Difference between revisions of "1986 INMO"
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\log_{4} z + \log_{16} x + \log_{16} y = 2 | \log_{4} z + \log_{16} x + \log_{16} y = 2 | ||
\end{cases} | \end{cases} | ||
| + | |||
| + | ==Problem 3== | ||
| + | Two circles with radii a and b respectively touch each other externally. Let c be the radius of a circle that touches these two circles as well as a common tangent to the two circles Prove that: | ||
| + | |||
| + | <cmath>\frac{1}{\sqrt{c}} = \frac{1}{\sqrt{a}} + \frac{1}{\sqrt{b}}</cmath> | ||
Revision as of 07:21, 2 October 2025
This was the first INMO conducted, it was conducted by the Indian Institute of Science, Department of Applied Mathematics
Problem 1
A person who left home between 4 p.m. and 5 p.m. returned between 5 p.m. and 6 p.m. and found that the hands of his watch had exchanged places. When did he go out ?
Problem 2
Solve:
\begin{cases} \log_{2} x + \log_{4} y + \log_{4} z = 2 \\[6pt] \log_{3} y + \log_{9} z + \log_{9} x = 2 \\[6pt] \log_{4} z + \log_{16} x + \log_{16} y = 2 \end{cases}
Problem 3
Two circles with radii a and b respectively touch each other externally. Let c be the radius of a circle that touches these two circles as well as a common tangent to the two circles Prove that:
![\[\frac{1}{\sqrt{c}} = \frac{1}{\sqrt{a}} + \frac{1}{\sqrt{b}}\]](http://latex.artofproblemsolving.com/6/f/a/6fa429a041db3675cc939c4d84aa176e54f9b696.png)
