Perimeters!

Geometry Level 2

Four ( 4 × 7 4 \times 7 ) rectangles are removed from the corners of a square having side length ( 8 n + 1 8n + 1 ), as shown in the Figure. Determine the largest integer n n for which the perimeter of Figure is less than 1000 1000 .

31 34 32 30 33

Hana Wehbi by Hana Wehbi , 51, USA

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1 solution

The removed rectangles, since they are rectangles, have no effect on the perimeter. Therefore, the perimeter is 4 ( 8 n + 1 ) ⇒ 32 n + 4 4(8\,n+1)\Rightarrow 32\,n+4 . Therefore the answer is ⌊ 996 32 ⌋ ⇒ 31 \lfloor\frac{996}{32}\rfloor \Rightarrow 31 .

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