Let $p$ and $q$ be two different primes. Prove that \[\left\lfloor\frac{p}{q}\right\rfloor+\left\lfloor\frac{2p}{q}\right\rfloor+\left\lfloor\frac{3p}{q}\right\rfloor+\ldots +\left\lfloor\frac{(q-1)p}{q}\right\rfloor=\frac{1}{2}(p-1)(q-1) \]

How many digits are in the product $4^5 \cdot 5^{10}$? $ \textbf{(A)} 8 \qquad\textbf{(B)} 9 \qquad\textbf{(C)} 10 \qquad\textbf{(D)} 11 \qquad\textbf{(E)} 12 $

In the acute-angled triangle $ABC$, let $M$ be the midpoint of the atlitude $h_b$ through $B$ and $N$ be the midpoint of the height $h_c$ through $C$. Further let $P$ be the intersection of $AM$ and $h_c$ and $Q$ be the intersection of $AN$ and $h_b$. Show that $M, N, P$ and $Q$ lie on a circle.

A quadrilateral \( ABCD \) with no parallel sides is inscribed in a circle \( \Omega \). Circles \( \omega_a, \omega_b, \omega_c, \omega_d \) are inscribed in triangles \( DAB, ABC, BCD, CDA \), respectively. Common external tangents are drawn between \( \omega_a \) and \( \omega_b \), \( \omega_b \) and \( \omega_c \), \( \omega_c \) and \( \omega_d \), and \( \omega_d \) and \( \omega_a \), not containing any sides of quadrilateral \( ABCD \). A quadrilateral whose consecutive sides lie on these four lines is inscribed in a circle \( \Gamma \). Prove that the lines joining the centers of \( \omega_a \) and \( \omega_c \), \( \omega_b \) and \( \omega_d \), and the centers of \( \Omega \) and \( \Gamma \) all intersect at one point. \\

Let be a natural number $ n\ge 2, $ the real numbers $ a_1,a_2,\ldots ,a_n,b_1,b_2,\ldots, b_n, $ and the matrix defined as $$ A=\left( a_i+b_j \right)_{1\le j\le n}^{1\le i\le n} . $$ [b]a)[/b] Show that $ n=2 $ if $ A $ is invertible. [b]b)[/b] Prove that the pair of numbers $ a_1,a_2 $ and $ b_1,b_2 $ are both consecutive (not necessarily in this order), if $ A $ is an invertible matrix of integers whose inverse is a matrix of integers. [i]Costică Ambrinoc[/i]

There are $ 7$ boxes arranged in a row and numbered $1$ through $7$. You have a stack of $2015$ cards, which you place one by one in the boxes. The first card is placed in box #$1$, the second in box #$2$, and so forth up to the seventh card which is placed in box #$7$. You then start working back in the other direction, placing the eighth card in box #$6$, the ninth in box #$5$, up to the thirteenth card being placed in box #$1$. The fourteenth card is then placed in box #$2$, and this continues until every card is distributed. What box will the last card be placed in?

For what is the smallest natural number $n$ there is a polynomial $P(x)$ with integer coefficients, having $m$ different integer roots, and at the same time the equation $P(x) = n$ has at least one integer solution if: a) $m = 5$, b) $ m = 6$?

Consider a convex $15$- gon with perimeter $21$. Show that there one can select three distinct pairs of vertices that form a triangle with area less than $1$. [hide=original wording of second sentence]Zeige, dass man davon drei paarweise verschiedene Eckpunkte auswählen kann, die ein Dreieck mit Fläche kleiner als 1 bilden.[/hide]

There are $n$ people standing on a circular track. We want to perform a number of [i]moves[/i] so that we end up with a situation where the distance between every two neighbours is the same. The [i]move[/i] that is allowed consists in selecting two people and asking one of them to walk a distance $d$ on the circular track clockwise, and asking the other to walk the same distance on the track anticlockwise. The two people selected and the quantity $d$ can vary from move to move. Prove that it is possible to reach the desired situation (where the distance between every two neighbours is the same) after at most $n-1$ moves.

To $m$ ounces of a $m\%$ solution of acid, $x$ ounces of water are added to yield a $(m-10)\%$ solution. If $m>25$, then $x$ is $\textbf{(A)}\ \frac{10m}{m-10} \qquad \textbf{(B)}\ \frac{5m}{m-10} \qquad \textbf{(C)}\ \frac{m}{m-10} \qquad \textbf{(D)}\ \frac{5m}{m-20} \\ \textbf{(E)}\ \text{not determined by the given information}$