Determine the smallest possible value of the sum $S (a, b, c) = \frac{ab}{c}+\frac{bc}{a}+\frac{ca}{b}$ where $a, b, c$ are three positive real numbers with $a^2 + b^2 + c^2 = 1$

Let $x, y, z$ be reals summing to $1$ which minimizes $2x^2 + 3y^2 + 4z^2$. Find $x$.

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 triangle $ABC$, the lengths of all sides are integers, $\angle B=2 \angle A$ and $\angle C> 90^o$. Find the smallest possible perimeter of this triangle.

Given an integer $n > 2$, consider all sequences $x_1,x_2,...,x_n$ of nonnegative real numbers such that $$x_1+ 2x_2 + ... + nx_n = 1.$$ Find the maximum value and the minimum value of $x_1^2+x_2^2+...+x_n^2$ and determine all the sequences $x_1,x_2,...,x_n$ for which these values are obtained.

For positive integers $n \ge 3$ and $r \ge 1$, define $$P(n, r) = (n - 2)\frac{r^2}{2} - (n - 4) \frac{r}{2}$$ We call a triple $(a, b, c)$ of natural numbers, with $a \le b \le c$, an $n$-gonal Pythagorean triple if $P(n, a)+P(n, b) = P(n, c)$. (For $n = 4$, we get the usual Pythagorean triple.) (a) Find an $n$-gonal Pythagorean triple for each $n \ge 3$. (b) Consider all triangles $ABC$ whose sides are $n$-gonal Pythagorean triples for some $n \ge 3$. Find the maximum and the minimum possible values of angle $C$.

Define $g(x)$ as the largest value of$ |y^2 - xy|$ for $y$ in $[0, 1]$. Find the minimum value of $g$ (for real $x$).

Find the smallest value of the expression $(x-1) (x -2) (x -3) (x - 4) + 10$.

The lengths, $a$ and $b$, of two sides of a triangle are known. (a) What length should the third side be, in order for the largest angle of the triangle to be of the least possible value? (b) What length should the third side be in order for the smallest angle of the triangle to be of the greatest possible value?

Let a triangle $ABC$ be given. For any point $X$ of the triangle denote $m(X)=\min\{XA,XB,XC\}.$ Find all points $X$ (of triangle $ABC$) such that $m(X)$ is maximal.

Let $x, y$ be positive real numbers. Find the minimum of $$x^2 + xy +\frac{y^2}{2}+\frac{2^6}{x + y}+\frac{3^4}{x^3}$$

Determine the minimum number of lines that can be drawn on the plane so that they intersect in exactly $200$ distinct points. (Note that for $3$ distinct points, the minimum number of lines is $3$ and for $4$ distinct points, the minimum is $4$)

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.

Peeter, Juri, Kati and Mari are standing at the entrance of a dark tunnel. They have one torch and none of them dares to be in the tunnel without it, but the tunnel is so narrow that at most two people can move together. It takes $1$ minute for Peeter, $2$ minutes for Juri, $5$ for Kati and $10$ for Mari to pass the tunnel. Find the minimum time in which they can all pass through the tunnel.

On the midline of the isosceles trapezoid $ABCD$ ($BC \parallel AD$) find the point $K$, for which the sum of the angles $\angle DAK + \angle BCK$ will be the smallest.

