MathDB

p1. Prove that

x = 2

is the only real number satisfying

3^x + 4^x = 5^x

.
p2. Show that

\sqrt{9 + 4\sqrt5} -\sqrt{9 - 4\sqrt5}

is an integer.
p3. Two players

A

and

B

play a game on a round table. Each time they take turn placing a round coin on the table. The coin has a uniform size, and this size is at least

10

times smaller than the table size. They cannot place the coin on top of any part of other coins, and the whole coin must be on the table. If a player cannot place a coin, he loses. Suppose

A

starts first. If both of them plan their moves wisely, there will be one person who will always win. Determine whether

A

B

will win, and then determine his winning strategy.
p4. Suppose you are given

4

pegs arranged in a square on a board. A “move” consists of picking up a peg, reflecting it through any other peg, and placing it down on the board. For how many integers

1 \le n \le 2013

is it possible to arrange the

4

pegs into a larger square using exactly

n

moves? Justify your answers.
p5. Find smallest positive integer that has a remainder of

1

when divided by

2

, a remainder of

2

when divided by

3

, a remainder of

3

when divided by

5

, and a remainder of

5

when divided by

7

.
p6. Find the value of

\sum_{m|496,m>0} \frac{1}{m},

where

m|496

means

496

is divisible by

m

.
p7. What is the value of

{100 \choose 0}+{100 \choose 4}+{100 \choose 8}+ ... +{100 \choose 100}?

p8. An

n

-term sequence

a_0, a_1, ...,a_n

will be called sweet if, for each

0 \le i \le n -1

a_i

is the number of times that the number

i

appears in the sequence. For example,

1, 2, 1,0

is a sweet sequence with

4

terms. Given that

a_0

a_1

...

a_{2013}

is a sweet sequence, find the value of

a^2_0+ a^2_1+ ... + a^2_{2013}.

PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement