[b]N[/b]ew this year at HMNT: the exciting game of RNG baseball! In RNG baseball, a team of infinitely many people play on a square field, with a base at each vertex; in particular, one of the bases is called the home base. Every turn, a new player stands at home base and chooses a number n uniformly at random from $\{0, 1, 2, 3, 4\}$. Then, the following occurs: • If $n>0$, then the player and everyone else currently on the field moves (counterclockwise) around the square by n bases. However, if in doing so a player returns to or moves past the home base, he/she leaves the field immediately and the team scores one point. • If $n=0$ (a strikeout), then the game ends immediately; the team does not score any more points. What is the expected number of points that a given team will score in this game?

For a positive integer $n$, let $6^{(n)}$ be the natural number whose decimal representation consists of $n$ digits $6$. Let us define, for all natural numbers $m$, $k$ with $1 \leq k \leq m$ \[\left[\begin{array}{ccc}m\\ k\end{array}\right] =\frac{ 6^{(m)} 6^{(m-1)}\cdots 6^{(m-k+1)}}{6^{(1)} 6^{(2)}\cdots 6^{(k)}} .\] Prove that for all $m, k$, $ \left[\begin{array}{ccc}m\\ k\end{array}\right] $ is a natural number whose decimal representation consists of exactly $k(m + k - 1) - 1$ digits.

A group of students play the following game: they are counting one by one from $00$ to $99$ taking turns, but instead of every number they only say one of its digits. (The numbers in order are $00$, $01$, $02$, $...$., meaning that one-digit numbers are regarded as two-digit numbers with a first digit $0$.) One way of starting the counting could be for example $0$, $1$, $2$, $0$, $4$, $0$, $6$, $7$, $8$, $9,$ $1$, $1$, $2$, $1$, $1$, $5$, $6$, $1$, $8$, $1$, $0$, $2$ etc. When they reach $99$, the counting restarts from $00$. At some point Csongor enters the room and after listening to the counting for a while, he discovers that he is able to tell what number the counting is at. How many digits has Csongor heard at least?

The load for the space station "Salute" is packed in containers. There are more than $35$ containers, and the total weight is $18$ metric tons. There are $7$ one-way transport spaceships "Progress", each able to bring $3$ metric tons to the station. It is known that they are able to take an arbitrary subset of $35$ containers. Prove that they are able to take all the load.

Trapezoid $ ABCD$ has bases $ \overline{AB}$ and $ \overline{CD}$ and diagonals intersecting at $ K$. Suppose that $ AB\equal{}9$, $ DC\equal{}12$, and the area of $ \triangle AKD$ is $ 24$. What is the area of trapezoid $ ABCD$? $ \textbf{(A)}\ 92 \qquad \textbf{(B)}\ 94 \qquad \textbf{(C)}\ 96 \qquad \textbf{(D)}\ 98 \qquad \textbf{(E)}\ 100$

Let $S$ be a set of $n$ points in the plane such that for any two points $(a, b), (c, d) \in S$, we have that $| a - c | \cdot | b - d | \ge 1$. Show that [list] [*] (a) If $S = \{ Q_1, Q_2, Q_3\}$ such that for any point $Q_i$ in $S$, this point doesn't lie in the axis-aligned rectangle with corners as the other two points, show that the area of $\triangle Q_1Q_2Q_3$ is at least $\frac{\sqrt{5}}{2}$. [*] (b) If all points in $S$ lie in an axis-aligned square with sidelength $a$, then $|S| \le \frac{a^2}{\sqrt{5}} + 2a + 1$. [/list]

Let $ABCD$ be a non-cyclic convex quadrilateral. The feet of perpendiculars from $A$ to $BC,BD,CD$ are $P,Q,R$ respectively, where $P,Q$ lie on segments $BC,BD$ and $R$ lies on $CD$ extended. The feet of perpendiculars from $D$ to $AC,BC,AB$ are $X,Y,Z$ respectively, where $X,Y$ lie on segments $AC,BC$ and $Z$ lies on $BA$ extended. Let the orthocenter of $\triangle ABD$ be $H$. Prove that the common chord of circumcircles of $\triangle PQR$ and $\triangle XYZ$ bisects $BH$.

Let $p$ be a prime number. For any $\sigma \in S_p$ (the permutation group of $\{1,2,...,p \}),$ define the matrix $A_{\sigma}=(a_{ij}) \in \mathcal{M}_p(\mathbb{Z})$ as $a_{ij} = \sigma^{i-1}(j),$ where $\sigma^0$ is the identity permutation and $\sigma^k = \underbrace{\sigma \circ \sigma \circ ... \circ \sigma}_k.$ Prove that $D = \{ |\det A_{\sigma}| : \sigma \in S_p \}$ has at most $1+ (p-2)!$ elements.

