Given two circles intersecting at points $A$ and $B$. A certain circle touches the first at point $A$, intersects the second at point $M$ and intersects the straight line $AB$ at point $P$ ($M$ and $P$ are different from $B$). Prove that the straight line $MP$ passes through a fixed point of the plane (for any change in the third circle).

Determine all sequences of integers $a_1, a_2,. . .,$ such that: (i) $1 \le a_i \le n$ for all $1 \le i \le n$. (ii) $| a_i - a_j| = | i - j |$ for any $1 \le i, j \le n$

Let $[a, b] = ab - a - b$. Shaq sees the numbers $2, 3, \dots , 101$ written on a blackboard. Let $V$ be the largest number that Shaq can obtain by repeatedly choosing two numbers $a, b$ on the board and replacing them with $[a, b]$ until there is only one number left. Suppose $N$ is the integer with $N!$ nearest to $V$. Find the nearest integer to $10^6 \cdot \tfrac{|V-N!|}{N!}$.

Four circles of radius $ 1$ are each tangent to two sides of a square and externally tangent to a circle of radius $ 2$, as shown. What is the area of the square? [asy]unitsize(5mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); real h=3*sqrt(2)/2; pair O0=(0,0), O1=(h,h), O2=(-h,h), O3=(-h,-h), O4=(h,-h); pair X=O0+2*dir(30), Y=O2+dir(45); draw((-h-1,-h-1)--(-h-1,h+1)--(h+1,h+1)--(h+1,-h-1)--cycle); draw(Circle(O0,2)); draw(Circle(O1,1)); draw(Circle(O2,1)); draw(Circle(O3,1)); draw(Circle(O4,1)); draw(O0--X); draw(O2--Y); label("$2$",midpoint(O0--X),NW); label("$1$",midpoint(O2--Y),SE);[/asy]$ \textbf{(A)}\ 32 \qquad \textbf{(B)}\ 22 \plus{} 12\sqrt {2}\qquad \textbf{(C)}\ 16 \plus{} 16\sqrt {3}\qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 36 \plus{} 16\sqrt {2}$

For any real numbers sequence $\{x_n\}$ ,suppose that $\{y_n\}$ is a sequence such that: $y_1=x_1, y_{n+1}=x_{n+1}-(\sum\limits_{i = 1}^{n} {x^2_i})^{ \frac{1}{2}}$ ${(n \ge 1})$ . Find the smallest positive number $\lambda$ such that for any real numbers sequence $\{x_n\}$ and all positive integers $m$ , have $\frac{1}{m}\sum\limits_{i = 1}^{m} {x^2_i}\le\sum\limits_{i = 1}^{m} {\lambda^{m-i}y^2_i} .$ (High School Affiliated to Nanjing Normal University )

We observe that number $10001=73\cdot137$ is not prime. Show that every member of infinite sequence $10001, 100010001, 1000100010001,...$ is not prime

Consider an $N\times N$ matrix, where $N$ is an odd positive integer, such that all its entries are $-1,0$ or $1$. Consider the sum of the numbers in every line and every column. Prove that at least two of the $2N$ sums are equal.

In a convex quadrilateral $PQRS$, $PQ =RS$, $(\sqrt{3} +1 )QR = SP$ and $\angle RSP - \angle SQP = 30^{\circ}$. Prove that $\angle PQR - \angle QRS = 90^{\circ}.$

Prove that, for all $n\in\mathbb{N}$, on $ [n-4\sqrt{n}, n+4\sqrt{n}]$ exists natural number $k=x^3+y^3$ where $x$, $y$ are nonnegative integers.

In the octagon below all sides have the length $1$ and all angles are equal. Determine the distance between the corners $A$ and $B$. [img]https://1.bp.blogspot.com/-i6TAFDvcQ8w/XzXCRhnV_kI/AAAAAAAAMVw/rKrQMfPYYJIaCwl8hhdVHdqO4fIn8O7cwCLcBGAsYHQ/s0/2011%2BMogh%2Bp2.png[/img]

If $n$ is a natural number, prove that the number $(n+1)(n+2)\cdots(n+10)$ is not a perfect square.

Length of sides of regular hexagon $ABCDEF$ is $a$. Two moving points $M,N$ moves on sides $BC,DE$, satisfy that $\angle MAN=\frac{\pi}{3}$. Prove that $AM\cdot AN-BM\cdot DN$ is a definite value.

