If $ a,b,c$ are positive integers less than $ 10$, then $ (10a \plus{} b)(10a \plus{} c) \equal{} 100a(a \plus{} 1) \plus{} bc$ if: $ \textbf{(A)}\ b \plus{} c \equal{} 10 \qquad\textbf{(B)}\ b \equal{} c \qquad\textbf{(C)}\ a \plus{} b \equal{} 10 \qquad\textbf{(D)}\ a \equal{} b \\ \textbf{(E)}\ a \plus{} b \plus{} c \equal{} 10$

Quadrilateral $ABCD$ has both an inscribed and a circumscribed circle and sidelengths $BC = 4, CD = 5, DA = 6$. Find the area of $ABCD$.

If $ a \equal{} \log_8 225$ and $ b \equal{} \log_2 15,$ then $ a$, in terms of $ b,$ is: $ \textbf{(A)}\ \frac{b}{2} \qquad \textbf{(B)}\ \frac{2b}{3}\qquad \textbf{(C)}\ b \qquad \textbf{(D)}\ \frac{3b}{2} \qquad \textbf{(E)}\ 2b$

There are 12 people in a line in a bank. When the desk closes, the people form a new line at a newly opened desk. In how many ways can they do this in such a way that none of the 12 people changes his/her position in the line by more than one?

Let one of the intersection points of two circles with centres $O_1,O_2$ be $P$. A common tangent touches the circles at $A,B$ respectively. Let the perpendicular from $A$ to the line $BP$ meet $O_1O_2$ at $C$. Prove that $AP\perp PC$.

We have some cards that have the same look, but at the back of some of them is written $0$ and for the others $1$.(We can't see the back of a card so we can't know what's the number on it's back). we have a machine. we give it two cards and it gives us the product of the numbers on the back of the cards. if we have $m$ cards with $0$ on their back and $n$ cards with $1$ on their back, at least how many times we must use the machine to be sure that we get the number $1$? (15 points)

For any positive integer $n$, show that there exists a polynomial $P(x)$ of degree $n$ with integer coefficients such that $P(0),P(1), \ldots, P(n)$ are all distinct powers of $2$.

Let $Oxy$ be a fixed rectangular coordinate system in the plane. Each ordered pair of points $A_1, A_2$ from the same plane which are different from O and have coordinates $x_1, y_1$ and $x_2, y_2$ respectively is associated with real number $f(A_1,A_2)$ in such a way that the following conditions are satisfied: (a) If $OA_1 = OB_1$, $OA_2 = OB_2$ and $A_1A_2 = B_1B_2$ then $f(A_1,A_2) = f(B_1,B_2)$. (b) There exists a polynomial of second degree $F(u,v,w,z)$ such that $f(A_1,A_2)=F(x_1,y_1,x_2,y_2)$. (c) There exists such a number $\phi \in (0,\pi)$ that for every two points $A_1, A_2$ for which $\angle A_1OA_2 = \phi$ is satisfied $f(A_1,A_2) = 0$. (d) If the points $A_1, A_2$ are such that the triangle $OA_1A_2$ is equilateral with side $1$ then$ f(A_1,A_2) = \frac12$. Prove that $f(A_1,A_2) = \overrightarrow{OA_1} \cdot \overrightarrow{OA_2}$ for each ordered pair of points $A_1, A_2$.

Let $W = \ldots x_{-1}x_0x_1x_2 \ldots$ be an infinite periodic word consisting of only the letters $a$ and $b$. The minimal period of $W$ is $2^{2016}$. Say that a word $U$ [i]appears[/i] in $W$ if there are indices $k \le \ell$ such that $U = x_kx_{k+1} \ldots x_{\ell}$. A word $U$ is called [i]special[/i] if $Ua, Ub, aU, bU$ all appear in $W$. (The empty word is considered special) You are given that there are no special words of length greater than 2015. Let $N$ be the minimum possible number of special words. Find the remainder when $N$ is divided by $1000$. [i]Proposed by Yang Liu[/i]

Does there exist a finite set $A$ of positive integers of at least two elements and an infinite set $B$ of positive integers, such that any two distinct elements in $A+B$ are coprime, and for any coprime positive integers $m,n$, there exists an element $x$ in $A+B$ satisfying $x\equiv n \pmod m$ ? Here $A+B=\{a+b|a\in A, b\in B\}$.

In a triangle $ABC$ with $AC = b \ne BC = a$, points $E,F$ are taken on the sides $AC,BC$ respectively such that $AE = BF =\frac{ab}{a+b}$. Let $M$ and $N$ be the midpoints of $AB$ and $EF$ respectively, and $P$ be the intersection point of the segment $EF$ with the bisector of $\angle ACB$. Find the ratio of the area of $CPMN$ to that of $ABC$.

There are several discs whose radii are no more that $1$, and whose centers all lie on a segment with length ${l}$. Prove that the union of all the discs has a perimeter not exceeding $4l+8$. [i]Proposed by Morteza Saghafian - Iran[/i]

$\frac{2^1+2^0+2^{-1}}{2^{-2}+2^{-3}+2^{-4}}$ equals $\text{(A)} \ 6 \qquad \text{(B)} \ 8 \qquad \text{(C)} \ \frac{31}{2} \qquad \text{(D)} \ 24 \qquad \text{(E)} \ 512$

Prove that there exist $C>0$, which satisfies the following conclusion: For any infinite positive arithmetic integer sequence $a_1, a_2, a_3,\cdots$, if the greatest common divisor of $a_1$ and $a_2$ is squarefree, then there exists a positive integer $m\le C\cdot {a_2}^2$, such that $a_m$ is squarefree. Note: A positive integer $N$ is squarefree if it is not divisible by any square number greater than $1$. [i]Proposed by Qu Zhenhua[/i]

Let $\omega$ be a circle. For each $n$, let $A_n$ be the area of a regular $n$-sided polygon circumscribed to $\omega$ and $B_n$ the area of a regular $n$-sided polygon inscribed in $\omega$ . Try that $3A_{2015} + B_{2015}> 4A_{4030}$

Let $z = a + bi$ be the complex number with $|z| = 5$ and $b > 0$ such that the distance between $(1 + 2i)z^3$ and $z^5$ is maximized, and let $z^4 = c + di$. Find $c+d$.