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.

A convex hexagon is circumscribed about a circle of radius $1$. Consider the three segments joining the midpoints of its opposite sides. Find the greatest real number $r$ such that the length of at least one segment is at least $r.$

Let $p = 1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\frac{1}{2^4}+\frac{1}{2^5}. $ For nonnegative reals $x, y,z$ satisfying $(x-1)^2 + (y-1)^2 + (z-1)^2 = 27,$ find the maximum value of $x^p + y^p + z^p.$

The world and the european champion are determined in the same tournament carried in one round. There are $20$ teams and $k$ of them are european. The european champion is determined according to the results of the games only between those $k$ teams. What is the greatest $k$ such that the situation, when the single european champion is the single world outsider, is possible if: a) it is hockey (draws allowed)? b) it is volleyball (no draws)?

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 $abcd$.

The five sides of a regular pentagon are extended to lines $\ell_1, \ell_2, \ell_3, \ell_4$, and $\ell_5$. Denote by $d_i$ the distance from a point $P$ to $\ell_i$. For which point(s) in the interior of the pentagon is the product $d_1d_2d_3d_4d_5$ maximal?

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 $n$?

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

Board with dimesions $2018 \times 2018$ is divided in unit cells $1 \times 1$. In some cells of board are placed black chips and in some white chips (in every cell maximum is one chip). Firstly we remove all black chips from columns which contain white chips, and then we remove all white chips from rows which contain black chips. If $W$ is number of remaining white chips, and $B$ number of remaining black chips on board and $A=min\{W,B\}$, determine maximum of $A$

Let $x,y$ be positive numbers, $s$ -- the least of $$\{ x, (y+ 1/x), 1/y\}$$ What is the greatest possible value of $s$? To what $x$ and $y$ does it correspond?

Determine the largest integer $N$, for which there exists a $6\times N$ table $T$ that has the following properties: $*$ Every column contains the numbers $1,2,\ldots,6$ in some ordering. $*$ For any two columns $i\ne j$, there exists a row $r$ such that $T(r,i)= T(r,j)$. $*$ For any two columns $i\ne j$, there exists a row $s$ such that $T(s,i)\ne T(s,j)$. (Proposed by Gerhard Woeginger, Austria)

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?

16 students took part in a competition. All problems were multiple choice style. Each problem had four choices. It was said that any two students had at most one answer in common, find the maximum number of problems.

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.

There are $9$ cards with the numbers $1, 2, 3, 4, 5, 6, 7, 8$ and $9$. What is the largest number of these cards can be decomposed in a certain order in a row, so that in any two adjacent cards, one of the numbers is divided by the other?

What is the largest number of acute angles that a convex polygon can have?

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 $D_n = 3^k$.

A unit square has a circle of radius $r$ with center at it's midpoint. The four quarter circles are centered on the vertices of the square and are tangent to the central circle (see figure). Find the maximum and minimum possible value of the area of the striped figure in the figure and the corresponding values of $r$ such these, the maximum and minimum are achieved. [img]https://2.bp.blogspot.com/-DOT4_B5Mx-8/XnmsTlWYfyI/AAAAAAAALgs/TVYkrhqHYGAeG8eFuqFxGDCTnogVbQFUwCK4BGAYYCw/s400/96%2Bestonia%2Bopen%2Bs1.4.png[/img]

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?

Let $X$ be the set of natural numbers whose all digits in the decimal representation are different. For $n \in N$, denote by $A_n$ the set of numbers whose digits are a permutation of the digits of $n$, and $d_n$ be the greatest common divisor of the numbers in $A_n$. (For example, $A_{1120} =\{112,121,...,2101,2110\}$, so $d_{1120} = 1$.) Find the maximum possible value of $d_n$.

We write $\{a,b,c\}$ for the set of three different positive integers $a, b$, and $c$. By choosing some or all of the numbers a, b and c, we can form seven nonempty subsets of $\{a,b,c\}$. We can then calculate the sum of the elements of each subset. For example, for the set $\{4,7,42\}$ we will find sums of $4, 7, 42,11, 46, 49$, and $53$ for its seven subsets. Since $7, 11$, and $53$ are prime, the set $\{4,7,42\}$ has exactly three subsets whose sums are prime. (Recall that prime numbers are numbers with exactly two different factors, $1$ and themselves. In particular, the number $1$ is not prime.) What is the largest possible number of subsets with prime sums that a set of three different positive integers can have? Give an example of a set $\{a,b,c\}$ that has that number of subsets with prime sums, and explain why no other three-element set could have more.

Let $a,b, c$ denote the real numbers such that $1 \le a, b, c\le 2$. Consider $T = (a - b)^{2018} + (b - c)^{2018} + (c - a)^{2018}$. Determine the largest possible value of $T$.

The senior coefficient $a$ in the square polynomial $$P(x) = ax^2 + bx + c$$ is more than $100$. What is the maximal number of integer values of $x$, such that $|P(x)|<50$.

If $x$ and $y$ are nonnegative real numbers such that $x+y=1$, determine minimal and maximal value of $$A=x\sqrt{1+y}+y\sqrt{1+x}$$

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.