Olympiad problems

2016 India PRMO, 10

Let $M$ be the maximum value of $(6x-3y-8z)$, subject to $2x^2+3y^2+4z^2 = 1$. Find $[M]$.

2009 Kyiv Mathematical Festival, 6

Tags: inequalities , maximum , set

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 $X\le 2Y$.

2015 JBMO Shortlist, NT1

Tags: integer , number theory , consecutive , divisible , difference , maximum

What is the greatest number of integers that can be selected from a set of $2015$ consecutive numbers so that no sum of any two selected numbers is divisible by their difference?

2015 Dutch IMO TST, 5

Tags: divisor , maximum , exponential , number theory

For a positive integer $n$, we dene $D_n$ as the largest integer that is a divisor of $a^n + (a + 1)^n + (a + 2)^n$ for all positive integers $a$. 1. Show that for all positive integers $n$, the number $D_n$ is of the form $3^k$ with $k \ge 0$ an integer. 2. Show that for all integers $k \ge 0$ there exists a positive integer n such that $D_n = 3^k$.

2017 Bosnia and Herzegovina Junior BMO TST, 4

Tags: combinatorics , board , positioned , maximum

In each cell of $5 \times 5$ table there is one number from $1$ to $5$ such that every number occurs exactly once in every row and in every column. Number in one column is [i]good positioned[/i] if following holds: - In every row, every number which is left from [i]good positoned[/i] number is smaller than him, and every number which is right to him is greater than him, or vice versa. - In every column, every number which is above from [i]good positoned[/i] number is smaller than him, and every number which is below to him is greater than him, or vice versa. What is maximal number of good positioned numbers that can occur in this table?

2017 Hanoi Open Mathematics Competitions, 5

Tags: number theory , maximum , digit

Let $a, b, c$ be two-digit, three-digit, and four-digit numbers, respectively. Assume that the sum of all digits of number $a+b$, and the sum of all digits of $b + c$ are all equal to $2$. The largest value of $a + b + c$ is (A): $1099$ (B): $2099$ (C): $1199$ (D): $2199$ (E): None of the above.

2018 Thailand Mathematical Olympiad, 4

Tags: maximum value , maximum , inequalities , algebra

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

1954 Moscow Mathematical Olympiad, 263

Tags: maximum , ratio , 3-digit , sum of digits

Define the maximal value of the ratio of a three-digit number to the sum of its digits.

1986 Tournament Of Towns, (114) 1

Tags: maximum , algebra , inequalities

For which natural number $k$ does $\frac{k^2}{1.001^k}$ attain its maximum value?

2018 Bosnia and Herzegovina Team Selection Test, 4

Tags: combinatorics , board , maximum , absolute value

Every square of $1000 \times 1000$ board is colored black or white. It is known that exists one square $10 \times 10$ such that all squares inside it are black and one square $10 \times 10$ such that all squares inside are white. For every square $K$ $10 \times 10$ we define its power $m(K)$ as an absolute value of difference between number of white and black squares $1 \times 1$ in square $K$. Let $T$ be a square $10 \times 10$ which has minimum power among all squares $10 \times 10$ in this board. Determine maximal possible value of $m(T)$

2007 Sharygin Geometry Olympiad, 20

Tags: geometry , 3d geometry , pyramid , volume , maximum

The base of a pyramid is a regular triangle having side of size $1$. Two of three angles at the vertex of the pyramid are right. Find the maximum value of the volume of the pyramid.

2011 Hanoi Open Mathematics Competitions, 5

Tags: algebra , maximum , inequalities

Let $a, b, c$ be positive integers such that $a + 2b +3c = 100$. Find the greatest value of $M = abc$

1986 Tournament Of Towns, (108) 2

Tags: number theory , maximum , digit , divide

A natural number $N$ is written in its decimal representation . It is known that for each digit in this representation , this digit divides exactly into the number $N$ (the digit $0$ is not encountered). What is the maximum number of different digits which there can be in such a representation of $N$? (S . Fomin, Leningrad)

1965 All Russian Mathematical Olympiad, 062

Tags: geometry , maximum , incircle

What is the maximal possible length of the segment, being cut out by the sides of the triangle on the tangent to the inscribed circle, being drawn parallel to the base, if the triangle's perimeter equals $2p$?

2015 Dutch IMO TST, 5

Tags: number theory , divisor , maximum , exponential

For a positive integer $n$, we dene $D_n$ as the largest integer that is a divisor of $a^n + (a + 1)^n + (a + 2)^n$ for all positive integers $a$. 1. Show that for all positive integers $n$, the number $D_n$ is of the form $3^k$ with $k \ge 0$ an integer. 2. Show that for all integers $k \ge 0$ there exists a positive integer n such that $D_n = 3^k$.

2013 Hanoi Open Mathematics Competitions, 10

Tags: geometry , perimeter , maximum , area

Consider the set of all rectangles with a given area $S$. Find the largest value o $ M = \frac{16-p}{p^2+2p}$ where $p$ is the perimeter of the rectangle.

2019 Dutch IMO TST, 3

Tags: maximum , gcd , greatest common divisor , number theory

Let $n$ be a positive integer. Determine the maximum value of $gcd(a, b) + gcd(b, c) + gcd(c, a)$ for positive integers $a, b, c$ such that $a + b + c = 5n$.

2015 Dutch Mathematical Olympiad, 2

Tags: domino , combinatorics , combinatorial geometry , maximum

On a $1000\times 1000$-board we put dominoes, in such a way that each domino covers exactly two squares on the board. Moreover, two dominoes are not allowed to be adjacent, but are allowed to touch in a vertex. Determine the maximum number of dominoes that we can put on the board in this way. [i]Attention: you have to really prove that a greater number of dominoes is impossible. [/i]

1980 All Soviet Union Mathematical Olympiad, 289

Tags: geometry , chord , maximum , area , circles

Given a point $E$ on the diameter $AC$ of the certain circle. Draw a chord $BD$ to maximise the area of the quadrangle $ABCD$.

2010 Dutch IMO TST, 2

Tags: functional equation , maximum , algebra

Find all functions $f : R \to R$ which satisfy $f(x) = max_{y\in R} (2xy - f(y))$ for all $x \in R$.

1995 Korea National Olympiad, Problem 3

Tags: geometry , inradius , equilateral triangle , maximum

Let $ABC$ be an equilateral triangle of side $1$, $D$ be a point on $BC$, and $r_1, r_2$ be the inradii of triangles $ABD$ and $ADC$. Express $r_1r_2$ in terms of $p = BD$ and find the maximum of $r_1r_2$.

1988 All Soviet Union Mathematical Olympiad, 463

Tags: combinatorics , algebra , maximum

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?

1993 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: geometry , perimeter , maximum , equilateral triangle , equilateral

Given three points $A, B$ and $C$ (not collinear) construct the equilateral triangle of greater perimeter such that each of its sides passes through one of the given points.

2005 Czech And Slovak Olympiad III A, 1

Tags: arithmetic sequence , sequence , algebra , 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 $k$.

2021 Regional Olympiad of Mexico Southeast, 3

Tags: minimum , maximum , inequalities

Let $a, b, c$ positive reals such that $a+b+c=1$. Prove that $$\min\{a(1-b),b(1-c),c(1-a)\}\leq \frac{1}{4}$$ $$\max\{a(1-b),b(1-c),c(1-a)\}\geq \frac{2}{9}$$