INMO 2016 Practice Set 1 (Number Theory only)

Hello friends, try these problems and post solutions :

1)1) Prove that for any set {a1,a2,...,an{ a }_{ 1 },{ a }_{ 2 },...,{ a }_{ n }} of positive integers there exists a positive
integer bb such that the set {ba1,ba2,...,ban{ ba }_{ 1 },b{ a }_{ 2 },...,b{ a }_{ n }} consists of perfect power.


2)2) Prove that for any integer k2k \ge 2, the equation  110n=1n1!+1n2!+...+1nk!\large\ \frac { 1 }{ { 10 }^{ n } } = \frac { 1 }{ { n }_{ 1 }! } + \frac { 1 }{ { n }_{ 2 }! } +...+ \frac { 1 }{ { n }_{ k }! } does not have integer solutions such that 1n1<n2<...<nk1 \le { n }_{ 1 } < { n }_{ 2 } <...< { n }_{ k }.


3)3) Prove that for every integer nn there is a positive integer kk such that kk appears in exactly nn non-trivial Pythagorean triples.


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


5)5) Let mm, nn be positive integers such that  A=(m+3)n+13m\large\ A = \frac { { (m + 3) }^{ n } + 1 }{ 3m }. Prove that AA is odd.


6)6) Let  a1,a2,...,a106\large\ { a }_{ 1 },{ a }_{ 2 },...,{ a }_{ { 10 }^{ 6 } } be integers between 11 and 99, inclusive. Prove that at most 100100 of the numbers  a1a2...ak(1k106)\large\ \overline { { a }_{ 1 }{ a }_{ 2 }...{ a }_{ k } } (1 \le k \le { 10 }^{ 6 }) are perfect squares.

Note by Priyanshu Mishra
3 years, 10 months ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

Sort by:

Top Newest

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 - 3 years, 10 months ago

Log in to reply

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

Priyanshu Mishra - 3 years, 10 months ago

Log in to reply

please tell the solution of question no. 1

onkar tiwari - 3 years, 9 months ago

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

Shrihari B - 3 years, 9 months ago

Log in to reply

Thanks for your solution.

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

Priyanshu Mishra - 3 years, 9 months ago

Log in to reply

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

Shrihari B - 3 years, 9 months ago

Log in to reply

@Shrihari B Absolutely yes.

Priyanshu Mishra - 3 years, 9 months ago

Log in to reply

You need to elaborate more.

Priyanshu Mishra - 3 years, 9 months ago

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 ?

Shrihari B - 3 years, 9 months ago

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.

Priyanshu Mishra - 3 years, 9 months ago

Log in to reply

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

Shrihari B - 3 years, 9 months ago

Log in to reply

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

Priyanshu Mishra - 3 years, 9 months ago

Log in to reply

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

INMO 2016 Practice SET -II (ALGEBRA ONLY)

Priyanshu Mishra - 3 years, 9 months ago

Log in to reply

I can' t find the algebra set pls help

himanshu singh - 3 years, 9 months ago

Log in to reply

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

Priyanshu Mishra - 3 years, 9 months ago

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

himanshu singh - 3 years, 9 months ago

Log in to reply

Fine solution.

Priyanshu Mishra - 3 years, 9 months ago

Log in to reply

How can we prove that in3 it appears in EXACTLYn times

himanshu singh - 3 years, 9 months ago

Log in to reply

Which question?

Priyanshu Mishra - 3 years, 9 months ago

Log in to reply

Ques3

himanshu singh - 3 years, 9 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...