Euclid places a morsel of food at the point $(0,0)$ and an ant at the point $(1,2)$. Every second, the ant walks one unit in one of the four coordinate directions. However, whenever the ant moves to $(x,\pm 3)$, Euclid's malicious brother Mobius picks it up and puts it at $(-x,\mp 2)$, and whenever it moves to $(\pm 2,y)$, his cousin Klein puts it at $(\mp 1,y)$. If $p$ and $q$ are relatively prime positive integers such that $\tfrac pq$ is the expected number of steps the ant takes before reaching the food, find $p+q$.

* The quadratic expression $ax^2 + bx + c$ is a square (of an integer) for any integer $x$. Prove that $ax^2 + bx + c = (dx + e)^2$ for some integers d and e.

Let $ABCD$ be a trapezoid of bases $AB$ and $CD$, and non-parallel sides $BC$ and $DA$. The angles $\angle BCD$ and $\angle CDA$ are acute. The lines $BC$ and $DA$ are cut at a point $E$. It is known that $AE = 2$, $AC = 6$, $CD =\sqrt{72}$ and area $( \vartriangle BCD)= 18$. (a) Find the height of the trapezoid $ABCD$. (b) Find the area of $\vartriangle ABC$.

Prove that for any real numbers $ a,b,c, $ the function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ defined as $$ f(x)=\sqrt{(x-c)^2+b^2} +\sqrt{(x+c)^2+b^2} $$ is decreasing on $ (-\infty ,0] $ and increasing on $ [0,\infty ) . $

Given sequence $ \{ c_n \}$ satisfying the conditions that $ c_0\equal{}1$, $ c_1\equal{}0$, $ c_2\equal{}2005$, and $ c_{n\plus{}2}\equal{}\minus{}3c_n \minus{} 4c_{n\minus{}1} \plus{}2008$, ($ n\equal{}1,2,3, \cdots$). Let $ \{ a_n \}$ be another sequence such that $ a_n\equal{}5(c_{n\plus{}1} \minus{} c_n) \cdot (502 \minus{} c_{n\minus{}1} \minus{} c_{n\minus{}2}) \plus{} 4^n \times 2004 \times 501$, ($ n\equal{}2,3, \cdots$). Is $ a_n$ a perfect square for every $ n > 2$?

Let be a natural number $ k $ and let be two infinite sequences $ \left( x_n \right)_{n\ge 1} ,\left( y_n \right)_{n\ge 1} $ such that $$ \{1\}\cap\{ x_1,x_2,\ldots ,x_k\}=\{1\}\cap\{ y_1,y_2,\ldots ,y_k\} =\{ x_1,x_2,\ldots ,x_k\}\cap\{ y_1,y_2,\ldots ,y_k\} =\emptyset , $$ and defined by the following recurrence relations: $$ x_{n+k}=\frac{y_n}{x_n} ,\quad y_{n+k} =\frac{y_n-1}{x_n-1} $$ Prove that $ \left( x_n \right)_{n\ge 1} $ and $ \left( y_n \right)_{n\ge 1} $ are periodic. [i]Dumitru Acu[/i]

Let $ABC$ be a isosceles triangle with $ AC = BC > AB$. Let $ E, F $ be the midpoints of segments $ AC, AB$, and let $l$ be the perpendicular bisector of $AC$. Let $ l $ meets $ AB$ at $K$, the line through $B$ parallel to $KC$ meets $AC$ at point $L$, and line $FL$ meets $ l$ at $W$. Let $ P $ be a point on segment $BF$. Let $H$ be the orthocenter of triangle $ACP$ and line $BH$ and $CP$ meet at point $J$. Line $FJ$ meets $l$ at $M$. Prove that $ AW = PW $ if and only if $B$ lies on the circumcircle of $EFM$.

Let $A_1A_2...A_{101}$ be a regular $101$-gon, and colour every vertex red or blue. Let $N$ be the number of obtuse triangles satisfying the following: The three vertices of the triangle must be vertices of the $101$-gon, both the vertices with acute angles have the same colour, and the vertex with obtuse angle have different colour. $(1)$ Find the largest possible value of $N$. $(2)$ Find the number of ways to colour the vertices such that maximum $N$ is acheived. (Two colourings a different if for some $A_i$ the colours are different on the two colouring schemes).

A cylinder is inscribed within a sphere of radius 10 such that its volume is [i]almost-half[/i] that of the sphere. If [i]almost-half[/i] is defined such that the cylinder has volume $\frac12+\frac{1}{250}$ times the sphere’s volume, find the sum of all possible heights for the cylinder.

In a triangle $ABC ~(\overline{AB} < \overline{AC})$, points $D (\neq A, B)$ and $E (\neq A, C)$ lies on side $AB$ and $AC$ respectively. Point $P$ satisfies $\overline{PB}=\overline{PD}, \overline{PC}=\overline{PE}$. $X (\neq A, C)$ is on the arc $AC$ of the circumcircle of triangle $ABC$ not including $B$. Let $Y (\neq A)$ be the intersection of circumcircle of triangle $ADE$ and line $XA$. Prove that $\overline{PX} = \overline{PY}$.

