MathDB

1

Part of 2011 IMO Shortlist

Problems(2)

IMO Shortlist 2011, G1

Source: IMO Shortlist 2011, G1

7/13/2012
Let ABCABC be an acute triangle. Let ω\omega be a circle whose centre LL lies on the side BCBC. Suppose that ω\omega is tangent to ABAB at B′B' and ACAC at C′C'. Suppose also that the circumcentre OO of triangle ABCABC lies on the shorter arc B′C′B'C' of ω\omega. Prove that the circumcircle of ABCABC and ω\omega meet at two points.
Proposed by Härmel Nestra, Estonia
geometrycircumcircletrigonometrygeometric transformationIMO Shortlist
IMO Shortlist 2011, Number Theory 1

Source: IMO Shortlist 2011, Number Theory 1

7/11/2012
For any integer d>0,d > 0, let f(d)f(d) be the smallest possible integer that has exactly dd positive divisors (so for example we have f(1)=1,f(5)=16,f(1)=1, f(5)=16, and f(6)=12f(6)=12). Prove that for every integer k≥0k \geq 0 the number f(2k)f\left(2^k\right) divides f(2k+1).f\left(2^{k+1}\right).
Proposed by Suhaimi Ramly, Malaysia
algorithmnumber theoryprime numbersDivisibilityIMO Shortlistnumber of divisors