April 22, 2019

Define x @ y as (x3 − y)/x, for distinct positive integers x and y. What is the value of 5 @ 10?

Evaluating 5 @ 10 yields (53 − 10)/5 = (125 − 10)/5 = 115/5 = 23.

Define a # b as a2 − b2 − ab, for real numbers a and b. What is the value of 5 # (4 # 3)?

Let's first evaluate the expression inside the parentheses. We have 4 # 3 = 42 − 32 − 4(3) = 16 − 9 − 12 = −5. We now evaluate 5 # (−5) to get 52 − (−5)2 − 5(−5) = 25 − 25 + 25 = 25.

Define m $ n as m2 + 2m/n and define m & n as (m2n2 + mn)/(2n). What is the value of (4 $ 2) & 10?

Again, let’s first evaluate the expression inside the parentheses. We have (4 $ 2) = 42 + 2(4)/2 = 16 + 8/2 = 16 + 4 = 20. We now evaluate 20 & 10 to get [202 − 102 + 20(10)]/[2(10)] = (400 − 100 + 200)/20 = 500/20 = 25.

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Page 2 contains ONLY PROBLEMS. ♦