MathDB

2

Part of Mathley 2014-15

Problems(4)

area of triangle with integer sidelenghts, t_{n + 2} = 14t_{n + 1} - t_n,

Source: Mathley 2014.1 p2

8/19/2020
Given the sequence (tn)(t_n) defined as t0=0t_0 = 0, t1=6t_1 = 6, tn+2=14tn+1−tnt_{n + 2} = 14t_{n + 1} - t_n. Prove that for every number n≥1n \ge 1, tnt_n is the area of a triangle whose lengths are all numbers integers.
Dang Hung Thang, University of Natural Sciences, Hanoi National University.
geometryTriangleIntegersSequencealgebraarea of a triangle
area of K_aK_bK_c does not exceed that of ABC, mixtilinears related

Source: Mathley 2014.2 p2

8/20/2020
Let ABCABC be a triangle with a circumcircle (K)(K). A circle touching the sides AB,ACAB,AC is internally tangent to (K)(K) at KaK_a; two other points Kb,KcK_b,K_c are defined in the same manner. Prove that the area of triangle KaKbKcK_aK_bK_c does not exceed that of triangle ABCABC.
Nguyen Minh Ha, Hanoi University of Education, Xuan Thuy, Cau Giay, Hanoi.
mixtilineargeometryareasgeometric inequality
1+x/(n + 1)+x^2/ (2n + 1)+ ...+ x^p/(pn + 1) = 0 no integer solution when p>n+1

Source: Mathley 2014.3 p2

8/18/2020
Let nn be a positive integer and pp a prime number p>n+1p > n + 1. Prove that the following equation does not have integer solution 1+xn+1+x22n+1+...+xppn+1=01 + \frac{x}{n + 1} + \frac{x^2}{2n + 1} + ...+ \frac{x^p}{pn + 1} = 0
Luu Ba Thang, Department of Mathematics, College of Education
Diophantine equationnumber theorydiophantine
MN = MH wanted, ME //AC, MF //BD , cyclic ABCD given

Source: Mathley 2015 p2

8/18/2020
A quadrilateral ABCDABCD is inscribed in a circle and its two diagonals AC,BDAC,BD meet at GG. Let MM be the center of CD,E,FCD, E,F be the points on BC,ADBC, AD respectively such that ME∥ACME \parallel AC and MF∥BDMF \parallel BD. Point HH is the projection of GG onto CDCD. The circumcircle of MEFMEF meets CDCD at NN distinct from MM. Prove that MN=MHMN = MH
Tran Quang Hung, Nguyen Le Phuoc, Thanh Xuan, Hanoi
cyclic quadrilateralparallelequal segments