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# 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.

Note by Naitik Sanghavi
1 year, 8 months ago

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Hints 1) Try Pigeonhole

2) Mods. · 1 year, 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! · 3 months, 1 week ago