AMC12 2019 B

AMC12 2019 B · Q9

AMC12 2019 B · Q9. It mainly tests Logarithms (rare), Triangles (properties).

For how many integral values of $x$ can a triangle of positive area be formed having side lengths $\log_2 x$, $\log_4 x$, and 3?

对于多少个整数值 $x$，可以形成一个正面积的三角形，其边长为 $\log_2 x$、$\log_4 x$ 和 3？

(A) 57 57

(B) 59 59

(D) 62 62

(E) 63 63

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): A triangle of positive area can be formed if and only if all three Triangle Inequalities are satisfied: $\log_2 x + \log_4 x > 3,$ $3 + \log_4 x > \log_2 x,$ and $3 + \log_2 x > \log_4 x.$ By the Change of Base Formula, $\log_4 x=\dfrac{\log_2 x}{\log_2 4}=\dfrac{1}{2}\log_2 x.$ Therefore the inequalities are equivalent, respectively, to $\log_2 x>2,$ $6>\log_2 x,$ and $\log_2 x>-6.$ These are in turn equivalent to $x>4$, $x<64$, and $x>\dfrac{1}{64}$. Because $x$ must be an integer, it must lie between $5$ and $63$, inclusive, and there are $63-4=59$ such numbers.

答案（B）：当且仅当满足三条三角形不等式时，才能构成面积为正的三角形： $\log_2 x + \log_4 x > 3,$ $3 + \log_4 x > \log_2 x,$ 且 $3 + \log_2 x > \log_4 x.$ 由换底公式， $\log_4 x=\dfrac{\log_2 x}{\log_2 4}=\dfrac{1}{2}\log_2 x.$ 因此，上述不等式分别等价于 $\log_2 x>2,$ $6>\log_2 x,$ 且 $\log_2 x>-6.$ 进一步分别等价于 $x>4$、$x<64$、$x>\dfrac{1}{64}$。由于 $x$ 必须是整数，所以 $x$ 必须在 $5$ 到 $63$（含）之间，共有 $63-4=59$ 个这样的整数。

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