Figure 1 is called a "stack map." The numbers tell how many cubes are stacked in each position. Fig. 2 shows these cubes, and Fig. 3 shows the view of the stacked cubes as seen from the front. Which of the following is the front view for the stack map in Fig. 4? [asy] unitsize(24); draw((0,0)--(2,0)--(2,2)--(0,2)--cycle); draw((1,0)--(1,2)); draw((0,1)--(2,1)); draw((5,0)--(7,0)--(7,1)--(20/3,4/3)--(20/3,13/3)--(19/3,14/3)--(16/3,14/3)--(16/3,11/3)--(13/3,11/3)--(13/3,2/3)--cycle); draw((20/3,13/3)--(17/3,13/3)--(17/3,10/3)--(14/3,10/3)--(14/3,1/3)); draw((20/3,10/3)--(17/3,10/3)--(17/3,7/3)--(20/3,7/3)); draw((17/3,7/3)--(14/3,7/3)); draw((7,1)--(6,1)--(6,2)--(5,2)--(5,0)); draw((5,1)--(6,1)--(6,0)); draw((20/3,4/3)--(6,4/3)); draw((17/3,13/3)--(16/3,14/3)); draw((17/3,10/3)--(16/3,11/3)); draw((14/3,10/3)--(13/3,11/3)); draw((5,2)--(13/3,8/3)); draw((5,1)--(13/3,5/3)); draw((6,2)--(17/3,7/3)); draw((9,0)--(11,0)--(11,4)--(10,4)--(10,3)--(9,3)--cycle); draw((11,3)--(10,3)--(10,0)); draw((11,2)--(9,2)); draw((11,1)--(9,1)); draw((13,0)--(16,0)--(16,2)--(13,2)--cycle); draw((13,1)--(16,1)); draw((14,0)--(14,2)); draw((15,0)--(15,2)); label("Figure 1",(1,0),S); label("Figure 2",(17/3,0),S); label("Figure 3",(10,0),S); label("Figure 4",(14.5,0),S); label("$1$",(1.5,.2),N); label("$2$",(.5,.2),N); label("$3$",(.5,1.2),N); label("$4$",(1.5,1.2),N); label("$1$",(13.5,.2),N); label("$3$",(14.5,.2),N); label("$1$",(15.5,.2),N); label("$2$",(13.5,1.2),N); label("$2$",(14.5,1.2),N); label("$4$",(15.5,1.2),N);[/asy] [asy] unitsize(18); draw((0,0)--(3,0)--(3,2)--(1,2)--(1,4)--(0,4)--cycle); draw((0,3)--(1,3)); draw((0,2)--(1,2)--(1,0)); draw((0,1)--(3,1)); draw((2,0)--(2,2)); draw((5,0)--(8,0)--(8,4)--(7,4)--(7,3)--(6,3)--(6,2)--(5,2)--cycle); draw((8,3)--(7,3)--(7,0)); draw((8,2)--(6,2)--(6,0)); draw((8,1)--(5,1)); draw((10,0)--(12,0)--(12,4)--(11,4)--(11,3)--(10,3)--cycle); draw((12,3)--(11,3)--(11,0)); draw((12,2)--(10,2)); draw((12,1)--(10,1)); draw((14,0)--(17,0)--(17,4)--(16,4)--(16,2)--(14,2)--cycle); draw((17,3)--(16,3)); draw((17,2)--(16,2)--(16,0)); draw((17,1)--(14,1)); draw((15,0)--(15,2)); draw((19,0)--(22,0)--(22,4)--(20,4)--(20,1)--(19,1)--cycle); draw((22,3)--(20,3)); draw((22,2)--(20,2)); draw((22,1)--(20,1)--(20,0)); draw((21,0)--(21,4)); label("(A)",(1.5,0),S); label("(B)",(6.5,0),S); label("(C)",(11,0),S); label("(D)",(15.5,0),S); label("(E)",(20.5,0),S);[/asy]

The product $$\frac12 \cdot \frac24 \cdot \frac38 \cdot \frac{4}{16} \cdot ... \cdot \frac{99}{2^{99}} \cdot \frac{100}{2^{100}}$$ is written in its most simplified form. What is the last digit of the denominator?

