2001 AMC 10 Problem 22

Below is the professionally curated solution for Problem 22 of the 2001 AMC 10, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2001 AMC 10 solutions, or check the answer key.

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Concepts:magic square linear equation

Difficulty rating: 1530

22.

In the magic square shown, the sums of the numbers in each row, column, and diagonal are the same. Five of these numbers are represented by $v, w, x, y,$ and $z.$ Find $y+z.$

$43$

$44$

$45$

$46$

$47$

Solution:

Since $v$ sits in the first row, first column, and main diagonal, the remaining two entries of each of those lines have equal sums: $25+18=24+w=21+x.$ So $w=19$ and $x=22.$

The anti-diagonal $25, 22, 19$ sums to $66,$ so the magic sum is $66.$ Then $v=66-24-19=23,$ $y=66-18-22=26,$ and $z=66-25-21=20.$

Hence $y+z=46.$ Thus, the correct answer is D.

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