MathDB

3

Part of 1996 National High School Mathematics League

Problems(2)

Problem about Prime

Source: 1996 National High School Mathematics League, Exam One, Problem 3

3/3/2020
For a prime number pp, there exists n∈Z+n\in\mathbb{Z}_+, p+n+n\sqrt{p+n}+\sqrt{n} is an integer, then (A)\text{(A)} there is no such pp (B)\text{(B)} there in only one such pp (C)\text{(C)} there is more than one such pp, but finitely many (D)\text{(D)} there are infinitely many such pp
number theoryprime numbers
Escribed Circle

Source: 1996 National High School Mathematics League, Exam Two, Problem 3

3/4/2020
⊙O1\odot O_1 and ⊙O2\odot O_2 are escribed circles of △ABC\triangle ABC (⊙O1\odot O_1 is in ∠ACB\angle ACB, ⊙O2\odot O_2 is in ∠ABC\angle ABC). ⊙O1\odot O_1 touches CB,CACB,CA at E,GE,G; ⊙O1\odot O_1 touches BC,BABC,BA at F,HF,H. EG∩FG=PEG\cap FG=P, prove that AP⊥BCAP\perp BC.