In trapezoid $ ABCD$, $ AB \parallel CD$, $ \angle CAB < 90^\circ$, $ AB \equal{} 5$, $ CD \equal{} 3$, $ AC \equal{} 15$. What are the sum of different integer values of possible $ BD$? $\textbf{(A)}\ 101 \qquad\textbf{(B)}\ 108 \qquad\textbf{(C)}\ 115 \qquad\textbf{(D)}\ 125 \qquad\textbf{(E)}\ \text{None}$

Let $n_1, n_2$ be positive integers. Consider in a plane $E$ two disjoint sets of points $M_1$ and $M_2$ consisting of $2n_1$ and $2n_2$ points, respectively, and such that no three points of the union $M_1 \cup M_2$ are collinear. Prove that there exists a straightline $g$ with the following property: Each of the two half-planes determined by $g$ on $E$ ($g$ not being included in either) contains exactly half of the points of $M_1$ and exactly half of the points of $M_2.$

Find maximum value of $|a^2-bc+1|+|b^2-ac+1|+|c^2-ba+1|$ when $a,b,c$ are reals in $[-2;2]$.

Let $a_1,a_2,\ldots a_n$ be real numbers such that their sum is equal to zero. Find the value of \[ \sum_{j=1}^{n} \frac{1}{a_j (a_j +a _{j+1}) (a_j + a_{j+1} + a_{j+2}) \ldots (a_j + \ldots a_{j+n-2})}. \] where the subscripts are taken modulo $n$ assuming none of the denominators is zero.

It is given a circle with center $O$ and radius $r$. $AB$ and $MN$ are two diameters. The lines $MB$ and $NB$ are tangent to the circle at the points $M'$ and $N'$ and intersect at point $A$. $M''$ and $N''$ are the midpoints of the segments $AM'$ and $AN'$. Prove that: (a) the points $M,N,N',M'$ are concyclic. (b) the heights of the triangle $M''N''B$ intersect in the midpoint of the radius $OA$.

Consider the sequences of six positive integers $a_1,a_2,a_3,a_4,a_5,a_6$ with the properties that $a_1=1$, and if for some $j > 1$, $a_j = m > 1$, then $m-1$ appears in the sequence $a_1,a_2,\dots,a_{j-1}$. Such sequences include $1,1,2,1,3,2$ and $1,2,3,1,4,1$ but not $1,2,2,4,3,2$. How many such sequences of six positive integers are there?

Convex quadrilateral $ABCD$ has side-lengths $AB=7$, $BC=9$, $CD=15$, and there exists a circle, lying inside the quadrilateral and having center $I$, that is tangent to all four sides of the quadrilateral. Points $M$ and $N$ are the midpoints of $AC$ and $BD$ respectively. It can be proven that point $I$ always lies on segment $MN$. Supposing further that $I$ is the midpoint of $MN$, the area of quadrilateral $ABCD$ may be expressed as $p\sqrt q$, where $p$ and $q$ are positive integers and $q$ is not divisible by the square of any prime. Compute $p\cdot q$.

Let be given a semi-sphere $\Sigma$ whose base-circle lies on plane $p$. A variable plane $Q$, parallel to a fixed plane non-perpendicular to $P$, cuts $\Sigma$ at a circle $C$. We denote by $C'$ the orthogonal projection of $C$ onto $P$. Find the position of $Q$ for which the cylinder with bases $C$ and $C'$ has the maximum volume.

[b]Q[/b] Let $n\geq 2$ be a fixed positive integer and let $d_1,d_2,...,d_m$ be all positive divisors of $n$. Prove that: $$\frac{d_1+d_2+...+d_m}{m}\geq \sqrt{n+\frac{1}{4}}$$Also find the value of $n$ for which the equality holds. [i]Proposed by dangerousliri [/i]

We wish to find the sum of $40$ given numbers utilizing $40$ processors. Initially, we have the number $0$ on the screen of each processor. Each processor adds the number on its screen with a number entered directly (only the given numbers could be entered directly to the processors) or transferred from another processor in a unit time. Whenever a number is transferred from a processor to another, the former processor resets. Find the least time needed to find the desired sum.

A tournament of order $n$, $n \in \mathbb{N}$, consists of $2^n$ players, which are numbered with $1, 2, \ldots, 2^n$, and has $n$ rounds. In each round, the remaining players paired with each other to play a match and the winner from each match advances to the next round. The winner of the $n$-th round is considered the winner of the tournament. Two tournaments are considered different if there is a match that took place in the $k$-th round of one tournament, but not in the $k$-th round of the other, or if the tournaments have different winners. Determine how many different tournaments of order $n$ there are with the property that in each round, the sum of the numbers of the players in each match is the same (but not necessarily the same for all rounds).

