[u]Set 4[/u]


[b]p16.[/b] Albert, Beatrice, Corey, and Dora are playing a card game with two decks of cards numbered $1-50$ each. Albert, Beatrice, and Corey draw cards from the same deck without replacement, but Dora draws from the other deck. What is the probability that the value of Corey’s card is the highest value or is tied for the highest value of all $4$ drawn cards?


[b]p17.[/b] Suppose that $s$ is the sum of all positive values of $x$ that satisfy $2016\{x\} = x+[x]$. Find $\{s\}$. (Note: $[x]$ denotes the greatest integer less than or equal to $x$ and $\{x\}$ denotes $x - [x]$.)


[b]p18.[/b] Let $ABC$ be a triangle such that $AB = 41$, $BC = 52$, and $CA = 15$. Let H be the intersection of the $B$ altitude and $C$ altitude. Furthermore let $P$ be a point on $AH$. Both $P$ and $H$ are reflected over $BC$ to form $P'$ and $H'$ . If the area of triangle $P'H'C$ is $60$, compute $PH$.


[b]p19.[/b] A random integer $n$ is chosen between $1$ and $30$, inclusive. Then, a random positive divisor of $n, k$, is chosen. What is the probability that $k^2 > n$?


[b]p20.[/b] What are the last two digits of the value $3^{361}$?


[u]Set 5[/u]


[b]p21.[/b] Let $f(n)$ denote the number of ways a $3 \times n$ board can be completely tiled with $1 \times 3$ and $1 \times 4$ tiles, without overlap or any tiles hanging over the edge. The tiles may be rotated. Find $\sum^9_{i=0} f(i) = f(0) + f(1) + ... + f(8) + f(9)$. By convention, $f(0) = 1$.


[b]p22.[/b] Find the sum of all $5$-digit perfect squares whose digits are all distinct and come from the set $\{0, 2, 3, 5, 7, 8\}$.


[b]p23.[/b] Mary is flipping a fair coin. On average, how many flips would it take for Mary to get $4$ heads and $2$ tails?


[b]p24.[/b] A cylinder is formed by taking the unit circle on the $xy$-plane and extruding it to positive infinity. A plane with equation $z = 1 - x$ truncates the cylinder. As a result, there are three surfaces: a surface along the lateral side of the cylinder, an ellipse formed by the intersection of the plane and the cylinder, and the unit circle. What is the total surface area of the ellipse formed and the lateral surface? (The area of an ellipse with semi-major axis $a$ and semi-minor axis $b$ is $\pi ab$.)


[b]p25.[/b] Let the Blair numbers be defined as follows: $B_0 = 5$, $B_1 = 1$, and $B_n = B_{n-1} + B_{n-2}$ for all $n \ge 2$. Evaluate $$\sum_{i=0}^{\infty}
\frac{B_i}{51^i}= B_0 +\frac{B_1}{51} +\frac{B_2}{51^2} +\frac{B_3}{51^3} +...$$


[u]Estimation[/u]


[b]p26.[/b] Choose an integer between $1$ and $10$, inclusive. Your score will be the number you choose divided by the number of teams that chose your number.


[b]p27.[/b] $2016$ blind people each bring a hat to a party and leave their hat in a pile at the front door. As each partier leaves, they take a random hat from the ones remaining in a pile. Estimate the probability that at least $1$ person gets their own hat back.


[b]p28.[/b] Estimate how many lattice points lie within the graph of $|x^3| + |y^3| < 2016$.


[b]p29.[/b] Consider all ordered pairs of integers $(x, y)$ with $1 \le x, y \le 2016$. Estimate how many such ordered pairs are relatively prime.


[b]p30.[/b] Estimate how many times the letter “e” appears among all Guts Round questions.


PS. You should use hide for answers. First sets have been posted [url=https://artofproblemsolving.com/community/c3h2779594p24402189]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
To find the last two digits of \(3^{361}\), we can use Euler's theorem. Euler's theorem states that if \(a\) and \(n\) are coprime, then \(a^{\phi(n)} \equiv 1 \pmod{n}\), where \(\phi(n)\) is the Euler's totient function.

1. **Calculate \(\phi(100)\):**
   \[
   \phi(100) = \phi(2^2 \cdot 5^2) = 100 \left(1 - \frac{1}{2}\right) \left(1 - \frac{1}{5}\right) = 100 \cdot \frac{1}{2} \cdot \frac{4}{5} = 40
   \]

2. **Apply Euler's theorem:**
   Since \((3, 100) = 1\), we have:
   \[
   3^{40} \equiv 1 \pmod{100}
   \]

3. **Express \(3^{361}\) in terms of \(3^{40}\):**
   \[
   3^{361} = 3^{360 + 1} = (3^{40})^9 \cdot 3^1
   \]
   Using Euler's theorem:
   \[
   (3^{40})^9 \equiv 1^9 \equiv 1 \pmod{100}
   \]
   Therefore:
   \[
   3^{361} \equiv 1 \cdot 3 \equiv 3 \pmod{100}
   \]

Thus, the last two digits of \(3^{361}\) are \(03\).

The final answer is \(\boxed{03}\).