Let $ABC$ be a triangle. Take points $D$, $E$, $F$ on the perpendicular bisectors of $BC$, $CA$, $AB$ respectively. Show that the lines through $A$, $B$, $C$ perpendicular to $EF$, $FD$, $DE$ respectively are concurrent.

How many of the first ten numbers of the sequence $121$, $11211$, $1112111$, ... are prime numbers? $\textbf{(A) } 0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3 \qquad \textbf{(E) }4$

The side lengths $a,b,c$ of a triangle $ABC$ are integers with $\gcd(a,b,c)=1$. The bisector of angle $BAC$ meets $BC$ at $D$. (a) show that if triangles $DBA$ and $ABC$ are similar then $c$ is a square. (b) If $c=n^2$ is a square $(n\ge 2)$, find a triangle $ABC$ satisfying (a).

[b]a)[/b] Find all real numbers $a$ such that the parabola $y=x^2-a$ and the hyperbola $y=1/x$ intersect each other in three different points. [b]b)[/b] Find the locus of centers of circumcircles of such triples of intersection points when $a$ takes all possible values. [i](I. Gorodnin)[/i]

Determine all integer $n \ge 2$ such that it is possible to construct an $n * n$ array where each entry is either $-1, 0, 1$ so that the sums of elements in every row and every column are distinct

Mary thought of a positive two-digit number. She multiplied it by $3$ and added $11.$ Then she switched the digits of the result, obtaining a number between $71$ and $75$, inclusive. What was Mary's number? $\textbf{(A)} \text{ 11} \qquad \textbf{(B)} \text{ 12} \qquad \textbf{(C)} \text{ 13} \qquad \textbf{(D)} \text{ 14} \qquad \textbf{(E)} \text{ 15}$

Find all functions $f : Z_{>0} \to Z_{>0}$ for which $f(n) | f(m) - n$ if and only if $n | m$ for all natural numbers $m$ and $n$.

$ OA, OB, OC, OD$ are 4 rays in space such that the angle between any two is the same. Show that for a variable ray $ OX,$ the sum of the cosines of the angles $ XOA, XOB, XOC, XOD$ is constant and the sum of the squares of the cosines is also constant.

$m$ and $n$ are positive integers. Set $A=\{ 1, 2, \cdots, n \}$. Let set $B_{n}^{m}=\{ (a_1, a_2 \cdots, a_m) \mid a_i \in A, i= 1, 2, \cdots, m \}$ satisfying: (1) $|a_i - a_{i+1}| \neq n-1$, $i=1,2, \cdots, m-1$; and (2) at least three of $a_1, a_2, \cdots, a_m$ ($m \geq 3$) are pairwise distince. Find $|B_n^m|$ and $|B_6^3|$.

A graph $G$ has $n$ vertices ($n>1$). For each edge $e$ let $c(e)$ be the number of vertices of the largest complete subgraph containing $e$. Prove that the inequality (the summation is over all edges of $G$): \[\sum_{e} \frac{c(e)}{c(e)-1}\le \frac{n^2}{2}.\]

Let $a, b, c, d, e \in \mathbb{R}^+$ and $f:\{(x, y) \in (\mathbb{R}^+)^2|c-dx-ey > 0\}\to \mathbb{R}^+$ be given by $f(x, y) = (ax)(by)(c- dx- ey)$. Find the maximum value of $f$.

The positive numbers $a, b,c, d, p, q$ are such that $(x+a)(x+b)(x+c)(x+d) = x^4+4px^3+6x^2+4qx+1$ holds for all real numbers $x$. Find the smallest value of $p$ or the largest value of $q$.

Let $a,b,c,d$ be pairwise distinct positive integers such that $$\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+a}$$ is an integer. Prove that $a+b+c+d$ is [b]not[/b] a prime number.

Sammy and Tammy run laps around a circular track that has a radius of $1$ kilometer. They begin and end at the same point and at the same time. Sammy runs $3$ laps clockwise while Tammy runs $4$ laps counterclockwise. How many times during their run is the straight-line distance between Sammy and Tammy exactly $1$ kilometer? $\text{(A) }7\qquad\text{(B) }8\qquad\text{(C) }13\qquad\text{(D) }14\qquad\text{(E) }21$