MathDB

Set 6
p16. Let

n! = n \times (n - 1) \times ... \times 2 \times 1

. Find the maximum positive integer value of

x

such that the quotient

\frac{160!}{160^x}

is an integer.
p17. Let

\vartriangle OAB

be a triangle with

\angle OAB = 90^o

. Draw points

C, D, E, F, G

in its plane so that

\vartriangle OAB \sim \vartriangle OBC \sim \vartriangle OCD \sim \vartriangle ODE \sim \vartriangle OEF \sim \vartriangle OFG,

and none of these triangles overlap. If points

O, A, G

lie on the same line, then let

x

be the sum of all possible values of

\frac{OG}{OA }

. Then,

x

can be expressed in the form

m/n

for relatively prime positive integers

m, n

. Compute

m + n

.
p18. Let

f(x)

denote the least integer greater than or equal to

x^{\sqrt{x}}

. Compute

f(1)+f(2)+f(3)+f(4)

.
Set 7
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

.
p19. Find the least odd prime factor of

(F_3)^{20} + (F_4)^{20} + (F_5)^{20}

.
p20. Let

S = \frac{1}{F_3F_5}+\frac{1}{F_4F_6}+\frac{1}{F_5F_7}+\frac{1}{F_6F_8}+...

Compute

420S

.
p21. Consider the number

Q = 0.000101020305080130210340550890144... ,

the decimal created by concatenating every Fibonacci number and placing a 0 right after the decimal point and between each Fibonacci number. Find the greatest integer less than or equal to

\frac{1}{Q}

.
Set 8
p22. In five dimensional hyperspace, consider a hypercube

C_0

of side length

2

. Around it, circumscribe a hypersphere

S_0

, so all

32

vertices of

C_0

are on the surface of

S_0

. Around

S_0

, circumscribe a hypercube

C_1

, so that

S_0

is tangent to all hyperfaces of

C_1

. Continue in this same fashion for

S_1

C_2

S_2

, and so on. Find the side length of

C_4

.
p23. Suppose

\vartriangle ABC

satisfies

AC = 10\sqrt2

BC = 15

\angle C = 45^o

. Let

D, E, F

be the feet of the altitudes in

\vartriangle ABC

, and let

U, V , W

be the points where the incircle of

\vartriangle DEF

is tangent to the sides of

\vartriangle DEF

. Find the area of

\vartriangle UVW

.
p24. A polynomial

P(x)

is called spicy if all of its coefficients are nonnegative integers less than

9

. How many spicy polynomials satisfy

P(3) = 2019

?
The next set will consist of three estimation problems.
Set 9
Points will be awarded based on the formulae below. Answers are nonnegative integers that may exceed

1,000,000

.
p25. Suppose a circle of radius

20192019

has area

A

. Let s be the side length of a square with area

A

. Compute the greatest integer less than or equal to

s

.
If

n

is the correct answer, an estimate of

e

gives

\max \{ 0, \left\lfloor 1030 ( min \{ \frac{n}{e},\frac{e}{n}\}^{18}\right\rfloor -1000 \}

points.
p26. Given a

50 \times 50

grid of squares, initially all white, define an operation as picking a square and coloring it and the four squares horizontally or vertically adjacent to it blue, if they exist. If a square is already colored blue, it will remain blue if colored again. What is the minimum number of operations necessary to color the entire grid blue?
If

n

is the correct answer, an estimate of

e

gives

\left\lfloor \frac{180}{5|n-e|+6}\right\rfloor

points.
p27. The sphere packing problem asks what percent of space can be filled with equally sized spheres without overlap. In three dimensions, the answer is

\frac{\pi}{3\sqrt2} \approx 74.05\%

of space (confirmed as recently as

2017!

), so we say that the packing density of spheres in three dimensions is about

0.74

. In fact, mathematicians have found optimal packing densities for certain other dimensions as well, one being eight-dimensional space. Let d be the packing density of eight-dimensional hyperspheres in eightdimensional hyperspace. Compute the greatest integer less than

10^8 \times d

.
If

n

is the correct answer, an estimate of e gives

\max \left\{ \lfloor 30-10^{-5}|n - e|\rfloor, 0 \right\}

points.

PS. You had better use hide for answers. First sets have be posted [url=https://artofproblemsolving.com/community/c4h2777330p24370124]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement