MathDB

1

a [ b n] = b [ a n ] , floor function equation

(a, b)

of real numbers such that

a \cdot \lfloor b \cdot n\rfloor = b \cdot \lfloor a \cdot n \rfloor

applies to all positive integers

n

. (For a real number

x, \lfloor x\rfloor

denotes the largest integer that is less than or equal to

x

.)

3

1

winning strategy with residue classes modulo n

Let

n\ge 2

be an integer. Ariane and Bérénice play a game on the number of the residue classes modulo

n

. At the beginning there is the residue class

1

on each piece of paper. It is the turn of the player whose turn it is to replace the current residue class

x

with either

x + 1

or by

2x

. The two players take turns, with Ariane starting. Ariane wins if the residue class

0

is reached during the game. Bérénice wins if she can prevent that permanently. Depending on

n

, determine which of the two has a winning strategy.

2

1

equal segments wanted, circles passing through incenter related

Let

ABC

be a triangle and

I

its incenter. The circle passing through

A, C

and

I

intersect the line

BC

for second time at point

X

. The circle passing through

B, C

and

I

intersects the line

AC

for second time at point

Y

. Show that the segments

AY

and

BX

have equal length.

1

a_n = 14a_{n-1} + a_{n-2} , b_n = 6b_{n-1}-b_{n-2}, infinite common elements

We consider the two sequences

(a_n)_{n\ge 0}

and

(b_n) _{n\ge 0}

of integers, which are given by

a_0 = b_0 = 2

and

a_1= b_1 = 14

and for

n\ge 2

they are defined as

a_n = 14a_{n-1} + a_{n-2}

,

b_n = 6b_{n-1}-b_{n-2}

. Determine whether there are infinite numbers that occur in both sequences

2019 Federal Competition For Advanced Students, P1

Subcontests

a [ b n] = b [ a n ] , floor function equation

winning strategy with residue classes modulo n

equal segments wanted, circles passing through incenter related

a_n = 14a_{n-1} + a_{n-2} , b_n = 6b_{n-1}-b_{n-2}, infinite common elements