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.

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.

Let $a, b$ be two real numbers that satisfy $a^3 + b^3 = 8-6ab$. Find the maximum value and the minimum value that $a + b$ can take.

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 $n$?

The number $n$ is called a composite number if it can be written in the form $n = a\times b$, where $a, b$ are positive integers greater than $1$. Write number $2013$ in a sum of $m$ composite numbers. What is the largest value of $m$? (A): $500$, (B): $501$, (C): $502$, (D): $503$, (E): None of the above.

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]

The pupil is training in the square equation solution. Having the recurrent equation solved, he stops, if it doesn't have two roots, or solves the next equation, with the free coefficient equal to the greatest root, the coefficient at $x$ equal to the least root, and the coefficient at $x^2$ equal to $1$. Prove that the process cannot be infinite. What maximal number of the equations he will have to solve?

The goal of this problem is to show that the maximum area of a polygon with a fixed number of sides and a fixed perimeter is achieved by a regular polygon. (a) Prove that the polygon with maximum area must be convex. (Hint: If any angle is concave, show that the polygon’s area can be increased.) (b) Prove that if two adjacent sides have different lengths, the area of the polygon can be increased without changing the perimeter. (c) Prove that the polygon with maximum area is equilateral, that is, has all the same side lengths. It is true that when given all four side lengths in order of a quadrilateral, the maximum area is achieved in the unique configuration in which the quadrilateral is cyclic, that is, it can be inscribed in a circle. (d) Prove that in an equilateral polygon, if any two adjacent angles are different then the area of the polygon can be increased without changing the perimeter. (e) Prove that the polygon of maximum area must be equiangular, or have all angles equal. (f ) Prove that the polygon of maximum area is a regular polygon. PS. You had better use hide for answers.

Let the numbers $a, b,c$ satisfy the relation $a^2+b^2+c^2 \le 8$. Determine the maximum value of $M = 4(a^3 + b^3 + c^3) - (a^4 + b^4 + c^4)$

From the set of numbers $1 , 2, 3, . . . , 1985$ choose the largest subset such that the difference between any two numbers in the subset is not a prime number (the prime numbers are $2, 3 , 5 , 7,... , 1$ is not a prime number) .

Let $x, y,z$ satisfy the following inequalities $\begin{cases} | x + 2y - 3z| \le 6 \\ | x - 2y + 3z| \le 6 \\ | x - 2y - 3z| \le 6 \\ | x + 2y + 3z| \le 6 \end{cases}$ Determine the greatest value of $M = |x| + |y| + |z|$.

For a positive integer $n$, we dene $D_n$ 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 $D_n = 3^k$.

If all $x_k$ ($k = 1, 2, 3, 4, 5)$ are positive reals, and $\{a_1,a_2, a_3, a_4, a_5\} = \{1, 2,3 , 4, 5\}$, find the maximum of $$\frac{(\sqrt{s_1x_1} +\sqrt{s_2x_2}+\sqrt{s_3x_3}+\sqrt{s_4x_4}+\sqrt{s_5x_5})^2}{a_1x_1 + a_2x_2 + a_3x_3 + a_4x_4 + a_5x_5}$$ ($s_k = a_1 + a_2 +... + a_k$)

a) Given a big square consisting of $7\times 7$ squares. You should mark the centres of $k$ points in such a way, that no quadruple of the marked points will be the vertices of a rectangle with the sides parallel to the sides of the given squares. What is the greatest $k$ such that the problem has solution? b) The same problem for $13\times 13$ square.

The quadrilateral $ABCD$ is inscribed in a circle of radius $1$, so that $AB$ is a diameter of the circumference and $CD = 1$. A variable point $X$ moves along the semicircle determined by $AB$ that does not contain $C$ or $D$. Determine the position of $X$ for which the sum of the distances from $X$ to lines $BC, CD$ and $DA$ is maximum.

Let the sum of positive numbers $x_1, x_2, ... , x_n$ be $1$. Let $s$ be the greatest of the numbers $$\left\{\frac{x_1}{1+x_1}, \frac{x_2}{1+x_1+x_2}, ..., \frac{x_n}{1+x_1+...+x_n}\right\}$$ What is the minimal possible $s$? What $x_i $correspond it?

Find all odd natural numbers $n$ such that $d(n)$ is the largest divisor of the number $n$ different from $n$. ($d(n)$ is the number of divisors of the number n including $1$ and $n$ ).

Let $x\in\mathbb{R}$. What is the maximum value of the following expression: $\sqrt{x-2018} + \sqrt{2020-x}$ ?

Among all orthogonal projections of a regular tetrahedron to all possible planes, find the projection of the greatest area.

1 - Prove that $55 < (1+\sqrt{3})^4 < 56$ . 2 - Find the largest power of $2$ that divides $\lceil(1+\sqrt{3})^{2n}\rceil$ for the positive integer $n$

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?

Find maximum of the expression $(a -b^2)(b - a^2)$, where $0 \le a,b \le 1$.

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?

Given two positive numbers $a,b$ such that $a^3 +b^3 = a^5 +b^5$, then the greatest value of $M = a^2 + b^2 - ab$ is (A): $\frac14$ (B): $\frac12$ (C): $2$ (D): $1$ (E): None of the above.

The natural numbers $ a_1 $, $ a_2 $, $ \dots $, $ a_n $ satisfy the condition $ 1 / a_1 + 1 / a_2 + \ldots + 1 / a_n = 1 $. Prove that all these numbers do not exceed $$ n ^ {2 ^ n} $$