Find all functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that admit a primitive $ F $ defined as $ F(x)=\left\{\begin{matrix} f(x)/x, & x\neq 0 \\ 2007, & x=0 \end{matrix}\right. . $

An $8'\text{ X }10'$ table sits in the corner of a square room, as in Figure 1 below. The owners desire to move the table to the position shown in Figure 2. The side of the room is $S$ feet. What is the smallest integer value of $S$ for which the table can be moved as desired without tilting it or taking it apart? [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair A=(0,0), B=(16,0), C=(16,16), D=(0,16), E=(32,0), F=(48,0), G=(48,16), H=(32,16), I=(0,8), J=(10,8), K=(10,16), L=(32,6), M=(40,6), N=(40,16); draw(A--B--C--D--A^^E--F--G--H--E^^I--J--K^^L--M--N); label("S", (18,8)); label("S", (50,8)); label("Figure 1", (A+B)/2, 2*S); label("Figure 2", (E+F)/2, 2*S); label("10'", (I+J)/2, S); label("8'", (12,12)); label("8'", (L+M)/2, S); label("10'", (42,11)); label("table", (5,12)); label("table", (36,11)); [/asy] $ \textbf{(A)}\ 11\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 13\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 15 $

[b]5.[/b] Denote by $c_n$ the $n$th positive integer which can be represented in the form $c_n = k^{l} (k,l = 2,3, \dots )$. Prove that $\sum_{n=1}^{\infty}\frac{1}{c_n-1}=1$ [b](N. 18)[/b]

Given is a trapezoid $ABCD$, $AD \parallel BC$. The angle bisectors of the two pairs of opposite angles meet at $X, Y$. Prove that $AXYD$ and $BXYC$ are cyclic.

In a rectangular box are a number of rectangular blocks, not necessarily identical to one another. Each block has one of its dimensions reduced. Is it always possible to pack these blocks in a smaller rectangular box, with the sides of the blocks parallel to the sides of the box? [i](6 points)[/i]

$11$ people are sitting around a circle table, orderly (means that the distance between two adjacent persons is equal to others) and $11$ cards with numbers $1$ to $11$ are given to them. Some may have no card and some may have more than $1$ card. In each round, one [and only one] can give one of his cards with number $ i $ to his adjacent person if after and before the round, the locations of the cards with numbers $ i-1,i,i+1 $ don’t make an acute-angled triangle. (Card with number $0$ means the card with number $11$ and card with number $12$ means the card with number $1$!) Suppose that the cards are given to the persons regularly clockwise. (Mean that the number of the cards in the clockwise direction is increasing.) Prove that the cards can’t be gathered at one person.

A circle through the vertices $A$ and $B$ of $\triangle ABC$ intersects segments $AC$ and $BC$ at points $P$ and $Q$ respectively. If $AQ=AC$, $\angle BAQ=\angle CBP$ and $AB=(\sqrt{3}+1)PQ$, find the measures of the angles of $\triangle ABC$.

We are given tiles in the form of right angled triangles having perpendicular sides of length $1$ cm and $2$ cm. Is it possible to form a square from $20$ such tiles? ( S . Fomin , Leningrad)

Determine all pairs $(x,y)$ of natural numbers, such that the numbers $\frac{x+1}{y}$ and $\frac{y+1}{x}$ are natural.

If $ m,n\in Z$, then $ m^{2} \plus{} 3mn \minus{} 4n^{2}$ cannot be $\textbf{(A)}\ 69 \qquad\textbf{(B)}\ 76 \qquad\textbf{(C)}\ 91 \qquad\textbf{(D)}\ 94 \qquad\textbf{(E)}\ \text{None}$

Let $ n \ge 1$ and $ k \ge 3$ be integers. A circle is divided into $ n$ sectors $ a_1,a_2,\dots,a_n$. We will color the $ n$ sectors with $ k$ different colors such that $ a_i$ and $ a_{i \plus{} 1}$ have different color for each $ i \equal{} 1,2,\dots,n$ where $ a_{n \plus{} 1}\equal{}a_1$. Find the number of ways to do such coloring.

For each $1\leq i\leq 9$ and $T\in\mathbb N$, define $d_i(T)$ to be the total number of times the digit $i$ appears when all the multiples of $1829$ between $1$ and $T$ inclusive are written out in base $10$. Show that there are infinitely many $T\in\mathbb N$ such that there are precisely two distinct values among $d_1(T)$, $d_2(T)$, $\dots$, $d_9(T)$.

A jar contains $97$ marbles that are either red, green, or blue. Neil draws two marbles from the jar without replacement and notes that the probability that they would be the same color is $\tfrac{5}{12}.$ After Neil puts his marbles back, Jerry draws two marbles from the jar with replacement. Compute the probability that the marbles that Jerry draws are the same color.