Projector emits rays in space. Consider all acute angles between the rays. It is known that no matter what ray we remove, the number of acute angles decreases by exactly $2$ What is the maximal number of rays the projector can emit? [i]M. Karpuk, E. Barabanov[/i]

Six points in general position are given in the space. For each two of them color red the common points (if they exist) of the segment between these points and the surface of the tetrahedron formed by four remaining points. Prove that the number of red points is even.

Let $0<t<\frac{\pi}{2}$. Evaluate \[\lim_{t\rightarrow \frac{\pi}{2}} \int_0^t \tan \theta \sqrt{\cos \theta}\ln (\cos \theta)d\theta\]

A man with mass $m$ jumps off of a high bridge with a bungee cord attached to his ankles. The man falls through a maximum distance $H$ at which point the bungee cord brings him to a momentary rest before he bounces back up. The bungee cord is perfectly elastic, obeying Hooke's force law with a spring constant $k$, and stretches from an original length of $L_0$ to a final length $L = L_0 + h$. The maximum tension in the Bungee cord is four times the weight of the man. Find the maximum extension of the bungee cord $h$. $\textbf{(A) } h = \frac{1}{2}H \\ \\ \textbf{(B) } h = \frac{1}{4}H\\ \\ \textbf{(C) } h = \frac{1}{5}H\\ \\ \textbf{(D) } h = \frac{2}{5}H\\ \\ \textbf{(E) } h = \frac{1}{8}H$

[b]1)[/b] Show that there exist quadratic polynoms $ P\in\mathbb{R}[X] $ whose composition with themselves have $ 1,2 $ and $ 3 $ as their fixed points. [b]2)[/b] Prove that the polynoms referred to at [b]1)[/b] are not integer. [i]Gheorghe Iurea[/i]

Does there exist an increasing arithmetic progression of (a) $11$ (b) $10000$ (c) infinitely many positive integers such that the sums of their digits in base $10$ also form an increasing arithmetic progression? (A Shapovalov)

Let $ABC$ be a triangle with $AB = 13$, $BC = 14$, $CA = 15$. Let $H$ be the orthocenter of $ABC$. Find the radius of the circle with nonzero radius tangent to the circumcircles of $AHB$, $BHC$, $CHA$.

In a trapezoid $ABCD$ with $AB\parallel CD$, the diagonals $AC$ and $BD$ meet at $O$. Let $M$ and $N$ be the centers of the regular hexagons constructed on the sides $AB$ and $CD$ in the exterior of the trapezoid. Prove that $M,O$ and $N$ are collinear.

Let $ ABC$ be an acute-angled triangle. The lines $ L_{A}$, $ L_{B}$ and $ L_{C}$ are constructed through the vertices $ A$, $ B$ and $ C$ respectively according the following prescription: Let $ H$ be the foot of the altitude drawn from the vertex $ A$ to the side $ BC$; let $ S_{A}$ be the circle with diameter $ AH$; let $ S_{A}$ meet the sides $ AB$ and $ AC$ at $ M$ and $ N$ respectively, where $ M$ and $ N$ are distinct from $ A$; then let $ L_{A}$ be the line through $ A$ perpendicular to $ MN$. The lines $ L_{B}$ and $ L_{C}$ are constructed similarly. Prove that the lines $ L_{A}$, $ L_{B}$ and $ L_{C}$ are concurrent.

Compute the number of ordered triples $(a, b, c)$ of integers between $-100$ and $100$ inclusive satisfying the simultaneous equations $$a^3 - 2a = abc - b - c$$ $$b^3 - 2b = bca - c - a$$ $$c^3 - 2c = cab - a - b.$$

Fifty points are chosen inside a convex polygon having eighty sides such that no three of the fifty points lie on the same straight line. The polygon is cut into triangles such that the vertices of the triangles are just the fifty points and the eighty vertices of the polygon. How many triangles are there?

Twenty percent less than $ 60$ is one-third more than what number? $ \textbf{(A)}\ 16\qquad \textbf{(B)}\ 30\qquad \textbf{(C)}\ 32\qquad \textbf{(D)}\ 36\qquad \textbf{(E)}\ 48$

Let the sequence $\{a_n\}$ satisfy $a_{n+1}=a_n^3+103,n=1,2,...$. Prove that at most one integer $n$ such that $a_n$ is a perfect square.

