Given that there are $24$ primes between $3$ and $100$, inclusive, what is the number of ordered pairs $(p, a)$ with $p$ prime, $3 \le p < 100$, and $1 \le a < p$ such that $p \mid (a^{p-2} - a)$?

Emma's calculator has ten buttons: one for each digit $1, 2, \ldots, 9$, and one marked ``clear''. When Emma presses one of the buttons marked with a digit, that digit is appended to the right of the display. When she presses the ``clear'' button, the display is completely erased. If Emma starts with an empty display and presses five (not necessarily distinct) buttons at random, where all ten buttons have equal probability of being chosen, the expected value of the number produced is $\frac{m}{n}$, for relatively prime positive integers $m$ and $n$. Find $100m+n$. (Take an empty display to represent the number 0.) [i]Proposed by Michael Tang[/i]

Let $ABC$ be an acute triangle and let $A_{1}, B_{1}, C_{1}$ be the feet of its altitudes. The incircle of the triangle $A_{1}B_{1}C_{1}$ touches its sides at the points $A_{2}, B_{2}, C_{2}$. Prove that the Euler lines of triangles $ABC$ and $A_{2}B_{2}C_{2}$ coincide.

In the accompanying figure, segments $AB$ and $CD$ are parallel, the measure of angle $D$ is twice the measure of angle $B$, and the measures of segments $AB$ and $CD$ are $a$ and $b$ respectively. Then the measure of $AB$ is equal to $\textbf{(A) }\dfrac{1}{2}a+2b\qquad\textbf{(B) }\dfrac{3}{2}b+\dfrac{3}{4}a\qquad\textbf{(C) }2a-b\qquad\textbf{(D) }4b-\dfrac{1}{2}a\qquad \textbf{(E) }a+b$ [asy] size(175); defaultpen(linewidth(0.8)); real r=50, a=4,b=2.5,c=6.25; pair A=origin,B=c*dir(r),D=(a,0),C=shift(b*dir(r))*D; draw(A--B--C--D--cycle); label("$A$",A,SW); label("$B$",B,N); label("$C$",C,E); label("$D$",D,S); label("$a$",D/2,N); label("$b$",(C+D)/2,NW); //Credit to djmathman for the diagram[/asy]

In a blackboard there are $K$ circles in a row such that one of the numbers $1,...,K$ is assigned to each circle from the left to the right. Change of situation of a circle is to write in it or erase the number which is assigned to it.At the beginning no number is written in its own circle. For every positive divisor $d$ of $K$ ,$1\leq d\leq K$ we change the situation of the circles in which their assigned numbers are divisible by $d$,performing for each divisor $d$ $K$ changes of situation. Determine the value of $K$ for which the following holds;when this procedure is applied once for all positive divisors of $K$ ,then all numbers $1,2,3,...,K$ are written in the circles they were assigned in.

Prove that for each $n \ge 3$ there exist $n$ distinct positive divisors $d_1,d_2, ...,d_n$ of $n!$ such that $n! = d_1 +d_2 +...+d_n$.

Four consecutive natural numbers are divided into two groups of $2$ numbers. It is known that the product of numbers in one group is $1995$ greater than the product of numbers in another group. Find these numbers.

[b]T[/b]wo ordered pairs $(a,b)$ and $(c,d)$, where $a,b,c,d$ are real numbers, form a basis of the coordinate plane if $ad \neq bc$. Determine the number of ordered quadruples $(a,b,c)$ of integers between $1$ and $3$ inclusive for which $(a,b)$ and $(c,d)$ form a basis for the coordinate plane.

Find the 3-digit positive integer that has the most divisors.

What is the largest integer $n$ with no repeated digits that is relatively prime to $6$? Note that two numbers are considered relatively prime if they share no common factors besides $1$. [i]Proposed by Jacob Stavrianos[/i]

Let $\triangle ABC$ be a triangle and let $D$ be a point on side $\overline{BC}$. Let $U$ and $V$ be the circumcenters of triangles $\triangle ABD$ and $\triangle ADC$, respectively. Show, that $\triangle ABC$ and $\triangle AUV$ are similar. [i](41th Austrian Mathematical Olympiad, regional competition, problem 3)[/i]

Prove that the equation $x^{2006} - 4y^{2006} -2006 = 4y^{2007} + 2007y$ has no solution in the set of the positive integers.

Let $ABCD$ be a trapezoid with $AB \parallel CD.$ Point $X$ is placed on segment $BC$ such that $\angle{BAX} = \angle{XDC}.$ Given that $AB = 5, BX =3, CX =4,$ and $CD =12,$ compute $AX.$

Prove that for all positive integers $n$ there exists a single positive integer divisible with $5^n$ which in decimal base is written using $n$ digits from the set $\{1,2,3,4,5\}$.

