Olympiad problems

2011 Tournament of Towns, 2

$49$ natural numbers are written on the board. All their pairwise sums are different. Prove that the largest of the numbers is greater than $600$. [hide=original wording in Russian]На доске написаны 49 натуральных чисел. Все их попарные суммы различны. Докажите, что наибольшее из чисел больше 600[/hide]

2008 Swedish Mathematical Competition, 6

Tags: algebra , inequalities , max , product

A [i]sum decomposition[/i] of the number 100 is given by a positive integer $n$ and $n$ positive integers $x_1<x_2<\cdots <x_n$ such that $x_1 + x_2 + \cdots + x_n = 100$. Determine the largest possible value of the product $x_1x_2\cdots x_n$, and $n$ , as $x_1, x_2,\dots, x_n$ vary among all sum decompositions of the number $100$.

2007 Oral Moscow Geometry Olympiad, 2

Tags: geometry , circles , area of triangle , max

Two circles intersect at points $P$ and $Q$. Point $A$ lies on the first circle, but outside the second. Lines $AP$ and $AQ$ intersect the second circle at points $B$ and $C$, respectively. Indicate the position of point $A$ at which triangle $ABC$ has the largest area. (D. Prokopenko)

2021 Bosnia and Herzegovina Team Selection Test, 1

Tags: algebra , inequalities , max

Let $x,y,z$ be real numbers from the interval $[0,1]$. Determine the maximum value of expression $$W=y\cdot \sqrt{1-x}+z\cdot\sqrt{1-y}+x\cdot\sqrt{1-z}$$

2006 MOP Homework, 5

Tags: inequalities , sum , maximal , max , algebra

Let $a_1, a_2,...,a_{2005}, b_1, b_2,...,b_{2005}$ be real numbers such that $(a_ix - b_i)^2 \ge \sum_{j\ne i,j=1}^{2005} (a_jx - b_j)$ for all real numbers x and every integer $i$ with $1 \le i \le 2005$. What is maximal number of positive $a_i$'s and $b_i$'s?

1997 Austrian-Polish Competition, 7

Tags: inequalities , max , algebra

(a) Prove that $p^2 + q^2 + 1 > p(q + 1)$ for any real numbers $p, q$, . (b) Determine the largest real constant $b$ such that the inequality $p^2 + q^2 + 1 \ge bp(q + 1)$ holds for all real numbers $p, q$ (c) Determine the largest real constant c such that the inequality $p^2 + q^2 + 1 \ge cp(q + 1)$ holds for all integers $p, q$.

1976 Czech and Slovak Olympiad III A, 2

Tags: inequalities , max , algebra

Show that for any real $x\in[0,1]$ the inequality \[\frac{(1-x)x^2}{(1+x)^3}<\frac{1}{25}\] holds.

Cono Sur Shortlist - geometry, 1993.10

Tags: geometry , max , geometric inequality , analytic geometry

Let $\omega$ be the unit circle centered at the origin of $R^2$. Determine the largest possible value for the radius of the circle inscribed to the triangle $OAP$ where $ P$ lies the circle and $A$ is the projection of $P$ on the axis $OX$.

1971 Swedish Mathematical Competition, 5

Tags: algebra , trigonometry , inequalities , max

Show that \[ \max\limits_{|x|\leq t} |1 - a \cos x| \geq \tan^2 \frac{t}{2} \] for $a$ positive and $t \in (0, \frac{\pi}{2})$.

2001 Estonia National Olympiad, 5

Tags: combinatorics , table , max

A table consisting of $9$ rows and $2001$ columns is filfed with integers $1,2,..., 2001$ in such a way that each of these integers occurs in the table exactly $9$ times and the integers in any column differ by no more than $3$. Find the maximum possible value of the minimal column sum (sum of the numbers in one column).

1998 Tournament Of Towns, 4

Tags: inequalities , algebra , max

Among all sets of real numbers $\{ x_1 , x_2 , ... , x_{20} \}$ from the open interval $(0, 1 )$ such that $$x_1x_2...x_{20}= ( 1 - x_1 ) ( 1 -x_2 ) ... (1 - x_{20} )$$ find the one for which $x_1 x_2... x_{20}$ is maximal. (A Cherniatiev)

2001 Abels Math Contest (Norwegian MO), 3a

Tags: max , area , geometry

What is the largest possible area of a quadrilateral with sidelengths $1, 4, 7$ and $8$ ?

2003 Olympic Revenge, 7

Tags: subset , max , combinatorics , positive integer

Let $X$ be a subset of $R_{+}^{*}$ with $m$ elements. Find $X$ such that the number of subsets with the same sum is maximum.

2004 Estonia National Olympiad, 2

Tags: min , max , difference , algebra

The positive differences $a_i-a_j$ of five different positive integers $a_1, a_2, a_3, a_4, a_5$ are all different (there are altogether $10$ such differences). Find the least possible value of the largest number among the $a_i$.

2014 Estonia Team Selection Test, 3

Tags: max , area , inequalities , algebra

Three line segments, all of length $1$, form a connected figure in the plane. Any two different line segments can intersect only at their endpoints. Find the maximum area of the convex hull of the figure.

2002 Croatia Team Selection Test, 1

Tags: combinatorics , max

In a certain language there are $n$ letters. A sequence of letters is a word, if there are no two equal letters between two other equal letters. Find the number of words of the maximum length.

2019 Dutch IMO TST, 4

Tags: combinatorics , max , graph theory

There are $300$ participants to a mathematics competition. After the competition some of the contestants play some games of chess. Each two contestants play at most one game against each other. There are no three contestants, such that each of them plays against each other. Determine the maximum value of $n$ for which it is possible to satisfy the following conditions at the same time: each contestant plays at most $n$ games of chess, and for each $m$ with $1 \le m \le n$, there is a contestant playing exactly $m$ games of chess.

2013 Saudi Arabia Pre-TST, 2.2

Tags: algebra , max , quadratic polynomial

The quadratic equation $ax^2 + bx + c = 0$ has its roots in the interval $[0, 1]$. Find the maximum of $\frac{(a - b)(2a - b)}{a(a - b + c)}$.

2017 Puerto Rico Team Selection Test, 5

Tags: algebra , inequalities , maximum , max , minimum , min

Let $a, b$ be two real numbers that satisfy $a^3 + b^3 = 8-6ab$. Find the maximum value and the minimum value that $a + b$ can take.

2000 Switzerland Team Selection Test, 9

Tags: intersecting circles , geometric construction , construction , geometry , segment , max

Two given circles $k_1$ and $k_2$ intersect at points $P$ and $Q$. Construct a segment $AB$ through $P$ with the endpoints at $k_1$ and $k_2$ for which $AP \cdot PB$ is maximal.

Ukrainian TYM Qualifying - geometry, IV.7

Tags: max , min , geometric inequality , inscribed , perimeter , geometry

Let $ABCD$ be the quadrilateral whose area is the largest among the quadrilaterals with given sides $a, b, c, d$, and let $PORS$ be the quadrilateral inscribed in $ABCD$ with the smallest perimeter. Find this perimeter.

2002 Junior Balkan Team Selection Tests - Romania, 3

Tags: geometry , max , area , midpoint , sidelengths

Let $ABC$ be a triangle and $a = BC, b = CA$ and $c = AB$ be the lengths of its sides. Points $D$ and $E$ lie in the same halfplane determined by $BC$ as $A$. Suppose that $DB = c, CE = b$ and that the area of $DECB$ is maximal. Let $F$ be the midpoint of $DE$ and let $FB = x$. Prove that $FC = x$ and $4x^3 = (a^2+b^2 + c^2)x + abc$.