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Source: 2018 Canadian Open Math Challenge Part C Problem 1 —--
At Math-

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-Mart, cans of cat food are arranged in an pentagonal pyramid of 15 layers high, with 1 can in the top layer, 5 cans in the second layer, 12 cans in the third layer, 22 cans in the fourth layer etc, so that the

k^{\text{th}}

layer is a pentagon with

k

cans on each side. https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNC9lLzA0NTc0MmM2OGUzMWIyYmE1OGJmZWQzMGNjMGY1NTVmNDExZjU2LnBuZw==&rn=YzFhLlBORw==https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYS9hLzA1YWJlYmE1ODBjMzYwZDFkYWQyOWQ1YTFhOTkzN2IyNzJlN2NmLnBuZw==&rn=YzFiLlBORw==

\text{(a)}

How many cans are on the bottom,

15^{\text{th}}

, (A.)layer of this pyramid?

\text{(b)}

The pentagonal pyramid is rearranged into a prism consisting of 15 identical layers. (B.)How many cans are on the bottom layer of the prism?

\text{(c)}

A triangular prism consist of indentical layers, each of which has a shape of a triangle. (C.)(the number of cans in a triangular layer is one of the triangular numbers: 1,3,6,10,...) (C.)For example, a prism could be composed of the following layers: https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvMi85L2NlZmE2M2Y3ODhiN2UzMTRkYzIxY2MzNjFmMDJkYmE0ZTJhMTcwLnBuZw==&rn=YzFjLlBORw== Prove that a pentagonal pyramid of cans with any number of layers

l\ge 2

can be rearranged (without a deficit or leftover) into a triangluar prism of cans with the same number of layers

l

C1

Problems(1)