In $\vartriangle ABC$, we have $AC = BC = 7$ and $AB = 2$. Suppose that $D$ is a point on line $AB$ such that $B$ lies between $A$ and $D$ and $CD = 8$. What is the length of the segment $BD$?

A prime number $p \ge 5$ is given. Write $\frac13+\frac24+... +\frac{p -3}{p - 1}=\frac{a}{b}$ for natural numbers $a$ and $b$. Show that $p$ divides $a$.

Let $a$, $b$ and $c$ be rational numbers for which $a+bc$, $b+ac$ and $a+b$ are all non-zero and for which we have \[\frac{1}{a+bc}+\frac{1}{b+ac}=\frac{1}{a+b}.\] Prove that $\sqrt{(c-3)(c+1)}$ is rational.

Let $n\geq 3$ a positive integer. In each cell of a $n\times n$ chessboard one must write $1$ or $2$ in such a way the sum of all written numbers in each $2\times 3$ and $3\times 2$ sub-chessboard is even. How many different ways can the chessboard be completed?

Consider a grid board of $n \times n$, with $n \ge 5$. Two unit squares are said to be far [i]apart [/i] if they are neither on the same row nor on consecutive rows and neither in the same column nor in consecutive columns. Take $3$ rectangles with vertices and sides on the points and lines of board so that if two unit squares belong to different rectangles, then they are [i]apart [/i]. In how many ways is it possible to do this?

p1. One day, a researcher placed two groups of species that were different, namely amoeba and bacteria in the same medium, each in a certain amount (in unit cells). The researcher observed that on the next day, which is the second day, it turns out that every cell species divide into two cells. On the same day every cell amoeba prey on exactly one bacterial cell. The next observation carried out every day shows the same pattern, that is, each cell species divides into two cells and then each cell amoeba prey on exactly one bacterial cell. Observation on day $100$ shows that after each species divides and then each amoeba cell preys on exactly one bacterial cell, it turns out kill bacteria. Determine the ratio of the number of amoeba to the number of bacteria on the first day. p2. It is known that $n$ is a positive integer. Let $f(n)=\frac{4n+\sqrt{4n^2-1}}{\sqrt{2n+1}+\sqrt{2n-1}}$. Find $f(13) + f(14) + f(15) + ...+ f(112).$ p3. Budi arranges fourteen balls, each with a radius of $10$ cm. The first nine balls are placed on the table so that form a square and touch each other. The next four balls placed on top of the first nine balls so that they touch each other. The fourteenth ball is placed on top of the four balls, so that it touches the four balls. If Bambang has fifty five balls each also has a radius of $10$ cm and all the balls are arranged following the pattern of the arrangement of the balls made by Budi, calculate the height of the center of the topmost ball is measured from the table surface in the arrangement of the balls done by Bambang. p4. Given a triangle $ABC$ whose sides are $5$ cm, $ 8$ cm, and $\sqrt{41}$ cm. Find the maximum possible area of ​​the rectangle can be made in the triangle $ABC$. p5. There are $12$ people waiting in line to buy tickets to a show with the price of one ticket is $5,000.00$ Rp.. Known $5$ of them they only have $10,000$ Rp. in banknotes and the rest is only has a banknote of $5,000.00$ Rp. If the ticket seller initially only has $5,000.00$ Rp., what is the probability that the ticket seller have enough change to serve everyone according to their order in the queue?

Define a domino to be an ordered pair of distinct positive integers. A proper sequence of dominos is a list of distinct dominos in which the first coordinate of each pair after the first equals the second coordinate of the immediately preceding pair, and in which $(i,j)$ and $(j,i)$ do not both appear for any $i$ and $j.$ Let $D_{40}$ be the set of all dominos whose coordinates are no larger than 40. Find the length of the longest proper sequence of dominos that can be formed using the dominos of $D_{40}.$

Inside the inscribed quadrilateral $ABCD$ there is a point $K$, the distances from which to the sides $ABCD$ are proportional to these sides. Prove that $K$ is the intersection point of the diagonals of $ABCD$.

A point $ M$ is chosen on the side $ AC$ of the triangle $ ABC$ in such a way that the radii of the circles inscribed in the triangles $ ABM$ and $ BMC$ are equal. Prove that \[ BM^{2} \equal{} X \cot \left( \frac {B}{2}\right) \] where X is the area of triangle $ ABC.$

A convex polygon on a plane contains at least $m^2+1$ points with integer coordinates. Prove that it contains $m+1$ points with integer coordinates that lie on the same line.

A real function $f$ is defined for $0\le x\le 1$, with its first derivative $f'$ defined for $0\le x\le 1$ and its second derivative $f''$ defined for $0<x<1$. Prove that if $f(0)=f'(0)=f'(1)=f(1)-1 =0$, then there exists a number $0<y<1$ such that $|f''(y)|\ge 4$.

There are $11$ members in the competetion committee. The problem set is kept in a safe having several locks. The committee members have been provided with keys in such a way that every six members can open the safe, but no five members can do that. What is the smallest possible number of locks, and how many keys are needed in that case?

What is the largest number of cells that can be colored in a $7\times7$ table in such a way that any $2\times2$ subtable has at most 2 colored cells?

A convex polyhedron $ Q$ has vertices $ V_1,V_2,\ldots,V_n$, and $ 100$ edges. The polyhedron is cut by planes $ P_1,P_2,\ldots,P_n$ in such a way that plane $ P_k$ cuts only those edges that meet at vertex $ V_k$. In addition, no two planes intersect inside or on $ Q$. The cuts produce $ n$ pyramids and a new polyhedron $ R$. How many edges does $ R$ have? $ \textbf{(A)}\ 200\qquad \textbf{(B)}\ 2n\qquad \textbf{(C)}\ 300\qquad \textbf{(D)}\ 400\qquad \textbf{(E)}\ 4n$

On the circle with center 0 and radius 1 the point $A_0$ is fixed and points $A_1, A_2, \ldots, A_{999}, A_{1000}$ are distributed in such a way that the angle $\angle A_00A_k = k$ (in radians). Cut the circle at points $A_0, A_1, \ldots, A_{1000}.$ How many arcs with different lengths are obtained. ?

Let $n>1$ be an integer and let $a_0,a_1,\ldots,a_n$ be non-negative real numbers. Definite $S_k=\sum_{i\equal{}0}^k \binom{k}{i}a_i$ for $k=0,1,\ldots,n$. Prove that\[\frac{1}{n} \sum_{k\equal{}0}^{n-1} S_k^2-\frac{1}{n^2}\left(\sum_{k\equal{}0}^{n} S_k\right)^2\le \frac{4}{45} (S_n-S_0)^2.\]