RMO Part -9

1)Let $1<a_1<a_2<a_3<.....a_{51}<142$ for positive integers $a_1,a_2,a_3,.....a_{51}$ .

Prove that among the 50 consecutive differences some value must occur at least 12 times.

2)Prove that in any perfect square the three digits immediately to the left of the unit digit cannot be 101.

Try to solve these 2 problems in 1 hour.

Also try my set RMO.

#Algebra #NumberTheory #RMO #INMO

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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

Hints 1) Try Pigeonhole

2) Mods.

Alan Yan - 5 years, 8 months ago

@Alan Yan @Nihar Mahajan @Swapnil Das @Mehul Arora @Svatejas Shivakumar @Adarsh Kumar @Sharky Kesa @Shivam Jadhav @Dev Sharma Try this guys!!

naitik sanghavi - 5 years, 8 months ago

For question 2

The last four digits will be :

1010 - Not possible because perfect squares can't end in odd number of zeroes.

1011 - Not possible because perfect squares aren't of the form 4k + 3.

1012 - Not possible because perfect squares aren't of the form 4k + 2.

1013 - Not possible because perfect squares aren't of the form 8k + 5.

1014 - Not possible because perfect squares aren't of the form 4k + 2.

1015 - Not possible because perfect squares aren't of the form 4k + 3.

1016 - Not possible because perfect squares aren't of the form 16k + 8.

1017 - Not possible because perfect squares aren't of the form 5k + 2.

1018 - Not possible because perfect squares aren't of the form 4k + 2.

1019 - Not possible because perfect squares aren't of the form 4k + 3.

@naitik sanghavi Hope this works!

Ankit Kumar Jain - 4 years, 4 months ago

@Rajdeep Dhingra ,I want to ask you something can u please give me your email or whatsapp no.

My email - [email protected]. Reply asap.

naitik sanghavi - 3 years, 6 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}`	$\boxed{123}$