MathDB

Set 1
1.1 Compute the number of real numbers x such that the sequence

x

x^2

x^3

x^4

x^5

...

eventually repeats. (To be clear, we say a sequence “eventually repeats” if there is some block of consecutive digits that repeats past some point—for instance, the sequence

1

2

3

4

5

6

5

6

5

6

...

is eventually repeating with repeating block

5

6

.)
1.2 Let

T

be the answer to the previous problem. Nicole has a broken calculator which, when told to multiply

a

b

, starts by multiplying

a

b

, but then multiplies that product by b again, and then adds

b

to the result. Nicole inputs the computation “

k \times k

” into the calculator for some real number

k

and gets an answer of

10T

. If she instead used a working calculator, what answer should she have gotten?
1.3 Let

T

be the answer to the previous problem. Find the positive difference between the largest and smallest perfect squares that can be written as

x^2 + y^2

for integers

x, y

satisfying

\sqrt{T} \le x \le T

and

\sqrt{T} \le y \le T

.
PS. You should use hide for answers.

Problem Statement