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AwesomeMath Admission Test Cover Sheet

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Admission Test A B C Check one

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AwesomeMath Test C

May 3 May 17, 2013

Do not be discouraged if you cannot solve all of the questions: the test is not made to be

easy. We want to see the solutions you come up with no matter how many problems yousolve.

Include all significant steps in your reasoning and computation. We are interested inyour ability to present your work, so unsupported answers will receive much less credit

than well-reasoned progress towards a solution without a correct answer.

In this document, you will find a cover sheet and an answer sheet. Print out each oneand make several copies of the blank answer sheet. Fill out the top of each answer sheet

as you go, and then fill out the cover sheet when you are finished. Start each problem ona new answer sheet.

All the work you present must be your own.

Do not be intimidated! Some of the problems involve complex mathematical ideas, but

all can be solved using only elementary techniques, admittedly combined in clever ways.

Be patient and persistent. Learning comes more from struggling with problems thanfrom solving them. Problem-solving becomes easier with experience. Success is not a

function of cleverness alone.

Submit your solutions by e-mail (preferred) by Friday, May 17, 2013.

Make sure that the cover sheet is the first page of your submission, and that it iscompletely filled out. Solutions are to be mailed to the following address:

Dr. Titu Andreescu

3425 Neiman RoadPlano TX 75025

If you e-mail your solutions, please send them to

tandreescu@gmail.com

E-mailed solutions may be written and scanned or typed in TeX. They should be sent as

an attachment in either .doc or .pdf format. If you write and scan your solutions, insert thescans into a .doc or .pdf file and send just the one file.

Please go to the next page for the problems

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Test C

May 3 May 17

1. Let pk be the kth prime number. Find the least n for which

(p21

+ 1)(p22

+ 1) . . . (p2n

+ 1)

is divisible by 106.

2. Solve the system of equations

x!y! = 6!y!z! = 7!

z!x! = 10!.

3. Let a, b, c be non-zero real numbers such that a + b + c = 0. Evaluate

a2

a2 (b c)2+

b2

b2 (c a)2+

c2

c2 (a b)2.

4. Prove that for any real numbers a and b,

(a2

b2

)2

ab(2a 3b)(3a 2b).

5. Find all primes p, q, r such that 7p3 q3 = r6.

6. Prove that for any odd integer n, 24 divides nn n.

7. Find all pairs (m, n) of positive integers such that m(n + 1) + n(m 1) = 2013.

8. Ifa, b, c are positive real numbers such that1

a+

1

b+

1

c=

2013

a + b + c, evaluate

1 +

a

b

1 +

b

c

1 +

c

a.

9. Write 1717 + 177 as a sum of two perfect squares.

10. Let a, b, c be non-zero real numbers, not all equal, such that1

a+

1

b+

1

c= 1 and

a3 + b3 + c3 = 3(a2 + b2 + c2). Prove that a + b + c = 3.

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AwesomeMath Answer Sheet

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Write neatly! All work should be inside the box. Do NOT write on the back of the page!