Question

Wilma, Xavier, Yaska and Zakir are four young friends, who have a passion for integers. One day, each of them selects one integer and writes it on a wall. The writing on the wall shows that Xavier and Zakir picked positive integers ( $X, Z \in \mathbb{Z}^+$ ), Yaska picked a negative one ( $Y < 0$ ), while Wilma's integer $W$ is either negative, zero or positive.

The following equations hold: $W^4 + X^3 + Y^2 + Z \le 4$ $X^3 + Z \ge 2$ $W^4 + Y^2 \le 2$ $Y^2 + Z \ge 3$

Given the above, which of these can $W^2 + X^2 + Y^2 + Z^2$ possibly evaluate to?

Accepted Answer

D. Since $Y < 0$, $Y^2 \ge 1$. $Y^2 + Z \ge 3 \implies Z \ge 2$ (since $Y^2 \le 2$). Since $Z \ge 2$, $X^3 \ge 1$, $Y^2 = 1 \implies Y = -1$. For $W^4 + X^3 + 1 + Z \le 4$ with $X \ge 1, Z \ge 2$, we must have $X = 1, Z = 2, W = 0$. Then $W^2 + X^2 + Y^2 + Z^2 = 0^2 + 1^2 + (-1)^2 + 2^2 = 0 + 1 + 1 + 4 = 6$.

Integer Inequalities System

Explanation

Practise this under exam conditions