10.10

Part of VI Soros Olympiad 1999 - 2000 (Russia)

Problems(2)

fixed point for circumcircles - - VI Soros Olympiad 1999-00 Round 1 10.10

Source:

Take an arbitrary point

D

BC

ABC

and draw a circle through point

D

and the centers of the circles inscribed in triangles

ABD

ACD

. Prove that all circles obtained for different points

D

BC

have a common point.

geometryfixedFixed point

[a^m] + 1 is divisible by n

Source: VI Soros Olympiad 1990-00 R1 10.10 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

Prove that for every integer

n \ge 1

there exists a real number

a

such that for any integer

m \ge 1

[a^m] + 1

is divisible by

n

[x]

denotes the largest integer that does not exceed

x