MathDB

Round 1
p1. A circle is inscribed inside a square such that the cube of the radius of the circle is numerically equal to the perimeter of the square. What is the area of the circle?
p2. If the coefficient of

z^ky^k

252

in the expression

(z + y)^{2k}

, find

k

.
p3. Let

f(x) = \frac{4x^4-2x^3-x^2-3x-2}{x^4-x^3+x^2-x-1}

be a function defined on the real numbers where the denominator is not zero. The graph of

f

has a horizontal asymptote. Compute the sum of the x-coordinates of the points where the graph of

f

intersects this horizontal asymptote. If the graph of f does not intersect the asymptote, write

0

.
Round 2
p4. How many

5

-digit numbers have strictly increasing digits? For example,

23789

has strictly increasing digits, but

23889

and

23869

do not.
p5. Let

y =\frac{1}{1 +\frac{1}{9 +\frac{1}{5 +\frac{1}{9 +\frac{1}{5 +...}}}}}

y

can be represented as

\frac{a\sqrt{b} + c}{d}

, where

b

is not divisible by any squares, and the greatest common divisor of

a

and

d

1

, find the sum

a + b + c + d

.
p6. “Counting” is defined as listing positive integers, each one greater than the previous, up to (and including) an integer

n

. In terms of

n

, write the number of ways to count to

n

.
Round 3
p7. Suppose

p

q

2p^2 + q^2

, and

p^2 + q^2

are all prime numbers. Find the sum of all possible values of

p

.
p8. Let

r(d)

be a function that reverses the digits of the

2

-digit integer

d

. What is the smallest

2

-digit positive integer

N

such that for some

2

-digit positive integer

n

and

2

-digit positive integer

r(n)

N

is divisible by

n

and

r(n)

, but not by

11

?
p9. What is the period of the function

y = (\sin(3\theta) + 6)^2 - 10(sin(3\theta) + 7) + 13

?
Round 4
p10. Three numbers

a, b, c

are given by

a = 2^2 (\sum_{i=0}^2 2^i)

b = 2^4(\sum_{i=0}^4 2^i)

, and

c = 2^6(\sum_{i=0}^6 2^i)

u, v, w

are the sum of the divisors of a, b, c respectively, yet excluding the original number itself. What is the value of

a + b + c -u - v - w

?
p11. Compute

\sqrt{6- \sqrt{11}} - \sqrt{6+ \sqrt{11}}

.
p12. Let

a_0, a_1,..., a_n

be such that

a_n\ne 0

and

(1 + x + x^3)^{341}(1 + 2x + x^2 + 2x^3 + 2x^4 + x^6)^{342} =\sum_{i=0}^n a_ix^i.

Find the number of odd numbers in the sequence

a_0, a_1,..., a_n

.

PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2781343p24424617]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Round 5
p13. How many ways can we form a group with an odd number of members (plural) from

99

people? Express your answer in the form

a^b + c

, where

a, b

, and

c

are integers and

a

is prime.
p14. A cube is inscibed in a right circular cone such that the ratio of the height of the cone to the radius is

2:1

. Compute the fraction of the cone’s volume that the cube occupies.
p15. Let

F_0 = 1

F_1 = 1

and

F_k = F_{k-1} + F_{k-2}

. Let

P(x) = \sum^{99}_{k=0} x^{F_k}

. The remainder when

P(x)

is divided by

x^3 - 1

can be expressed as

ax^2 + bx + c

. Find

2a + b

.
Round 6
p16. Ankit finds a quite peculiar deck of cards in that each card has n distinct symbols on it and any two cards chosen from the deck will have exactly one symbol in common. The cards are guaranteed to not have a certain symbol which is held in common with all the cards. Ankit decides to create a function f(n) which describes the maximum possible number of cards in a set given the previous constraints. What is the value of

f(10)

?
p17. If

|x| <\frac14

and

X = \sum^{\infty}_{N=0} \sum^{N}_{n=0} {N \choose n}x^{2n}(2x)^{N-n}.

then write

X

in terms of

x

without any summation or product symbols (and without an infinite number of ‘

+

’s, etc.).
p18. Dietrich is playing a game where he is given three numbers

a, b, c

which range from

[0, 3]

in a continuous uniform distribution. Dietrich wins the game if the maximum distance between any two numbers is no more than

1

. What is the probability Dietrich wins the game?
Round 7
p19. Consider f defined by

f(x) = x^6 + a_1x^5 + a_2x^4 + a_3x^3 + a_4x^2 + a_5x + a_6.

How many tuples of positive integers

(a_1, a_2, a_3, a_4, a_5, a_6)

exist such that

f(-1) = 12

and

f(1) = 30

?
p20. Let

a_n

be the number of permutations of the numbers

S = \{1, 2, ... , n\}

such that for all

k

with

1 \le k \le n

, the sum of

k

and the number in the

k

th position of the permutation is a power of

2

. Compute

a_1 + a_2 + a_4 + a_8 + ... + a_{1048576}

.
p21. A

4

-dimensional hypercube of edge length

1

is constructed in

4

-space with its edges parallel to the coordinate axes and one vertex at the origin. Its coordinates are given by all possible permutations of

(0, 0, 0, 0)

(1, 0, 0, 0)

(1, 1, 0, 0)

(1, 1, 1, 0)

, and

(1, 1, 1, 1)

. The

3

-dimensional hyperplane given by

x+y+z+w = 2

intersects the hypercube at

6

of its vertices. Compute the 3-dimensional volume of the solid formed by the intersection.
PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2781335p24424563]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

2014

Problems(2)