Olympiad question

1) I want a correct explanation please: A semi-circle is drawn outwardly on chord AB of the circle with center O and unit radius. The perpendicular from O to AB, meets the semi-circle on AB at C. Determine the measure of angle( AOB ) and length AB so that OC has maximum length.

2) Find the number of numbers <= $10^{8}$ which are neither perfect squares,nor perfect cubes, nor perfect fifth powers.

#Geometry #NumberTheory #OlympiadMath #MathProblem #Math

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
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`\boxed{123}`	$\boxed{123}$

Comments

jmo questions........... answer to 1 is 90 degree . Did you complete rest of the questions?

Kumar Ashutosh - 7 years, 9 months ago

The answer to the first is a bit obvious but I like the second one. What is the relation between squares? How can we find the difference between two squared numbers without squaring them? I hope its clear.

Addismiraph Abebe - 7 years, 9 months ago

V easy. It's an elementary problem of inclusion n exclusion. First u take out the no of squares ie 10000 n similarily cubes n fifth powers n subtract the no of 6th, 15th ,10th powers and add the 30th power nos. I hope u got it.. if u want my written solution I ll mail it to u.. but the answer is 10^8 - 10474.

Priyanshi Somani - 7 years, 9 months ago

angle90

abhimanyu manu - 7 years, 9 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$