[b]p1.[/b] What is $\sqrt[2015]{2^01^5}$?


[b]p2.[/b] What is the ratio of the area of square $ABCD$ to the area of square $ACEF$?


[b]p3.[/b] $2015$ in binary is $11111011111$, which is a palindrome. What is the last year which also had this property?


[b]p4.[/b] What is the next number in the following geometric series: $1020100$, $10303010$, $104060401$?


[b]p5.[/b] A circle has radius $A$ and area $r$. If $A = r^2\pi$, then what is the diameter, $C$, of the circle?


[b]p6.[/b] If
$$O + N + E = 1$$
$$T + H + R + E + E = 3$$
$$N + I + N + E = 9$$
$$T + E + N = 10$$
$$T + H + I + R + T + E + E + N = 13$$
Then what is the value of $O$?


[b]p7.[/b] By shifting the initial digit, which is $6$, of the positive integer $N$ to the end (for example, $65$ becomes $56$), we obtain a number equal to $\frac{N}{4}$ . What is the smallest such $N$?


[b]p8.[/b] What is $\sqrt[3]{\frac{2015!(2013!)+2014!(2012!)}{2013!(2012!)}}$ ?


[b]p9.[/b] How many permutations of the digits of $1234$ are divisible by $11$?


[b]p10.[/b] If you choose $4$ cards from a normal $52$ card deck (with replacement), what is the probability that you will get exactly one of each suit (there are $4$ suits)?


[b]p11.[/b] If $LMT$ is an equilateral triangle, and $MATH$ is a square, such that point $A$ is in the triangle, then what is $HL/AL$?


[b]p12.[/b] If
$$\begin{tabular}{cccccccc}
 &  &  &  &  & L & H & S\\
+ &  &  &  & H & I & G & H \\
+ &  & S  & C & H & O & O & L \\
\hline
= &  & S  & O & C & O & O & L \\
\end{tabular}$$ and $\{M, A, T,H, S, L,O, G, I,C\} = \{0, 1, 2, 3,4, 5, 6, 7, 8, 9\} $, then what is the ordered pair $(M + A +T + H, [T + e + A +M])$ where $e$ is $2.718...$and $[n]$ is the greatest integer less than or equal to $n$ ?


[b]p13.[/b] There are $5$ marbles in a bag. One is red, one is blue, one is green, one is yellow, and the last is white. There are $4$ people who take turns reaching into the bag and drawing out a marble without replacement. If the marble they draw out is green, they get to draw another marble out of the bag. What is the probability that the $3$rd person to draw a marble gets the white marble?



[b]p14.[/b] Let a "palindromic product" be a product of numbers which is written the same when written back to front, including the multiplication signs. For example, $234 * 545 * 432$, $2 * 2 *2 *2$, and $14 * 41$ are palindromic products whereas $2 *14 * 4 * 12$, $567 * 567$, and $2* 2 * 3* 3 *2$ are not. 2015 can be written as a "palindromic product" in two ways, namely $13 * 5 * 31$ and $31 * 5 * 13$. How many ways can you write $2016$ as a palindromic product without using 1 as a factor?


[b]p15.[/b] Let a sequence be defined as $S_n = S_{n-1} + 2S_{n-2}$, and $S_1 = 3$ and $S_2 = 4$. What is $\sum_{n=1}^{\infty}\frac{S_n}{3^n}$ ?


[b]p16.[/b] Put the numbers $0-9$ in some order so that every $2$-digit substring creates a number which is either a multiple of $7$, or a power of $2$.


[b]p17.[/b] Evaluate 
$\dfrac{8+ \dfrac{8+ \dfrac{8+...}{3+...}}{3+ \dfrac{8+...}{3+...}}}{3+\dfrac{8+ \dfrac{8+...}{3+...}}{
3+ \dfrac{8+...}{3+...}}}$, assuming that it is a positive real number.


[b]p18.[/b] $4$ non-overlapping triangles, each of area $A$, are placed in a unit circle. What is the maximum value of $A$?


[b]p19.[/b] What is the sum of the reciprocals of all the (positive integer) factors of $120$ (including $1$ and $120$ itself).


[b]p20.[/b] How many ways can you choose $3$ distinct elements of $\{1, 2, 3,...,4000\}$ to make an increasing arithmetic series?


PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1. To find the last year before 2015 that has a binary representation which is a palindrome, we need to find the largest palindromic binary number less than \(11111011111_2\).

2. The next largest palindrome made out of only ones and zeroes is \(11110101111_2\).

3. We convert \(11110101111_2\) to decimal:
   \[
   11110101111_2 = 2^{10} + 2^9 + 2^8 + 2^7 + 2^5 + 2^3 + 2^2 + 2^1 + 2^0
   \]
   \[
   = 1024 + 512 + 256 + 128 + 32 + 8 + 4 + 2 + 1
   \]
   \[
   = 1967
   \]

The final answer is \(\boxed{1967}\).