
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
**p1.** What is  $\sqrt[2015]{2^01^5}$ ?**p2.** What is the ratio of the area of square  $ABCD$  to the area of square  $ACEF$ ?**p3.**  $2015$  in binary is  $11111011111$ , which is a palindrome. What is the last year which also had this property?**p4.** What is the next number in the following geometric series:  $1020100$ ,  $10303010$ ,  $104060401$ ?**p5.** A circle has radius  $A$  and area  $r$ . If  $A = r^2\pi$ , then what is the diameter,  $C$ , of the circle?**p6.** 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  $O$ ?**p7.** 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  $N$ ?**p8.** What is  $\sqrt[3]{\frac{2015!(2013!)+2014!(2012!)}{2013!(2012!)}}$  ?**p9.** How many permutations of the digits of  $1234$  are divisible by  $11$ ?**p10.** 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)?**p11.** If  $LMT$  is an equilateral triangle, and  $MATH$  is a square, such that point  $A$  is in the triangle, then what is  $HL/AL$ ?**p12.** 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  $n$  ?**p13.** 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?**p14.** 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?**p15.** Let a sequence be defined as  $S_n = S_{n-1} + 2S_{n-2}$ , and  $S_1 = 3$  and  $S_2 = 4$ . What is  $\sum_{n=1}^{\infty}\frac{S_n}{3^n}$  ?**p16.** 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  $2$ .**p17.** 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.**p18.**  $4$  non-overlapping triangles, each of area  $A$ , are placed in a unit circle. What is the maximum value of  $A$ ?**p19.** What is the sum of the reciprocals of all the (positive integer) factors of  $120$  (including  $1$  and  $120$  itself).**p20.** 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 [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

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