There are $12$ members in a club. The members created some small groups, which satisfy the following: - The small group consists of $3$ or $4$ people. - Also, for two arbitrary members, there exists exactly one small group that has both members. Prove that all members are in the same number of small groups.

Let $f(x) = x^3 + 3x^2 - 1$ have roots $a,b,c$. (a) Find the value of $a^3 + b^3 + c^3$ (b) Find all possible values of $a^2b + b^2c + c^2a$

Let $BC$ be a chord of a circle $(O)$ such that $BC$ is not a diameter. Let $AE$ be the diameter perpendicular to $BC$ such that $A$ belongs to the larger arc $BC$ of $(O)$. Let $D$ be a point on the larger arc $BC$ of $(O)$ which is different from $A$. Suppose that $AD$ intersects $BC$ at $S$ and $DE$ intersects $BC$ at $T$. Let $F$ be the midpoint of $ST$ and $I$ be the second intersection point of the circle $(ODF)$ with the line $BC$. 1. Let the line passing through $I$ and parallel to $OD$ intersect $AD$ and $DE$ at $M$ and $N$, respectively. Find the maximum value of the area of the triangle $MDN$ when $D$ moves on the larger arc $BC$ of $(O)$ (such that $D \ne A$). 2. Prove that the perpendicular from $D$ to $ST$ passes through the midpoint of $MN$

Kostya marked the points $A(0, 1), B(1, 0), C(0, 0)$ in the coordinate plane. On the legs of the triangle ABC he marked the points with coordinates $(\frac{1}{2},0), (\frac{1}{3},0), \cdots, (\frac{1}{n+1},0)$ and $(0,\frac{1}{2}), (0,\frac{1}{3}), \cdots, (0,\frac{1}{n+1}).$ Then Kostya joined each pair of marked points with a segment. Sasha drew a $1 \times n$ rectangle and joined with a segment each pair of integer points on its border. As a result both the triangle and the rectangle are divided into polygons by the segments drawn. Who has the greater number of polygons: Sasha or Kostya? [i](M. Alekseyev )[/i]

A collector offers to buy state quarters for $2000\%$ of their face value. At that rate how much will Bryden get for his four state quarters? $ \text{(A)}\ 20\text{ dollars}\qquad\text{(B)}\ 50\text{ dollars}\qquad\text{(C)}\ 200\text{ dollars}\qquad\text{(D)}\ 500\text{ dollars}\qquad\text{(E)}\ 2000\text{ dollars} $

Any three vertices of the cube $PQRSTUVW,$ shown in the figure below, can be connected to form a triangle. $($For example, vertices $P, Q,$ and $R$ can be connected to form $\triangle{PQR}.)$ How many of these triangles are equilateral and contain $P$ as a vertex? $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }6$ [asy] unitsize(4); pair P,Q,R,S,T,U,V,W; P=(0,30); Q=(30,30); R=(40,40); S=(10,40); T=(10,10); U=(40,10); V=(30,0); W=(0,0); draw(W--V); draw(V--Q); draw(Q--P); draw(P--W); draw(T--U); draw(U--R); draw(R--S); draw(S--T); draw(W--T); draw(P--S); draw(V--U); draw(Q--R); dot(P); dot(Q); dot(R); dot(S); dot(T); dot(U); dot(V); dot(W); label("$P$",P,NW); label("$Q$",Q,NW); label("$R$",R,NE); label("$S$",S,N); label("$T$",T,NE); label("$U$",U,NE); label("$V$",V,SE); label("$W$",W,SW); [/asy]

Regular hexagon $ABCDEF$ has side length $2$ and center $O$. The point $P$ is defined as the intersection of $AC$ and $OB$. Find the area of quadrilateral $OPCD$.

Let $\mathbb{A}$ be the least subset of finite sequences of nonnegative integers that satisfies the following two properties: -$(0,0) \in \mathbb{A}$ - If $(a_1,\ldots,a_n)\in \mathbb{A}$ then $(a_1,\ldots,a_{i-2},a_{i-1}+1,1,a_{i}+1,a_{i+1},\ldots,a_n)\in \mathbb{A}$ for all $i\in \{2,\ldots,n\}$. For each $n\geq 2$, let $\mathbb{B}(n)$ be the set of sequences in $\mathbb{A}$ with $n$ terms. Find the number of elements of $\mathbb{B}$.

