MathDB

1

Part of 2014 China National Olympiad

Problems(2)

Circumcentre is Incentre

Source: China Mathematical Olympiad 2014

12/21/2013
Let ABCABC be a triangle with AB>ACAB>AC. Let DD be the foot of the internal angle bisector of AA. Points FF and EE are on AC,ABAC,AB respectively such that B,C,F,EB,C,F,E are concyclic. Prove that the circumcentre of DEFDEF is the incentre of ABCABC if and only if BE+CF=BCBE+CF=BC.
geometryincenterratiosymmetrycircumcircletrigonometryangle bisector
No. of distinct prime factors and no. of prime factors

Source: China Mathematical Olympiad 2014 Q4

12/22/2013
Let n=p1a1p2a2⋯ptatn=p_1^{a_1}p_2^{a_2}\cdots p_t^{a_t} be the prime factorisation of nn. Define ω(n)=t\omega(n)=t and Ω(n)=a1+a2+…+at\Omega(n)=a_1+a_2+\ldots+a_t. Prove or disprove: For any fixed positive integer kk and positive reals α,β\alpha,\beta, there exists a positive integer n>1n>1 such that i) ω(n+k)ω(n)>α\frac{\omega(n+k)}{\omega(n)}>\alpha ii) Ω(n+k)Ω(n)<β\frac{\Omega(n+k)}{\Omega(n)}<\beta.
inductionlogarithmsalgebranumber theory proposednumber theoryChina