MathDB

1

Part of 2006 Iran Team Selection Test

Problems(2)

Question 1

Source: Iran TST 2006

4/17/2006
Suppose that pp is a prime number. Find all natural numbers nn such that p∣φ(n)p|\varphi(n) and for all aa such that (a,n)=1(a,n)=1 we have n∣aφ(n)p−1 n|a^{\frac{\varphi(n)}{p}}-1
number theoryleast common multiplefunctionnumber theory proposed
Question 1

Source: Iran TST 2006

4/29/2006
We have nn points in the plane, no three on a line. We call kk of them good if they form a convex polygon and there is no other point in the convex polygon. Suppose that for a fixed kk the number of kk good points is ckc_k. Show that the following sum is independent of the structure of points and only depends on nn : ∑i=3n(−1)ici \sum_{i=3}^n (-1)^i c_i
algebrapolynomialcombinatorial geometryIMO ShortlistRing Theorycombinatorics proposedcombinatorics