MathDB

4

Part of 1993 Bundeswettbewerb Mathematik

Problems(2)

ratio of areas, hexagon / triangle : G/F>13

Source: 1993 German Federal - Bundeswettbewerb Mathematik - BWM - Round 1 p4

11/20/2022
Given is a triangle ABCABC with side lengths a,b,ca, b, c (a=BC‾a = \overline{BC}, b=CA‾b = \overline{CA}, c=AB‾c = \overline{AB}) and area FF. The side ABAB is extended beyond AA by a and beyond BB by bb. Correspondingly, BCBC is extended beyond BB and CC by bb and cc, respectively. Eventually CACA is extended beyond CC and AA by cc and aa, respectively. Connecting the outer endpoints of the extensions , a hexagon if formed with area GG. Prove that GF>13\frac{G}{F}>13.
ratiogeometric inequalityareasgeometry
Existence of First Four Digits of a Factorial

Source: 1993 German Maths Olympiad

7/25/2020
Does there exist a non-negative integer n, such that the first four digits of n! is 1993?
number theoryfactorialnumber theory unsolvednational olympiad