In $ ABC$ we have $ AB \equal{} 25$, $ BC \equal{} 39$, and $ AC \equal{} 42$. Points $ D$ and $ E$ are on $ AB$ and $ AC$ respectively, with $ AD \equal{} 19$ and $ AE \equal{} 14$. What is the ratio of the area of triangle $ ADE$ to the area of quadrilateral $ BCED$? $ \textbf{(A)}\ \frac{266}{1521}\qquad \textbf{(B)}\ \frac{19}{75}\qquad \textbf{(C)}\ \frac{1}{3}\qquad \textbf{(D)}\ \frac{19}{56}\qquad \textbf{(E)}\ 1$

Denote $U$ as the set of $20$ diagonals of the regular polygon $P_1P_2P_3P_4P_5P_6P_7P_8$. Find the number of sets $S$ which satisfies the following conditions. 1. $S$ is a subset of $U$. 2. If $P_iP_j \in S$ and $P_j P_k \in S$, and $i \neq k$, $P_iP_k \in S$.

If $x$ cows give $x+1$ cans of milk in $x+2$ days, how many days will it take $x+3$ cows to give $x+5$ cans of milk? $\textbf{(A) }\frac{x(x+2)(x+5)}{(x+1)(x+3)}\qquad\textbf{(B) }\frac{x(x+1)(x+5)}{(x+2)(x+3)}\qquad$ $\textbf{(C) }\frac{(x+1)(x+3)(x+5)}{x(x+2)}\qquad\textbf{(D) }\frac{(x+1)(x+3)}{x(x+2)(x+5)}\qquad \textbf{(E) }\text{none of these}$

Let $n$ and $t$ be positive integers. What is the number of ways to place $t$ dominoes $(1\times 2$ or $2\times 1$ rectangles) in a $2\times n$ table so that there is no $2\times 2$ square formed by $2$ dominoes and each $2\times 3$ rectangle either does not have a horizontal domino in the middle and last cell in the first row or does not have a horizontal domino in the first and middle cell in the second row (or both)?

Find all integers $a,b,c $ such that $a+b+c=abc$

In the adjoining figure, CDE is an equilateral triangle and ABCD and DEFG are squares. The measure of $\angle GDA$ is $\text{(A)} \ 90^\circ \qquad \text{(B)} \ 105^\circ \qquad \text{(C)} \ 120^\circ \qquad \text{(D)} \ 135^\circ \qquad \text{(E)} \ 150^\circ$ [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair D=origin, C=D+dir(240), E=D+dir(300), F=E+dir(30), G=D+dir(30), A=D+dir(150), B=C+dir(150); draw(E--D--G--F--E--C--D--A--B--C); pair point=(0,0.5); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(-15)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F)); label("$G$", G, dir(point--G));[/asy]

The Fibonacci sequence $F_n$ is defined by $F_0=0,F_1=1$ and the recurrence relation $F_n=F_{n-1}+F_{n-2}$ for all integers $n\geq2$. Let $p\geq3$ be a prime number. [list=a] [*] Prove that $F_{p-1}+F_{p+1}-1$ is divisible by $p$. [*] Prove that $F_{p^{k+1}-1}+F_{p^{k+1}+1}-\left(F_{p^k-1}+F_{p^k+1}\right)$ is divisible by $p^{k+1}$ for any positive integer $k$. [/list]

Let $f(x)=a_0x^3+a_1x^2+a_2x+a_3$ be a polynomial with real coefficients ($a_0\ne0$) such that $|f(x)|\le1$ for every $x\in[-1,1]$. Prove that (a) there exist a constant $c$ (one and the same for all polynomials with the given property), for which (b) $|a_0|\le4$. [i]V. Petkov[/i]

Let $0 <a <b$ be two rational numbers. Let $M$ be a set of positive real numbers with the properties: (i) $a \in M$ and $b \in M$; (ii) if $x$ $\in M$ and $y \in M$, then $\sqrt{xy} \in M$. Let $M^*$denote the set of all irrational numbers in $M$. prove that every $c,d$ such that $a <c <d<b$, $M^*$ contains an element $m$ with property $c<m<d$

In triangle $ABC$, find the smallest possible value of $$|(\cot A + \cot B)(\cot B +\cot C)(\cot C + \cot A)|$$

Find all non-negative real numbers $x, y, z$ satisfying $x^2y^2 + 1 = x^2 + xy$, $y^2z^2 + 1 = y^2 + yz$ and $z^2x^2 + 1 = z^2 + xz$.

