UKMT - Intermediate Mathematical Challenge - 2023 - 22 | Exact Science

22

The numbers x and y satisfy both of the equations.

23x + 977y = 2023 and 977x + 23y = 2977.

What is the value of x² − y²?

A. 1

B. 2

C. 3

D. 4

E. 5

Answer

E. 5

‍

SOLUTIOn

Adding both equations gives: 1000x + 1000y = 5000. So x + y = 5.Subtracting the first equation from the second gives: 954x − 954y = 954. Therefore x − y = 1.

Hence x² − y² = (x + y) (x − y) = 5 × 1 = 5.

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