Olympiad problems

2004 Mexico National Olympiad, 6

Tags: maximum , combinatorics

What is the maximum number of possible change of directions in a path traveling on the edges of a rectangular array of $2004 \times 2004$, if the path does not cross the same place twice?.

2009 Kyiv Mathematical Festival, 6

Tags: maximum , set , inequalities

Let $\{a_1,...,a_n\}\subset \{-1,1\}$ and $a>0$ . Denote by $X$ and $Y$ the number of collections $\{\varepsilon_1,...,\varepsilon_n\}\subset \{-1,1\}$, such that $$max_{1\le k\le n}(\varepsilon_1a_1+...+\varepsilon_ka_k) >\alpha$$ and $$\varepsilon_1a_1+...+\varepsilon_na_n>a$$ respectively. Prove that $X\le 2Y$.

1988 All Soviet Union Mathematical Olympiad, 463

Tags: maximum , combinatorics , algebra

A book contains $30$ stories. Each story has a different number of pages under $31$. The first story starts on page $1$ and each story starts on a new page. What is the largest possible number of stories that can begin on odd page numbers?

2015 Caucasus Mathematical Olympiad, 2

Tags: maximum , number theory , divisible

There are $9$ cards with the numbers $1, 2, 3, 4, 5, 6, 7, 8$ and $9$. What is the largest number of these cards can be decomposed in a certain order in a row, so that in any two adjacent cards, one of the numbers is divided by the other?

2018 Thailand Mathematical Olympiad, 4

Tags: maximum value , inequalities , algebra , maximum

Let $a, b, c$ be nonzero real numbers such that $a + b + c = 0$. Determine the maximum possible value of $\frac{a^2b^2c^2}{ (a^2 + ab + b^2)(b^2 + bc + c^2)(c^2 + ca + a^2)}$ .

2014 Czech-Polish-Slovak Junior Match, 6

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

Determine the largest and smallest fractions $F = \frac{y-x}{x+4y}$ if the real numbers $x$ and $y$ satisfy the equation $x^2y^2 + xy + 1 = 3y^2$.

2009 Stars Of Mathematics, 3

Tags: lattice points , maximum , lattice , geometry , ratio

Let $A,B,C$ be nodes of the lattice $Z\times Z$ such that inside the triangle $ABC$ lies a unique node $P$ of the lattice. Denote $E = AP \cap BC$. Determine max $\frac{AP}{PE}$ , over all such configurations.

2017 Federal Competition For Advanced Students, P2, 4

Tags: maximum , three variable inequality , rational , algebra , inequalities

(a) Determine the maximum $M$ of $x+y +z$ where $x, y$ and $z$ are positive real numbers with $16xyz = (x + y)^2(x + z)^2$. (b) Prove the existence of infinitely many triples $(x, y, z)$ of positive rational numbers that satisfy $16xyz = (x + y)^2(x + z)^2$ and $x + y + z = M$. Proposed by Karl Czakler

2013 Hanoi Open Mathematics Competitions, 11

Tags: minimum , maximum , polynomial

The positive numbers $a, b,c, d, p, q$ are such that $(x+a)(x+b)(x+c)(x+d) = x^4+4px^3+6x^2+4qx+1$ holds for all real numbers $x$. Find the smallest value of $p$ or the largest value of $q$.

2001 Bosnia and Herzegovina Team Selection Test, 3

Tags: subset , maximum , combinatorics

Find maximal value of positive integer $n$ such that there exists subset of $S=\{1,2,...,2001\}$ with $n$ elements, such that equation $y=2x$ does not have solutions in set $S \times S$

2013 Nordic, 2

Tags: maximum , arithmetic sequence , combinatorics , game

In a football tournament there are n teams, with ${n \ge 4}$, and each pair of teams meets exactly once. Suppose that, at the end of the tournament, the final scores form an arithmetic sequence where each team scores ${1}$ more point than the following team on the scoreboard. Determine the maximum possible score of the lowest scoring team, assuming usual scoring for football games (where the winner of a game gets ${3}$ points, the loser ${0}$ points, and if there is a tie both teams get ${1}$ point).

1985 All Soviet Union Mathematical Olympiad, 406

Tags: line , maximum , combinatorial geometry

$n$ straight lines are drawn in a plane. They divide the plane onto several parts. Some of the parts are painted. Not a pair of painted parts has non-zero length common bound. Prove that the number of painted parts is not more than $\frac{n^2 + n}{3}$.

