MathDB

4

Part of 2015 Iran MO (3rd round)

Problems(3)

Constant polynomial

Source: Iranian third round algebra 2015 problem 4

9/8/2015
p(x)∈C[x]p(x)\in \mathbb{C}[x] is a polynomial such that: ∀z∈C,∣z∣=1⟹p(z)∈R\forall z\in \mathbb{C}, |z|=1\Longrightarrow p(z)\in \mathbb{R} Prove that p(x)p(x) is constant.
algebrapolynomialcomplex numbers
the same prime factors

Source: Iranian third round number theory P4

9/6/2015
a,b,c,d,k,la,b,c,d,k,l are positive integers such that for every natural number nn the set of prime factors of nk+an+c,nl+bn+dn^k+a^n+c,n^l+b^n+d are same. prove that k=l,a=b,c=dk=l,a=b,c=d.
number theoryfermat's little theorem
Three collinear points

Source: Iranian third round geometry problem 4

9/10/2015
Let ABCABC be a triangle with incenter II. Let KK be the midpoint of AIAI and BI∩⊙(△ABC)=M,CI∩⊙(△ABC)=NBI\cap \odot(\triangle ABC)=M,CI\cap \odot(\triangle ABC)=N. points P,QP,Q lie on AM,ANAM,AN respectively such that ∠ABK=∠PBC,∠ACK=∠QCB\angle ABK=\angle PBC,\angle ACK=\angle QCB. Prove that P,Q,IP,Q,I are collinear.
geometryincenter