HMMT November

Algebra Difficulty 4.8 AIME Find the answer

Real numbers $x$ and $y$ satisfy the following equations: $x=\log_{10}(10^{y-1}+1)-1$ and $y=\log_{10}(10^{x}+1)-1$ . Compute $10^{x-y}$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Taking 10 to the power of both sides in each equation, these equations become: $10^{x}=\left(10^{y-1}+1\right) \cdot 10^{-1}$ and $10^{y}=\left(10^{x}+1\right) \cdot 10^{-1}$ . Let $a=10^{x}$ and $b=10^{y}$ . Our equations become: $10a=b/10+1$ and $10b=a+1$ and we are asked to compute $a/b$ . Subtracting the equations gives $10a-10b=b/10-a \Longrightarrow 11a=101b/10$ giving an answer of $101/110$ .

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