MathDB

2

Part of 2018 Romania Team Selection Tests

Problems(3)

Concyclic points from mixtilinear circle

Source: Romanian 2018 TST Day 1 Problem 2

5/25/2020
Let ABCABC be a triangle, let II be its incenter, let Ω\Omega be its circumcircle, and let ω\omega be the AA- mixtilinear incircle. Let D,ED,E and TT be the intersections of ω\omega and AB,ACAB,AC and Ω\Omega, respectively, let the line ITIT cross ω\omega again at PP, and let lines PDPD and PEPE cross the line BCBC at MM and NN respectively. Prove that points D,E,M,ND,E,M,N are concyclic. What is the center of this circle?
geometrymixtilinear incircleincentercircumcircle
2n^2+2n+1 is composite

Source: Romanian 2018 TST Problem 2 Day 2

5/25/2020
Show that a number n(n+1)n(n+1) where nn is positive integer is the sum of 2 numbers k(k+1)k(k+1) and m(m+1)m(m+1) where mm and kk are positive integers if and only if the number 2n2+2n+12n^2+2n+1 is composite.
number theoryquadratic reciprocitySum of Squarescomposite numbers
Sum [kn^(1/3)]

Source: Romanian 2018 TST Problem 2 Day 3

5/25/2020
Given a square-free integer n>2n>2, evaluate the sum ∑k=1(n−2)(n−1)⌊(kn)1/3⌋\sum_{k=1}^{(n-2)(n-1)} \lfloor ({kn})^{1/3} \rfloor.
number theoryfloor functionseries summation