Olympiad problems

1964 Swedish Mathematical Competition, 5

Tags: trigonometry , min , max , inequalities , algebra

$a_1, a_2, ... , a_n$ 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 $n$.

2016 Peru IMO TST, 4

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

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

1987 Austrian-Polish Competition, 6

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

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 $D(A) = D_n$.

1998 Rioplatense Mathematical Olympiad, Level 3, 2

Tags: algebra , inequalities , min , max

Given an integer $n > 2$, consider all sequences $x_1,x_2,...,x_n$ of nonnegative real numbers such that $$x_1+ 2x_2 + ... + nx_n = 1.$$ Find the maximum value and the minimum value of $x_1^2+x_2^2+...+x_n^2$ and determine all the sequences $x_1,x_2,...,x_n$ for which these values are obtained.

Ukrainian TYM Qualifying - geometry, I.10

Tags: ratio , max , geometric inequality , geometry

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.

Kyiv City MO 1984-93 - geometry, 1990.10.5

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

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 $ABCD$.

1999 Ukraine Team Selection Test, 3

Tags: subset , combinatorics , max

Let $m,n$ be positive integers with $m \le n$, and let $F$ be a family of $m$-element subsets of $\{1,2,...,n\}$ satisfying $A \cap B \ne \varnothing$ for all $A,B \in F$. Determine the maximum possible number of elements in $F$.

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 $x + y$.

2015 Dutch IMO TST, 5

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

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

1962 Swedish Mathematical Competition, 5

Tags: geometry , 3d geometry , max , cube , tetrahedron

Find the largest cube which can be placed inside a regular tetrahedron with side $1$ so that one of its faces lies on the base of the tetrahedron.

1963 Czech and Slovak Olympiad III A, 2

Tags: number theory , max , relatively prime

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

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.

Ukrainian TYM Qualifying - geometry, 2010.6

Tags: geometry , max , geometric inequality

Find inside the triangle $ABC$, points $G$ and $H$ for which, respectively, the geometric mean and the harmonic mean of the distances to the sides of the triangle acquire maximum values. In which cases is the segment $GH$ parallel to one of the sides of the triangle? Find the length of such a segment $GH$.

2019 Dutch IMO TST, 2

Tags: algebra , system of equations , max and min , max , min

Determine all $4$-tuples $(a,b, c, d)$ of positive real numbers satisfying $a + b +c + d = 1$ and $\max (\frac{a^2}{b},\frac{b^2}{a}) \cdot \max (\frac{c^2}{d},\frac{d^2}{c}) = (\min (a + b, c + d))^4$

2005 Bosnia and Herzegovina Junior BMO TST, 1

Tags: min , max , algebra , inequalities

Non-negative real numbers $x, y, z$ satisfy the following relations: $3x + 5y + 7z = 10$ and $x + 2y + 5z = 6$. Find the minimum and maximum of $w = 2x - 3y + 4z$.

2015 Indonesia MO Shortlist, C5

Tags: combinatorics , max

A meeting was attended by $n$ people. They are welcome to occupy the $k$ table provided $\left( k \le \frac{n}{2} \right)$. Each table is occupied by at least two people. When the meeting begins, the moderator selects two people from each table as representatives for talk to. Suppose that $A$ is the number of ways to choose representatives to speak. Determine the maximum value of $A$ that is possible.

1996 Israel National Olympiad, 8

Tags: function , algebra , max

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.

1987 Polish MO Finals, 4

Tags: geometry , 3d geometry , tetrahedron , max , volume

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.

2015 Czech-Polish-Slovak Junior Match, 1

Tags: geometry , max , area of a triangle , angle , right triangle

In the right triangle $ABC$ with shorter side $AC$ the hypotenuse $AB$ has length $12$. Denote $T$ its centroid and $D$ the feet of altitude from the vertex $C$. Determine the size of its inner angle at the vertex $B$ for which the triangle $DTC$ has the greatest possible area.

1963 Swedish Mathematical Competition., 2

Tags: geometry , chessboard , max , square

The squares of a chessboard have side $4$. What is the circumference of the largest circle that can be drawn entirely on the black squares of the board?

1982 Polish MO Finals, 1

Tags: combinatorics , max

Find a way of arranging $n$ girls and $n$ boys around a round table for which $d_n-c_n$ is maximum, where dn is the number of girls sitting between two boys and $c_n$ is the number of boys sitting between two girls.

Ukrainian TYM Qualifying - geometry, VI.18

Tags: geometry , convex , convex polygon , max , area , geometric inequality

The convex polygon $A_1A_2...A_n$ is given in the plane. Denote by $T_k$ $(k \le n)$ the convex $k$-gon of the largest area, with vertices at the points $A_1, A_2, ..., A_n$ and by $T_k(A+1)$ the convex k-gon of the largest area with vertices at the points $A_1, A_2, ..., A_n$ in which one of the vertices is in $A_1$. Set the relationship between the order of arrangement in the sequence $A_1, A_2, ..., A_n$ vertices: 1) $T_3$ and $T_3 (A_2)$ 2) $T_k$ and $T_k (A_1) $ 3) $T_k$ and $T_{k+1}$

2011 Junior Balkan Team Selection Tests - Romania, 3

Tags: max , inequalities , algebra

a) Find the largest possible value of the number $x_1x_2 + x_2x_3 + ... + x_{n-1}x_n$, if $x_1, x_2, ... , x_n$ ($n \ge 2$) are non-negative integers and their sum is $2011$. b) Find the numbers $x_1, x_2, ... , x_n$ for which the maximum value determined at a) is obtained

2003 Denmark MO - Mohr Contest, 4

Tags: geometry , max , circles

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.

2003 Switzerland Team Selection Test, 4

Tags: divide , max , number theory

Find the largest natural number $n$ that divides $a^{25} -a$ for all integers $a$.