Consider the shape formed from taking equilateral triangle $ABC$ with side length $6$ and tracing out the arc $BC$ with center $A$. Set the shape down on line $l$ so that segment $AB$ is perpendicular to $l$, and $B$ touches $l$. Beginning from arc $BC$ touching $l$, we roll $ABC$ along $l$ until both points $A$ and $C$ are on the line. The area traced out by the roll can be written in the form $n\pi$, where $n$ is an integer. Find $n$.

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 the equality $$\frac{2}{x^2-1}+\frac{4}{x^2-4} +\frac{6}{x^2-9}+...+\frac{20}{x^2-100} =\frac{11}{(x-1)(x+10)}+\frac{11}{(x-2)(x+9)}+...+\frac{11}{(x-10)(x+1)}$$

Find all positive integers $ n$ such that there exists sequence consisting of $ 1$ and $ - 1: a_{1},a_{2},\cdots,a_{n}$ satisfying $ a_{1}\cdot1^2 + a_{2}\cdot2^2 + \cdots + a_{n}\cdot n^2 = 0.$

Let $M$ be the set of palindromic integers of the form $5n+4$ where $n\ge 0$ is an integer. [list=a] [*]If we write the elements of $M$ in increasing order, what is the $50^{\text{th}}$ number? [*]Among all numbers in $M$ with nonzero digits which sum up to $2014$ which is the largest and smallest one?[/list]

The doctor instructed a person to take $48$ pills for next $30$ days. Every day he take at least $1$ pill and at most $6$ pills. Show that exist the numbers of conscutive days such that the total numbers of pills he take is equal with $11$.

Prove that the medians from the vertices $A$ and $B$ of a triangle $ABC$ are orthogonal if and only if $BC^2+AC^2=5AB^2$.

The faces of a cubical die are marked with the numbers $ 1$, $ 2$, $ 2$, $ 3$, $ 3$, and $ 4$. The faces of a second cubical die are marked with the numbers $ 1$, $ 3$, $ 4$, $ 5$, $ 6$, and $ 8$. Both dice are thrown. What is the probability that the sum of the two top numbers will be $ 5$, $ 7$, or $ 9$ ? $ \textbf{(A)}\ \frac {5}{18} \qquad \textbf{(B)}\ \frac {7}{18} \qquad \textbf{(C)}\ \frac {11}{18} \qquad \textbf{(D)}\ \frac {3}{4} \qquad \textbf{(E)}\ \frac {8}{9}$

Let $ABCD$ be a cyclic quadrilateral such that $AB$ is a diameter of it's circumcircle. Suppose that $AB$ and $CD$ intersect at $I$, $AD$ and $BC$ at $J$, $AC$ and $BD$ at $K$, and let $N$ be a point on $AB$. Show that $IK$ is perpendicular to $JN$ if and only if $N$ is the midpoint of $AB$.

Let $ABCD$ be a tetrahedron such that edges $AB$, $AC$, and $AD$ are mutually perpendicular. Let the areas of triangles $ABC$, $ACD$, and $ADB$ be denoted by $x$, $y$, and $z$, respectively. In terms of $x$, $y$, and $z$, find the area of triangle $BCD$.

Let $ABC$ be an acute-angled triangle and $P$ be a point in its interior. Let $P_A,P_B$ and $P_c$ be the images of $P$ under reflection in the sides $BC,CA$, and $AB$, respectively. If $P$ is the orthocentre of the triangle $P_AP_BP_C$ and if the largest angle of the triangle that can be formed by the line segments$ PA, PB$. and $PC$ is $x^o$, determine the value of $x$.

Find all pairs $(a,b)$ of positive integers such that $a!+b$ and $b!+a$ are both powers of $5$. [i]Nikola Velov, North Macedonia[/i]

For positive integers $n, k$ with $1 \le k \le n$, define $$L(n, k) = Lcm \,(n, n - 1, n -2, ..., n - k + 1)$$ Let $f(n)$ be the largest value of $k$ such that $L(n, 1) < L(n, 2) < ... < L(n, k)$. Prove that $f(n) < 3\sqrt{n}$ and $f(n) > k$ if $n > k! + k$.

[asy]size(150); draw((0,0)--(1,0)--(1,1.5)--(0,1.5)--cycle); draw((2,0)--(3,0)--(3,1.5)--(2,1.5)--cycle); draw((4,0)--(5,0)--(5,1.5)--(4,1.5)--cycle); draw((2,2.5)--(3,2.5)--(3,4)--(2,4)--cycle); draw((4,2.5)--(5,2.5)--(5,4)--(4,4)--cycle); label("3",(0.5,0.5),N); label("4",(2.5,0.5),N); label("6",(4.5,0.5),N); label("P",(2.5,3),N); label("Q",(4.5,3),N);[/asy] Five cards are lying on a table as shown. Each card has a letter on one side and a whole number on the other side. Jane said, "If a vowel is on one side of any card, then an even number is on the other side." Mary showed Jane was wrong by turning over one card. Which card did Mary turn over? \[ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ \text{P} \qquad \textbf{(E)}\ \text{Q} \qquad \]

At a party attended by $n$ married couples, each person talks to everyone else at the party except his or her spouse. The conversations involve sets of persons or cliques $C_1, C_2, \cdots, C_k$ with the following property: no couple are members of the same clique, but for every other pair of persons there is exactly one clique to which both members belong. Prove that if $n \geq 4$, then $k \geq 2n$. [i]Proposed by USA.[/i]

Determine all positive integers $a,b,c$ satisfying $a^{(b^c)}=(b^a)^c$