Given three parallel straight lines. Construct a square three of whose vertices belong to these lines.

Let $d$ be any positive integer not equal to $2, 5$ or $13$. Show that one can find distinct $a,b$ in the set $\{2,5,13,d\}$ such that $ab-1$ is not a perfect square.

Some cells of a checkered plane are marked so that figure $A$ formed by marked cells satisfies the following condition:$1)$ any cell of the figure $A$ has exactly two adjacent cells of $A$; and $2)$ the figure $A$ can be divided into isosceles trapezoids of area $2$ with vertices at the grid nodes (and acute angles of trapezoids are equal to $45$) . Prove that the number of marked cells is divisible by $8$.

Four whole numbers, when added three at a time, give the sums $180$, $197$, $208$, and $222$. What is the largest of the four numbers? $\text{(A)} \ 77 \qquad \text{(B)} \ 83 \qquad \text{(C)} \ 89 \qquad \text{(D)} \ 95 \qquad \text{(E)} \ \text{cannot be determined}$

The sequence of positive integers $a_0, a_1, a_2, \ldots$ is recursively defined such that $a_0$ is not a power of $2$, and for all nonnegative integers $n$: (i) if $a_n$ is even, then $a_{n+1} $ is the largest odd factor of $a_n$ (ii) if $a_n$ is odd, then $a_{n+1} = a_n + p^2$ where $p$ is the smallest prime factor of $a_n$ Prove that there exists some positive integer $M$ such that $a_{m+2} = a_m $ for all $m \geq M$. [i]Proposed by Andrew Wen[/i]

Consider the configuration of six squares as shown on the picture. Prove that the sum of the area of the three outer squares ($ I,II$ and $ III$) equals three times the sum of the areas of the three inner squares ($ IV,V$ and $ VI$).

All $ 2n\ge 2 $ squares of a $ 2\times n $ rectangle are colored with three colors. We say that a color has a [i]cut[/i] if there is some column (from all $ n $) that has both squares colored with it. Determine: [b]a)[/b] the number of colorings that have no cuts. [b]b)[/b] the number of colorings that have a single cut.

[b]11.[/b] Let $a_n = (-1)^n (n= 1, 2, \dots , 2N)$. Denote by $A_{N}(x)$ the number of the sequences $1 \leq i_1 < i_2< \dots <i_N \leq 2N$ such that $a_{i_1}+a_{i_2}+ \dots +a_{i_N}< x \sqrt{\frac{N}{2}} (-\infty < x < \infty)$. Show that $\lim_{N \to \infty} \frac{A_{N}(x)}{\binom{2N}{N}} = \frac {1}{\sqrt {2\pi}} \int_{-\infty}^{\infty} e^{-\frac{u^2}{2}} du$. [b](N. 16)[/b]

The incircle of triangle $ABC$ touches sides $BC, CA$ and $AB$ at points $D, E$ and $F$, respectively. Let $P$ be the intersection of lines $AD$ and $BE$. The reflections of $P$ with respect to $EF, FD$ and $DE$ are $X,Y$ and $Z$, respectively. Prove that lines $AX, BY$ and $CZ$ are concurrent at a point on line $IO$, where $I$ and $O$ are the incenter and circumcenter of triangle $ABC$.

What is the positive difference between the largest possible two-digit integer and the smallest possible three-digit integer? $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }5\qquad\textbf{(E) }9$

Determine all triples $(a,b,c)$ of real numbers with \[a+\frac{4}{b}=b+\frac{4}{c}=c+\frac{4}{a}.\]

How many triangles formed by three vertices of a regular $17$-gon are obtuse? $\text{(A) }156\qquad\text{(B) }204\qquad\text{(C) }357\qquad\text{(D) }476\qquad\text{(E) }524$

Suppose $S= \{x|x=a^2+ab+b^2,a,b \in Z\}$. Prove that: (1) If $m \in S$, $3|m$ , then $\frac{m}{3} \in S$ (2) If $m,n \in S$ , then $mn\in S$.

Let $ABCDEF$ be a convex hexagon. The diagonal $AC$ is cut by $BF$ and $BD$ at points $P$ and $Q$, respectively. The diagonal $CE$ is cut by $DB$ and $DF$ at points $R$ and $S$, respectively. The diagonal $EA$ is cut by $FD$ and $FB$ at points $T$ and $U$, respectively. It is known that each of the seven triangles $APB, PBQ, QBC, CRD, DRS, DSE$ and $AUF$ has area $1$. Find the area of the hexagon $ABCDEF$.