The length of the altitude of equilateral triangle $ ABC$ is $3$. A circle with radius $2$, which is tangent to $ \left[BC\right]$ at its midpoint, meets other two sides. If the circle meets $ AB$ and $ AC$ at $ D$ and $ E$, at the outer of $\triangle ABC$ , find the ratio $ \frac {Area\, \left(ABC\right)}{Area\, \left(ADE\right)}$. $\textbf{(A)}\ 2\left(5 \plus{} \sqrt {3} \right) \qquad\textbf{(B)}\ 7\sqrt {2} \qquad\textbf{(C)}\ 5\sqrt {3} \\ \qquad\textbf{(D)}\ 2\left(3 \plus{} \sqrt {5} \right) \qquad\textbf{(E)}\ 2\left(\sqrt {3} \plus{} \sqrt {5} \right)$

Let $n$ be a positive integer. A frog starts on the number line at $0$. Suppose it makes a finite sequence of hops, subject to two conditions: [list] [*]The frog visits only points in $\{1, 2, \dots, 2^n-1\}$, each at most once. [*]The length of each hop is in $\{2^0, 2^1, 2^2, \dots\}$. (The hops may be either direction, left or right.) [/list] Let $S$ be the sum of the (positive) lengths of all hops in the sequence. What is the maximum possible value of $S$? [i]Ashwin Sah[/i]

Each one of three friends, Mário, João and Filipe, does one, and only one, of the following sports: football, basketball and swimming. None of these sports is done by more than one of the friends. Each one of the friends likes a certain kind of fruit: one likes oranges, another likes bananas and the other likes papayas. Find, for each one, which sport he plays and which fruit he prefers, given that: * Mário doesn't like oranges; * João doesn't play football; * The swimmer hates bananas; * The swimmer and the one who likes oranges do different sports; * The one who likes papayas and the footballer visit Filipe every Saturday.

$ABCD$ - convex quadrilateral. $P$ is such point on $AC$ and inside $\triangle ABD$, that $$\angle ACD+\angle BDP = \angle ACB+ \angle DBP = 90-\angle BAD$$. Prove that $\angle BAD+ \angle BCD =90$ or $\angle BDA + \angle CAB = 90$

Given quadrilateral $ABCD$. $AC$ and $BD$ meets at $E$, and $M, N$ are the midpoints of $AC, BD$, respectively. Let the circumcircles of $ABE$ and $CDE$ meets again at $X\neq E$. Prove that $E, M, N, X$ are concyclic. [i]Proposed by chengbilly[/i]

Tatjana imagined a polynomial $P(x)$ with nonnegative integer coefficients. Danica is trying to guess the polynomial. In each step, she chooses an integer $k$ and Tatjana tells her the value of $P(k)$. Find the smallest number of steps Danica needs in order to find the polynomial Tatjana imagined.

Let $f(x)=\frac{P(x)}{Q(x)}$, where $P(x), Q(x)$ are two non-constant polynomials with no common zeros and $P(0)=P(1)=0$. Suppose $f(x)f\left(\frac{1}{x}\right)=f(x)+f\left(\frac{1}{x}\right)$ for infinitely many values of $x$. a) Show that $\text{deg}(P)<\text{deg}(Q)$. b) Show that $P'(1)=2Q'(1)-\text{deg}(Q)\cdot Q(1)$. Here, $P'(x)$ denotes the derivative of $P(x)$ as usual.

Let $a$ be a fixed positive integer and $(e_n)$ the sequence, which is defined by $e_0=1$ and $$ e_n=a + \prod_{k=0}^{n-1} e_k$$ for $n \geq 1$. Prove that (a) There exist infinitely many prime numbers that divide one element of the sequence. (b) There exists one prime number that does not divide an element of the sequence. (Theresia Eisenkölbl)

Solve the quasi-homogeneous equation \[\frac{dy}{dx}=x+\frac{x^3}{y}\]

Write either $1$ or $-1$ in each of the cells of a $(2n) \times (2n)$-table, in such a way that there are exactly $2n^2$ entries of each kind. Let the minimum of the absolute values of all row sums and all column sums be $M$. Determine the largest possible value of $M$.

Let $a$ and $b$ be real numbers with $a<b,$ and let $f$ and $g$ be continuous functions from $[a,b]$ to $(0,\infty)$ such that $\int_a^b f(x)\,dx=\int_a^b g(x)\,dx$ but $f\ne g.$ For every positive integer $n,$ define \[I_n=\int_a^b\frac{(f(x))^{n+1}}{(g(x))^n}\,dx.\] Show that $I_1,I_2,I_3,\dots$ is an increasing sequence with $\displaystyle\lim_{n\to\infty}I_n=\infty.$

In a circle we enscribe a regular $2001$-gon and inside it a regular $667$-gon with shared vertices. Prove that the surface in the $2001$-gon but not in the $667$-gon is of the form $k.sin^3\left(\frac{\pi}{2001}\right).cos^3\left(\frac{\pi}{2001}\right)$ with $k$ a positive integer. Find $k$.

Let $a,b,c$ be positive real numbers such that $ab+bc+ca\le 3abc$. Prove that \[\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}+3\le \sqrt{2} (\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a})\]

Let $C_1$ be a circle inside the circle $C_2$ and let $P$ in the interior of $C_1$, $Q$ in the exterior of $C_2$. One draws variable lines $l_i$ through $P$, not passing through $Q$. Let $l_i$ intersect $C_1$ in $A_i,B_i$, and let the circumcircle of $QA_iB_i$ intersect $C_2$ in $M_i,N_i$. Show that all lines $M_i,N_i$ are concurrent.