In a scalene triangle $ABC$, the points $E$ and $F$ are the foot of altitudes drawn from $B$ and $C$, respectively. The points $X$ and $Y$ are the reflections of the vertices $B$ and $C$ to the line $EF$, respectively. Let the circumcircles of the $\triangle ABC$ and $\triangle AEF$ intersect at $T$ for the second time. Show that the four points $A, X, Y, T$ lie on a single circle.

Let $a=2019^{1009}, b=2019!$ and $c=1010^{2019}$, then which of the following is true? (A)$c<b<a$ (B)$a<b<c$ (C)$b<a<c$ (D)$b<c<a$

A subset $S$ of the set $\{1, 2, \ldots, 10\}$ is chosen randomly, with all possible subsets being equally likely. Compute the expected number of positive integers which divide the product of the elements of $S.$ (By convention, the product of the elements of the empty set is $1.$)

One hundred sages play the following game. They are waiting in some fixed order in front of a room. The sages enter the room one after another. When a sage enters the room, the following happens - the guard in the room chooses two arbitrary distinct numbers from the set {$1,2,3$}, and announces them to the sage in the room. Then the sage chooses one of those numbers, tells it to the guard, and leaves the room, and the next enters, and so on. During the game, before a sage chooses a number, he can ask the guard what were the chosen numbers of the previous two sages. During the game, the sages cannot talk to each other. At the end, when everyone has finished, the game is considered as a failure if the sum of the 100 chosen numbers is exactly $200$; else it is successful. Prove that the sages can create a strategy, by which they can win the game.

Let $q$ be a real number. Suppose there are three distinct positive integers $a, b,c$ such that $q + a$, $q + b$,$q + c$ is a geometric progression. Show that $q$ is rational.

Points $A, B, C$ and $D$ lie on a circle, in that order clockwise, such that there is a point $E$ on segment $CD$ with the property that $AD = DE$ and $BC = EC$. Prove that the intersection point of the bisectors of the angles $\angle DAB$ and $\angle ABC$ is on the line $CD$.

The sequence $(a_n)$ is determined by $a_1 = 0$ and $(n+1)^3a_{n+1} = 2n^2(2n+1)a_n+2(3n+1)$ for $n \geq 1$. Prove that infinitely many terms of the sequence are positive integers.

$G$ is a group that order of each element of it Commutator group is finite. Prove that subset of all elemets of $G$ which have finite order is a subgroup og $G$.

What is the correct ordering of the three numbers $\frac5{19}$, $\frac7{21}$, and $\frac9{23}$, in increasing order? $\textbf{(A)}\hspace{.05in}\dfrac9{23} < \dfrac7{21} < \dfrac5{19}$ $\textbf{(B)}\hspace{.05in}\dfrac5{19} < \dfrac7{21} < \dfrac9{23}$ $\textbf{(C)}\hspace{.05in}\dfrac9{23} < \dfrac5{19} < \dfrac7{21}$ $\textbf{(D)}\hspace{.05in}\dfrac5{19} < \dfrac9{23} < \dfrac7{21}$ $\textbf{(E)}\hspace{.05in}\dfrac7{21} < \dfrac5{19} < \dfrac9{23}$

A trapezoid with bases $AD$ and $BC$ is circumscribed about a circle, $E$ is the intersection point of the diagonals. Prove that $\angle AED$ is not acute.

Let $ a,b,c $ be three real numbers such that $ \cos a+\cos b+\cos c=\sin a+\sin b+\sin c=0. $ Prove that [b]i)[/b] $ \cos 6a+\cos 6b+\cos 6c=3\cos (2a+2b+2c) $ [b]ii)[/b] $ \sin 6a+\sin 6b+\sin 6c=3\sin (2a+2b+2c) $ [i]Vasile Pop[/i]

Several players try out for the USAMTS basketball team, and they all have integer heights and weights when measured in centimeters and pounds, respectively. In addition, they all weigh less in pounds than they are tall in centimeters. All of the players weigh at least $190$ pounds and are at most $197$ centimeters tall, and there is exactly one player with every possible height-weight combination. The USAMTS wants to field a competitive team, so there are some strict requirements. [list] [*] If person $P$ is on the team, then anyone who is at least as tall and at most as heavy as $P$ must also be on the team. [*] If person $P$ is on the team, then no one whose weight is the same as $P$’s height can also be on the team. [/list] Assuming the USAMTS team can have any number of members (including zero), how many different basketball teams can be constructed?