Let $f(x) = x^2 + 6x + 7$. Determine the smallest possible value of $f(f(f(f(x))))$ over all real numbers $x.$

Six unit disks $C_1$, $C_2$, $C_3$, $C_4$, $C_5$, $C_6$ are in the plane such that they don't intersect each other and $C_i$ is tangent to $C_{i+1}$ for $1 \le i \le 6$ (where $C_7 = C_1$). Let $C$ be the smallest circle that contains all six disks. Let $r$ be the smallest possible radius of $C$, and $R$ the largest possible radius. Find $R - r$.

Find the sum of all real numbers $x$ such that $5x^4-10x^3+10x^2-5x-11=0$.

34. Find the sum of the ages of everyone who wrote a problem for this year’s HMMT November contest. If your answer is $X$ and the actual value is $Y$ , your score will be $\text{max}(0, 20 - |X - Y|)$ 35. Find the total number of occurrences of the digits $0, 1 \ldots , 9$ in the entire guts round (the official copy). If your answer is $X$ and the actual value is $Y$ , your score will be $\text{max}(0, 20 - \frac{|X-Y|}{2})$ 36. Find the number of positive integers less than $1000000$ which are less than or equal to the sum of their proper divisors. If your answer is $X$ and the actual value is $Y$, your score will be $\text{max}(0, 20 - 80|1 - \frac{X}{Y}|)$ rounded to the nearest integer.

Let $S = \{a_1, \ldots, a_n \}$ be a finite set of positive integers of size $n \ge 1$, and let $T$ be the set of all positive integers that can be expressed as sums of perfect powers (including $1$) of distinct numbers in $S$, meaning \[ T = \left\{ \sum_{i=1}^n a_i^{e_i} \mid e_1, e_2, \dots, e_n \ge 0 \right\}. \] Show that there is a positive integer $N$ (only depending on $n$) such that $T$ contains no arithmetic progression of length $N$. [i]Yang Liu[/i]

Your math friend Steven rolls five fair icosahedral dice (each of which is labelled $1,2, \dots,20$ on its sides). He conceals the results but tells you that at least half the rolls are $20$. Suspicious, you examine the first two dice and find that they show $20$ and $19$ in that order. Assuming that Steven is truthful, what is the probability that all three remaining concealed dice show $20$?

Let $ABCDEF$ be a regular hexagon of area $1$. Let $M$ be the midpoint of $DE$. Let $X$ be the intersection of $AC$ and $BM$, let $Y$ be the intersection of $BF$ and $AM$, and let $Z$ be the intersection of $AC$ and $BF$. If $[P]$ denotes the area of a polygon $P$ for any polygon $P$ in the plane, evaluate $[BXC] + [AYF] + [ABZ] - [MXZY]$.

Find all ordered triples $(a,b,c)$ of positive reals that satisfy: $\lfloor a\rfloor bc=3,a\lfloor b\rfloor c=4$, and $ab\lfloor c\rfloor=5$, where $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$.