Given is an array $A$ of $2n$ numbers, where $n$ is a positive integer. Give an algorithm to create an array $prod$ of length $2n$ where $$prod[i] \, = \, A[i] \times A[i+1] \times \cdots \times A[i+n-1],$$ ($A[x]$ means $A[x \ \text{mod}\ 2n]$) in $O(n)$ time [b]withou[/b]t using division. Assume that all binary arithmetic operations are $O(1)$

Let us denote by $b(n)$ the number of ways to represent $n$ in the form $$n = a_0+a_1 \cdot 2 +a_2 \cdot 2^2+...+ a_k \cdot 2^k,$$ where the coefficients at, $r = 1$,$2$,$...$, $k$ can be equal to $0$, $1$ or $2$. Find $b(1996)$.

A pair of numbers are [i]twin primes[/i] if they differ by two, and both are prime. Prove that, except for the pair $\{3, 5\}$, the sum of any pair of twin primes is a multiple of $ 12$.

A convex polyhedron has $2n$ faces ($n\ge 3$), and all faces are triangles. What is the largest number of vertices at which converges exactly $3$ edges at a such a polyhedron ?

The pairwise greatest common divisors of five positive integers are $$2, 3, 4, 5, 6, 7, 8, p, q, r$$ in some order, for some positive integers $p, q, r$. Compute the minimum possible value of $p + q + r$.

Carl is given three distinct non-parallel lines $\ell_1, \ell_2, \ell_3$ and a circle $\omega$ in the plane. In addition to a normal straightedge, Carl has a special straightedge which, given a line $\ell$ and a point $P$, constructs a new line passing through $P$ parallel to $\ell$. (Carl does not have a compass.) Show that Carl can construct a triangle with circumcircle $\omega$ whose sides are parallel to $\ell_1,\ell_2,\ell_3$ in some order. [i]Proposed by Vincent Huang[/i]

Gordon has the least number of coins (half-dollars, quarters, dimes, nickels, pennies) needed to make $99\cent$. He randomly chooses one. What is the probability that it is a penny? $\textbf{(A) } \dfrac15\qquad\textbf{(B) } \dfrac13\qquad\textbf{(C) } \dfrac12\qquad\textbf{(D) } \dfrac23\qquad\textbf{(E) } \dfrac34$

Prove that, if a pentagon (five-sided polygon) inscribed in a circle has equal angles, then its sides are equal.

Show that among any seven coplanar unit vectors there are at least two of them such that the magnitude of their sum is greater than $ \sqrt 3. $ [i]Ion Tecu[/i] and [i]Teodor Radu[/i]

Given real numbers $x_1,x_2,y_1,y_2,z_1,z_2$ satisfying $x_1>0,x_2>0,x_1y_1>z_1^2$, and $x_2y_2>z_2^2$, prove that: \[ {8\over(x_1+x_2)(y_1+y_2)-(z_1+z_2)^2}\le{1\over x_1y_1-z_1^2}+{1\over x_2y_2-z_2^2}. \] Give necessary and sufficient conditions for equality.

A triangle has sides with lengths $a,b,c$ such that \[ a^2 + b^2 = 5c^2 \] Prove that medians to the sides of lengths $a$ and $b$ are perpendicular.

Which of the following sets could NOT be the lengths of the external diagonals of a right rectangular prism [a "box"]? (An [i]external diagonal[/i] is a diagonal of one of the rectangular faces of the box.) $ \textbf{(A)}\ \{4, 5, 6\} \qquad\textbf{(B)}\ \{4, 5, 7\} \qquad\textbf{(C)}\ \{4, 6, 7\} \qquad\textbf{(D)}\ \{5, 6, 7\} \qquad\textbf{(E)}\ \{5, 7, 8\} $

Let $n$ be a positive integer. A regular hexagon with side length $n$ is divided into equilateral triangles with side length $1$ by lines parallel to its sides. Find the number of regular hexagons all of whose vertices are among the vertices of those equilateral triangles. [i]UK - Sahl Khan[/i]

Arrange the following four numbers from smallest to largest $ a \equal{} (10^{100})^{10}$, $ b \equal{} 10^{(10^{10})}$, $ c \equal{} 1000000!$, $ d \equal{} (100!)^{10}$

