AMC10 2003 A

AMC10 2003 A · Q6

AMC10 2003 A · Q6. It mainly tests Absolute value.

Define $x\heartsuit y$ to be $|x - y|$ for all real numbers $x$ and $y$. Which of the following statements is not true?

定义对于所有实数 $x$ 和 $y$，$x\heartsuit y$ 为 $|x - y|$。以下哪个陈述不正确？

(A) $x\heartsuit y = y\heartsuit x$ for all $x$ and $y$ 对所有 $x$ 和 $y$，$x\heartsuit y = y\heartsuit x$

(B) $2(x\heartsuit y) = (2x)\heartsuit(2y)$ for all $x$ and $y$ 对所有 $x$ 和 $y$，$2(x\heartsuit y) = (2x)\heartsuit(2y)$

(D) $x\heartsuit x = 0$ for all $x$ 对所有 $x$，$x\heartsuit x = 0$

(E) $x\heartsuit y > 0$ if $x \ne y$ 若 $x \ne y$，则 $x\heartsuit y > 0$

Answer

Correct choice: (C)

正确答案：（C）

Solution

(C) For example, $-1 \heartsuit 0 = |-1-0| = 1 \ne -1$. All the other statements are true: (A) $x \heartsuit y = |x-y| = |-(y-x)| = |y-x| = y \heartsuit x$ for all $x$ and $y$. (B) $2(x \heartsuit y) = 2|x-y| = |2x-2y| = (2x) \heartsuit (2y)$ for all $x$ and $y$. (D) $x \heartsuit x = |x-x| = 0$ for all $x$. (E) $x \heartsuit y = |x-y| > 0$ if $x \ne y$.

（C）例如，$-1 \heartsuit 0 = |-1-0| = 1 \ne -1$。其他所有陈述都为真：（A）对所有 $x$ 和 $y$，有 $x \heartsuit y = |x-y| = |-(y-x)| = |y-x| = y \heartsuit x$。（B）对所有 $x$ 和 $y$，有 $2(x \heartsuit y) = 2|x-y| = |2x-2y| = (2x) \heartsuit (2y)$。（D）对所有 $x$，有 $x \heartsuit x = |x-x| = 0$。（E）若 $x \ne y$，则 $x \heartsuit y = |x-y| > 0$。

Topics

AL.004 Absolute value