# INMO 2017 Practice SET-II (Algebra only)

Hello friends, try these problems and post solutions :

$$(1)$$ For each positive integer $$n$$, show that there exists a positive integer $$k$$ such that

$k = f(x){ (x + 1) }^{ 2n }+ g(x)({ x }^{ 2n }+ 1)$

for some polynomials $f$, $g$ with integer coefficients, and find the smallest such $k$ as a function of $n$.

($2)$ Prove that there exists a polynomial $P(x, y)$ with real coefficients such that $P(x, y) \ge 0$ for all real numbers $x$ and $y$, which cannot be written as the sum of squares of polynomials with real coefficients.

$(3)$ Let ${ x }_{ 1 } = 2$ and ${ x }_{ n + 1 } = { x }^{ 2 }n - { x }_{ n } + 1$ for $n\ge 1$.

Prove that

$\large\ 1 - \frac { 1 }{ { { 2 }^{ 2 } }^{ n - 1 } } < \frac { 1 }{ { x }_{ 1 } } + \frac { 1 }{ { x }_{ 2 } } +...+ \frac { 1 }{ { x }_{ n } } < 1 - \frac { 1 }{ { { 2 }^{ 2 } }^{ n } }$.

$(4)$ Find, with proof, all nonzero polynomials $f(z)$ such that $f({ z }^{ 2 }) + f(z)f(z + 1) = 0$.

$(5)$ Find the least real number $r$ such that for each triangle with side lengths $a$, $b$, $c$,

$\large\ \frac { max(a, b, c) }{ \sqrt [ 3 ]{ { a }^{ 3 } + { b }^{ 3 } + { c }^{ 3 } + 3abc } } < r$.

$(6)$ Let $a$, $b$, $c$ be positive real numbers. Prove that

$\large\ { \left( \sum { \frac { a }{ b+c } } \right) }^{ 2016 } \le \left( \sum { \frac { { a }^{ 2016 } }{ { b }^{ 2016 } + { b }^{ 2015 }c } } \right) { \left( \sum { \frac { a }{ c + a } } \right) }^{ 2015 }$.

Note by Priyanshu Mishra
4 years, 11 months ago

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The answer to4 is f(x)=0 &f(x)=(-1)^n+1(x(1-x))^n

- 4 years, 11 months ago

Correct. Can you show the solution?

- 4 years, 10 months ago

In the he eq put x0 such that f(x0)=0giving that f(x0^2)also equals to zerozero thus by induction there are infinitely many roots of funless x0=0/1 infinitely many roots gives f(x)=0The other case gives after some calc that f(x)=0 for 0,1 only thus the other function is of the formx^m(x-1)^nkeeping in mind that the leading coefficient must be negative keep the above poly in the given funct eq to obtain n=m some thought to the leading coefficient gives the -1at the beginning

- 4 years, 10 months ago

I am in desperate need of geometry and combinatorics sets pls hlp

- 4 years, 10 months ago

I will post it before INMO.

- 4 years, 10 months ago

Is the ans to Q5 2^1/3 i m doubtful

- 4 years, 10 months ago

Yes I think you forgot to write it as 1/cube root 2.

- 4 years, 10 months ago

Will the other discussions also be present on Brilliant Discussion link as this one

- 4 years, 10 months ago

Yes.

- 4 years, 10 months ago

I just got the 3 one.its as follows prove by induction that the given expressions equals1-( (x1x2.......xn)^-1) then make the reqd changes in the inequality. By the way the above induction also showds that xn=(x1x2.......xn-1)+1 Now again induction on the transformed ineq completes the sol

- 4 years, 10 months ago

Nice, concise solution.

HAVE you solved the problem 6?

- 4 years, 10 months ago

Not started yet. But i ve no idea of the first one pls post the sol or give a hint

- 4 years, 10 months ago

I cant understand what u say

- 4 years, 10 months ago

I suppose that 6 uses the chebyshev's once

- 4 years, 10 months ago

This is the proposal so I don't know the answer.

- 4 years, 10 months ago

I have solution of Q5 Is answer (-1)^2k +1 {( z(z-1)}^n

- 2 years, 5 months ago