# 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
5 years, 7 months 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...

- 5 years, 7 months ago

Thanks for giving your thoughts. It will be more nice if you post solutions here also.

- 5 years, 7 months ago

please tell the solution of question no. 1

- 5 years, 7 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

- 5 years, 7 months ago

You can also post solutions for others also if you want.

- 5 years, 7 months ago

Yea I am trying the others. Does the solution seem satisfactory ?

- 5 years, 7 months ago

Absolutely yes.

- 5 years, 7 months ago

You need to elaborate more.

- 5 years, 7 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 ?

- 5 years, 7 months 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.

- 5 years, 7 months ago

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

- 5 years, 7 months ago

Wait for 5 days and then you will get other ones also.

- 5 years, 7 months ago

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

INMO 2016 Practice SET -II (ALGEBRA ONLY)

- 5 years, 7 months ago

I can' t find the algebra set pls help

- 5 years, 6 months ago

Type this in search " INMO 2016 PRACTICE SET-II (ALGEBRA ONLY)". It is by me.

- 5 years, 6 months 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

- 5 years, 6 months ago

Fine solution.

- 5 years, 6 months ago

How can we prove that in3 it appears in EXACTLYn times

- 5 years, 6 months ago

Which question?

- 5 years, 6 months ago

Ques3

- 5 years, 6 months ago