Olympiad problems

2011 Junior Balkan Team Selection Tests - Romania, 3

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

2001 Estonia National Olympiad, 2

Tags: number theory , max , divide

Find the maximum value of $k$ for which one can choose $k$ integers out of $1,2... ,2n$ so that none of them divides another one.

2018 Estonia Team Selection Test, 3

Tags: algebra , sum , max , min , 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.

2021 Bosnia and Herzegovina Team Selection Test, 1

Tags: inequalities , max , algebra

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

2018 Thailand Mathematical Olympiad, 8

Tags: max , algebra , sum , maximum value

There are $2n + 1$ tickets, each with a unique positive integer as the ticket number. It is known that the sum of all ticket numbers is more than $2330$, but the sum of any $n$ ticket numbers is at most $1165$. What is the maximum value of $n$?

2012 Czech And Slovak Olympiad IIIA, 4

Tags: area , parallelogram , max , geometry

Inside the parallelogram $ABCD$ is a point $X$. Make a line that passes through point $X$ and divides the parallelogram into two parts whose areas differ from each other the most.

1976 Czech and Slovak Olympiad III A, 2

Tags: max , algebra , inequalities

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

1962 Swedish Mathematical Competition, 5

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

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.

2007 Portugal MO, 6

Tags: geometry , min , max , distance

In a village, the maximum distance between two houses is $M$ and the minimum distance is $m$. Prove that if the village has $6$ houses, then $\frac{M}{m} \ge \sqrt3$.

2009 Postal Coaching, 2

Tags: geometry , polygon , max , cyclic , diagonal , area , perpendicular

Let $n \ge 4$ be an integer. Find the maximum value of the area of a $n$-gon which is inscribed in the circle of radius $1$ and has two perpendicular diagonals.

1964 Swedish Mathematical Competition, 4

Tags: combinatorial geometry , min , max , geometric inequalities , inequalities , geometry

Points $H_1, H_2, ... , H_n$ are arranged in the plane so that each distance $H_iH_j \le 1$. The point $P$ is chosen to minimise $\max (PH_i)$. Find the largest possible value of $\max (PH_i)$ for $n = 3$. Find the best upper bound you can for $n = 4$.

2001 Abels Math Contest (Norwegian MO), 3a

Tags: geometry , area , max

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

Ukrainian TYM Qualifying - geometry, VI.18

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

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 $T_{k+1}$

1999 Kazakhstan National Olympiad, 8

Tags: permutation , max , n-variable inequality , inequalities , algebra

Let $ {{a} _ {1}}, {{a} _ {2}}, \ldots, {{a} _ {n}} $ be permutation of numbers $ 1,2, \ldots, n $, where $ n \geq 2 $. Find the maximum value of the sum $$ S (n) = | {{a} _ {1}} - {{a} _ {2}} | + | {{a} _ {2}} - {{a} _ {3}} | + \cdots + | {{a} _ {n-1}} - {{a} _ {n}} |. $$

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 $\frac{(a - b)(2a - b)}{a(a - b + c)}$.

1971 Swedish Mathematical Competition, 5

Tags: trigonometry , algebra , max , inequalities

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

2014 Junior Balkan Team Selection Tests - Moldova, 6

Tags: max , inequalities

The non-negative real numbers $x, y, z$ satisfy the equality $x + y + z = 1$. Determine the highest possible value of the expression $E (x, y, z) = (x + 2y + 3z) (6x +3y + 2z)$.

1998 Tournament Of Towns, 4

Tags: algebra , max , inequalities

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)

1994 Czech And Slovak Olympiad IIIA, 2

Tags: cube , polyhedron , volume , max , 3d geometry , projection , geometry

A cuboid of volume $V$ contains a convex polyhedron $M$. The orthogonal projection of $M$ onto each face of the cuboid covers the entire face. What is the smallest possible volume of polyhedron $M$?

Durer Math Competition CD Finals - geometry, 2015.C4

Tags: geometry , perimeter , arc , max

On a circumference of a unit radius, take points $A$ and $B$ such that section $AB$ has length one. $C$ can be any point on the longer arc of the circle between $A$ and $B$. How do we take $C$ to make the perimeter of the triangle $ABC$ as large as possible?

1999 Ukraine Team Selection Test, 3

Tags: max , combinatorics , subset

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 $F$.

2009 Ukraine Team Selection Test, 11

Tags: combinatorics , coloring , min , max

Suppose that integers are given $m <n $. Consider a spreadsheet of size $n \times n $, whose cells arbitrarily record all integers from $1 $ to ${{n} ^ {2}} $. Each row of the table is colored in yellow $m$ the largest elements. Similarly, the blue colors the $m$ of the largest elements in each column. Find the smallest number of cells that are colored yellow and blue at a time

2003 Singapore MO Open, 2

Tags: algebra , max , inequalities

Find the maximum value of $\frac{xyz}{(1 + 5x)(4x + 3y)(5y + 6z)(z + 18)}$ as $x, y$ and $z$ range over the set of all positive real numbers. Justify your answer.

1998 Czech And Slovak Olympiad IIIA, 4

Tags: max , number theory , power of 3

For each date of year $1998$, we calculate day$^{month}$ −year and determine the greatest power of $3$ that divides it. For example, for April $21$ we get $21^4 - 1998 =192483 = 3^3 \cdot 7129$, which is divisible by $3^3$ and not by $3^4$ . Find all dates for which this power of $3$ is the greatest.

2012 Estonia Team Selection Test, 5

Tags: algebra , inequalities , max

Let $x, y, z$ be positive real numbers whose sum is $2012$. Find the maximum value of $$ \frac{(x^2 + y^2 + z^2)(x^3 + y^3 + z^3)}{(x^4 + y^4 + z^4)}$$