Olympiad problems

Denmark (Mohr) - geometry, 2003.4

Georg and his mother love pizza. They buy a pizza shaped as an equilateral triangle. Georg demands to be allowed to divide the pizza by a straight cut and then make the first choice. The mother accepts this reluctantly, but she wants to choose a point of the pizza through which the cut must pass. Determine the largest fraction of the pizza which the mother is certain to get by this procedure.

2018 Hanoi Open Mathematics Competitions, 7

Tags: power of 2 , sum , max , number theory

Some distinct positive integers were written on a blackboard such that the sum of any two integers is a power of $2$. What is the maximal possible number written on the blackboard?

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 $p, q$.

1987 Austrian-Polish Competition, 6

Tags: geometry , combinatorics , combinatorial geometry , circles , subset , min , max

Let $C$ be a unit circle and $n \ge 1$ be a fixed integer. For any set $A$ of $n$ points $P_1,..., P_n$ on $C$ define $D(A) = \underset{d}{max}\, \underset{i}{min}\delta (P_i, d)$, where $d$ goes over all diameters of $C$ and $\delta (P, \ell)$ denotes the distance from point $P$ to line $\ell$. Let $F_n$ be the family of all such sets $A$. Determine $D_n = \underset{A\in F_n}{min} D(A)$ and describe all sets $A$ with $D(A) = D_n$.

1997 Poland - Second Round, 6

Tags: combinatorial geometry , geometry , 3d geometry , cube , distance , max , point

Let eight points be given in a unit cube. Prove that two of these points are on a distance not greater than $1$.

2007 Singapore Senior Math Olympiad, 5

Tags: algebra , inequalities , min , max

Find the maximum and minimum of $x + y$ such that $x + y = \sqrt{2x-1}+\sqrt{4y+3}$

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 $100$.

2018 Malaysia National Olympiad, A2

Tags: inequalities , max , algebra

The product of $10$ integers is $1024$. What is the greatest possible sum of these $10$ integers?

2013 Saudi Arabia IMO TST, 1

Tags: inequalities , algebra , min , max

Find the maximum and the minimum values of $S = (1 - x_1)(1 -y_1) + (1 - x_2)(1 - y_2)$ for real numbers $x_1, x_2, y_1,y_2$ with $x_1^2 + x_2^2 = y_1^2 + y_2^2 = 2013$.

2015 Dutch IMO TST, 5

Tags: number theory , positive integer , divide , min , max

Let $N$ be the set of positive integers. Find all the functions $f: N\to N$ with $f (1) = 2$ and such that $max \{f(m)+f(n), m+n\}$ divides $min\{2m+2n,f (m+ n)+1\}$ for all $m, n$ positive integers

2004 Thailand Mathematical Olympiad, 18

Tags: algebra , inequalities , max , product

Find positive reals $a, b, c$ which maximizes the value of $abc$ subject to the constraint that $b(a^2 + 2) + c(a + 2) = 12$.

2018 Costa Rica - Final Round, 2

Tags: algebra , inequalities , max

Let $a, b, c$, and $d$ be real numbers. The six sums of two numbers $x$ and $y$, different from the previous four, are $117$, $510$, $411$, $252$, in no particular order. Determine the maximum possible value of $x + y$.

Kyiv City MO 1984-93 - geometry, 1986.10.5

Tags: geometric inequality , max , area , geometry

Let $E$ be a point on the side $AD$ of the square $ABCD$. Find such points $M$ and $K$ on the sides $AB$ and $BC$ respectively, such that the segments $MK$ and $EC$ are parallel, and the quadrilateral $MKCE$ has the largest area.

1983 Swedish Mathematical Competition, 4

Tags: triangle inequality , concentric , max , triangle area , rectangle , geometry

$C$, $C'$ are concentric circles with radii $R$, $R'$. A rectangle has two adjacent vertices on $C$ and the other two vertices on $C'$. Find its sides if its area is as large as possible.

1968 Swedish Mathematical Competition, 1

Tags: inequalities , algebra , min , max

Find the maximum and minimum values of $x^2 + 2y^2 + 3z^2$ for real $x, y, z$ satisfying $x^2 + y^2 + z^2 = 1$.

1996 Singapore Senior Math Olympiad, 2

Tags: geometry , trigonometry , angle , max , semicircle , triangle area , sum

Let $180^o < \theta_1 < \theta_2 <...< \theta_n = 360^o$. For $i = 1,2,..., n$, $P_i = (\cos \theta_i^o, \sin \theta_i^o)$ is a point on the circle $C$ with centre $(0,0)$ and radius $1$. Let $P$ be any point on the upper half of $C$. Find the coordinates of $P$ such that the sum of areas $[PP_1P_2] + [PP_2P_3] + ...+ [PP_{n-1}P_n]$ attains its maximum.

2019 Dutch IMO TST, 4

Tags: max , graph theory , combinatorics

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.

2004 Thailand Mathematical Olympiad, 19

Tags: inequalities , algebra , sum , max

Find positive reals $a, b, c$ which maximizes the value of $a+ 2b+ 3c$ subject to the constraint that $9a^2 + 4b^2 + c^2 = 91$

Ukrainian TYM Qualifying - geometry, I.10

Tags: geometry , max , geometric inequality , ratio

Given a circle of radius $R$. Find the ratio of the largest area of the circumscribed quadrilateral to the smallest area of the inscribed one.

2002 Junior Balkan Team Selection Tests - Romania, 3

Tags: geometry , area , max , 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 $4x^3 = (a^2+b^2 + c^2)x + abc$.

1997 Czech And Slovak Olympiad IIIA, 5

Tags: function , trigonometry , max , algebra , inequalities

For a given integer $n \ge 2$, find the maximum possible value of $V_n = \sin x_1 \cos x_2 +\sin x_2 \cos x_3 +...+\sin x_n \cos x_1$, where $x_1,x_2,...,x_n$ are real numbers.

1973 Chisinau City MO, 67

Tags: inequalities , algebra , max , sum , product

The product of $10$ natural numbers is equal to $10^{10}$. What is the largest possible sum of these numbers?

1990 Czech and Slovak Olympiad III A, 4

Tags: max , inequalities , algebra , parameter

Determine the largest $k\ge0$ such that the inequality \[\left(\sum_{j=1}^n x_j\right)^2\left(\sum_{j=1}^n x_jx_{j+1}\right)\ge k\sum_{j=1}^n x_j^2x_{j+1}^2\] holds for every $n\ge2$ and any $n$-tuple $x_1,\ldots,x_n$ of non-negative numbers (given that $x_{n+1}=x_1$)

1991 Austrian-Polish Competition, 7

Tags: function , algebra , max

For a given positive integer $n$ determine the maximum value of the function $f (x) = \frac{x + x^2 +...+ x^{2n-1}}{(1 + x^n)^2}$ over all $x \ge 0$ and find all positive $x$ for which the maximum is attained.

2006 Belarusian National Olympiad, 6

Tags: table , sum , max , combinatorics

An $n \times m$ table ( $n \le m$ ) is filled in accordance with the rules of the game "Minesweeper": mines are placed at some cells (not more than one mine at the cell) and in the remaining cells one writes the number of the mines in the neighboring (by side or by vertex) cells. Then the sum of allnumbers in the table is computed (this sum is equal to $9$ for the picture). What is the largest possible value of this sum? (V. Lebed) [img]https://cdn.artofproblemsolving.com/attachments/2/9/726ccdbc57807788a5f6e88a5acb42b10a6cc0.png[/img]