Three positive real numbers $a,b,c$ are such that $a^2+5b^2+4c^2-4ab-4bc=0$. Can $a,b,c$ be the lengths of te sides of a triangle? Justify your answer.

Given triangle $ ABC $ with points $ M $ and $ N $ are in the sides $ AB $ and $ AC $ respectively. If $ \dfrac{BM}{MA} +\dfrac{CN}{NA} = 1 $ , then prove that the centroid of $ ABC $ lies on $ MN $ .

In the adjoining figure, the circle meets the sides of an equilateral triangle at six points. If $AG=2$, $GF=13$, $FC=1$, and $HJ=7$, then $DE$ equals [asy] size(200); defaultpen(fontsize(10)); real r=sqrt(22); pair B=origin, A=16*dir(60), C=(16,0), D=(10-r,0), E=(10+r,0), F=C+1*dir(120), G=C+14*dir(120), H=13*dir(60), J=6*dir(60), O=circumcenter(G,H,J); dot(A^^B^^C^^D^^E^^F^^G^^H^^J); draw(Circle(O, abs(O-D))^^A--B--C--cycle, linewidth(0.7)); label("$A$", A, N); label("$B$", B, dir(210)); label("$C$", C, dir(330)); label("$D$", D, SW); label("$E$", E, SE); label("$F$", F, dir(170)); label("$G$", G, dir(250)); label("$H$", H, SE); label("$J$", J, dir(0)); label("2", A--G, dir(30)); label("13", F--G, dir(180+30)); label("1", F--C, dir(30)); label("7", H--J, dir(-30));[/asy] $\textbf {(A) } 2\sqrt{22} \qquad \textbf {(B) } 7\sqrt{3} \qquad \textbf {(C) } 9 \qquad \textbf {(D) } 10 \qquad \textbf {(E) } 13$

There is a billiard table in shape of rectangle $2 \times 1$, with pockets at its corners and at midpoints of its two largest sizes. Find the minimal number of balls one has to place on the table interior so that any pocket is on a straight line with some two balls. (Assume that pockets and balls are points). [i](4 points)[/i]

Let $L$ and $K$ be the feet of the internal and the external bisector of angle $A$ of a triangle $ABC$. Let $P$ be the common point of the tangents to the circumcircle of the triangle at $B$ and $C$. The perpendicular from $L$ to $BC$ meets $AP$ at point $Q$. Prove that $Q$ lies on the medial line of triangle $LKP$.

Let $2\mathbb{Z} + 1$ denote the set of odd integers. Find all functions $f:\mathbb{Z} \mapsto 2\mathbb{Z} + 1$ satisfying \[ f(x + f(x) + y) + f(x - f(x) - y) = f(x+y) + f(x-y) \] for every $x, y \in \mathbb{Z}$.

An international society has its members from six different countries. The list of members contain $1978$ names, numbered $1, 2, \dots, 1978$. Prove that there is at least one member whose number is the sum of the numbers of two members from his own country, or twice as large as the number of one member from his own country.

Let $A$ denote the set of all the positive integer divisors of $30.$ For each nonempty subset $s \subseteq A,$ define $p(s)$ to be the product of the elements in $s.$ Finally, let $B$ denote the set of all possible remainders when $p(s)$ is divided by $30.$ How many (distinct) elements are in $B?$

Prove that for all $n\in\mathbb{Z}^+$, we have \[ \sum\limits_{p=1}^n\sum\limits_{q=1}^p\left\lfloor -\frac{1+\sqrt{8q+(2p-1)^2}}{2}\right\rfloor =-\frac{n(n+1)(n+2)}{3} \]

Madoka chooses $4$ random numbers $a, b, c, d$ between $0$ and $1$. She notices that $a+b+c = 1$. If the probability that $d > a, b, c$ can be written in simplest form as $\frac{m}{n}$, find $m + n$.

Let $P(x)$ be a polynomial with integer coefficients. Show that if $Q(x) = P(x) +12$ has at least six distinct integer roots, then $P(x)$ has no integer roots.