The [i]distance[/i] between two vertices in a connected graph is defined to be the length of the shortest path between them. How many graphs with the vertex set $\{0,1,2,\dots,6\}$ satisfy the following property: there are $3$ vertices of distance $1$ away from vertex $0$, $2$ of distance $2$ away, and $1$ of distance $3$ away?

Find all the continuous functions $f : \mathbb{R} \mapsto\mathbb{R}$ such that $\forall x,y \in \mathbb{R}$, $(1+f(x)f(y))f(x+y)=f(x)+f(y)$.

In a mathematical contest, three problems, $A,B,C$ were posed. Among the participants ther were 25 students who solved at least one problem each. Of all the contestants who did not solve problem $A$, the number who solved $B$ was twice the number who solved $C$. The number of students who solved only problem $A$ was one more than the number of students who solved $A$ and at least one other problem. Of all students who solved just one problem, half did not solve problem $A$. How many students solved only problem $B$?

Let be four positive integers $m, n, p, q$, with $p < m$ given and $q < n$. Take four points $A(0; 0), B(p; 0), C (m; q)$ and $D(m; n)$ in the coordinate plane. Consider the paths $f$ from $A$ to $D$ and the paths $g$ from $B$ to $C$ such that when going along $f$ or $g$, one goes only in the positive directions of coordinates and one can only change directions (from the positive direction of one axe coordinate into the the positive direction of the other axe coordinate) at the points with integral coordinates. Let $S$ be the number of couples $(f, g)$ such that $f$ and $g$ have no common points. Prove that \[S = \binom{n}{m+n} \cdot \binom{q}{m+q-p} - \binom{q}{m+q} \cdot \binom{n}{m+n-p}.\]

Let $n_1,n_2,n_3,...,n_{2000}$ be $2000$ positive integers satisfying $n_1<n_2<n_3<...<n_{2000}$. Prove that $$\frac{n_1}{[n_1,n_2]}+\frac{n_1}{[n_2,n_3]}+\frac{n_1}{[n_3,n_4]}+...+\frac{n_1}{[n_{1999},n_{2000}]} \le 1 - \frac{1}{2^{1999}}$$ where $[a, b]$ denotes the least common multiple of $a$ and $b$.

In the creation of the world there is a lonely island inhabited by dragons, snakes and crocodiles. Every inhabitant eats once a day: every snake eats one dragon for breakfast, every dragon eats one crocodile for lunch and every crocodile eats a snake for dinner. Find the total number of dragons, snakes and crocodiles on the island immediately after the creation of the world (at the beginning of the first day), when, at the end of the sixth day, there is only one inhabitant alive on the island, only one crocodile and during these six days none of the inhabitants of the island considered any to give up their meals due to lack of food.

If $a,b,x,y$ are real numbers then find all solutions of $a,b$ such that $\left \lfloor ax+by \right \rfloor + \left \lfloor bx+ay \right \rfloor = (a+b) \left \lfloor x+y \right \rfloor$

Let be three finite and nonempty sets $ A,B,C $ such that $ |A|=|C|\le |B| , $ and a bijection $ A\stackrel{\beta }{\longrightarrow } C. $ How many pairs of functions $ A\stackrel{f_2 }{\longrightarrow } B\stackrel{f_1 }{\longrightarrow } C $ that satisfy $ f_1\circ f_2=\beta $ are there?

$ABC$ is an equilateral triangle with side length $1$. Point $D$ lies on $\overline{AB}$, point $E$ lies on $\overline{AC}$, and points $G$ and $F$ lie on $\overline{BC}$, such that $DEFG$ is a square. What is the area of $DEFG$?

Determine all $n \in \mathbb{Z}^+$ such that a regular hexagon (i.e. all sides equal length, all interior angles same size) can be partitioned in finitely many $n-$gons such that they can be composed into $n$ congruent regular hexagons in a non-overlapping way upon certain rotations and translations.

Group of $10$ people are buying books. We know the following: $i)$ Every person bought four different books $ii)$ Every two persons bought at least one book common for both of them Taking in consideration book which was bought by maximum number of people, determine minimal value of that number

Let $k$ be a positive integer. Two players $A$ and $B$ play a game on an infinite grid of regular hexagons. Initially all the grid cells are empty. Then the players alternately take turns with $A$ moving first. In his move, $A$ may choose two adjacent hexagons in the grid which are empty and place a counter in both of them. In his move, $B$ may choose any counter on the board and remove it. If at any time there are $k$ consecutive grid cells in a line all of which contain a counter, $A$ wins. Find the minimum value of $k$ for which $A$ cannot win in a finite number of moves, or prove that no such minimum value exists.

