1

Part of 1992 Vietnam Team Selection Test

Problems(2)

difference is not divided by n

Source: Vietnam TST 1992 for the 33nd IMO, problem 1

Let two natural number

n > 1

m

be given. Find the least positive integer

k

which has the following property: Among

k

arbitrary integers

a_1, a_2, \ldots, a_k

satisfying the condition

a_i - a_j

1 \leq i < j \leq k

) is not divided by

n

, there exist two numbers

a_p, a_s

p \neq s

m + a_p - a_s

n

floor functionnumber theory unsolvednumber theory

every two circles touch each other

Source: Vietnam TST 1992 for the 33nd IMO, problem 4

In the plane let a finite family of circles be given satisfying the condition: every two circles, either are outside each other, either touch each other from outside and each circle touch at most 6 other circles. Suppose that every circle which does not touch 6 other circles be assigned a real number. Show that there exist at most one assignment to each remaining circle a real number equal to arithmetic mean of 6 numbers assigned to 6 circles which touch it.

geometry unsolvedgeometry