MathDB

Round 5
p13. Let

\{a\} _{n\ge 1}

be an arithmetic sequence, with

a_ 1 = 0

, such that for some positive integers

k

and

x

we have

a_{k+1} = {k \choose x}

a_{2k+1} ={k \choose {x+1}}

, and

a_{3k+1} ={k \choose {x+2}}

. Let

\{b\}_{n\ge 1}

be an arithmetic sequence of integers with

b_1 = 0

. Given that there is some integer

m

such that

b_m ={k \choose x}

, what is the number of possible values of

b_2

?
p14. Let

A = arcsin \left( \frac14 \right)

and

B = arcsin \left( \frac17 \right)

. Find

\sin(A + B) \sin(A - B)

.
p15. Let

\{f_i\}^{9}_{i=1}

be a sequence of continuous functions such that

f_i : R \to Z

is continuous (i.e. each

f_i

maps from the real numbers to the integers). Also, for all

i

f_i(i) = 3^i

. Compute

\sum^{9}_{k=1} f_k \circ f_{k-1} \circ ... \circ f_1(3^{-k})

.
Round 6
p16. If

x

and

y

are integers for which

\frac{10x^3 + 10x^2y + xy^3 + y^4}{203}= 1134341

and

x - y = 1

, then compute

x + y

.
p17. Let

T_n

be the number of ways that n letters from the set

\{a, b, c, d\}

can be arranged in a line (some letters may be repeated, and not every letter must be used) so that the letter a occurs an odd number of times. Compute the sum

T_5 + T_6

.
p18. McDonald plays a game with a standard deck of

52

cards and a collection of chips numbered

1

52

. He picks

1

card from a fully shuffled deck and

1

chip from a bucket, and his score is the product of the numbers on card and on the chip. In order to win, McDonald must obtain a score that is a positive multiple of

6

. If he wins, the game ends; if he loses, he eats a burger, replaces the card and chip, shuffles the deck, mixes the chips, and replays his turn. The probability that he wins on his third turn can be written in the form

\frac{x^2 \cdot y}{z^3}

such that

x, y

, and

z

are relatively prime positive integers. What is

x + y + z

?
(NOTE: Use Ace as

1

, Jack as

11

, Queen as

12

, and King as

13

)
Round 7
p19. Let

f_n(x) = ln(1 + x^{2^n}+ x^{2^{n+1}}+ x^{3\cdot 2^n})

. Compute

\sum^{\infty}_{k=0} f_{2k} \left( \frac12 \right)

.
p20.

ABCD

is a quadrilateral with

AB = 183

BC = 300

CD = 55

DA = 244

, and

BD = 305

. Find

AC

.
p21. Define

\overline{xyz(t + 1)} = 1000x + 100y + 10z + t + 1

as the decimal representation of a four digit integer. You are given that

3^x5^y7^z2^t = \overline{xyz(t + 1)}

where

x, y, z

, and t are non-negative integers such that

t

is odd and

0 \le x, y, z,(t + 1) \le 9

. Compute

3^x5^y7^z

PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2782822p24445934]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement