MathDB

4

Part of 2020 Iran MO (3rd Round)

Problems(4)

Best geo problem of the exam.

Source: Iranian Third Round 2020 Geometry exam Problem4

11/18/2020
Triangle ABCABC is given. Let OO be it's circumcenter. Let II be the center of it's incircle.The external angle bisector of AA meet BCBC at DD. And IAI_A is the AA-excenter . The point KK is chosen on the line AIAI such that AK=2AIAK=2AI and AA is closer to KK than II. If the segment DFDF is the diameter of the circumcircle of triangle DKIADKI_A, then prove OF=3OIOF=3OI.
geometry
Interesting problem

Source: Iranian Third Round 2020 Algebra exam Problem4

11/20/2020
We call a polynomial P(x)P(x) intresting if there are 13981398 distinct positive integers n1,...,n1398n_1,...,n_{1398} such that P(x)=∑xni+1P(x)=\sum_{}{x^{n_i}}+1 Does there exist infinitly many polynomials P1(x),P2(x),...P_1(x),P_2(x),... such that for each distinct i,ji,j the polynomial Pi(x)Pj(x)P_i(x)P_j(x) is interesting.
polynomialequationalgebra
something to fill the void of P4

Source: Iranian Third Round 2020 Combinatorics exam Problem4

11/18/2020
What is the maximum number of subsets of size 55, taken from the set A={1,2,3,...,20}A=\{1,2,3,...,20\} such that any 22 of them share exactly 11 element.
combinatoricsSetsSubsets
Tricky problem about the difference of power of b

Source: Iranian Third Round 2020 Number Theory exam Problem4

11/21/2020
Prove that for every two positive integers a,ba,b greater than 11. there exists infinitly many nn such that the equation ϕ(an−1)=bm−bt\phi(a^n-1)=b^m-b^t can't hold for any positive integers m,tm,t.
number theorytotient function