MathDB

1

Part of 2010 Gheorghe Vranceanu

Problems(5)

interesting broken lines geometry

Source:

2/7/2021
Let A1,B1,C1 A_1,B_1,C_1 be the middlepoints of the sides of a triangle ABC ABC and let A2,B2,C2 A_2,B_2,C_2 be on the middle of the paths CAB,ABC,BCA, CAB,ABC,BCA, respectively. Prove that A1A2,B1B2,C1C2 A_1A_2,B_1B_2,C_1C_2 are concurrent.
geometryMediansconcurrence
well known? decomposition into a 5 perfect cubes

Source:

2/7/2021
a) Prove that any multiple of 6 6 is the sum of four cubes. b) Show that any integer is the sum of five cubes.
number theory
function problem hiding behind a geometry problem

Source:

2/7/2021
Let be a number x x and three positive numbers a,b,c a,b,c such that ax+bx=cx. a^x+b^x=c^x. Prove that ay,by,cy a^y,b^y,c^y are the lenghts of the sides of an obtuse triangle if and only if y<x<2y. y<x<2y.
geometryanalytic geometrymonotonyfunction
straightforward limit with a bit of discussion

Source:

2/7/2021
lim⁡n→∞n(n3−6n2+6n+13−n2−an+5) \lim_{n\to\infty } n\left( \sqrt[3]{n^3-6n^2+6n+1}-\sqrt{n^2-an+5} \right)
calculus
Characterization of groups with property on semigroups

Source:

2/7/2021
Let be a semigroup with the property that for any two elements of it a,b, a,b, there is another element c c such that axa=b. axa=b. Prove that it's a group.
group theory