1

Part of 1994 IberoAmerican

Problems(2)

9th ibmo - brazil 1994/q1.

Source: Spanish Communities

n

is said to be nice if it exists an integer

r>0

such that the expression of

n

r

has all its digits equal. For example, 62 and 15 are \emph{nice} because 62 is 222 in base 5, and 15 is 33 in base 4. Show that 1993 is not nice, but 1994 is.

number theory unsolvednumber theory

9th ibmo - brazil 1994/q4.

Source: Spanish Communities

A,\ B

C

be given points on a circumference

K

such that the triangle

\triangle{ABC}

P

be a point in the interior of

K

X,\ Y

Z

be the other intersection of

AP, BP

CP

with the circumference. Determine the position of

P

\triangle{XYZ}

is equilateral.

geometrycircumcirclegeometry unsolved