Omar has four fair standard six-sided dice. Omar invented a game where he rolls his four dice over and over again until the number 1 does not appear on the top face of any of the dice. Omar wins the game if on that roll the top faces of his dice show at least one 2 and at least one 5. The probability that Omar wins the game is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

The weight of $\frac 13$ of a large pizza together with $3 \frac 12$ cups of orange slices is the same as the weight of $\frac 34$ of a large pizza together with $\frac 12$ cup of orange slices. A cup of orange slices weighs $\frac 14$ of a pound. What is the weight, in pounds, of a large pizza? $\textbf{(A)}~1\frac45\qquad\textbf{(B)}~2\qquad\textbf{(C)}~2\frac25\qquad\textbf{(D)}~3\qquad\textbf{(E)}~3\frac35$

The function $f:\mathbb R^{\ge 0} \longrightarrow \mathbb R^{\ge 0}$ satisfies the following properties for all $a,b\in \mathbb R^{\ge 0}$: [b]a)[/b] $f(a)=0 \Leftrightarrow a=0$ [b]b)[/b] $f(ab)=f(a)f(b)$ [b]c)[/b] $f(a+b)\le 2 \max \{f(a),f(b)\}$. Prove that for all $a,b\in \mathbb R^{\ge 0}$ we have $f(a+b)\le f(a)+f(b)$. [i]Proposed by Masoud Shafaei[/i]

It is known that from segments of lengths $a$, $b$ and $c$, a triangle can be formed. Can it happen that from segments of lengths$$\sqrt{a^2 + bc}, \quad \sqrt{b^2 + ca}, \quad \sqrt{c^2 + ab}$$an obtuse triangle can be formed?

The fraction $\frac23$ can be eypressed as a sum of two distinct unit fractions: $\frac12 + \frac16$ . Show that the fraction $\frac{p-1}{p}$ where $p\ge 5$ is a prime cannot be expressed as a sum of two distinct unit fractions.

Let $L$ be a subset in the coordinate plane defined by $L = \{(41x + 2y, 59x + 15y) | x, y \in \mathbb Z \}$, where $\mathbb Z$ is set of integers. Prove that for any parallelogram with center in the origin of coordinate and area $1990$, there exist at least two points of $L$ located in it.

A pool table has the shape of a triangle whose angles are in a rational ratio. A ball positioned at an interior point of the table is hit by a stick. The ball reflects from the sides of the triangle according to the law of reflection. Prove that the ball will move only along a finite number of segments. (It is assumed that the ball does not reach the vertices of the triangle.)

Solve in $\Bbb{N}^*$ the equation $$ 4^a \cdot 5^b - 3^c \cdot 11^d = 1.$$

In a round-robin tournament with $n$ players in which there are no draws, the numbers of wins scored by the players are $s_1 , s_2 , \ldots, s_n$. Prove that a necessary and sufficient condition for the existence of three players $A,B,C$ such that $A$ beats $B$, $B$ beats $C$, and $C$ beats $A$ is $$s_{1}^{2} +s_{2}^{2} + \ldots +s_{n}^{2} < \frac{(2n-1)(n-1)n}{6}.$$

Knights, who always tell truth, and liars, who always lie, live on an island. They have been distributed into two teams $A$ and $B$ for a game of tennis, and team $A$ had more members than team $B$. Two players from different teams started the game, whenever a player loses the game, he leaves it forever and he is replaces by a member of his team (that has never played before). The team, all of whose members left the game, loses. After the tournament, every member of team $A$ was asked: "Is it true that you have lost to a liar in some game?", and every member of team $B$ was asked: "Is it true that you have won at least two games, in which your opponent was a knight?". It turns out that every single answer was positive. Which team won?

Let \(n\) be a non-negative integer and define \(a_n = 2^n - n\). Determine all non-negative integers \(m\) such that \(s_m = a_0 + a_1 + \dots + a_m\) is a power of 2.

In a school there are $2007$ male and $2007$ female students. Each student joins not more than $100$ clubs in the school. It is known that any two students of opposite genders have joined at least one common club. Show that there is a club with at least $11$ male and $11$ female members.

Let $M$ be a set of $n \ge 4$ points in the plane, no three of which are collinear. Initially these points are connected with $n$ segments so that each point in $M$ is the endpoint of exactly two segments. Then, at each step, one may choose two segments $AB$ and $CD$ sharing a common interior point and replace them by the segments $AC$ and $BD$ if none of them is present at this moment. Prove that it is impossible to perform $n^3 /4$ or more such moves. [i]Proposed by Vladislav Volkov, Russia[/i]

In triangle $ABC$ $(b \geq c)$, point $E$ is the midpoint of shorter arc $BC$. If $D$ is the point such that $ED$ is the diameter of circumcircle $ABC$, prove that $\angle DEA = \frac{1}{2}(\beta-\gamma)$

Let $a,b,c$ be the lengths of the sides of a triangle, and $\alpha, \beta, \gamma$ respectively, the angles opposite these sides. Prove that if \[ a+b=\tan{\frac{\gamma}{2}}(a\tan{\alpha}+b\tan{\beta}) \] the triangle is isosceles.

Maxi chose $3$ digits, and by writing down all possible permutations of these digits, he obtained $6$ distinct $3$-digit numbers. If exactly one of those numbers is a perfect square and exactly three of them are prime, find Maxi’s $3$ digits. Give all of the possibilities for the $3$ digits.

None of the $n$ (not necessarily distinct) digits selected are equal to $0$ or $7$. It turns out that every $n$-digit number formed using these digits is divisible by $7$. Prove that $n$ is divisible by $6$.