MathDB

p1. Compute the value of

N

, where

N = 818^3 - 6 \cdot 818^2 \cdot 209 + 12 \cdot 818 \cdot 209^2 - 8 \cdot 209^3

p2. Suppose

x \le 2019

is a positive integer that is divisible by

2

and

5

, but not

3

. If

7

is one of the digits in

x

, how many possible values of

x

are there?
p3. Find all non-negative integer solutions

(a,b)

to the equation

b^2 + b + 1 = a^2.

p4. Compute the remainder when

\sum^{2019}_{n=1} n^4

is divided by

53

.
p5. Let

ABC

be an equilateral triangle and

CDEF

a square such that

E

lies on segment

AB

and

F

on segment

BC

. If the perimeter of the square is equal to

4

, what is the area of triangle

ABC

? https://cdn.artofproblemsolving.com/attachments/1/6/52d9ef7032c2fadd4f97d7c0ea051b3766b584.png
p6.

S = \frac{4}{1\times 2\times 3}+\frac{5}{2\times 3\times 4} +\frac{6}{3\times 4\times 5}+ ... +\frac{101}{98\times 99\times 100}

Let

T = \frac54 - S

. If

T = \frac{m}{n}

, where

m

and

n

are relatively prime integers, find the value of

m + n

.
p7. Find the sum of

\sum^{2019}_{i=0}\frac{2^i}{2^i + 2^{2019-i}}

p8. Let

A

and

B

be two points in the Cartesian plane such that

A

lies on the line

y = 12

, and

B

lies on the line

y = 3

. Let

C_1

C_2

be two distinct circles that intersect both

A

and

B

and are tangent to the

x

-axis at

P

and

Q

, respectively. If

PQ = 420

, determine the length of

AB

.
p9. Zion has an average

2

out of

3

hit rate for

2

-pointers and

1

out of

3

hit rate for

3

-pointers. In a recent basketball match, Zion scored

18

points without missing a shot, and all the points came from

2

3

-pointers. What is the probability that all his shots were

3

-pointers?
p10. Let

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

. Find the number of non-constant functions

f : S \to S

such that

f(k) = f(f(k + 1)) \le f(k + 1) \,\,\,\, for \,\,\,\, all \,\,\,\, 1 \le k \le 2018.

Express your answer in the form

{m \choose n}

, where

m

and

n

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

Problem Statement