MathDB

1

Part of 2012 Czech-Polish-Slovak Match

Problems(2)

One of the three is the AM of other two

Source: Czech-Polish-Slovak 2012, P1

4/12/2013
Given a positive integer nn, let τ(n)\tau(n) denote the number of positive divisors of nn and φ(n)\varphi(n) denote the number of positive integers not exceeding nn that are relatively prime to nn. Find all nn for which one of the three numbers n,τ(n),φ(n)n,\tau(n), \varphi(n) is the arithmetic mean of the other two.
number theoryrelatively primenumber theory unsolved
Ratio of area is independent of the position of point

Source: Czech-Polish-Slovak 2012, P4

4/12/2013
Let ABCABC be a right angled triangle with hypotenuse ABAB and PP be a point on the shorter arc ACAC of the circumcircle of triangle ABCABC. The line, perpendicuar to CPCP and passing through CC, intersects APAP, BPBP at points KK and LL respectively. Prove that the ratio of area of triangles BKLBKL and ACPACP is independent of the position of point PP.
ratiogeometrycircumcirclearea of a trianglegeometry proposed