Olympiad problems

2013 India PRMO, 18

What is the maximum possible value of $k$ for which $2013$ can be written as a sum of $k$ consecutive positive integers?

2012 BAMO, 1

Tags: combinatorics , chessboard , maximum

Hugo places a chess piece on the top left square of a $20 \times 20$ chessboard and makes $10$ moves with it. On each of these $10$ moves, he moves the piece either one square horizontally (left or right) or one square vertically (up or down). After the last move, he draws an $X$ on the square that the piece occupies. When Hugo plays the game over and over again, what is the largest possible number of squares that could eventually be marked with an $X$? Prove that your answer is correct.

2021 JBMO Shortlist, N2

Tags: shortlist , number theory , maximum , minimum

The real numbers $x, y$ and $z$ are such that $x^2 + y^2 + z^2 = 1$. a) Determine the smallest and the largest possible values of $xy + yz - xz$. b) Prove that there does not exist a triple $(x, y, z)$ of rational numbers, which attains any of the two values in a).

2002 Cono Sur Olympiad, 5

Tags: combinatorics , number theory , maximum , difference

Consider the set $A = \{1, 2, ..., n\}$. For each integer $k$, let $r_k$ be the largest quantity of different elements of $A$ that we can choose so that the difference between two numbers chosen is always different from $k$. Determine the highest value possible of $r_k$, where $1 \le k \le \frac{n}{2}$

2013 Dutch Mathematical Olympiad, 1

Tags: maximum , combinatorics , table , coloring

In a table consisting of $n$ by $n$ small squares some squares are coloured black and the other squares are coloured white. For each pair of columns and each pair of rows the four squares on the intersections of these rows and columns must not all be of the same colour. What is the largest possible value of $n$?

2022 3rd Memorial "Aleksandar Blazhevski-Cane", P1

Tags: combinatorics , board , coloring , chesslike , maximum

A $6 \times 6$ board is given such that each unit square is either red or green. It is known that there are no $4$ adjacent unit squares of the same color in a horizontal, vertical, or diagonal line. A $2 \times 2$ subsquare of the board is [i]chesslike[/i] if it has one red and one green diagonal. Find the maximal possible number of chesslike squares on the board. [i]Proposed by Nikola Velov[/i]

2014 Hanoi Open Mathematics Competitions, 15

Tags: algebra , sum , inequalities , maximum

Let $a_1,a_2,...,a_9 \ge - 1$ and $a^3_1+a^3_2+...+a^3_9= 0$. Determine the maximal value of $M = a_1 + a_2 + ... + a_9$.

2015 Dutch Mathematical Olympiad, 1

Tags: combinatorics , maximum , set

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?

2016 India Regional Mathematical Olympiad, 2

Tags: maximum , algebra , combinatorics

On a stormy night ten guests came to dinner party and left their shoes outside the room in order to keep the carpet clean. After the dinner there was a blackout, and the gusts leaving one by one, put on at random, any pair of shoes big enough for their feet. (Each pair of shoes stays together). Any guest who could not find a pair big enough spent the night there. What is the largest number of guests who might have had to spend the night there?

2018 Hanoi Open Mathematics Competitions, 2

Tags: area , maximum , hexagon , equilateral , geometry

What is the largest area of a regular hexagon that can be drawn inside the equilateral triangle of side $3$? A. $3\sqrt7$ B. $\frac{3 \sqrt3}{2}$ C. $2\sqrt5$ D. $\frac{3\sqrt3}{8}$ E. $3\sqrt5$

2009 Stars Of Mathematics, 3

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

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.

2018 India PRMO, 9

Tags: trinomial , maximum , integer , root

Suppose $a, b$ are integers and $a+b$ is a root of $x^2 +ax+b = 0$. What is the maximum possible value of $b^2$?

1984 Brazil National Olympiad, 2

Tags: combinatorics , maximum , group

Each day $289$ students are divided into $17$ groups of $17$. No two students are ever in the same group more than once. What is the largest number of days that this can be done?

