MathDB

p1. How many nonzero digits are in the number

(5^{94} + 5^{92})(2^{94} + 2^{92})

?
p2. Suppose

A

is a set of

2013

distinct positive integers such that the arithmetic mean of any subset of

A

is also an integer. Find an example of

A

.
p3. How many minutes until the smaller angle formed by the minute and hour hands on the face of a clock is congruent to the smaller angle between the hands at

5:15

pm? Round your answer to the nearest minute.
p4. Suppose

a

and

b

are positive real numbers,

a + b = 1

, and

1 +\frac{a^2 + 3b^2}{2ab}=\sqrt{4 +\frac{a}{b}+\frac{3b}{a}}.

Find

a

.
p5. Suppose

f(x) = \frac{e^x- 12e^{-x}}{ 2}

. Find all

x

such that

f(x) = 2

.
p6. Let

P_1

P_2

...

P_n

be points equally spaced on a unit circle. For how many integer

n \in \{2, 3, ... , 2013\}

is the product of all pairwise distances:

\prod_{1\le i<j\le n} P_iP_j

a rational number? Note that

\prod

means the product. For example,

\prod_{1\le i\le 3} i = 1\cdot 2 \cdot 3 = 6

.
p7. Determine the value

a

such that the following sum converges if and only if

r \in (-\infty, a)

\sum^{\infty}_{n=1}(\sqrt{n^4 + n^r} - n^2).

Note that

\sum^{\infty}_{n=1}\frac{1}{n^s}

converges if and only if

s > 1

.
p8. Find two pairs of positive integers

(a, b)

with

a > b

such that

a^2 + b^2 = 40501

.
p9. Consider a simplified memory-knowledge model. Suppose your total knowledge level the night before you went to a college was

100

units. Each day, when you woke up in the morning you forgot

1\%

of what you had learned. Then, by going to lectures, working on the homework, preparing for presentations, you had learned more and so your knowledge level went up by

10

units at the end of the day. According to this model, how long do you need to stay in college until you reach the knowledge level of exactly

1000

?
p10. Suppose

P(x) = 2x^8 + x^6 - x^4 +1

, and that

P

has roots

a_1

a_2

...

a_8

(a complex number

z

is a root of the polynomial

P(x)

P(z) = 0

). Find the value of

(a^2_1-2)(a^2_2-2)(a^2_3-2)...(a^2_8-2).

p11. Find all values of

x

satisfying

(x^2 + 2x-5)^2 = -2x^2 - 3x + 15

.
p12. Suppose

x, y

and

z

are positive real numbers such that

x^2 + y^2 + xy = 9,

y^2 + z^2 + yz = 16,

x^2 + z^2 + xz = 25.

Find

xy + yz + xz

(the answer is unique).
p13. Suppose that

P(x)

is a monic polynomial (i.e, the leading coefficient is

1

) with

20

roots, each distinct and of the form

\frac{1}{3^k}

for

k = 0,1,2,..., 19

. Find the coefficient of

x^{18}

P(x)

.
p14. Find the sum of the reciprocals of all perfect squares whose prime factorization contains only powers of

3

5

7

(i.e.

\frac{1}{1} + \frac{1}{9} + \frac{1}{25} + \frac{1}{419} + \frac{1}{811} + \frac{1}{215} + \frac{1}{441} + \frac{1}{625} + ...

).
p15. Find the number of integer quadruples

(a, b, c, d)

which also satisfy the following system of equations:

1+b + c^2 + d^3 =0,

a + b^2 + c^3 + d^4 =0,

a^2 + b^3 + c^4 + d^5 =0,

a^3+b^4+c^5+d^6 =0.

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

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