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A paper triangle has its vertices cut off. Each cut is along a straight line
parallel to the side opposite the vertex and tangent to the triangle's incircle.

Prove that the triangle's inradius is equal to the sum of inradii of the
three triangles cut off.
  
(No Solution Yet, 0 Comments) Submitted on 2017-07-28 by Bractals    

The number, 1406357289, is a 0 to 9 pandigital number because it is made up of each of the digits 0 to 9 in some order, but it also has a rather interesting sub-string divisibility property:

d2d3d4=406 is divisible by 2
d3d4d5=063 is divisible by 3
d4d5d6=635 is divisible by 5
d5d6d7=357 is divisible by 7
d6d7d8=572 is divisible by 11
d7d8d9=728 is divisible by 13
d8d9d10=289 is divisible by 17

Find the sum of all 0 to 9 pandigital numbers with this property.

(No Solution Yet, 3 Comments) Submitted on 2017-07-27 by Ady TZIDON    

Leonardo of Pisa, also known as Fibonacci, recounts that he was given this problem by John of Palermo as part of a mathematical tournament.

Three men possess a pile of money, their shares being 1/2, 1/3, 1/6. Each man takes some money from the pile until nothing is left. The first man returns 1/2 of what he took, the second 1/3 and third 1/6. When the total so returned is divided equally among the men it is found that each then possesses what he is entitled to. How much money was in the original pile, and how much did each man take from the pile?

Note: This is a Diophantine problem so you may just give the smallest whole number solution.

(No Solution Yet, 3 Comments) Submitted on 2017-07-26 by Jer    

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