Consider the following $2 \times 3$ arrangement of pegs on a board. Jane places three rubber bands on the pegs on the board such that the following conditions are satisfied: $~$ [center] (I) No two rubber bands cross each other. (II) Each peg has a rubber band wrapped around it [/center]$~$ How many distinct arrangements could Jane create exist? One acceptable arrangement is shown below. [asy] size(100); filldraw(circle((0,0),0.2),black); filldraw(circle((1,0),0.2),black); filldraw(circle((2,0),0.2),black); filldraw(circle((0,1),0.2),black); filldraw(circle((1,1),0.2),black); filldraw(circle((2,1),0.2),black); draw((0,1.2)--(1,1.2)); draw((0,0.8)--(1,0.8)); draw((1,0.2)--(2,0.2)); draw((1,-0.2)--(2,-0.2)); draw((0,0.2)--(2,1.2)); draw((0,-0.2)--(2,0.8)); [/asy] $\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }8$

Find the number of ordered pairs $(m, n)$ such that $m$ and $n$ are positive integers in the set $\{1, 2, ..., 30\}$ and the greatest common divisor of $2^m + 1$ and $2^n - 1$ is not $1.$

Let $\{a_n\}$ be a sequence satisfying: $a_1=1$ and $a_{n+1}=2a_n+n\cdot (1+2^n),(n=1,2,3,\cdots)$. Determine the general term formula of $\{a_n\}$.

In a quadrilateral $ABCD$ let $E$ be the intersection of the two diagonals, I the center of the parallelogram whose vertices are the midpoints of the four sides of the quadrilateral, and K the center of the parallelogram whose sides pass through the points. divide the four sides of the quadrilateral into three equal parts (see illustration ). [img]https://cdn.artofproblemsolving.com/attachments/1/c/8f2617103edd8361b8deebbee13c6180fa848b.png[/img] a) Prove that $\overrightarrow{EK} =\frac43 \overrightarrow{EI}$. b) Prove that $$\lambda_A \overrightarrow{KA} +\lambda_B \overrightarrow{KB} + \lambda_C \overrightarrow{KC} + \lambda_D \overrightarrow{KD} = \overrightarrow{0}$$ , where $$\lambda_A=1+\frac{S(ADB)}{S(ABCD)},\lambda_B=1+\frac{S(BCA)}{S(ABCD)},\lambda_C=1+\frac{S(CDB)}{S(ABCD)},\lambda_D=1+\frac{S(DAC)}{S(ABCD)}$$ , where $S$ is the area symbol.

A set $A$ contains exactly $n$ integers, each of which is greater than $1$ and every of their prime factors is less than $10$. Determine the smallest $n$ such that $A$ must contain at least two distinct elements $a$ and $b$ such that $ab$ is the square of an integer.

Let $p$ and $q$ be different prime numbers. Solve the following system in integers: \[\frac{z+ p}x+\frac{z-p}y= q,\\ \frac{z+ p}y -\frac{z-p}x= q.\]

On the plane, a Cartesian coordinate system is chosen. Given points $ A_1,A_2,A_3,A_4$ on the parabola $ y \equal{} x^2$, and points $ B_1,B_2,B_3,B_4$ on the parabola $ y \equal{} 2009x^2$. Points $ A_1,A_2,A_3,A_4$ are concyclic, and points $ A_i$ and $ B_i$ have equal abscissas for each $ i \equal{} 1,2,3,4$. Prove that points $ B_1,B_2,B_3,B_4$ are also concyclic.

Compute the remainder when $222!^{111} + 111^{222!} + 111!^{222} + 222^{111!}$ is divided by $2007$.

When Walter drove up to the gasoline pump, he noticed that his gasoline tank was $\frac{1}{8}$ full. He purchased $7.5$ gallons of gasoline for $ \$10$. With this additional gasoline, his gasoline tank was then $\frac{5}{8}$ full. The number of gallons of gasoline his tank holds when it is full is $\text{(A)}\ 8.75 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 11.5 \qquad \text{(D)}\ 15 \qquad \text{(E)}\ 22.5$

Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.

Let $N$ be a positive integer. Brothers Michael and Kylo each select a positive integer less than or equal to $N$, independently and uniformly at random. Let $p_N$ denote the probability that the product of these two integers has a units digit of $0$. The maximum possible value of $p_N$ over all possible choices of $N$ can be written as $\tfrac ab,$ where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$.

Each vertex of a regular $17$-gon is colored red, blue, or green in such a way that no two adjacent vertices have the same color. Call a triangle “multicolored” if its vertices are colored red, blue, and green, in some order. Prove that the $17$-gon can be cut along nonintersecting diagonals to form at least two multicolored triangles. (A diagonal of a polygon is a a line segment connecting two nonadjacent vertices. Diagonals are called nonintersecting if each pair of them either intersect in a vertex or do not intersect at all.)

Let $ AB$ be a diameter of a circle, $ C$ be any fixed point between $ A$ and $ B$ on this diameter, and $ Q$ be a variable point on the circumference of the circle. Let $ P$ be the point on the line determined by $ Q$ and $ C$ for which $ \frac{AC}{CB}\equal{}\frac{QC}{CP}$. Describe, with proof, the locus of the point $ P$.

In an acute-angled triangle $ABC$, $AB>AC$. $M,N$ are distinct points on side $BC$ such that $\angle BAM=\angle CAN$. Let $O_1,O_2$ be the circumcentres of $\triangle ABC, \triangle AMN$, respectively. Prove that $O_1,O_2,A$ are collinear.

In an acute-angled isosceles triangle $ABC$, altitudes $CC_1$ and $BB_1$ intersect the line passing through the vertex $A$ and parallel to the line $BC$, at points $P$ and $Q$. Let $A_0$ be the midpoint of side $BC$, and $AA_1$ the altitude. Lines $A_0C_1$ and $A_0B_1$ intersect line $PQ$ at points $K$ and $L$. Prove that the circles circumscribed around triangles $PQA_1, KLA_0, A_1B_1C_1$ and a circle with a diameter $AA_1$ intersect at one point.