Let $\frac{35x-29}{x^2-3x+2}=\frac{N_1}{x-1}+\frac{N_2}{x-2}$ be an identity in $x$. The numerical value of $N_1N_2$ is: $\text{(A)} \ -246 \qquad \text{(B)} \ -210 \qquad \text{(C)} \ -29 \qquad \text{(D)} \ 210 \qquad \text{(E)} \ 246$

Let $A$ and $B$ be two non-infinite sets of natural numbers, each of which contains at least 3 elements. Two numbers $a\in A$ and $b\in B$ are called [i]"harmonious"[/i], if they are not coprime. It is known that each element from $A$ is not [i]harmonious[/i] with at least one element from $B$ and each element from $B$ is harmonious with at least one from $A$. Prove that there exist $a_1,a_2\in A$ and $b_1,b_2\in B$ such that $(a_1,b_1)$ and $(a_2,b_2)$ are [i]harmonious[/i] but $(a_1,b_2)$ and $(a_2,b_1)$ are not.

Is it possible to represent the number $1986$ as the sum of squares of $6$ odd integers?

Let $a,b,c$ be real numbers, which are not all equal, such that $$a+b+c=\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=3.$$ Prove that at least one of $a, b, c$ is negative.

Given a square whose side measures $a$, consider the set of all points of its plane through which passes a circumference of radius whose circle contains to the quoted square. You are asked to prove that the contour of the figure formed by the points with this property is formed by arcs of circumference, and determine the positions, their centers, their radii and their lengths.

Determine the number of positive integers $N = \overline{abcd}$, with $a, b, c, d$ nonzero digits, which satisfy $(2a -1) (2b -1) (2c- 1) (2d - 1) = 2abcd -1$.

Let $\mathfrak F=\{A_1,A_2,\ldots,A_n\}$ be a collection of subsets of the set $S=\{1,2,\ldots,n\}$ satisfying the following conditions: (a) Any two distinct sets from $\mathfrak F$ have exactly one element in common; (b) each element of $S$ is contained in exactly $k$ of the sets in $\mathfrak F$. Can $n$ be equal to $1996$?

From an $n \times n $ square, $n \ge 2,$ the unit squares situated on both odd numbered rows and odd numbers columns are removed. Determine the minimum number of rectangular tiles needed to cover the remaining surface.

Find the least positive integer ending in $7$ with exactly $12$ positive divisors.

The set $M = \{1, 2, . . . , 2n\}$ is partitioned into $k$ nonintersecting subsets $M_1,M_2, \dots, M_k,$ where $n \ge k^3 + k.$ Prove that there exist even numbers $2j_1, 2j_2, \dots, 2j_{k+1}$ in $M$ that are in one and the same subset $M_i$ $(1 \le i \le k)$ such that the numbers $2j_1 - 1, 2j_2 - 1, \dots, 2j_{k+1} - 1$ are also in one and the same subset $M_j (1 \le j \le k).$

Prove that the modulus of an integer root of a polynomial with integer coefficients cannot exceed the maximum of the moduli of the coefficients.

Rectangle $ABCD$ has side lengths $AB = 3$ and $BC = 7$. Let $E$ be a point on $BC$, and let $F$ be the intersection of $DE$ and $AC$. Given that $[CDF] = 4$, find $\frac{DF}{FE}$ .

Let $ f(n)$ be a function defined on the set of all positive integers and having its values in the same set. Suppose that $ f(f(n) \plus{} f(m)) \equal{} m \plus{} n$ for all positive integers $ n,m.$ Find the possible value for $ f(1988).$

In the figure, $ ABCD$ is a $ 2\times 2$ square, $ E$ is the midpoint of $ \overline{AD}$, and $ F$ is on $ \overline{BE}$. If $ \overline{CF}$ is perpendicular to $ \overline{BE}$, then the area of quadrilateral $ CDEF$ is [asy]defaultpen(linewidth(.8pt)); dotfactor=4; pair A = (0,2); pair B = origin; pair C = (2,0); pair D = (2,2); pair E = midpoint(A--D); pair F = foot(C,B,E); dot(A);dot(B);dot(C);dot(D);dot(E);dot(F); label("$A$",A,N);label("$B$",B,S);label("$C$",C,S);label("$D$",D,N);label("$E$",E,N);label("$F$",F,NW); draw(A--B--C--D--cycle); draw(B--E); draw(C--F); draw(rightanglemark(B,F,C,4));[/asy]$ \textbf{(A)}\ 2\qquad \textbf{(B)}\ 3 \minus{} \frac {\sqrt {3}}{2}\qquad \textbf{(C)}\ \frac {11}{5}\qquad \textbf{(D)}\ \sqrt {5}\qquad \textbf{(E)}\ \frac {9}{4}$

Find the least positive integer, which may not be represented as ${2^a-2^b\over 2^c-2^d}$, where $a,\,b,\,c,\,d$ are positive integers.