MathDB

p1. Trung took five tests this semester. For his first three tests, his average was

60

, and for the fourth test he earned a

50

. What must he have earned on his fifth test if his final average for all five tests was exactly

60

?
p2. Find the number of pairs of integers

(a, b)

such that

20a + 16b = 2016 - ab

.
p3. Let

f : N \to N

be a strictly increasing function with

f(1) = 2016

and

f(2t) = f(t) + t

for all

t \in N

. Find

f(2016)

.
p4. Circles of radius

7

7

18

, and

r

are mutually externally tangent, where

r = \frac{m}{n}

for relatively prime positive integers

m

and

n

. Find

m + n

.
p5. A point is chosen at random from within the circumcircle of a triangle with angles

45^o

75^o

60^o

. What is the probability that the point is closer to the vertex with an angle of

45^o

than either of the two other vertices?
p6. Find the largest positive integer

a

less than

100

such that for some positive integer

b

a - b

is a prime number and

ab

is a perfect square.
p7. There is a set of

6

parallel lines and another set of six parallel lines, where these two sets of lines are not parallel with each other. If Blythe adds

6

more lines, not necessarily parallel with each other, find the maximum number of triangles that could be made.
p8. Triangle

ABC

has sides

AB = 5

AC = 4

, and

BC = 3

. Let

O

be any arbitrary point inside

ABC

, and

D \in BC

E \in AC

F \in AB

, such that

OD \perp BC

OE \perp AC

OF \perp AB

. Find the minimum value of

OD^2 + OE^2 + OF^2

.
p9. Find the root with the largest real part to

x^4-3x^3+3x+1 = 0

over the complex numbers.
p10. Tony has a board with

2

rows and

4

columns. Tony will use

8

numbers from

1

8

to fill in this board, each number in exactly one entry. Let array

(a_1,..., a_4)

be the first row of the board and array

(b_1,..., b_4)

be the second row of the board. Let

F =\sum^{4}_{i=1}|a_i - b_i|

, calculate the average value of

F

across all possible ways to fill in.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement