisfying the following conditions: (1) $x + cy \le 36$,(2) $2x+ 3z \le 72$, where $c$ is a given positive number. Prove that if $c \ge 3$ then the maximum of the sum $x + y + z$ is $36$, while if $c < 3$, the maximum of the sum is $24 + \frac{36}{c}$ .

2011 Hanoi Open Mathematics Competitions, 4

Tags: algebra , inequalities

Among the five statements on real numbers below, how many of them are correct? "If $a < b < 0$ then $a < b^2$" , "If $0 < a < b$ then $a < b^2$", "If $a^3 < b^3$ then $a < b$", "If $a^2 < b^2$ then $a < b$", "If $|a| < |b|$ then $a < b$", (A) $0$, (B) $1$, (C) $2$, (D) $3$, (E) $4$

2005 Bulgaria Team Selection Test, 4

Tags: function , quadratic , inequalities

Let $a_{i}$ and $b_{i}$, where $i \in \{1,2, \dots, 2005 \}$, be real numbers such that the inequality $(a_{i}x-b_{i})^{2} \ge \sum_{j=1, j \not= i}^{2005} (a_{j}x-b_{j})$ holds for all $x \in \mathbb{R}$ and all $i \in \{1,2, \dots, 2005 \}$. Find the maximum possible number of positive numbers amongst $a_{i}$ and $b_{i}$, $i \in \{1,2, \dots, 2005 \}$.

2004 Indonesia MO, 4

Tags: geometry , inequalities

8. Sebuah lantai luasnya 3 meter persegi ditutupi lima buah karpet dengan ukuran masing-masing 1 meter persegi. Buktikan bahwa ada dua karpet yang tumpang tindih dengan luas tumpang tindih minimal 0,2 meter persegi. A floor of a certain room has a $ 3 \ m^2$ area. Then the floor is covered by 5 rugs, each has an area of $ 1 \ m^2$. Prove that there exists 2 overlapping rugs, with at least $ 0.2 \ m^2$ covered by both rugs.

2008 Swedish Mathematical Competition, 3

Tags: algebra , inequalities

The function $f(x)$ has the property that $\frac{f(x)}{x}$ is increasing for $x>0$. Show that \[ f(x)+f(y) \leq f(x+y) \qquad , \qquad \text{for all } x,y>0 \]

1969 Polish MO Finals, 4

Tags: inequalities , algebra

Show that if natural numbers $a,b, p,q,r,s$ satisfy the conditions $$qr- ps = 1 \,\,\,\,\, and \,\,\,\,\, \frac{p}{q}<\frac{a}{b}<\frac{r}{s},$$ then $b \ge q+s.$

2024 Indonesia TST, A

Tags: algebra , inequalities , absolute value

Given real numbers $x,y,z$ which satisfies $$|x+y+z|+|xy+yz+zx|+|xyz| \le 1$$ Show that $max\{ |x|,|y|,|z|\} \le 1$.

2010 Serbia National Math Olympiad, 1

Tags: inequalities , floor function , combinatorics

Some of $n$ towns are connected by two-way airlines. There are $m$ airlines in total. For $i = 1, 2, \cdots, n$, let $d_i$ be the number of airlines going from town $i$. If $1\le d_i \le 2010$ for each $i = 1, 2,\cdots, 2010$, prove that \[\displaystyle\sum_{i=1}^n d_i^2\le 4022m- 2010n\] Find all $n$ for which equality can be attained. [i]Proposed by Aleksandar Ilic[/i]

2005 MOP Homework, 7

Tags: inequalities

Let $n$ be a positive integer with $n>1$, and let $a_1$, $a_2$, ..., $a_n$ be positive integers such that $a_1<a_2<...<a_n$ and $\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n} \le 1$. Prove that $(\frac{1}{a_1^2+x^2}+\frac{1}{a_2^2+x^2}+...+\frac{1}{a_n^2+x^2})^2 \le \frac{1}{2} \cdot \frac{1}{a_1(a_1-1)+x^2}$ for all real numbers $x$.

2009 USAMTS Problems, 4

Tags: inequalities , pigeonhole principle

Let $S$ be a set of $10$ distinct positive real numbers. Show that there exist $x,y \in S$ such that \[0 < x - y < \frac{(1 + x)(1 + y)}{9}.\]

2018 District Olympiad, 4

Tags: number theory , inequalities

a) Consider the positive integers $a, b, c$ so that $a < b < c$ and $a^2+b^2 = c^2$. If $a_1 = a^2$, $a_2 = ab$, $a_3 = bc$, $a_4 = c^2$, prove that $a_1^2+a_2^2+a_3^2=a_4^2$ and $a_1 < a_2 < a_3 < a_4$. b) Show that for any $n \in N$, $n\ge 3$, there exist the positive integers $a_1, a_2,..., a_n$ so that $a_1^2+a_2^2+...+ a_{n-1}^2=a_n^2$ and $a_1 < a_2 < ...< a_{n-1} < a_n$