Olympiad problems

1987 Polish MO Finals, 4

Let $S$ be the set of all tetrahedra which satisfy: (1) the base has area $1$, (2) the total face area is $4$, and (3) the angles between the base and the other three faces are all equal. Find the element of $S$ which has the largest volume.

1981 Czech and Slovak Olympiad III A, 5

Tags: max , integer , inequalities , algebra

Let $n$ be a positive integer. Determine the maximum of the sum $x_1+\cdots+x_n$ where $x_1,\ldots,x_n$ are non-negative integers satisfying the condition \[x_1^3+\cdots+x_n^3\le7n.\]

2008 Postal Coaching, 3

Tags: max , median , angle bisector , geometry

Let $ABC$ be a triangle. For any point $X$ on $BC$, let $AX$ meet the circumcircle of $ABC$ in $X'$. Prove or disprove: $XX'$ has maximum length if and only if $AX$ lies between the median and the internal angle bisector from $A$.

2006 MOP Homework, 5

Tags: inequalities , max , sum , maximal , 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?

Kyiv City MO 1984-93 - geometry, 1990.10.5

Tags: parabola , area , max , geometric inequality , geometry , analytic geometry

A circle centered at a point $(0, 1)$ on the coordinate plane intersects the parabola $y = x^2$ at four points: $A, B, C, D.$ Find the largest possible value of the area of the quadrilateral $ABCD$.

2003 Denmark MO - Mohr Contest, 4

Tags: max , circles , geometry

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.

2005 Thailand Mathematical Olympiad, 20

Tags: max , algebra , inequalities

Let $a, b, c, d > 0$ satisfy $36a + 4b + 4c + 3d = 25$. What is the maximum possible value of $ab^{1/2}c^{1/3}d^{1/4}$ ?

2003 Olympic Revenge, 7

Tags: positive integer , subset , max , combinatorics

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

2017 Greece Junior Math Olympiad, 4

Tags: max , combinatorics

A group of $n$ people play a board game with the following rules: 1) In each round of the game exactly $3$ people play 2) The game ends after exactly $n$ rounds 3) Every pair of players has played together at least at one round Find the largest possible value of $n$

2018 Estonia Team Selection Test, 3

Tags: sum , min , max , algebra , inequalities

Given a real number $c$ and an integer $m, m \ge 2$. Real numbers $x_1, x_2,... , x_m$ satisfy the conditions $x_1 + x_2 +...+ x_m = 0$ and $\frac{x^2_1 + x^2_2 + ...+ x^2_m}{m}= c$. Find max $(x_1, x_2,..., x_m)$ if it is known to be as small as possible.

2001 Czech And Slovak Olympiad IIIA, 4

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.

2004 Estonia National Olympiad, 2

Tags: max , min , 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 $a_i$.

2019 Hanoi Open Mathematics Competitions, 14

Tags: max , algebra , inequalities

Let $a, b, c$ be nonnegative real numbers satisfying $a + b + c =3$. a) If $c > \frac32$, prove that $3(ab + bc + ca) - 2abc < 7$. b) Find the greatest possible value of $M =3(ab + bc + ca) - 2abc $.

Ukrainian TYM Qualifying - geometry, IV.7

Tags: min , geometric inequality , inscribed , max , 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.

1985 All Soviet Union Mathematical Olympiad, 397

Tags: chessboard , max

What maximal number of the men in checkers game can be put on the chess-board $8\times 8$ so, that every man can be taken by at least one other man ?

2016 Swedish Mathematical Competition, 1

Tags: geometry , max , area

In a garden there is an $L$-shaped fence, see figure. You also have at your disposal two finished straight fence sections that are $13$ m and $14$ m long respectively. From point $A$ you want to delimit a part of the garden with an area of at least $200$ m$^2$ . Is it possible to do this? [img]https://1.bp.blogspot.com/-VLWIImY7HBA/X0yZq5BrkTI/AAAAAAAAMbg/8CyP6DzfZTE5iX01Qab3HVrTmaUQ7PvcwCK4BGAYYCw/s400/sweden%2B16p1.png[/img]

1963 Czech and Slovak Olympiad III A, 2

Tags: max , relatively prime , number theory

Let an even positive integer $2k$ be given. Find such relatively prime positive integers $x, y$ that maximize the product $xy$.

1996 Israel National Olympiad, 8

Tags: max , function , algebra

Consider the function $f : N \to N$ given by (i) $f(1) = 1$, (ii) $f(2n) = f(n)$ for any $n \in N$, (iii) $f(2n+1) = f(2n)+1$ for any $n \in N$. (a) Find the maximum value of $f(n)$ for $1 \le n \le 1995$; (b) Find all values of $f$ on this interval.

2015 Dutch IMO TST, 5

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

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

2013 Thailand Mathematical Olympiad, 2

Tags: max , geometry , angle bisector , incircle

Let $\vartriangle ABC$ be a triangle with $\angle ABC > \angle BCA \ge 30^o$ . The angle bisectors of $\angle ABC$ and $\angle BCA$ intersect $CA$ and $AB$ at $D$ and $E$ respectively, and $BD$ and $CE$ intersect at $P$. Suppose that $P D = P E$ and the incircle of $\vartriangle ABC$ has unit radius. What is the maximum possible length of $BC$?

1964 Swedish Mathematical Competition, 5

Tags: inequalities , max , min , algebra , trigonometry

$a_1, a_2, ... , a_n$ are constants such that $f(x) = 1 + a_1 cos x + a_2 cos 2x + ...+ a_n cos nx \ge 0$ for all $x$. We seek estimates of $a_1$. If $n = 2$, find the smallest and largest possible values of $a_1$. Find corresponding estimates for other values of $n$.