2015 Hanoi Open Mathematics Competitions, 15

Tags: algebra , maximum , summation , inequalities

Let the numbers $a, b,c$ satisfy the relation $a^2+b^2+c^2+d^2 \le 12$. Determine the maximum value of $M = 4(a^3 + b^3 + c^3+d^3) - (a^4 + b^4 + c^4+d^4)$

2017 India PRMO, 15

Tags: maximum , average , arithmetic mean , algebra

Integers $1, 2, 3, ... ,n$, where $n > 2$, are written on a board. Two numbers $m, k$ such that $1 < m < n, 1 < k < n$ are removed and the average of the remaining numbers is found to be $17$. What is the maximum sum of the two removed numbers?

2016 Balkan MO Shortlist, A5

Tags: algebra , system of equations , inequalities , minimum , maximum

Let $a, b,c$ and $d$ be real numbers such that $a + b + c + d = 2$ and $ab + bc + cd + da + ac + bd = 0$. Find the minimum value and the maximum value of the product $abcd$.

1966 All Russian Mathematical Olympiad, 083

Tags: minimum , maximum , game strategy , combinatorics

$20$ numbers are written on the board $1, 2, ... ,20$. Two players are putting signs before the numbers in turn ($+$ or $-$). The first wants to obtain the minimal possible absolute value of the sum. What is the maximal value of the absolute value of the sum that can be achieved by the second player?

1976 All Soviet Union Mathematical Olympiad, 226

Tags: concyclic , maximum , midpoint

Given regular $1976$-gon. The midpoints of all the sides and diagonals are marked. What is the greatest number of the marked points lying on one circumference?

2018 India PRMO, 11

Tags: combinatorics , maximum

There are several teacups in the kitchen, some with handles and the others without handles. The number of ways of selecting two cups without a handle and three with a handle is exactly $1200$. What is the maximum possible number of cups in the kitchen?

2016 German National Olympiad, 6

Tags: cyclic function , function , inequality , maximum , algebra

Let \[ f(x_1,x_2,x_3,x_4,x_5,x_6,x_7)=x_1x_2x_4+x_2x_3x_5+x_3x_4x_6+x_4x_5x_7+x_5x_6x_1+x_6x_7x_2+x_7x_1x_3 \] be defined for non-negative real numbers $x_1,x_2,\dots,x_7$ with sum $1$. Prove that $f(x_1,x_2,\dots,x_7)$ has a maximum value and find that value.

1990 All Soviet Union Mathematical Olympiad, 517

Tags: inequalities , maximum , algebra , absolute value

What is the largest possible value of $|...| |a_1 - a_2| - a_3| - ... - a_{1990}|$, where $a_1, a_2, ... , a_{1990}$ is a permutation of $1, 2, 3, ... , 1990$?

2015 Balkan MO Shortlist, N5

Tags: number theory , maximum , power of 2 , divide , product

For a positive integer $s$, denote with $v_2(s)$ the maximum power of $2$ that divides $s$. Prove that for any positive integer $m$ that: $$v_2\left(\prod_{n=1}^{2^m}\binom{2n}{n}\right)=m2^{m-1}+1.$$ (FYROM)

1946 Moscow Mathematical Olympiad, 111

Tags: plane , maximum , geometry , 3d geometry , angle

Given two intersecting planes $\alpha$ and $\beta$ and a point $A$ on the line of their intersection. Prove that of all lines belonging to $\alpha$ and passing through $A$ the line which is perpendicular to the intersection line of $\alpha$ and $\beta$ forms the greatest angle with $\beta$.

1962 All Russian Mathematical Olympiad, 023

Tags: inequalities , geometry , maximum , area

What maximal area can have a triangle if its sides $a,b,c$ satisfy inequality $0\le a\le 1\le b\le 2\le c\le 3$ ?

2014 Contests, 1

Tags: number theory , prime , factor , maximum

A natural number $k$ is such that $k^2 < 2014 < (k +1)^2$. What is the largest prime factor of $k$?