MathDB

14

Part of 2017 Hanoi Open Mathematics Competitions

Problems(2)

trapezoid, 4 perpendiculars, collinear ? (2017 HOMC Junior Q14)

Source:

8/3/2019
Given trapezoid ABCDABCD with bases AB∥CDAB \parallel CD (AB<CDAB < CD). Let OO be the intersection of ACAC and BDBD. Two straight lines from DD and CC are perpendicular to ACAC and BDBD intersect at EE , i.e. CE⊥BDCE \perp BD and DE⊥ACDE \perp AC . By analogy, AF⊥BDAF \perp BD and BF⊥ACBF \perp AC . Are three points E,O,FE , O, F located on the same line?
geometrytrapezoidcollinearcollinearityperpendicular
P=m^{2003}n^{2017}-m^{2017}n^{2003} divisible by 24,7 (HOMC 2017 S Q14)

Source:

9/8/2019
Put P=m2003n2017−m2017n2003P = m^{2003}n^{2017} - m^{2017}n^{2003} , where m,n∈Nm, n \in N. a) Is PP divisible by 2424? b) Do there exist m,n∈Nm, n \in N such that PP is not divisible by 77?
divisiblenumber theory