MathDB

5

Part of Mathley 2014-15

Problems(3)

u_{n + 2} = u_{n + 1} ++u_ n + [(-1)^n-1]/ 2}

Source: Mathley 2014 .1 p5

8/20/2020
Given the sequence (un)n=1∞(u_n)_{n=1}^{\infty}, where u1=1,u2=2u_1 = 1, u_2 = 2, and un+2=un+1+un+(−1)n−12u_{n + 2} = u_{n + 1} +u_ n+ \frac{(-1)^n-1}{2} for any positive integers nn. Prove that every positive integers can be expressed as the sum of some distinguished numbers of the sequence of numbers (un)n=1∞(u_n)_{n=1}^{\infty}
Nguyen Duy Thai Son, The University of Danang, Da Nang.
Sequencerecurrence relationalgebra
concurrency wanted, cyclic ABCD, 4 more circles related

Source: Mathley 2014.2 p5

8/20/2020
A quadrilateral ABCDABCD is inscribed in a circle (O)(O). Another circle (I)(I) is tangent to the diagonals AC,BDAC, BD at M,NM, N respectively. Suppose that MNMN meets AB,CDAB,CD at P,QP, Q respectively. The circumcircle of triangle IMNIMN meets the circumcircles of IAB,ICDIAB, ICD at K,LK, L respectively, which are distinct from II. Prove that the lines PK,QLPK, QL, and OIOI are concurrent.
Tran Minh Ngoc, a student of Ho Chi Minh City College, Ho Chi Minh
geometryCycliccircles
Mathley 2014.3 p5 by Tran Quang Hung

Source:

8/18/2020
Triangle ABCABC has incircle (I)(I) and P,QP,Q are two points in the plane of the triangle. Let QA,QB,QCQA,QB,QC meet BA,CA,ABBA,CA,AB respectively at D,E,FD,E,F. The tangent at DD, other than BCBC, of the circle (I)(I) meets PAPA at XX. The points YY and ZZ are defined in the same manner. The tangent at XX, other than XDXD, of the circle (I)(I) meets (I) (I) at UU. The points V,WV,W are defined in the same way. Prove that three lines (AU,BV,CW)(AU,BV,CW) are concurrent.
Tran Quang Hung, Dean of the Faculty of Science, Thanh Xuan, Hanoi.
incirclegeometryconcurrent