MathDB

8

Part of Mathley 2014-15

Problems(2)

x_n/ y_n =\sum_{k=1}^{n}{\frac{1}{k {n \choose k}}}, y_n not divisible by 2^n

Source: Mathley 2014.1 p8

8/20/2020
For every nn positive integers we denote xnyn=∑k=1n1k(nk)\frac{x_n}{y_n}=\sum_{k=1}^{n}{\frac{1}{k {n \choose k}}} where xn,ynx_n, y_n are coprime positive integers. Prove that yny_n is not divisible by 2n2^n for any positive integers nn.
Ha Duy Hung, high school specializing in the Ha University of Education, Hanoi, Xuan Thuy, Cau Giay, Hanoi
number theorydivisiblepower of 2divides
circumcircle of triangle formed by t_Q, t_S, AB is also tangent to (W)

Source: Mathley 2014.3 p8

8/19/2020
Two circles (U)(U) and (V)(V) intersect at A,BA,B. A line d meets (U),(V)(U), (V) at P,QP, Q and R,SR,S respectively. Let tP,tQ,tR,tSt_P, t_Q, t_R,t_S be the tangents at P,Q,R,SP,Q,R, S of the two circles. Another circle (W)(W) passes through through A,BA, B. Prove that if the circumcircle of triangle that is formed by the intersections of tP,tR,ABt_P,t_R, AB is tangent to (W)(W) then the circumcircle of triangle formed by tQ,tS,ABt_Q, t_S, AB is also tangent to (W)(W).
Tran Minh Ngoc, a student of Ho Chi Minh City College, Ho Chi Minh
circlestangent circlesgeometry