MathDB

[hide=D stands for Descartes, L stands for Leibniz]they had two problem sets under those two names
D1. What is the solution to the equation

3 \cdot x \cdot 5 = 4 \cdot 5 \cdot 6

?
D2. Mr. Rose is making Platonic solids! If there are five different types of Platonic solids, and each Platonic solid can be one of three colors, how many different colored Platonic solids can Mr. Rose make?
D3. What fraction of the multiples of

5

between

1

and

100

inclusive are also multiples of

20

?
D4. What is the maximum number of times a circle can intersect a triangle?
D5 / L1. At an interesting supermarket, the nth apple you purchase costs

n

dollars, while pears are

3

dollars each. Given that Layla has exactly enough money to purchase either

k

apples or

2k

pears for

k > 0

, how much money does Layla have?
D6 / L3. For how many positive integers

1 \le n \le 10

does there exist a prime

p

such that the sum of the digits of

p

n

?
D7 / L2. Real numbers

a, b, c

are selected uniformly and independently at random between

0

and

1

. What is the probability that

a \ge b \le c

?
D8. How many ordered pairs of positive integers

(x, y)

satisfy

lcm(x, y) = 500

?
D9 / L4. There are

50

dogs in the local animal shelter. Each dog is enemies with at least

2

other dogs. Steven wants to adopt as many dogs as possible, but he doesn’t want to adopt any pair of enemies, since they will cause a ruckus. Considering all possible enemy networks among the dogs, find the maximum number of dogs that Steven can possibly adopt.
D10 / L7. Unit circles

a, b, c

satisfy

d(a, b) = 1

d(b, c) = 2

, and

d(c, a) = 3,

where

d(x, y)

is defined to be the minimum distance between any two points on circles

x

and

y

. Find the radius of the smallest circle entirely containing

a

b

, and

c

.
D11 / L8. The numbers

1

through

5

are written on a chalkboard. Every second, Sara erases two numbers

a

and

b

such that

a \ge b

and writes

\sqrt{a^2 - b^2}

on the board. Let M and m be the maximum and minimum possible values on the board when there is only one number left, respectively. Find the ordered pair

(M, m)

.
D12 / L9.

N

people stand in a line. Bella says, “There exists an assignment of nonnegative numbers to the

N

people so that the sum of all the numbers is

1

and the sum of any three consecutive people’s numbers does not exceed

1/2019

.” If Bella is right, find the minimum value of

N

possible.
D13 / L10. In triangle

\vartriangle ABC

D

is on

AC

such that

BD

is an altitude, and

E

is on

AB

such that

CE

is an altitude. Let F be the intersection of

BD

and

CE

. If

EF = 2FC

BF = 8DF

, and

DC = 3

, then find the area of

\vartriangle CDF

.
D14 / L11. Consider nonnegative real numbers

a_1, ..., a_6

such that

a_1 +... + a_6 = 20

. Find the minimum possible value of

\sqrt{a^2_1 + 1^2} +\sqrt{a^2_2 + 2^2} +\sqrt{a^2_3 + 3^2} +\sqrt{a^2_4 + 4^2} +\sqrt{a^2_5 + 5^2} +\sqrt{a^2_6 + 6^2}.

D15 / L13. Find an

a < 1000000

so that both

a

and

101a

are triangular numbers. (A triangular number is a number that can be written as

1 + 2 +... + n

for some

n \ge 1

.)
Note: There are multiple possible answers to this problem. You only need to find one.
L6. How many ordered pairs of positive integers

(x, y)

, where

x

is a perfect square and

y

is a perfect cube, satisfy

lcm(x, y) = 81000000

?
L12. Given two points

A

and

B

in the plane with

AB = 1

, define

f(C)

to be the incenter of triangle

ABC

, if it exists. Find the area of the region of points

f(f(X))

where

X

is arbitrary.
L14. Leptina and Zandar play a game. At the four corners of a square, the numbers

1, 2, 3

, and

4

are written in clockwise order. On Leptina’s turn, she must swap a pair of adjacent numbers. On Zandar’s turn, he must choose two adjacent numbers

a

and

b

with

a \ge b

and replace

a

with

a - b

. Zandar wants to reduce the sum of the numbers at the four corners of the square to

2

in as few turns as possible, and Leptina wants to delay this as long as possible. If Leptina goes first and both players play optimally, find the minimum number of turns Zandar can take after which Zandar is guaranteed to have reduced the sum of the numbers to

2

.
L15. There exist polynomials

P, Q

and real numbers

c_0, c_1, c_2, ... , c_{10}

so that the three polynomials

P, Q

, and

c_0P^{10} + c_1P^9Q + c_2P^8Q^2 + ... + c_{10}Q^{10}

are all polynomials of degree 2019. Suppose that

c_0 = 1

c_1 = -7

c_2 = 22

. Find all possible values of

c_{10}

.
Note: The answer(s) are rational numbers. It suffices to give the prime factorization(s) of the numerator(s) and denominator(s).
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement