MathDB

Set 1
p1. Find

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11

.
p2. Find

1 \cdot 11 + 2 \cdot 10 + 3 \cdot 9 + 4 \cdot 8 + 5 \cdot 7 + 6 \cdot 6

.
p3. Let

\frac{1}{1\cdot 2} +\frac{1}{2\cdot 3} +\frac{1}{3\cdot 4} +\frac{1}{4\cdot 5} +\frac{1}{5\cdot 6} +\frac{1}{6\cdot 7} +\frac{1}{7\cdot 8} +\frac{1}{8\cdot 9} +\frac{1}{9\cdot 10} +\frac{1}{10\cdot 11} =\frac{m}{n}

, where

m

and

n

are positive integers that share no prime divisors. Find

m + n

.
Set 2
p4. Define

0! = 1

and let

n! = n \cdot (n - 1)!

for all positive integers

n

. Find the value of

(2! + 0!)(1! + 8!)

.
p5. Rachel’s favorite number is a positive integer

n

. She gives Justin three clues about it:

\bullet

n

is prime.

\bullet

n^2 - 5n + 6 \ne 0

\bullet

n

is a divisor of

252

. What is Rachel’s favorite number?
p6. Shen eats eleven blueberries on Monday. Each day after that, he eats five more blueberries than the day before. For example, Shen eats sixteen blueberries on Tuesday. How many blueberries has Shen eaten in total before he eats on the subsequent Monday?
Set 3
p7. Triangle

ABC

satisfies

AB = 7

BC = 12

, and

CA = 13

. If the area of

ABC

can be expressed in the form

m\sqrt{n}

, where

n

is not divisible by the square of a prime, then determine

m + n

.
p8. Sebastian is playing the game Split! on a coordinate plane. He begins the game with one token at

(0, 0)

. For each move, he is allowed to select a token on any point

(x, y)

and take it off the plane, replacing it with two tokens, one at

(x + 1, y)

, and one at

(x, y + 1)

. At the end of the game, for a token on

(a, b)

, it is assigned a score

\frac{1}{2^{a+b}}

. These scores are summed for his total score. Determine the highest total score Sebastian can get in

100

moves.
p9. Find the number of positive integers

n

satisfying the following two properties:

\bullet

n

has either four or five digits, where leading zeros are not permitted,

\bullet

The sum of the digits of

n

is a multiple of

3

.
Set 4
p10. A unit square rotated

45^o

about a vertex, Sweeps the area for Farmer Khiem’s pen. If

n

is the space the pigs can roam, Determine the floor of

100n

.
If

n

is the area a unit square sweeps out when rotated 4

5

degrees about a vertex, determine

\lfloor 100n \rfloor

. Here

\lfloor x \rfloor

denotes the greatest integer less than or equal to

x

.
https://cdn.artofproblemsolving.com/attachments/b/1/129efd0dbd56dc0b4fb742ac80eaf2447e106d.png
p11. Michael is planting four trees, In a grid, three rows of three, If two trees are close, Then both are bulldozed, So how many ways can it be?
In a three by three grid of squares, determine the number of ways to select four squares such that no two share a side.
p12. Three sixty-seven Are the last three digits of

n

cubed. What is

n

?
If the last three digits of

n^3

are

367

for a positive integer

n

less than

1000

, determine

n

.
Set 5
p13. Determine

\sqrt[4]{97 + 56\sqrt{3}} + \sqrt[4]{97 - 56\sqrt{3}}

.
p14. Triangle

\vartriangle ABC

is inscribed in a circle

\omega

of radius

12

so that

\angle B = 68^o

and

\angle C = 64^o

. The perpendicular from

A

BC

intersects

\omega

D

, and the angle bisector of

\angle B

intersects

\omega

E

. What is the value of

DE^2

?
p15. Determine the sum of all positive integers

n

such that

4n^4 + 1

is prime.
Set 6
p16. Suppose that

p, q, r

are primes such that

pqr = 11(p + q + r)

such that

p\ge q \ge r

. Determine the sum of all possible values of

p

.
p17. Let the operation

\oplus

satisfy

a \oplus b =\frac{1}{1/a+1/b}

. Suppose

N = (...((2 \oplus 2) \oplus 2) \oplus ... 2),

where there are

2018

instances of

\oplus

. If

N

can be expressed in the form

m/n

, where

m

and

n

are relatively prime positive integers, then determine

m + n

.
p18. What is the remainder when

\frac{2018^{1001} - 1}{2017}

is divided by

2017

?

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Problem Statement