$2013$ users have registered on the social network "Graph". Some users are friends, and friendship in "Graph" is mutual. It is known that among network users there are no three, each of whom would be friends. Find the biggest one possible number of pairs of friends in "Graph".

Determine all functions $f : R \to R$ satisfying $f(f(x) + 2y)= 6x + f(f(y) -x)$ for all real numbers $x,y$

[b]a)[/b] For $ a,b\ge 0 $ and $ x,y>0, $ show that: $$ \frac{a^3}{x^2} +\frac{b^3}{y^2}\ge \frac{(a+b)^3}{(x+y)^2} . $$ [b]b)[/b] For $ a,b,c\ge 0 $ and $ x,y,z>0 $ under the condition $ a+b+c=x+y+z, $ prove that: $$ \frac{a^3}{x^2} +\frac{b^3}{y^2} +\frac{c^3}{z^2} \ge a+b+c. $$

p1. Given $N = 9 + 99 + 999 + ... +\underbrace{\hbox{9999...9}}_{\hbox{121\,\,numbers}}$. Determine the value of N. p2. The triangle $ABC$ in the following picture is isosceles, with $AB = AC =90$ cm and $BC = 108$ cm. The points $P$ and $Q$ are located on $BC$, respectively such that $BP: PQ: QC = 1: 2: 1$. Points $S$ and $R$ are the midpoints of $AB$ and $AC$ respectively. From these two points draw a line perpendicular to $PR$ so that it intersects at $PR$ at points $M$ and $N$ respectively. Determine the length of $MN$. [img]https://cdn.artofproblemsolving.com/attachments/7/1/e1d1c4e6f067df7efb69af264f5c8de5061a56.png[/img] p3. If eight equilateral triangles with side $ 12$ cm are arranged as shown in the picture on the side, we get a octahedral net. Define the volume of the octahedron. [img]https://cdn.artofproblemsolving.com/attachments/4/8/18cdb8b15aaf4d92f9732880784facf9348a84.png[/img] p4. It is known that $a^2 + b^2 = 1$ and $x^2 + y^2 = 1$. Continue with the following algebraic process. $(a^2 + b^2)(x^2 + y^2) – (ax + by)^2 = ...$ a. What relationship can be concluded between $ax + by$ and $1$? b. Why? p5. A set of questions consists of $3$ questions with a choice of answers True ($T$) or False ($F$), as well as $3$ multiple choice questions with answers $A, B, C$, or $D$. Someone answer all questions randomly. What is the probability that he is correct in only $2$ questions?

A positive integer $n$ is abundant if the sum of its proper divisors exceeds $n$. Show that every integer greater than $89 \times 315$ is the sum of two abundant numbers.

In $\triangle ABC$, $AB=17$, $AC=25$, and $BC=28$. Points $M$ and $N$ are the midpoints of $\overline{AB}$ and $\overline{AC}$ respectively, and $P$ is a point on $\overline{BC}$. Let $Q$ be the second intersection point of the circumcircles of $\triangle BMP$ and $\triangle CNP$. It is known that as $P$ moves along $\overline{BC}$, line $PQ$ passes through some fixed point $X$. Compute the sum of the squares of the distances from $X$ to each of $A$, $B$, and $C$.

The integers $ c_{m,n}$ with $ m \geq 0, \geq 0$ are defined by \[ c_{m,0} \equal{} 1 \quad \forall m \geq 0, c_{0,n} \equal{} 1 \quad \forall n \geq 0,\] and \[ c_{m,n} \equal{} c_{m\minus{}1,n} \minus{} n \cdot c_{m\minus{}1,n\minus{}1} \quad \forall m > 0, n > 0.\] Prove that \[ c_{m,n} \equal{} c_{n,m} \quad \forall m > 0, n > 0.\]

Let $a$ and $b$ be two positive integers satisfying the equation \[ 20\sqrt{12} = a\sqrt{b}. \] Compute the sum of all possible distinct products $ab$. [i]Proposed by Lewis Chen[/i]