Find the greatest real number $ \alpha$ for which there exists a sequence of infinitive integers $ (a_n)$, ($ n \equal{} 1, 2, 3, \ldots$) satisfying the following conditions: 1) $ a_n > 1997n$ for every $ n \in\mathbb{N}^{*}$; 2) For every $ n\ge 2$, $ U_n\ge a^{\alpha}_n$, where $ U_n \equal{} \gcd\{a_i \plus{} a_k | i \plus{} k \equal{} n\}$.

Let ${\cal C}_1$ and ${\cal C}_2$ be concentric circles, with ${\cal C}_2$ in the interior of ${\cal C}_1$. From a point $A$ on ${\cal C}_1$ one draws the tangent $AB$ to ${\cal C}_2$ ($B\in {\cal C}_2$). Let $C$ be the second point of intersection of $AB$ and ${\cal C}_1$, and let $D$ be the midpoint of $AB$. A line passing through $A$ intersects ${\cal C}_2$ at $E$ and $F$ in such a way that the perpendicular bisectors of $DE$ and $CF$ intersect at a point $M$ on $AB$. Find, with proof, the ratio $AM/MC$.

The cells of a $2021\times 2021$ table are filled with numbers using the following rule. The bottom left cell, which we label with coordinate $(1, 1)$, contains the number $0$. For every other cell $C$, we consider a route from $(1, 1)$ to $C$, where at each step we can only go one cell to the right or one cell up (not diagonally). If we take the number of steps in the route and add the numbers from the cells along the route, we obtain the number in cell $C$. For example, the cell with coordinate $(2, 1)$ contains $1 = 1 + 0$, the cell with coordinate $(3, 1)$ contains $3 = 2 + 0 + 1$, and the cell with coordinate $(3, 2)$ contains $7 = 3 + 0 + 1 + 3$. What is the last digit of the number in the cell $(2021, 2021)$?

Let $G$ be a simple graph with $n$ vertices and $m$ edges. Two vertices are called [i]neighbours[/i] if there is an edge between them. It turns out the $G$ does not contain any cycles of length from 3 to $2k$ (inclusive), where $k\geq2$ is a given positive integer. a) Prove that it is possible to pick a non-empty set $S$ of vertices of $G$ such that every vertex in $S$ has at least $\left\lceil \frac mn \right\rceil$ neighbours that are in $S$. ($\lceil x\rceil$ denotes the smallest integer larger than or equal to $x$.) b) Suppose a set $S$ as described in (a) is chosen. Let $H$ be the graph consisting of the vertices in $S$ and the edges between those vertices only. Let $v$ be a vertex of $H$. Prove that at least $\left\lceil \left(\frac mn -1\right)^k \right\rceil$ vertices of $H$ can be reached by starting at $v$ and travelling across the edges of $H$ for at most $k$ steps. (Note that $v$ itself satisfies this condition, since it can be reached by starting at $v$ and travelling along the edges of $H$ for 0 steps.)

a) In a convex $13$-gon all diagonals are drawn, dividing it into smaller polygons. What is the greatest number of sides can these polygons have? b) In a convex $1950$-gon all diagonals are drawn, dividing it into smaller polygons. What is the greatest number of sides can these polygons have?

Positive real numbers $x \neq 1$ and $y \neq 1$ satisfy $\log_2{x} = \log_y{16}$ and $xy = 64$. What is $(\log_2{\tfrac{x}{y}})^2$? $\textbf{(A) } \frac{25}{2} \qquad\textbf{(B) } 20 \qquad\textbf{(C) } \frac{45}{2} \qquad\textbf{(D) } 25 \qquad\textbf{(E) } 32$

A traffic light installed at a main junction of a road, in which you circulate in both directions, it remains red for $30$ s and green for another $30$ s, alternately. You want to install another traffic light on the same road, for a secondary crossing, located $400$ m away from the first, which works with the same period of $1$ min duration. It is wanted that the cars that circulate at $60$ km/h on the road in any of the two senses and that they do not have to stop if there was only the traffic light of the main intersection. They also don't have to stop after installing the secondary crossover. How many seconds can red be on at the secondary traffic light? Note: It is suggested to reason on a Cartesian representation of the march of the cars, taking an axis of distances and another of times.

