MathDB

Round 5
p13. Suppose

\vartriangle ABC

is an isosceles triangle with

\overline{AB} = \overline{BC}

, and

X

is a point in the interior of

\vartriangle ABC

. If

m \angle ABC = 94^o

m\angle ABX = 17^o

, and

m\angle BAX = 13^o

, then what is

m\angle BXC

(in degrees)?
p14. Find the remainder when

\sum^{2019}_{n=1} 1 + 2n + 4n^2 + 8n^3

is divided by

2019

.
p15. How many ways can you assign the integers

1

through

10

to the variables

a, b, c, d, e, f, g, h, i

, and

j

in some order such that

a < b < c < d < e, f < g < h < i

a < g, b < h, c < i

f < b, g < c

, and

h < d

?
Round 6
p16. Call an integer

n

equi-powerful if

n

and

n^2

leave the same remainder when divided by 1320. How many integers between

1

and

1320

(inclusive) are equi-powerful?
p17. There exists a unique positive integer

j \le 10

and unique positive integers

n_j

n_{j+1}

...

n_{10}

such that

j \le n_j < n_{j+1} < ... < n_{10}

and

{n_{10} \choose 10}+ {n_9 \choose 9}+ ... + {n_j \choose j}= 2019.

Find

n_j + n_{j+1} + ... + n_{10}

.
p18. If

n

is a randomly chosen integer between

1

and

390

(inclusive), what is the probability that

26n

has more positive factors than

6n

?
Round 7
p19. Suppose

S

is an

n

-element subset of

\{1, 2, 3, ..., 2019\}

. What is the largest possible value of

n

such that for every

2 \le k \le n

k

divides exactly

n - 1

of the elements of

S

?
p20. For each positive integer

n

, let

f(n)

be the fewest number of terms needed to write

n

as a sum of factorials. For example,

f(28) = 3

because

4! + 2! + 2! = 28

and 28 cannot be written as the sum of fewer than

3

factorials. Evaluate

f(1) + f(2) + ... + f(720)

.
p21. Evaluate

\sum_{n=1}^{\infty}\frac{\phi (n)}{101^n-1}

, where

\phi (n)

is the number of positive integers less than or equal to n that are relatively prime to

n

.
PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2788993p24519281]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement