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.

No vote yet

1 vote

×

Problem Loading...

Note Loading...

Set Loading...

Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

TopNewestFor 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...

Log in to reply

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

Log in to reply

please tell the solution of question no. 1

Log in to reply

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

Log in to reply

Thanks for your solution.

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

Log in to reply

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

Log in to reply

Log in to reply

You need to elaborate more.

Log in to reply

@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 ?

Log in to reply

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.

Log in to reply

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

Log in to reply

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

Log in to reply

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

INMO 2016 Practice SET -II (ALGEBRA ONLY)

Log in to reply

I can' t find the algebra set pls help

Log in to reply

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

Log in to reply

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

Log in to reply

Fine solution.

Log in to reply

How can we prove that in3 it appears in EXACTLYn times

Log in to reply

Which question?

Log in to reply

Ques3

Log in to reply