Given a prime number $p > 2$. Find the least $n\in Z_+$, for which every set of $n$ perfect squares not divisible by $p$ contains nonempty subset with product of all it's elements equal to $1\ (\text{mod}\ p)$

A building has some rooms and there is one or more than one doors between the rooms. We know that we can go from each room to another one. Two rooms $E,S$ has been labeled, and the room $S$ has exactly one door. Someone is in the room $S$ and wants to move to the room $E$. [list][list][list][list][list][list][img]http://s1.picofile.com/file/6475095570/image005.jpg[/img][/list][/list][/list][/list][/list][/list] A "[i]way[/i]" $P$ for moving between the rooms is an infinite sequence of $L$ and $R$. We say that someone moves according to the "[i]way[/i]" $P$, if he start moving from the room $S$, and after passing the $n$'th door, if $P_n$ is $R$, then he goes to the first door which is in the right side, and if $P_n$ is $L$, then he goes to the first door which is in the left side (obviously, if some room has exactly one door, then there is no difference between $L$ and $R$), and when he arrives to the room $E$, he stops moving. Prove that there exists a "[i]way[/i]" such that if the person move according to it, then he can arrive to the room $E$ of any building.

Find all functions $f : R \to R$ such that, for all real numbers $x, y,$ $$(x - y)(f(x) - f(y)) = f(x - f(y))f(f(x) - y).$$

In the figure, equilateral hexagon $ABCDEF$ has three nonadjacent acute interior angles that each measure $30^\circ$. The enclosed area of the hexagon is $6\sqrt{3}$. What is the perimeter of the hexagon? [asy] size(6cm); pen p=black+linewidth(1),q=black+linewidth(5); pair C=(0,0),D=(cos(pi/12),sin(pi/12)),E=rotate(150,D)*C,F=rotate(-30,E)*D,A=rotate(150,F)*E,B=rotate(-30,A)*F; draw(C--D--E--F--A--B--cycle,p); dot(A,q); dot(B,q); dot(C,q); dot(D,q); dot(E,q); dot(F,q); label("$C$",C,2*S); label("$D$",D,2*S); label("$E$",E,2*S); label("$F$",F,2*dir(0)); label("$A$",A,2*N); label("$B$",B,2*W); [/asy] $(\textbf{A})\: 4\qquad(\textbf{B}) \: 4\sqrt3\qquad(\textbf{C}) \: 12\qquad(\textbf{D}) \: 18\qquad(\textbf{E}) \: 12\sqrt3$

$(BUL 5)$ Let $Z$ be a set of points in the plane. Suppose that there exists a pair of points that cannot be joined by a polygonal line not passing through any point of $Z.$ Let us call such a pair of points unjoinable. Prove that for each real $r > 0$ there exists an unjoinable pair of points separated by distance $r.$

Coefficients $p$ and $q$ of the equation $x^2+px+q = 0$ are changed and the new ones differ from the old ones by $0.001$ or less. Can the greater root of the new equation differ from that of the old one by $1000$ or more?

What is the least possible sum of two positive integers $a$ and $b$ where $a \cdot b = 10! ?$

Denote by $S(n)$ the sum of digits of $n$. Given a positive integer $N$, we consider the following process: We take the sum of digits $S(N)$, then take its sum of digits $S(S(N))$, then its sum of digits $S(S(S(N)))$... We continue this until we are left with a one-digit number. We call the number of times we had to activate $S(\cdot)$ the [b]depth[/b] of $N$. For example, the depth of 49 is 2, since $S(49)=13\rightarrow S(13)=4$, and the depth of 45 is 1, since $S(45)=9$. [list=a] [*] Prove that every positive integer $N$ has a finite depth, that is, at some point of the process we get a one-digit number. [*] Define $x(n)$ to be the [u]minimal[/u] positive integer with depth $n$. Find the residue of $x(5776)\mod 6$. [*] Find the residue of $x(5776)-x(5708)\mod 2016$. [/list]

In tetrahedron $ABCD$, let $V$ be the volume of the tetrahedron and $R$ the radius of the sphere that it tangent to all four faces of the tetrahedron. Let $P$ be the surface area of the tetrahedron. Prove that $$r=\frac{3V}{P}.$$ [i]Proposed by CaptainFlint.[/i]

Let $ n$ be a positive integer. Prove that $ 2^{n\minus{}1}$ divides \[ \sum_{0\leq k < n/2} \binom{n}{2k\plus{}1}5^k.\]