2025 EGMO, 6

Tags: board , maximum

In each cell of a $2025 \times 2025$ board, a nonnegative real number is written in such a way that the sum of the numbers in each row is equal to $1$, and the sum of the numbers in each column is equal to $1$. Define $r_i$ to be the largest value in row $i$, and let $R = r_1 + r_2 + ... + r_{2025}$. Similarly, define $c_i$ to be the largest value in column $i$, and let $C = c_1 + c_2 + ... + c_{2025}$. What is the largest possible value of $\frac{R}{C}$? [i]Proposed by Paulius Aleknavičius, Lithuania, and Anghel David Andrei, Romania[/i]

2014 Korea Junior Math Olympiad, 6

Tags: maximum , inequalities , algebra

Let $p = 1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\frac{1}{2^4}+\frac{1}{2^5}. $ For nonnegative reals $x, y,z$ satisfying $(x-1)^2 + (y-1)^2 + (z-1)^2 = 27,$ find the maximum value of $x^p + y^p + z^p.$

2018 Greece JBMO TST, 3

Tags: sum , sum of powers , maximum , combinatorics , algebra

$12$ friends play a tennis tournament, where each plays only one game with any of the other eleven. Winner gets one points. Loser getos zero points, and there is no draw. Final points of the participants are $B_1, B_2, ..., B_{12}$. Find the largest possible value of the sum $\Sigma_3=B_1^3+B_2^3+ ... + B_{12}^3$ .

1983 Brazil National Olympiad, 6

Tags: 3d geometry , maximum , sphere , geometry

Show that the maximum number of spheres of radius $1$ that can be placed touching a fixed sphere of radius $1$ so that no pair of spheres has an interior point in common is between $12$ and $14$.

2015 Dutch IMO TST, 4

Tags: maximum value , sum , coloring , maximum , combinatorics

Each of the numbers $1$ up to and including $2014$ has to be coloured; half of them have to be coloured red the other half blue. Then you consider the number $k$ of positive integers that are expressible as the sum of a red and a blue number. Determine the maximum value of $k$ that can be obtained.

1972 All Soviet Union Mathematical Olympiad, 169

Tags: minimum , maximum , algebra , inequalities

Let $x,y$ be positive numbers, $s$ -- the least of $$\{ x, (y+ 1/x), 1/y\}$$ What is the greatest possible value of $s$? To what $x$ and $y$ does it correspond?

2014 India PRMO, 4

Tags: algebra , maximum , perimeter , geometry

In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is $17$. What is the greatest possible perimeter of the triangle?

1989 Austrian-Polish Competition, 5

Tags: 3d geometry , cube , product , sphere , maximum , geometry

Let $A$ be a vertex of a cube $\omega$ circumscribed about a sphere $k$ of radius $1$. We consider lines $g$ through $A$ containing at least one point of $k$. Let $P$ be the intersection point of $g$ and $k$ closer to $A$, and $Q$ be the second intersection point of $g$ and $\omega$. Determine the maximum value of $AP\cdot AQ$ and characterize the lines $g$ yielding the maximum.

2005 Czech And Slovak Olympiad III A, 1

Tags: arithmetic sequence , algebra , sequence , maximum , recurrence relation

Consider all arithmetical sequences of real numbers $(x_i)^{\infty}=1$ and $(y_i)^{\infty} =1$ with the common first term, such that for some $k > 1, x_{k-1}y_{k-1} = 42, x_ky_k = 30$, and $x_{k+1}y_{k+1} = 16$. Find all such pairs of sequences with the maximum possible $k$.

2015 Dutch Mathematical Olympiad, 1

Tags: maximum , set , combinatorics

We make groups of numbers. Each group consists of [i]five[/i] distinct numbers. A number may occur in multiple groups. For any two groups, there are exactly four numbers that occur in both groups. (a) Determine whether it is possible to make $2015$ groups. (b) If all groups together must contain exactly [i]six [/i] distinct numbers, what is the greatest number of groups that you can make? (c) If all groups together must contain exactly [i]seven [/i] distinct numbers, what is the greatest number of groups that you can make?

2015 Dutch IMO TST, 4

Tags: maximum , sum , coloring , maximum value , combinatorics

Each of the numbers $1$ up to and including $2014$ has to be coloured; half of them have to be coloured red the other half blue. Then you consider the number $k$ of positive integers that are expressible as the sum of a red and a blue number. Determine the maximum value of $k$ that can be obtained.

1977 Vietnam National Olympiad, 6

Tags: 3-dimensional geometry , distance , geometry , maximum , minimum

The planes $p$ and $p'$ are parallel. A polygon $P$ on $p$ has $m$ sides and a polygon $P'$ on $p'$ has $n$ sides. Find the largest and smallest distances between a vertex of $P$ and a vertex of $P'$.

1992 China Team Selection Test, 1

Tags: combinatorics , probabilistic method , double counting , maximum

16 students took part in a competition. All problems were multiple choice style. Each problem had four choices. It was said that any two students had at most one answer in common, find the maximum number of problems.