Some problems

1)How many times is the digit 2 used when writing the numbers 1,2,.......,1000?

2) Define the special mean of two numbers x and y as the average of their Arithmetic mean and geometric mean. For how many x and y in the set {1,2,......,2016} is the special mean of x and y a perfect square?

3) Let N be an integer. Add the digits of N to get a smaller number. Repeat this process until you get a single digital number. For how many integers < 1000 is the final single digit number 5?

4) Find the maximum value of 6x-3y-8z subject to 2(x^2) +3(y^2)+4(z^2)=1.

5) Find the number of integers satisfying the following condition: [(x/100)•[x/100]]=5.

All these problems are from Pre RMO 2016 West Bengal region.

Easy Math Editor

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`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
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1. numbered 2. list	numbered list
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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}`	$\boxed{123}$

Comments

Can anybody give me the source for finding out the answer keys?

Rakhi Bhattacharyya - 4 years, 8 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}`	$\boxed{123}$