A circle of radius 1 is tangent to a circle of radius 2. The sides of $ \triangle ABC$ are tangent to the circles as shown, and the sides $ \overline{AB}$ and $ \overline{AC}$ are congruent. What is the area of $ \triangle ABC$? [asy]defaultpen(black+linewidth(0.7)); size(7cm); real t=2^0.5; D((0,0)--(4*t,0)--(2*t,8)--cycle, black); D(CR((2*t,2),2), black); D(CR((2*t,5),1), black); dot(origin^^(4t,0)^^(2t,8)); label("B", (0,0), SW); label("C", (4*t,0), SE); label("A", (2*t,8), N); D((2*t,2)--(2*t,4), black); D((2*t,5)--(2*t,6), black); MP('2', (2*t,3), W); MP('1',(2*t, 5.5), W);[/asy] $ \textbf{(A) } \frac {35}2 \qquad \textbf{(B) } 15\sqrt {2} \qquad \textbf{(C) } \frac {64}3 \qquad \textbf{(D) } 16\sqrt {2} \qquad \textbf{(E) } 24$

Two given circles $\alpha$ and $\beta$ intersect each other at two points. Find the locus of the centers of all circles that are orthogonal to both $\alpha$ and $\beta$.

The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m,n)$ are there such that $1\leq m\leq 2012$ and \[5^n<2^m<2^{m+2}<5^{n+1}?\] $\textbf{(A) }278\qquad \textbf{(B) }279\qquad \textbf{(C) }280\qquad \textbf{(D) }281\qquad \textbf{(E) }282\qquad$

Let $\sqrt{23}>\frac{m}{n}$ where $ m,n$ are positive integers. i) Prove that $ \sqrt{23}>\frac{m}{n}\plus{}\frac{3}{mn}.$ ii) Prove that $ \sqrt{23}<\frac{m}{n}\plus{}\frac{4}{mn}$ occurs infinitely often, and give at least three such examples. [i]Dan Schwarz[/i]

Let $t$ be a line passing through the vertex $A$ of the equilateral $ABC$, parallel to the side $BC$. On the side $AC$ arbitrarily mark the point $D$. Bisector of the angle $ABD$ intersects the line $t$at the point $E$. Prove that $BD=CD+AE$.

Let $k\geq 0$ an integer. The sequence $a_0,\ a_1,\ a_2, \ a_3, \ldots$ is defined as follows: [LIST] [*] $a_0=k$ [/*] [*] For $n\geq 1$, we have that $a_n$ is the smallest integer greater than $a_{n-1}$ so that $a_n+a_{n-1}$ is a perfect square. [/*] [/LIST] Prove that there are exactly $\left \lfloor{\sqrt{2k}} \right \rfloor$ positive integers that cannot be written as the difference of two elements of such a sequence. [i]Note.[/i] If $x$ is a real number, $\left \lfloor{x} \right \rfloor$ denotes the greatest integer smaller or equal than $x$.

Let $D$ be a family of $s$-element subsets of $\{1.\ldots,n\}$ such that every $k$ members of $D$ have non-empty intersection. Denote by $D(n,s,k)$ the maximum cardinality of such a family. a) Find $D(n,s,4)$. b) Find $D(n,s,3)$.

Team Stanford has a $ \frac{1}{3}$ chance of winning any given math contest. If Stanford competes in 4 contests this quarter, what is the probability that the team will win at least once?

Sequence of integers $\{u_n\}_{n \in \mathbb{N}_0}$ is given as: $u_0=0$, $u_{2n}=u_n$, $u_{2n+1}=1-u_n$ for all $n \in \mathbb{N}_0$ $a)$ Find $u_{1998}$ $b)$ If $p$ is a positive integer and $m=(2^p-1)^2$, find $u_m$

Let $a_1<a_2<...<a_t$ be $t$ given positive integers where no three form an arithmetic progression. For $k=t,t+1,...$ define $a_{k+1}$ to be the smallest positive integer larger than $a_k$ satisfying the condition that no three of $a_1,a_2,...,a_{k+1}$ form an arithmetic progression. For any $x\in\mathbb{R}^+$ define $A(x)$ to be the number of terms in $\{a_i\}_{i\ge 1}$ that are at most $x$. Show that there exist $c>1$ and $K>0$ such that $A(x)\ge c\sqrt{x}$ for any $x>K$.

Let $ABCDA'B'C'D'$ be the rectangular parallelepiped. Let $M, N, P$ be midpoints of the edges $[AB], [BC],[BB']$ respectively . Let $\{O\} = A'N \cap C'M$. a) Prove that the points $D, O, P$ are collinear. b) Prove that $MC' \perp (A'PN)$ if and only if $ABCDA'B'C'D'$ is a cube.