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INMO 2016 Practice Set 1 (Number Theory only)

Hello friends, try these problems and post solutions :

\(1)\) Prove that for any set {\({ a }_{ 1 },{ a }_{ 2 },...,{ a }_{ n }\)} of positive integers there exists a positive
integer \(b\) such that the set {\({ ba }_{ 1 },b{ a }_{ 2 },...,b{ a }_{ n }\)} consists of perfect power.


\(2)\) Prove that for any integer \(k \ge 2\), the equation \(\large\ \frac { 1 }{ { 10 }^{ n } } = \frac { 1 }{ { n }_{ 1 }! } + \frac { 1 }{ { n }_{ 2 }! } +...+ \frac { 1 }{ { n }_{ k }! }\) does not have integer solutions such that \(1 \le { n }_{ 1 } < { n }_{ 2 } <...< { n }_{ k }\).


\(3)\) Prove that for every integer \(n\) there is a positive integer \(k\) such that \(k\) appears in exactly \(n\) non-trivial Pythagorean triples.


\(4)\) Determine all solutions \((x, y, z)\) of positive integers such that \(\large\ { (x + 1) }^{ y + 1 } + 1 = { (x + 2) }^{ z + 1 }\).


\(5)\) Let \(m\), \(n\) be positive integers such that \(\large\ A = \frac { { (m + 3) }^{ n } + 1 }{ 3m }\). Prove that \(A\) is odd.


\(6)\) Let \(\large\ { a }_{ 1 },{ a }_{ 2 },...,{ a }_{ { 10 }^{ 6 } }\) be integers between \(1\) and \(9\), inclusive. Prove that at most \(100\) of the numbers \(\large\ \overline { { a }_{ 1 }{ a }_{ 2 }...{ a }_{ k } } (1 \le k \le { 10 }^{ 6 })\) are perfect squares.

Note by Priyanshu Mishra
11 months, 3 weeks ago

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Ques3 Himanshu Singh · 10 months, 4 weeks ago

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How can we prove that in3 it appears in EXACTLYn times Himanshu Singh · 11 months ago

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@Himanshu Singh Which question? Priyanshu Mishra · 10 months, 4 weeks ago

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The4 th can be slvd as follows first prpve that a+1divides m and further that m is odd shpws that a is even.Moreover we show that n has the form 2k (i ve convtd the eq to a^m+1=a+1)^n )go on toshow that a^m is the product of 2consecutive even numbers and since it is of the forma^m it can be only 8or 4 is my sol satisfactory Himanshu Singh · 11 months ago

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@Himanshu Singh Fine solution. Priyanshu Mishra · 10 months, 4 weeks ago

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I can' t find the algebra set pls help Himanshu Singh · 11 months ago

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@Himanshu Singh Type this in search " INMO 2016 PRACTICE SET-II (ALGEBRA ONLY)". It is by me. Priyanshu Mishra · 11 months ago

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@Priyanshu Mishra ...... when are u posting the other sets ? i.e. Geometry, Algebra,Combinatorics ? Shrihari B · 11 months, 1 week ago

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@Shrihari B Shrihari B .... I have posted Algebra set.

INMO 2016 Practice SET -II (ALGEBRA ONLY) Priyanshu Mishra · 11 months ago

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@Shrihari B Wait for 5 days and then you will get other ones also. Priyanshu Mishra · 11 months, 1 week ago

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@Priyanshu Mishra Did you want to mention something else in the problem number 1? Its quite trivial i guess. If we take b=(ai)^k we are done isn't it ? Shrihari B · 11 months, 2 weeks ago

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@Shrihari B Yes you can take that but you have to show that the set contain s perfect power.

Hint: use congruences and CRT by assuming something for the set. Priyanshu Mishra · 11 months, 2 weeks ago

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Comment deleted 11 months ago

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@Shrihari B The solution indeed exists.

Hint:set \(a =x + 1\) and similarly for b and c. Then apply congruences. The only solution is \((1, 2, 1)\). Priyanshu Mishra · 11 months, 2 weeks ago

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For Q.1 we can take b to be any of a1, a2,..., aN and the condition is satisfied... I think I have made a mistake but where... Sarthak Behera · 11 months, 2 weeks ago

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@Sarthak Behera Thanks for giving your thoughts. It will be more nice if you post solutions here also. Priyanshu Mishra · 11 months, 2 weeks ago

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@Priyanshu Mishra please tell the solution of question no. 1 Onkar Tiwari · 11 months, 1 week ago

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My solution for question 3 is as follows Consider n primitive pythagorean triples (a1,b1,c1),(a2,b2,c2),......,(an,bn,cn) where ai^2+bi^2=ci^2. Now consider the numbers a1,a2,a3,.....an. Now let the LCM of these n numbers be L. Now multiply pythagorean triple (aj,bj,cj ) by (L/aj). Note that this too forms a pythagorean triple. And also see that each of these pythagorean triple contains the element L. Hence proved :) Please can someone rate my solution out of 10 ? I am bad at proof writing Sorry for not using latex as I am in a hurry. I will get back to the other sums Shrihari B · 11 months, 2 weeks ago

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@Shrihari B You need to elaborate more. Priyanshu Mishra · 11 months, 2 weeks ago

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@Shrihari B Thanks for your solution.

You can also post solutions for others also if you want. Priyanshu Mishra · 11 months, 2 weeks ago

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@Priyanshu Mishra Yea I am trying the others. Does the solution seem satisfactory ? Shrihari B · 11 months, 2 weeks ago

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@Shrihari B Absolutely yes. Priyanshu Mishra · 11 months, 2 weeks ago

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