Let $ a,b,c$ be nonnegative real numbers. Suppose that $ a\plus{}b\plus{}c\ge abc$. Prove that: $ a^2\plus{}b^2\plus{}c^2 \ge abc.$

An archery target can be represented as three concentric circles with radii $3$, $2$, and $1$ which split the target into $3$ regions, as shown in the figure below. What is the area of Region $1$ plus the area of Region $3$? [i]2015 CCA Math Bonanza Team Round #1[/i]

[center][u]Guts Round[/u] / [u]Set 7[/u][/center] [b]p19.[/b] Let $N \ge 3$ be the answer to Problem 21. A regular $N$-gon is inscribed in a circle of radius $1$. Let $D$ be the set of diagonals, where we include all sides as diagonals. Then, let $D'$ be the set of all unordered pairs of distinct diagonals in $D$. Compute the sum $$\sum_{\{d,d'\}\in D'} \ell (d)^2 \ell (d')^2,$$where $\ell (d)$ denotes the length of diagonal $d$. [b]p20.[/b] Let $N$ be the answer to Problem $19$, and let $M$ be the last digit of $N$. Let $\omega$ be a primitive $M$th root of unity, and define $P(x)$ such that$$P(x) = \prod^M_{k=1} (x - \omega^{i_k}),$$where the $i_k$ are chosen independently and uniformly at random from the range $\{0, 1, . . . ,M-1\}$. Compute $E \left[P\left(\sqrt{\rfloor \frac{1250}{N} \rfloor } \right)\right].$ [b]p21.[/b] Let $N$ be the answer to Problem $20$. Define the polynomial $f(x) = x^{34} +x^{33} +x^{32} +...+x+1$. Compute the number of primes $p < N$ such that there exists an integer $k$ with $f(k)$ divisible by $p$.

Let $ABCD$ be a convex quadrilateral with $\angle ABC>90$, $CDA>90$ and $\angle DAB=\angle BCD$. Denote by $E$ and $F$ the reflections of $A$ in lines $BC$ and $CD$, respectively. Suppose that the segments $AE$ and $AF$ meet the line $BD$ at $K$ and $L$, respectively. Prove that the circumcircles of triangles $BEK$ and $DFL$ are tangent to each other. $\emph{Slovakia}$

Nonzero real numbers $x_1, x_2, \ldots, x_n$ satisfy the following condition: $$x_1 - \frac{1}{x_2} = x_2 - \frac{1}{x_3} = \ldots = x_{n-1} - \frac{1}{x_n} = x_n - \frac{1}{x_1}$$ Determine all $n$ for which $x_1, x_2, \ldots, x_n$ have to be equal. [i](Proposed by Oleksii Masalitin, Anton Trygub)[/i]

The Lucas numbers $L_n$ are defined recursively as follows: $L_0=2,L_1=1,L_n=L_{n-1}+L_{n-2}$ for $n\geq2$. Let $r=0.21347\dots$, whose digits form the pattern of the Lucas numbers. When the numbers have multiple digits, they will "overlap," so $r=0.2134830\dots$, [b]not[/b] $0.213471118\dots$. Express $r$ as a rational number $\frac{p}{q}$, where $p$ and $q$ are relatively prime.

Let the lengths of the sides of triangle $ABC$ be denoted by $a,b,$ and $c,$ using the standard notations. Let $G$ denote the centroid of triangle $ABC.$ Prove that for an arbitrary point $P$ in the plane of the triangle the following inequality is true: \[a\cdot PA^3+b\cdot PB^3+c\cdot PC^3\geq 3abc\cdot PG.\][i]Proposed by János Schultz, Szeged[/i]

Leonard wrote three 3-digit numbers on the board whose sum is $1000$. All of the nine digits are different. Determine which digit does not appear on the board. [i]Proposed by Giorgi Arabidze, Georgia[/i]