[b]p1.[/b] Trung took five tests this semester. For his first three tests, his average was $60$, and for the fourth test he earned a $50$. What must he have earned on his fifth test if his final average for all five tests was exactly $60$?


[b]p2.[/b] Find the number of pairs of integers $(a, b)$ such that $20a + 16b = 2016 - ab$.


[b]p3.[/b] Let $f : N \to N$ be a strictly increasing function with $f(1) = 2016$ and $f(2t) = f(t) + t$ for all $t \in N$. Find $f(2016)$.


[b]p4.[/b] Circles of radius $7$, $7$,  $18$, and $r$ are mutually externally tangent, where $r = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m + n$.


[b]p5.[/b] A point is chosen at random from within the circumcircle of a triangle with angles $45^o$, $75^o$, $60^o$. What is the probability that the point is closer to the vertex with an angle of $45^o$ than either of the two other vertices?


[b]p6.[/b] Find the largest positive integer $a$ less than $100$ such that for some positive integer $b$, $a - b$ is a prime number and $ab$ is a perfect square.


[b]p7.[/b] There is a set of $6$ parallel lines and another set of six parallel lines, where these two sets of lines are not parallel with each other. If Blythe adds $6$ more lines, not necessarily parallel with each other, find the maximum number of triangles that could be made.


[b]p8.[/b] Triangle $ABC$ has sides $AB = 5$, $AC = 4$, and $BC = 3$. Let $O$ be any arbitrary point inside $ABC$, and $D \in BC$, $E \in AC$, $F \in AB$, such that $OD \perp BC$, $OE \perp AC$, $OF \perp AB$. Find the minimum value of $OD^2 + OE^2 + OF^2$.


[b]p9.[/b] Find the root with the largest real part to $x^4-3x^3+3x+1 = 0$ over the complex numbers.


[b]p10.[/b] Tony has a board with $2$ rows and $4$ columns. Tony will use $8$ numbers from $1$ to $8$ to fill in this board, each number in exactly one entry. Let array $(a_1,..., a_4)$ be the first row of the board and array $(b_1,..., b_4)$ be the second row of the board. Let $F =\sum^{4}_{i=1}|a_i - b_i|$, calculate the average value of $F$ across all possible ways to fill in.


PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1. We are given that \( f : \mathbb{N} \to \mathbb{N} \) is a strictly increasing function with \( f(1) = 2016 \) and \( f(2t) = f(t) + t \) for all \( t \in \mathbb{N} \).
2. We need to find \( f(2016) \).

First, let's find a general form for \( f \) by examining the given properties.

### Step 1: Base Case
We know:
\[ f(1) = 2016 \]

### Step 2: Inductive Step
We need to find \( f(2^k) \) for \( k \in \mathbb{N} \).

Using the given functional equation \( f(2t) = f(t) + t \):
- For \( t = 1 \):
  \[ f(2) = f(1) + 1 = 2016 + 1 = 2017 \]
- For \( t = 2 \):
  \[ f(4) = f(2) + 2 = 2017 + 2 = 2019 \]
- For \( t = 4 \):
  \[ f(8) = f(4) + 4 = 2019 + 4 = 2023 \]

We observe a pattern. Let's generalize this:
\[ f(2^k) = f(2^{k-1}) + 2^{k-1} \]

By induction, we can prove:
\[ f(2^k) = 2^k + f(1) - 1 \]

### Step 3: General Form
Since \( f \) is strictly increasing and we have the form for powers of 2, we can generalize for any \( n \):
\[ f(n) = n + f(1) - 1 \]

### Step 4: Calculate \( f(2016) \)
Using the general form:
\[ f(n) = n + f(1) - 1 \]
\[ f(2016) = 2016 + 2016 - 1 = 4031 \]

The final answer is \( \boxed{4031} \)