The cities of countries $A$ and $B$ are marked on the map, which has the form of a square with vertices at points $(0, 0)$ , $ (0, 1)$ , $(1, 1)$ , $(1, 0)$ of the plane. According to the trade agreement, country $A$ must ensure the delivery of $n$ kg of wheat to $n$ cities of country $B$, located at the points of the square with coordinates $y_1,..., y_n$, $1$ kg each city. Currently, $n$ kg of wheat are distributed among $n$ cities of country $A$, located at the points of the square with coordinates $x_1,... , x_n$, $1$ kg in each city. From each city of country $A$ to each city of the country $A$ any amount of wheat can be transported (of course, not more than $1$ kg). Transportation cost is for $t$ kg of wheat from a city with coordinates $x_i$ to a city with coordinates $y_j$ is equal to $tl_{ij}$, where $l_{ij }$is the length of the segment connecting the points $x_i$ and $y_j$. The government of country A is going to implement the optimal one (i,e. the cheapest) transportation plan. (a) Is it possible to implement the optimal transportation plan so that from each city of country $A$ to transport wheat only to one city of country $B$? (b) Will the response change if country $A$ is to deliver $n+1$ kg of wheat, in city $x_1$ is $2$ kg of wheat, and $2$ kg should be delivered to city $y_1$ (when for other cities the conditions remain the same)? [hide=original wording] Мiста країн A та B позначенi на мапi, що має вигляд квадрату з вершинами в точках (0, 0), (0, 1), (1, 1), (1, 0) площини. Згiдно торгової угоди, країна A має забез- печити доставку n кг пшеницi в n мiст країни B, що розташованi в точках квадрату з координатами y1, . . . , yn, по 1 кг в кожне мiсто. Наразi n кг пшеницi розподiленi серед n мiст країни A, що розташованi в точках квадрату з координатами x1, . . . , xn, по 1 кг в кожному мiстi. З кожного мiста країни A в кожне мiсто країни B можна перевезти довiльну кiлькiсть пшеницi (звичайно, не бiльше 1 кг). Вартiсть переве- зення t кг пшеницi з мiста з координатами xi в мiсто з координатами yj дорiвнює tlij , де lij – довжина вiдрiзку, що сполучає точки xi та yj . Уряд країни A збирається реалiзувати оптимальний (тобто найдешевший) план перевезення. 1. Чи можна реалiзувати оптимальний план перевезення таким чином, щоби з кожного мiста країни A перевозити пшеницю тiльке в одне мiсто країни B? 2. Чи змiниться вiдповiдь, якщо країна A має забезпечити доставку n + 1 кг пше- ницi, в мiстi x1 знаходиться 2 кг пшеницi, i в мiсто y1 має бути доставлено 2 кг пшеницi (щодо iнших мiст умови лишаються такими ж)?[/hide]

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 $\frac{M}{m} \ge \sqrt3$.

Let $a$ and $b$ be positive real numbers such that $3(a^2+b^2-1) = 4(a+b$). Find the minimum value of the expression $\frac{16}{a}+\frac{1}{b}$ .

Let $n$ be a positive integer. Determine the smallest value of the sum $a_1b_1+a_2b_2+...+a_{2n+2}b_{2n+2}$ where $(a_1,a_2,...,a_{2n+2})$ and $(b_1,b_2,...,b_{2n+2})$ are rearrangements of the binomial coefficients $2n+1 \choose 0$, $2n+1 \choose 1$,...,$2n+1 \choose 2n+1$. Justify your answer

Consider the cube with the vertices at the points $(\pm 1, \pm 1, \pm 1)$. Let $V_1,...,V_{95}$ be arbitrary points within this cube. Denote $v_i = \overrightarrow{OV_i}$, where $O = (0,0,0)$ is the origin. Consider the $2^{95}$ vectors of the form $s_1v_1 + s_2v_2 +...+ s_{95}v_{95}$, where $s_i = \pm 1$. (a) If $d = 48$, prove that among these vectors there is a vector $w = (a, b, c)$ such that $a^2 + b^2 + c^2 \le 48$. (b) Find a smaller $d$ (the smaller, the better) with the same property.

For a natural number $n$, let $w(n)$ denote the number of (positive) prime divisors of $n$. Find the smallest positive integer $k$ such that $2^{w(n)} \le k \sqrt[4]{ n}$ for each $n \in N$.

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 $n = 4$.

Let $m$ be a positive integer. Determine the smallest positive integer $n$ for which there exist real numbers $x_1, x_2,...,x_n \in (-1, 1)$ such that $|x_1| + |x_2| +...+ |x_n| = m + |x_1 + x_2 + ... + x_n|$.

Define $S_n=\sum_{j,k=1}^{n} \frac{1}{\sqrt{j^2+k^2}}$. Find a positive constant $C$ such that the inequality $n\le S_n \le Cn$ holds for all $n \ge 3$. (Note. The smaller $C$, the better the solution.)

A Christmas tree must be erected inside a convex rectangular garden and attached to the posts at the corners of the garden with four ropes running at the same height from the ground. At what point should the Christmas tree be placed, so that the sum of the lengths of these four cords is as small as possible?