MathDB

p1. What is

20\times 20 - 19\times 19

?
p2. Andover has a total of

1440

students and teachers as well as a

1 : 5

teacher-to-student ratio (for every teacher, there are exactly

5

students). In addition, every student is either a boarding student or a day student, and

70\%

of the students are boarding students. How many day students does Andover have?
p3. The time is

2:20

. If the acute angle between the hour hand and the minute hand of the clock measures

x

degrees, find

x

. https://cdn.artofproblemsolving.com/attachments/b/a/a18b089ae016b15580ec464c3e813d5cb57569.png
p4. Point

P

is located on segment

AC

of square

ABCD

with side length

10

such that

AP >CP

. If the area of quadrilateral

ABPD

70

, what is the area of

\vartriangle PBD

?
p5. Andrew always sweetens his tea with sugar, and he likes a

1 : 7

sugar-to-unsweetened tea ratio. One day, he makes a

100

ml cup of unsweetened tea but realizes that he has run out of sugar. Andrew decides to borrow his sister's jug of pre-made SUPERSWEET tea, which has a

1 : 2

sugar-to-unsweetened tea ratio. How much SUPERSWEET tea, in ml,does Andrew need to add to his unsweetened tea so that the resulting tea is his desired sweetness?
p6. Jeremy the architect has built a railroad track across the equator of his spherical home planet which has a radius of exactly

2020

meters. He wants to raise the entire track

6

meters off the ground, everywhere around the planet. In order to do this, he must buymore track, which comes from his supplier in bundles of

2

meters. What is the minimum number of bundles he must purchase? Assume the railroad track was originally built on the ground.
p7. Mr. DoBa writes the numbers

1, 2, 3,..., 20

on the board. Will then walks up to the board, chooses two of the numbers, and erases them from the board. Mr. DoBa remarks that the average of the remaining

18

numbers is exactly

11

. What is the maximum possible value of the larger of the two numbers that Will erased?
p8. Nathan is thinking of a number. His number happens to be the smallest positive integer such that if Nathan doubles his number, the result is a perfect square, and if Nathan triples his number, the result is a perfect cube. What is Nathan's number?
p9. Let

S

be the set of positive integers whose digits are in strictly increasing order when read from left to right. For example,

1

24

, and

369

are all elements of

S

, while

20

and

667

are not. If the elements of

S

are written in increasing order, what is the

100

th number written?
p10. Find the largest prime factor of the expression

2^{20} + 2^{16} + 2^{12} + 2^{8} + 2^{4} + 1

.
p11. Christina writes down all the numbers from

1

2020

, inclusive, on a whiteboard. What is the sum of all the digits that she wrote down?
p12. Triangle

ABC

has side lengths

AB = AC = 10

and

BC = 16

. Let

M

and

N

be the midpoints of segments

BC

and

CA

, respectively. There exists a point

P \ne A

on segment

AM

such that

2PN = PC

. What is the area of

\vartriangle PBC

?
p13. Consider the polynomial

P(x) = x^4 + 3x^3 + 5x^2 + 7x + 9.

Let its four roots be

a, b, c, d

. Evaluate the expression

(a + b + c)(a + b + d)(a + c + d)(b + c + d).

p14. Consider the system of equations

|y - 1| = 4 -|x - 1|

|y| =\sqrt{|k - x|}.

Find the largest

k

for which this system has a solution for real values

x

and

y

.
p16. Let

T_n = 1 + 2 + ... + n

denote the

n

th triangular number. Find the number of positive integers

n

less than

100

such that

n

and

T_n

have the same number of positive integer factors.
p17. Let

ABCD

be a square, and let

P

be a point inside it such that

PA = 4

PB = 2

, and

PC = 2\sqrt2

. What is the area of

ABCD

?
p18. The Fibonacci sequence

\{F_n\}

is defined as

F_0 = 0

F_1 = 1

, and

F_{n+2}= F_{n+1} + F_n

for all integers

n \ge 0

. Let

S =\dfrac{1}{F_6 + \frac{1}{F_6}}+\dfrac{1}{F_8 + \frac{1}{F_8}}+\dfrac{1}{F_{10} +\frac{1}{F_{10}}}+\dfrac{1}{F_{12} + \frac{1}{F_{12}}}+ ...

Compute

420S

.
p19. Let

ABCD

be a square with side length

5

. Point

P

is located inside the square such that the distances from

P

AB

and

AD

are

1

and

2

respectively. A point

T

is selected uniformly at random inside

ABCD

. Let

p

be the probability that quadrilaterals

APCT

and

BPDT

are both not self-intersecting and have areas that add to no more than

10

. If

p

can be expressed in the form

\frac{m}{n}

for relatively prime positive integers

m

and

n

, find

m + n

.
Note: A quadrilateral is self-intersecting if any two of its edges cross.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement