MathDB

2019 A/NT7: Sum of Phis and Squares of Floors

Source:

1/27/2019

For all positive integers

n

, let

f(n) = \sum_{k=1}^n\varphi(k)\left\lfloor\frac nk\right\rfloor^2.

Compute

f(2019) - f(2018)

. Here

\varphi(n)

denotes the number of positive integers less than or equal to

n

which are relatively prime to

n

.

2019algebranumber theory

2019 C/CS7: Languages Do Not Intersect

Source:

1/27/2019

Consider the set

L

of binary strings of length less than or equal to

9

, and for a string

w

define

w^{+}

to be the set

\{w,w^2,w^3,\ldots\}

where

w^k

represents

w

concatenated to itself

k

times. How many ways are there to pick an ordered pair of (not necessarily distinct) elements

x,y\in L

such that

x^{+}\cap y^{+}\neq \varnothing

?

2019combinatoricscomputer science

2019 G7: Tangent to Incircle of a Triangle

Source:

1/27/2019

Let

ABC

be a triangle with

AB=13

,

BC=14

, and

AC=15

. Denote by

\omega

its incircle. A line

\ell

tangent to

\omega

intersects

\overline{AB}

and

\overline{AC}

at

X

and

Y

respectively. Suppose

XY=5

. Compute the positive difference between the lengths of

\overline{AX}

and

\overline{AY}

.

geometry

2019 T7: Random Walk with Three People

Source:

1/27/2019

Suppose you start at

0

, a friend starts at

6

, and another friend starts at

8

on the number line. Every second, the leftmost person moves left with probability

\tfrac14

, the middle person with probability

\tfrac13

, and the rightmost person with probability

\tfrac12

. If a person does not move left, they move right, and if two people are on the same spot, they are randomly assigned which one of the positions they are. Determine the expected time until you all meet in one point.

2019team

7

Problems(4)