Given a circle $\gamma$ and a point $P$ inside it, find the maximum and minimum value of the sum of the lengths of two perpendicular chords of $\gamma$ passing through $P$.

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.

Find the maximum and minimum of $x + y$ such that $x + y = \sqrt{2x-1}+\sqrt{4y+3}$

Petya and Vasya live in neighboring houses (see the plan in the figure). Vasya lives in the fourth entrance. It is known that Petya runs to Vasya by the shortest route (it is not necessary walking along the sides of the cells) and it does not matter from which side he runs around his house. Determine in which entrance he lives Petya . [img]https://cdn.artofproblemsolving.com/attachments/b/1/741120341a54527b179e95680aaf1c4b98ff84.png[/img]

A circle is inscribed in an angle with vertex $O$, touching its sides at points $M$ and $N$. On an arc $MN$ nearest to point $O$, an arbitrary point $P$ is selected. At point $P$, a tangent is drawn to the circle $P$, intersecting the sides of the angle at points $A$ and $B$. Prove that that the length of the segment $AB$ is the smallest when $P$ is its midpoint.

Let $F$ be the boundary and $M,N$ be any interior points of a triangle $ABC$. Consider the function $f_{M,N}: F \to R$ defined by $f_{M,N}(P) = MP^2 +NP^2$ and let $\eta_{M,N}$ be the number of points $P$ for which $f{M,N}$ attains its minimum. (a) Prove that $1 \le \eta_{M,N} \le 3$. (b) If $M$ is fixed, find the locus of $N$ for which $\eta_{M,N} > 1$. (c) Prove that the locus of $M$ for which there are points $N$ such that $\eta_{M,N} = 3$ is the interior of a tangent hexagon.

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]

Find the maximum and minimum values of $x^2 + 2y^2 + 3z^2$ for real $x, y, z$ satisfying $x^2 + y^2 + z^2 = 1$.

The numbers $1,2,...,64 $ are written on the unit squares of a table $8 \times 8$. The two smallest numbers of every row are marked black and the two smaller numbers of every comlumn are marked white. Prove or disprove that there are at least k numbers on the table that are marked both black and white when: a) $k=3$ b) $k=4$ c) $k=5$ .

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 $C$.

Let $a, b,c$ be positive real numbers satisfying the condition $$(x + y + z) \left( \frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right)= 10$$ Find the greatest value and the least value of $$T = (x^2 + y^2 + z^2) \left(\frac{1}{x^2} + \frac{1}{y^2} + \frac{1}{z^2}\right)$$ Trần Nam Dũng

(a) Find the point $M$ in the plane of triangle $ABC$ for which the sum $MA + MB+ MC$ is minimal. (b) Given a parallelogram $ABCD$ whose angles do not exceed $120^o$, determine $min \{MA+ MB+NC+ND+ MN | M,N$ are in the plane $ABCD\}$ in terms of the sides and angles of the parallelogram.

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 $a^2 + b^2 + c^2 = 1$

Given that $n> 2$ is a positive integer and a sequence of positive integers $a_1 <a_2 <...<a_n$. In the subsets of the set $\{1,2,..., n\} $, there a subset $X$ such that $| \sum_{i \notin X} a_i -\sum_{i \in X} a_i |$ is the smallest . Prove that there exists a sequence of positive integers $0<b_1 <b_2 <...<b_n$ such that $\sum_{i \notin X} b_i= \sum_{i \in X} b_i$. In case this doesn't make sense, have a look at [url=https://drive.google.com/file/d/1xoBhJlG0xHwn6zAAA7AZDoaAqzZue-73/view]original wording in Vietnamese[/url].

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

We are allowed to put several brackets in the expression $$\frac{29 : 28 : 27 : 26 :... : 17 : 16}{15 : 14 : 13 : 12 : ... : 3 : 2}$$ always in the same places below each other. (a) Find the smallest possible integer value we can obtain in that way. (b) Find all possible integer values that can be obtained. Remark: in this problem, $$\frac{(29 : 28) : 27 : ... : 16}{(15 : 14) : 13 : ... : 2},$$ is valid position of parenthesis, on the other hand $$\frac{(29 : 28) : 27 : ... : 16}{15 : (14 : 13) : ... : 2}$$ is forbidden.

For an integer $n \ge 2$, find all real numbers $x$ for which the polynomial $f(x) = (x-1)^4 +(x-2)^4 +...+(x-n)^4$ takes its minimum value.

Prove that the sum of the squares of the distances from a point $P$ to the vertices of a triangle $ABC$ is minimum when $ P$ is the centroid of the triangle.

Find the smallest positive value of $36^k - 5^m$, where $k$ and $m$ are positive integers.

Find the maximum and the minimum values of $S = (1 - x_1)(1 -y_1) + (1 - x_2)(1 - y_2)$ for real numbers $x_1, x_2, y_1,y_2$ with $x_1^2 + x_2^2 = y_1^2 + y_2^2 = 2013$.

Let $n$ be an integer, $n \ge 3$. Select $n$ points on the plane, none of which are three on the same line. Consider all triangles with vertices at selected points, denote the smallest of all the interior angles of these triangles by the variable $\alpha$. Find the largest possible value of $\alpha$ and identify all the selected $n$ point placements for which the max occurs.

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 $n \in N$.

Find the smallest $n$ such that $n^2 -n+11$ is the product of four primes (not necessarily distinct).

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