2021 JMPSC Sprint Problems/Problem 20

Problem

For all integers $x$ and $y$ , define the operation $\Delta$ as $x \Delta y = x^3+y^2+x+y.$ Find $\sqrt{\dfrac{257 \Delta 256}{258}}.$

Solution

Let $258=a$ . Then, $257=a-1$ and $256=a-2$ . We substitute these values into expression $(1)$ to get $\sqrt{\frac{(a-1) \Delta (a-2)}{a}}.$ Recall the definition for the operation $\Delta$ ; using this, we simplify our expression to $\sqrt{\frac{(a-1)^3+(a-2)^2+(a-1)+(a-2)}{a}}.$ We have $(a-1)^3=a^3-3a^2+3a-1$ and $(a-2)^2=a^2-4a+4$ , so we can expand the numerator of the fraction within the square root as $a^3-3a^2+3a-1+a^2-4a+4+a-1+a-2=a^3-2a^2+a$ to get $\sqrt{\frac{a^3-2a^2+a}{a}}=\sqrt{a^2-2a+1}=\sqrt{(a-1)^2}=a-1=\boxed{257}.$ ~samrocksnature

Solution 2

Basically the same as above, but instead we can let $257 = 256 + 1$ . Then we have $\sqrt{\frac{(256+1)(256^2 + 256 + 1) + 1(256^2 + 257) + 256}{258}},$ $\sqrt{\frac{258(256^2 + 257) + 256}{258}},$ $\sqrt{256^2 + 256 + 256 + 1} =$ $\sqrt{256^2 + 2\cdot256 + 1} =$ $\sqrt{(256+1)^2} =$ $\sqrt{(257^2)}$