2. $k_1, k_2, k_3$ are three circles. $k_2$ and $k_3$ touch externally at $P$, $k_3$ and $k_1$ touch externally at $Q$, and $k_1$ and $k_2$ touch externally at $R$. The line $PQ$ meets $k_1$ again at $S$, the line $PR$ meets $k_1$ again at $T$. The line $RS$ meets $k_2$ again at $U$, and the line $QT$ meets $k_3$ again at $V$. Show that $P$, $U$, $V$ are collinear.
3. The edges of a cube are labeled from $1$ to $12$. Show that there must exist at least eight triples $(i, j, k)$ with $1 \leq i < j < k \leq 12$ so that the edges $i, j, k$ are consecutive edges of a path. Also show that there exists labeling in which we cannot find nine such triples.
4. $n, k$ are positive integers. $A_0$ is the set $\{1, 2, ... , n\}$. $A_i$ is a randomly chosen subset of $A_{i-1}$ (with each subset having equal probability). Show that the expected number of elements of $A_k$ is $\dfrac{n}{2^k}$.
5. Three circles of radius $a$ are drawn on the surface of a sphere of radius $r$. Each pair of circles touches externally and the three circles all lie in one hemisphere. Find the radius of a circle on the surface of the sphere which touches all three circles.
6. Show that for positive reals $a, b, c, d$ we have $\left(\dfrac{ab + ac + ad + bc + bd + cd}{6} \right)^3 \geq \left(\dfrac{abc + abd + acd + bcd}{4}\right)^2$
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