MathDB

2

Part of 1986 Bundeswettbewerb Mathematik

Problems(2)

x_n = n/(n+a)

Source: 1986 German Federal - Bundeswettbewerb Mathematik - BWM - Round 1 p2

11/21/2022
Let aa be a given natural number and x1,x2,x3,...x_1, x_2, x_3, ... the sequence with xn=nn+ax_n = \frac{n}{n+a} (n∈N∗n \in N^* ). Prove that for every n∈N∗n \in N^* , the term xnx_n can be represented as the product of two terms of this sequence , and determine the number of representations depending on nn and aa.
algebra
right triangle exradii criterions

Source: 1986 German Federal - Bundeswettbewerb Mathematik - BWM - Round 2 p2

11/21/2022
A triangle has sides a,b,ca, b,c, radius of the incircle rr and radii of the excircles ra,rb,rcr_a, r_b, r_c: Prove that: a) The triangle is right-angled if and only if: r+ra+rb+rc=a+b+cr + r_a + r_b + r_c = a + b + c. b) The triangle is right-angled if and only if: r2+ra2+rb2+rc2=a2+b2+c2r^2 + r^2_a + r^2_b + r^2_c = a^2 + b^2 + c^2.
geometryexcircleexradiusright triangle