All this year’s chapter competitions have officially concluded and let’s take a look at a few of our favorites from the 2017 Chapter Competition sprint round.

How many triangles of any size are in the Belgian truss shown?

*[Sprint #12] *

There are 12 “unit” triangles—six in the left half and six in the right half. The truss as a whole is 1 triangle. The only question left now is how many compound (bigger than unit) triangles but smaller than the whole truss. Starting from the left vertex of the truss, each segment connecting the bottom edge and the upper left edge of the truss combined with the bottom and upper left edges toward the left vertex form a triangle. The leftmost connecting segment forms a unit triangle so it must not be counted again. However, the other 5 connecting segments do form a triangle not yet counted; there are likewise 5 such compound triangles in the right half of the truss. Adding them together yields 12 + 1 + 5 + 5 =triangles in total.23

What is the largest sum that results when one of the arrangements of the digits of 2017 is added to one of the arrangements of the digits 2016, if none of the digits 0, 1 or 2 can occupy the same position in both numbers?

*[Sprint #24] *

First aim for the largest number of thousands—use 7 from 2017 and 6 from 2016. Then aim for the largest number of hundreds using what is left, which must be a 2 and a 1, since we are not allowed to use the same digit twice for one place value. Then aim for the largest number of tens using what is left—that might seem like 1 and 2, but then we would have both 0’s for the ones; the next best we can do is 0 and 2. That leaves 1 and 0 for the ones. Therefore: 7201 + 6120 =.13,321

What positive value should replace *x* in this statement to make it true?

*[Sprint #29] *

55 × 59 – 53 × 57 = (57 – 2)(57 + 2) – (55 – 2)(55 + 2) = 57^{2}– 2^{2}– (55^{2}– 2^{2}) = 57^{2}– 55^{2}= (57 + 55)(57 – 55) = 112 × 2 = 224 = 15^{2}– 1, so x =.15

**♦ Page 1 of the linked PDF contains PROBLEMS & SOLUTIONS . **

Page 2 contains ONLY PROBLEMS. ♦