Given is a convex quadrilateral $ ABCD$ with $ \angle ADC\equal{}\angle BCD>90^{\circ}$. Let $ E$ be the point of intersection of the line $ AC$ with the parallel line to $ AD$ through $ B$ and $ F$ be the point of intersection of the line $ BD$ with the parallel line to $ BC$ through $ A$. Show that $ EF$ is parallel to $ CD$

A pile of $n$ pebbles is placed in a vertical column. This configuration is modified according to the following rules. A pebble can be moved if it is at the top of a column which contains at least two more pebbles than the column immediately to its right. (If there are no pebbles to the right, think of this as a column with 0 pebbles.) At each stage, choose a pebble from among those that can be moved (if there are any) and place it at the top of the column to its right. If no pebbles can be moved, the configuration is called a [i]final configuration[/i]. For each $n$, show that, no matter what choices are made at each stage, the final configuration obtained is unique. Describe that configuration in terms of $n$. [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=119189]IMO ShortList 2001, combinatorics problem 7, alternative[/url]

Let $p$ be a prime number. Find all positive integers $a,b,c\ge 1$ such that: \[a^p+b^p=p^c.\]

Prove that the number $2^9 +2^{99}$ is divisible by $100$.

1.Let $ x \equal{} \alpha ,\ \beta \ (\alpha < \beta )$ are $ x$ coordinates of the intersection points of a parabola $ y \equal{} ax^2 \plus{} bx \plus{} c\ (a\neq 0)$ and the line $ y \equal{} ux \plus{} v$. Prove that the area of the region bounded by these graphs is $ \boxed{\frac {|a|}{6}(\beta \minus{} \alpha )^3}$. 2. Let $ x \equal{} \alpha ,\ \beta \ (\alpha < \beta )$ are $ x$ coordinates of the intersection points of parabolas $ y \equal{} ax^2 \plus{} bx \plus{} c$ and $ y \equal{} px^2 \plus{} qx \plus{} r\ (ap\neq 0)$. Prove that the area of the region bounded by these graphs is $ \boxed{\frac {|a \minus{} p|}{6}(\beta \minus{} \alpha )^3}$.

There is a safe that can be opened by entering a secret code consisting of $n$ digits, each of them is $0$ or $1$. Initially, $n$ zeros were entered, and the safe is closed (so, all zeros is not the secret code). In one attempt, you can enter an arbitrary sequence of $n$ digits, each of them is $0$ or $1$. If the entered sequence matches the secret code, the safe will open. If the entered sequence matches the secret code in more positions than the previously entered sequence, you will hear a click. In any other cases the safe will remain locked and there will be no click. Find the smallest number of attempts that is sufficient to open the safe in all cases.

(F.Nilov, A.Zaslavsky) Let $ CC_0$ be a median of triangle $ ABC$; the perpendicular bisectors to $ AC$ and $ BC$ intersect $ CC_0$ in points $ A'$, $ B'$; $ C_1$ is the meet of lines $ AA'$ and $ BB'$. Prove that $ \angle C_1CA \equal{} \angle C_0CB$.

A circle is circumscribed around an isosceles triangle whose two base angles are equal to $x^{\circ}$. Two points are chosen independently and randomly on the circle, and a chord is drawn between them. The probability that the chord intersects the triangle is $\frac{14}{25}.$ Find the sum of the largest and smallest possible value of $x$.

How many ten digit positive integers with distinct digits are multiples of $11111$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1264 \qquad\textbf{(C)}\ 2842 \qquad\textbf{(D)}\ 3456 \qquad\textbf{(E)}\ 11111 $

The summation $\sum_{k=1}^{360} \frac{1}{k \sqrt{k+1} + (k+1)\sqrt{k}}$ is the ratio of two relatively prime positive integers $m$ and $n$. Find $m + n$.

Consider the triangle $ABC$ with $\angle A= 90^{\circ}$ and $\angle B \not= \angle C$. A circle $\mathcal{C}(O,R)$ passes through $B$ and $C$ and intersects the sides $AB$ and $AC$ at $D$ and $E$, respectively. Let $S$ be the foot of the perpendicular from $A$ to $BC$ and let $K$ be the intersection point of $AS$ with the segment $DE$. If $M$ is the midpoint of $BC$, prove that $AKOM$ is a parallelogram.