$n$ integers are arranged along a circle in such a way that each number is equal to the absolute value of the difference of the two numbers following that number in clockwise direction. If the sum of all numbers is $278$, how many different values can $n$ take? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 139 \qquad\textbf{(E)}\ \text{None of the above} $

A not necessary nonplanar polygon in $\mathbb{R}^3$ is called [b]Grid Polygon[/b] if each of it's edges are parallel to one of the axes. (a) There's a right angle between each two neighbour sides of the grid polygon, the plane containing this angle could be parallel to either $xy$ plane, $yz$ plane, or $xz$ plane. Prove that parity of the number of the angles that the plane containing each of them is parallel to $xy$ plane is equal to parity of the number of the angles that the plane containing each of them is parallel to $yz$ plane and parity of the number of the angles that the plane containing each of them is parallel to $zx$ plane. (b) A grid polygon is called [b]Inscribed[/b] if there's a point in the space that has an equal distance from all of the vertices of the polygon. Prove that any nonplanar grid hexagon is inscribed. (c) Does there exist a grid 2014-gon without repeated vertices such that there exists a plane that intersects all of it's edges? (d) If $a,b,c \in \mathbb{N}-\{1\}$, prove that $a,b,c$ are sidelengths of a triangle iff there exists a grid polygon in which the number of it's edges that are parallel to $x$ axis is $a$, the number of it's edges that are parallel to $y$ axis is $b$ and the number of it's edges that are parallel to $z$ axis is $c$. Time allowed for this exam was 1 hour.

The Judgment of the Council of Sages proceeds as follows: the king arranges the sages in a line and places either a white hat or a black hat on each sage's head. Each sage can see the color of the hats of the sages in front of him, but not of his own hat or of the hats of the sages behind him. Then one by one (in an order of their choosing), each sage guesses a color. Afterward, the king executes those sages who did not correctly guess the color of their own hat. The day before, the Council meets and decides to minimize the number of executions. What is the smallest number of sages guaranteed to survive in this case? See also [url]http://www.artofproblemsolving.com/Forum/viewtopic.php?f=42&t=530553[/url]

$(NET 3)$ Let $x_1, x_2, x_3, x_4,$ and $x_5$ be positive integers satisfying \[x_1 +x_2 +x_3 +x_4 +x_5 = 1000,\] \[x_1 -x_2 +x_3 -x_4 +x_5 > 0,\] \[x_1 +x_2 -x_3 +x_4 -x_5 > 0,\] \[-x_1 +x_2 +x_3 -x_4 +x_5 > 0,\] \[x_1 -x_2 +x_3 +x_4 -x_5 > 0,\] \[-x_1 +x_2 -x_3 +x_4 +x_5 > 0\] $(a)$ Find the maximum of $(x_1 + x_3)^{x_2+x_4}$ $(b)$ In how many different ways can we choose $x_1, . . . , x_5$ to obtain the desired maximum?

Let be a triangle $ ABC $ with $ \angle BAC = 90^{\circ } . $ On the perpendicular of $ BC $ through $ B, $ consider $ D $ such that $ AD=BC. $ Find $ \angle BAD. $

Show that it is possible to write a $n \times n$ array of non-negative numbers (not necessarily distinct) such that the sums of entries on each row and each column are pairwise distinct perfect squares.

Problems 14, 15 and 16 involve Mrs. Reed's English assignment. A Novel Assignment The students in Mrs. Reed's English class are reading the same 760-page novel. Three friends, Alice, Bob and Chandra, are in the class. Alice reads a page in 20 seconds, Bob reads a page in 45 seconds and Chandra reads a page in 30 seconds. Before Chandra and Bob start reading, Alice says she would like to team read with them. If they divide the book into three sections so that each reads for the same length of time, how many seconds will each have to read? $ \textbf{(A)}\ 6400 \qquad \textbf{(B)}\ 6600 \qquad \textbf{(C)}\ 6800 \qquad \textbf{(D)}\ 7000 \qquad \textbf{(E)}\ 7200$

Given $n$ digits $a_{1}, a_{2},..., a_{n}$ in that order. Does there exist a positive integer such that the first $n$ decimal digits after the dot of that number's square root are exactly those given digits¿ Give reason for your answer.

Let $S$ be the set $\{1,2,3,...,19\}$. For $a,b \in S$, define $a \succ b$ to mean that either $0 < a - b \leq 9$ or $b - a > 9$. How many ordered triples $(x,y,z)$ of elements of $S$ have the property that $x \succ y$, $y \succ z$, and $z \succ x$? $ \textbf{(A)} \ 810 \qquad \textbf{(B)} \ 855 \qquad \textbf{(C)} \ 900 \qquad \textbf{(D)} \ 950 \qquad \textbf{(E)} \ 988$

Let $ABC$ be a triangle with incenter $I$, and suppose the incircle is tangent to $CA$ and $AB$ at $E$ and $F$. Denote by $G$ and $H$ the reflections of $E$ and $F$ over $I$. Let $Q$ be the intersection of $BC$ with $GH$, and let $M$ be the midpoint of $BC$. Prove that $IQ$ and $IM$ are perpendicular.

Find all polynomials $P(x)$ satisfying $P(x+1)-2P(x)+P(x-1)= x$ for all $x$

Let \(f:\mathbb R\to\mathbb R\) be a function such that for all real numbers \(x\neq1\), \[f(x-f(x))+f(x)=\frac{x^2-x+1}{x-1}.\] Find all possible values of \(f(2023)\). [i]Proposed by Linus Tang[/i]

Determine all integers $k\geqslant 1$ with the following property: given $k$ different colours, if each integer is coloured in one of these $k$ colours, then there must exist integers $a_1<a_2<\cdots<a_{2023}$ of the same colour such that the differences $a_2-a_1,a_3-a_2,\dots,a_{2023}-a_{2022}$ are all powers of $2$.