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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
9 months, 2 weeks ago

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Ques3 · 8 months, 3 weeks ago

How can we prove that in3 it appears in EXACTLYn times · 8 months, 3 weeks ago

Which question? · 8 months, 3 weeks ago

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 · 8 months, 3 weeks ago

Fine solution. · 8 months, 3 weeks ago

I can' t find the algebra set pls help · 8 months, 3 weeks ago

Type this in search " INMO 2016 PRACTICE SET-II (ALGEBRA ONLY)". It is by me. · 8 months, 3 weeks ago

@Priyanshu Mishra ...... when are u posting the other sets ? i.e. Geometry, Algebra,Combinatorics ? · 9 months ago

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

INMO 2016 Practice SET -II (ALGEBRA ONLY) · 8 months, 4 weeks ago

Wait for 5 days and then you will get other ones also. · 9 months ago

@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 ? · 9 months, 1 week ago

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. · 9 months, 1 week ago

Comment deleted 9 months ago

The solution indeed exists.

Hint:set $$a =x + 1$$ and similarly for b and c. Then apply congruences. The only solution is $$(1, 2, 1)$$. · 9 months, 1 week ago

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... · 9 months, 1 week ago

Thanks for giving your thoughts. It will be more nice if you post solutions here also. · 9 months, 1 week ago

please tell the solution of question no. 1 · 9 months ago

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 · 9 months, 1 week ago

You need to elaborate more. · 9 months, 1 week ago

You can also post solutions for others also if you want. · 9 months, 1 week ago

Yea I am trying the others. Does the solution seem satisfactory ? · 9 months, 1 week ago