AMC10 2016 A

AMC10 2016 A · Q23

AMC10 2016 A · Q23. It mainly tests Manipulating equations, Logic puzzles.

A binary operation $\diamond$ has the properties that $a\diamond(b\diamond c)=(a\diamond b)\cdot c$ and that $a\diamond a=1$ for all nonzero real numbers $a$, $b$, and $c$. (Here the dot $\cdot$ represents the usual multiplication operation.) The solution to the equation $2016\diamond(6\diamond x)=100$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p+q$?

一种二元运算 $\diamond$ 满足性质：对所有非零实数 $a,b,c$，有 $a\diamond(b\diamond c)=(a\diamond b)\cdot c$，且 $a\diamond a=1$。（这里的点号 $\cdot$ 表示通常的乘法运算。）方程 $2016\diamond(6\diamond x)=100$ 的解可写成 $\frac{p}{q}$，其中 $p$ 和 $q$ 是互素的正整数。求 $p+q$。

(A) 109 109

(B) 201 201

(D) 3049 3049

(E) 33,601 33,601

Answer

Correct choice: (A)

正确答案：（A）

Solution

Answer (A): From the given properties, $a \diamond 1 = a \diamond (a \diamond a) = (a \diamond a)\cdot a = 1\cdot a = a$ for all nonzero $a$. Then for nonzero $a$ and $b$, $a = a \diamond 1 = a \diamond (b \diamond b) = (a \diamond b)\cdot b$. It follows that $a \diamond b = \frac{a}{b}$. Thus $100 = 2016 \diamond (6 \diamond x) = 2016 \diamond \frac{6}{x} = \frac{2016}{6/x} = 336x,$ so $x = \frac{100}{336} = \frac{25}{84}$. The requested sum is $25+84=109$.

答案 (A)：由已知性质，对所有非零$a$，有 $a \diamond 1 = a \diamond (a \diamond a) = (a \diamond a)\cdot a = 1\cdot a = a$。然后对非零$a,b$，有 $a = a \diamond 1 = a \diamond (b \diamond b) = (a \diamond b)\cdot b$。因此 $a \diamond b = \frac{a}{b}$。于是 $100 = 2016 \diamond (6 \diamond x) = 2016 \diamond \frac{6}{x} = \frac{2016}{6/x} = 336x,$ 所以 $x = \frac{100}{336} = \frac{25}{84}$。所求和为 $25+84=109$。

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