AMC12 2016 A

AMC12 2016 A · Q20

AMC12 2016 A · Q20. 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 $p+q$?

二元运算$\diamond$满足：对所有非零实数$a,b,c$，都有$a\diamond(b\diamond c)=(a\diamond b)\cdot c$，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$p,q$为互质的正整数。求$p+q$。

(A) 109 109

(B) 201 201

(D) 3049 3049

(E) 33601 33601

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 $a\diamond b=\dfrac{a}{b}$. Thus $$ 100=2016\diamond(6\diamond x)=2016\diamond \dfrac{6}{x}=\dfrac{2016}{\dfrac{6}{x}}=336x, $$ so $x=\dfrac{100}{336}=\dfrac{25}{84}$. The requested sum is $25+84=109$.

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

Topics