
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
*20 problems for 20 minutes.***p1.** Euclid eats  $\frac17$  of a pie in  $7$  seconds. Euler eats  $\frac15$  of an identical pie in  $10$  seconds. Who eats faster?**p2.** Given that  $\pi = 3.1415926...$  , compute the circumference of a circle of radius 1. Express your answer as a decimal rounded to the nearest hundred thousandth (i.e.  $1.234562$  and  $1.234567$  would be rounded to  $1.23456$  and  $1.23457$ , respectively).**p3.** Alice bikes to Wonderland, which is  $6$  miles from her house. Her bicycle has two wheels, and she also keeps a spare tire with her. If each of the three tires must be used for the same number of miles, for how many miles will each tire be used?**p4.** Simplify  $\frac{2010 \cdot 2010}{2011}$  to a mixed number. (For example,  $2\frac12$  is a mixed number while  $\frac52$  and  $2.5$  are not.)**p5.** There are currently  $175$  problems submitted for  $EMC^2$ . Chris has submitted  $51$  of them. If nobody else submits any more problems, how many more problems must Chris submit so that he has submitted  $\frac13$  of the problems?**p6.** As shown in the diagram below, points  $D$  and  $L$  are located on segment  $AK$ , with  $D$  between  $A$  and  $L$ , such that  $\frac{AD}{DK}=\frac{1}{3}$  and  $\frac{DL}{LK}=\frac{5}{9}$ . What is  $\frac{DL}{AK}$ ?
![Image](https://cdn.artofproblemsolving.com/attachments/9/a/3f92bd33ffbe52a735158f7ebca79c4c360d30.png)**p7.** Find the number of possible ways to order the letters  $G, G, e, e, e$  such that two neighboring letters are never  $G$  and  $e$  in that order.**p8.** Find the number of odd composite integers between  $0$  and  $50$ .**p9.** Bob tries to remember his  $2$ -digit extension number. He knows that the number is divisible by  $5$  and that the first digit is odd. How many possibilities are there for this number?**p10.** Al walks  $1$  mile due north, then  $2$  miles due east, then  $3$  miles due south, and then  $4$  miles due west. How far, in miles, is he from his starting position? (Assume that the Earth is flat.)**p11.** When n is a positive integer,  $n!$  denotes the product of the first  $n$  positive integers; that is,  $n! = 1 \cdot 2  \cdot 3 \cdot ...  \cdot n$ . Given that  $7! = 5040$ , compute  $8! + 9! + 10!$ .**p12.** Sam's phone company charges him a per-minute charge as well as a connection fee (which is the same for every call) every time he makes a phone call. If Sam was charged  $\$ 4.88 $ for an $ 11 $-minute call and $ \ $6.00$  for a  $19$ -minute call, how much would he be charged for a  $15$ -minute call?**p13.** For a positive integer  $n$ , let  $s_n$  be the sum of the n smallest primes. Find the least  $n$  such that  $s_n$  is a perfect square (the square of an integer).**p14.** Find the remainder when  $2011^{2011}$  is divided by  $7$ .**p15.** Let  $a, b, c$ , and  $d$  be  $4$  positive integers, each of which is less than  $10$ , and let  $e$  be their least common multiple. Find the maximum possible value of  $e$ .**p16.** Evaluate  $100 - 1 + 99 - 2 + 98 - 3 + ... + 52 - 49 + 51 - 50$ .**p17.** There are  $30$  basketball teams in the Phillips Exeter Dorm Basketball League. In how ways can  $4$  teams be chosen for a tournament if the two teams Soule Internationals and Abbot United cannot be chosen at the same time?**p18.** The numbers  $1, 2, 3, 4, 5, 6$  are randomly written around a circle. What is the probability that there are four neighboring numbers such that the sum of the middle two numbers is less than the sum of the other two?**p19.** What is the largest positive  $2$ -digit factor of  $3^{2^{2011}} - 2^{2^{2011}}$ ?**p20.** Rhombus  $ABCD$  has vertices  $A = (-12,-4)$ ,  $B = (6, b)$ ,  $C = (c,-4)$  and  $D = (d,-28)$ , where  $b$ ,  $c$ , and  $d$  are integers. Find a constant  $m$  such that the line y =  $mx$  divides the rhombus into two regions of equal area.


PS. You should use hide for answers. Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

Present the answer in LaTex format: \boxed{Your answer}
user