A rubber band of negligible thickness encloses three pegs that lie in a perfect line, as shown. Each peg has a diameter of $4$ cm, as shown. What is the length of the rubber band used, in centimeters? All pegs shown are congruent circles. [asy] size(120); draw(circle((0,0),1));draw(circle((0,2),1));draw(circle((0,4),1)); dot((0,0)^^(0,2)^^(0,4)); draw((-1,0)--(-1,4)--arc((0,4),1,180,0)--(1,4)--(1,0)--arc((0,0),1,360,180),linewidth(2)); draw((-1,0)--(1,0),dotted); label("$4$ cm", (-0.38,0)--(1,0), N); [/asy] $\textbf{(A) }8\qquad\textbf{(B) }8+4\pi\qquad\textbf{(C) }16+4\pi\qquad\textbf{(D) }16+8\pi\qquad\textbf{(E) }16\pi$

Let $ABCDA'B'C'D'$ be a cube (where $ABCD$ is a square and $AA'\parallel BB'\parallel CC'\parallel DD'$). Furthermore, let $\mathcal R$ be a rotation (with respect some line) that maps vertex $A$ to $B.$ Find the set of all images $X=\mathcal R(C)$ such that $X$ lies on the surface of the cube for some rotation $\mathcal R(A)=B.$

Opposite sides of a regular hexagon are $12$ inches apart. The length of each side, in inches, is $\textbf{(A) }7.5\qquad\textbf{(B) }6\sqrt{2}\qquad\textbf{(C) }5\sqrt{2}\qquad\textbf{(D) }\frac{9}{2}\sqrt{3}\qquad \textbf{(E) }4\sqrt{3}$

A number $x_n$ of the form 10101...1 has $n$ ones. Find all $n$ such that $x_n$ is prime.

Let $n$ $\geq$ $2$ and $k$ $\geq$ $2$ be positive integers. A cat and a mouse are playing [i]Wim[/i], which is a stone removal game. The game starts with $n$ stones and they take turns removing stones, with the cat going first. On each turn they are allowed to remove $1$, $2$, $\dotsb$, or $k$ stones, and the player who cannot remove any stones on their turn loses. \\\\ A raccoon finds Wim very boring and creates [i]Wim 2[/i], which is Wim but with the following additional rule: [i]You cannot remove the same number of stones that your opponent removed on the previous turn[/i]. \\\\Find all values of $k$ such that for every $n$, the cat has a winning strategy in Wim if and only if it has a winning strategy in Wim 2.

For integers a, b, c, and d the polynomial $p(x) =$ $ax^3 + bx^2 + cx + d$ satisfies $p(5) + p(25) = 1906$. Find the minimum possible value for $|p(15)|$.

Find all functions $f : R \to R$ such that, for all real numbers $x, y,$ $$(x - y)(f(x) - f(y)) = f(x - f(y))f(f(x) - y).$$

Let $\mathbb{R}^{+}$ be the set of positive real numbers. Determine all non-negative real number $\alpha$ such that there exist a function $f:\mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that $$f(x^{\alpha}+y)=(f(x+y))^{\alpha}+f(y)$$ for any $x,y$ positive real numbers.

On the side $AB$ of the parallelogram $ABCD$ we take the points $X$ and $Y$ such that the points $A$, $X$, $Y$ and $B$ appear in this order. The lines $DX$ and $CY$ intersect at the point $Z$. Suppose that the area of the triangle $\triangle XYZ$ is equal to the sum of the areas of the triangles $\triangle AXD$ and $\triangle CYB$. Prove that the area of the quadrilateral $XYCD$ is equal to $3$ times the area of the triangle $\triangle XYZ$.

All faces of the tetrahedron $ABCD $ are acute-angled triangles.$AK$ and $AL$ -are altitudes in faces $ABC$ and $ABD$. Points $C,D,K,L$ lies on circle. Prove, that $AB \perp CD$

Prove that for positive integers $ x,y,z$ the number $ x^2 \plus{} y^2 \plus{} z^2$ is not divisible by $ 3(xy \plus{} yz \plus{} zx)$.

The triangle $ABC$ has perimeter $36$, and the length of $BC$ is $9$. Point $M$ is the midpoint of $AC$, and $I$ is the incenter. Find the angle $MIC$.

The circumference of a circle is divided into $45$ arcs, each of length $1.$ Initially, there are $15$ snakes, each of length $1,$ occupying every third arc. Every second, each snake independently moves either one arc left or one arc right, each with probability $\tfrac{1}{2}.$ If two snakes ever touch, they merge to form a single snake occupying the arcs of both of the previous snakes, and the merged snake moves as one snake. Compute the expected number of seconds until there is only one snake left.

A set $S$ is called [i]neighbouring [/i] if it has the following two properties: a) $S$ has exactly four elements b) for every element $x$ of $S$, at least one of the numbers $x - 1$ or $x+1$ belongs to $S$. Find the number of all [i]neighbouring [/i] subsets of the set $\{1,2,... ,n\}$.

Let $ABCD$ be a parallelogram with the angle at $A$ obtuse. Let $P$ be a point on segment $BD$. The circle with center $P$ passing through $A$ cuts line $AD$ at $A$ and $Y$ and cuts line $AB$ at $A$ and $X$. Line $AP$ intersects $BC$ at $Q$ and $CD$ at $R$. Prove $\angle XPY = \angle XQY + \angle XRY$.

Chandra pays an on-line service provider a fixed monthly fee plus an hourly charge for connect time. Her December bill was $\$12.48$, but in January her bill was $\$17.54$ because she used twice as much connect time as in December. What is the fixed monthly fee? $\mathrm{(A)} \$2.53 \qquad\mathrm{(B)} \$5.06 \qquad\mathrm{(C)} \$6.24 \qquad\mathrm{(D)} \$7.42 \qquad\mathrm{(E)} \$8.77$