Math Problems

As a high school student I attended a weekly Saturday's math club run by Zbigniew Romanowicz (Wroclaw, Poland). We got a weekly list of problems to prepare before the meeting and then discussed them together. I found those problems quite interesting and maybe different from, for example, a preparation to Mathematical Olympiad. More in the style of International competition for mathematical and logical games. Below I list some of the problems form the lists, I will try to make this as complete as I can  (according to my pile of pages of problems) as time goes by.

GIM stands for middle school level problems.
LO stands for high school level problems.
Sometimes lists have been joined and are flagged as GIM-LO.

GIM-LO/03/04/11/1

Is it possible to transform an arbitrary triangle into a rectangular triangle performing one of the following operations:

A. extending each side of the triangle by a segment of the same length 

B. shortening each side of the triangle by a segment of the same length

LO/04/05/1/1

A block with dimensions a x b x c (a,b,c are positive integers) is divided into abc unit blocks. If we pierce this block with a long needle along the diagonal, how many unit blocks would we pierce?

LO/04/05/1/2

Pawel had multiplied two integers on his computer. Unfortunately, he accidentally asked the program to sort the digits. After that he got the following 3 numbers:

multiplicand: 0246

multiplier: 1457

product: 11338899

Could you guess what the product was before sorting?

LO/04/05/1/3

Divide a given triangle using only calipers and a ruler into 3 triangles T₁, T₂ and T₃, such that you could choose one median from each triangle m₁, m₂, and m₃ having m₁ = m₂ = m₃.