MathDB

[hide=C stands for Cantor, G stands for Gauss]they had two problem sets under those two names
Set 4
G.16 A number

k

is the product of exactly three distinct primes (in other words, it is of the form

pqr

, where

p, q, r

are distinct primes). If the average of its factors is

66

, find

k

.
G.17 Find the number of lattice points contained on or within the graph of

\frac{x^2}{3} +\frac{y^2}{2}= 12

. Lattice points are coordinate points

(x, y)

where

x

and

y

are integers.
G.18 / C.23 How many triangles can be made from the vertices and center of a regular hexagon? Two congruent triangles with different orientations are considered distinct.
G.19 Cindy has a cone with height

15

inches and diameter

16

inches. She paints one-inch thick bands of paint in circles around the cone, alternating between red and blue bands, until the whole cone is covered with paint. If she starts from the bottom of the cone with a blue strip, what is the ratio of the area of the cone covered by red paint to the area of the cone covered by blue paint?
G.20 / C.25 An even positive integer

n

has an odd factorization if the largest odd divisor of

n

is also the smallest odd divisor of n greater than 1. Compute the number of even integers

n

less than

50

with an odd factorization.
Set 5
G.21 In the magical tree of numbers,

n

is directly connected to

2n

and

2n + 1

for all nonnegative integers n. A frog on the magical tree of numbers can move from a number

n

to a number connected to it in

1

hop. What is the least number of hops that the frog can take to move from

1000

2018

?
G.22 Stan makes a deal with Jeff. Stan is given 1 dollar, and every day for

10

days he must either double his money or burn a perfect square amount of money. At first Stan thinks he has made an easy

1024

dollars, but then he learns the catch - after

10

days, the amount of money he has must be a multiple of

11

or he loses all his money. What is the largest amount of money Stan can have after the

10

days are up?
G.23 Let

\Gamma_1

be a circle with diameter

2

and center

O_1

and let

\Gamma_2

be a congruent circle centered at a point

O_2 \in \Gamma_1

. Suppose

\Gamma_1

and

\Gamma_2

intersect at

A

and

B

. Let

\Omega

be a circle centered at

A

passing through

B

. Let

P

be the intersection of

\Omega

and

\Gamma_1

other than

B

and let

Q

be the intersection of

\Omega

and ray

\overrightarrow{AO_1}

. Define

R

to be the intersection of

PQ

with

\Gamma_1

. Compute the length of

O_2R

.
G.24

8

people are at a party. Each person gives one present to one other person such that everybody gets a present and no two people exchange presents with each other. How many ways is this possible?
G.25 Let

S

be the set of points

(x, y)

such that

y = x^3 - 5x

and

x = y^3 - 5y

. There exist four points in

S

that are the vertices of a rectangle. Find the area of this rectangle.

PS. You should use hide for answers. C1-15/ G1-10 have been posted [url=https://artofproblemsolving.com/community/c3h2790674p24540132]here and C16-30/G10-15, G25-30 [url=https://artofproblemsolving.com/community/c3h2790676p24540145]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here

Problem Statement