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imc

algebra intermediate

Problem

Mr. Zhou places all the integers from to into a by grid. He places in the middle square (eighth row and eighth column) and places other numbers one by one clockwise, as shown in part in the diagram below. What is the sum of the greatest number and the least number that appear in the second row from the top?
problem
(A)
(B)
(C)
(D)
Solution
In the diagram below, the red arrows indicate the progression of numbers. In the second row from the top, the greatest number and the least number are and respectively. Note that the numbers in the yellow cells are consecutive odd perfect squares, as we can prove by induction. By observations, we proceed as follows: \begin{alignat}{6} A=15^2=225, \ B=13^2=169 \quad &\implies \quad &C &= \hspace{1mm}&&A-14\hspace{1mm} &= 211& \\ \quad &\implies \quad &D &= &&C-1 &= 210& \\ \quad &\implies \quad &E &= &&B-12 &= 157&. \end{alignat} Therefore, the answer is
Final answer
A