Find the function $f(x)$ such that \[f(x)^2f\left(\frac{1-x}{x+1}\right) =64x \] for $x\not=0,x\not=1,x\not=-1$.

Let $a,b$ be positive integers.Prove that there are no positive integers on the interval $\bigg[\frac{b^2}{a^2+ab},\frac{b^2}{a^2+ab-1}\bigg)$.

The diagonals of a cyclic quadrilateral $ABCD$ meet at point $P$. Points $K$ and $L$ lie on $AC$, $BD$ respectively in such a way that $CK=AP$ and $DL=BP$. Prove that the line joining the common points of circles $ALC$ and $BKD$ passes through the mass-center of $ABCD$. Proposed by:V.Konyshev

Let $p_n$ be the $n$-th prime, so that $p_1 = 2, p_2 = 3,...$ and define $$X_n = \{0\} \cup \{p_1,...,p_n\}$$ for each positive integer $n$. Find all $n$ for which there exist $A,B \subseteq N$ such that$ |A|,|B| \ge 2$ and $$X_n = A + B$$, where $A + B :=\{a + b : a \in A; b \in B \}$ and $N := \{0,1, 2,...\}$. (Salvatore Tringali)

The flights map of air company $K_{r,r}$ presents several cities. Some cities are connected by a direct (two way) flight, the total number of flights is m. One must choose two non-intersecting groups of r cities each so that every city of the first group is connected by a flight with every city of the second group. Prove that number of possible choices does not exceed $2*m^r$ .

If $125 + n + 135 + 2n + 145 = 900,$ find $n.$

For positive integer $k$, let $p(k)$ denote the greatest odd divisor of $k$. Prove that for every positive integer $n$, $$\frac{2n}{3} < \frac{p(1)}{1}+ \frac{p(2)}{2}+... +\frac{ p(n)}{n}<\frac{2(n + 1)}{3}$$

Determine all positive integers $n$ such that the number $n(n+2)(n+4)$ has at most $15$ positive divisors.

For positive integers $t,a,b,$a $(t,a,b)$-[i]game[/i] is a two player game defined by the following rules. Initially, the number $t$ is written on a blackboard. At his first move, the 1st player replaces $t$ with either $t-a$ or $t-b$. Then, the 2nd player subtracts either $a$ or $b$ from this number, and writes the result on the blackboard, erasing the old number. After this, the first player once again erases either $a$ or $b$ from the number written on the blackboard, and so on. The player who first reaches a negative number loses the game. Prove that there exist infinitely many values of $t$ for which the first player has a winning strategy for all pairs $(a,b)$ with $a+b=2005$.

In a circle $C_1$ with centre $O$, let $AB$ be a chord that is not a diameter. Let $M$ be the midpoint of this chord $AB$. Take a point $T$ on the circle $C_2$ with $OM$ as diameter. Let the tangent to $C_2$ at $T$ meet $C_1$ at $P$. Show that $PA^2 + PB^2 = 4 \cdot PT^2$.

Show that there are infinitely many triples $(x, y, z)$ of real numbers such that $$\displaystyle{x^2+y = y^2+z= z^2 + x}$$ and $x\ne y\ne z \ne x.$

Peeter, Juri, Kati and Mari are standing at the entrance of a dark tunnel. They have one torch and none of them dares to be in the tunnel without it, but the tunnel is so narrow that at most two people can move together. It takes $1$ minute for Peeter, $2$ minutes for Juri, $5$ for Kati and $10$ for Mari to pass the tunnel. Find the minimum time in which they can all pass through the tunnel.

Solve the equation in natural numbers $$ xy+yz+zx = xyz + 2. $$

The diagonals of convex quadrilateral $ABCD$ intersect at point $E$. Triangles $ABE$ and $CED$ have a common excircle $\Omega$, tangent to segments $AE$ and $DE$ at points $B_1$ and $C_1$, respectively. Denote by $I$ and $J$ the centers of the incircles of these triangles, respectively. Segments $IC_1$ and $JB_1$ intersect at point $S$. It is known that $S$ lies on $\Omega$. Prove that the circumcircle of triangle $AED$ is tangent to $\Omega$. [i]Proposed by David Brodsky[/i]