MathDB

1

Part of 2012 Dutch IMO TST

Problems(2)

line passes through the incentre

Source: All-Russian 2007

5/4/2007
A line, which passes through the incentre II of the triangle ABCABC, meets its sides ABAB and BCBC at the points MM and NN respectively. The triangle BMNBMN is acute. The points K,LK,L are chosen on the side ACAC such that ∠ILA=∠IMB\angle ILA=\angle IMB and ∠KC=∠INB\angle KC=\angle INB. Prove that AM+KL+CN=ACAM+KL+CN=AC. S. Berlov
geometryincentergeometry unsolved
prime power criterion with a @ b = \frac{a - b}{gcd(a, b)}

Source: Dutch IMO TST2 2012 p1

1/10/2020
For all positive integers aa and bb, we de ne a@b=a−bgcd(a,b)a @ b = \frac{a - b}{gcd(a, b)} . Show that for every integer n>1n > 1, the following holds: nn is a prime power if and only if for all positive integers mm such that m<nm < n, it holds that gcd(n,n@m)=1gcd(n, n @m) = 1.
number theoryPerfect powerprimeGCDgreatest common divisor