If \(n\) is a positive integer . Then find values of \(n\) such that \(\sqrt{n-1}+\sqrt{n+1}\) is a rational number.

If a define a sequence of natural numbers such that \(F_{0}=F_{1}=1,F_{n}=F_{n-1}+F_{n-2}\) for all \(n\geq2\) . Then prove that \(F_{n+3}\) divides \(7(F_{n+2})^{3}-(F_{n})^{3}-(F_{n+1})^{3}\).

If \(x,y,z,a,b,c\) are non-zero real numbers such that \(\frac{x^{2}(y+z)}{a^{3}}=\frac{y^{2}(x+z)}{b^{3}}=\frac{z^{2}(y+x)}{c^{3}}=\frac{xyz}{abc}=1\) . Prove that \(a^{3}+b^{3}+c^{3}+abc=0\).

If \(x,y,z>1\) prove that \(\sum_{cyclic}^{x,y,z}\frac{x^{4}}{(y-1)^{2}}\geq48\).

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## Comments

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TopNewestJust algebra: \[ \begin{align*} 7F_{n+2}^3-F_{n+1}^3-F_n^3 &= 7F_{n+2}^3-(F_{n+1}+F_n)(F_{n+1}^2-F_{n+1}F_{n}+F_n^2)\\ &= F_{n+2}(7F_{n+2}^2-F_{n+1}^2+F_{n+1}F_n-F_n^2)\\ &= F_{n+2}(7(F_{n+1}+F_n)^2-F_{n+1}^2+F_{n+1}F_n-F_n^2)\\ &= F_{n+2}(6F_{n+1}^2+15F_{n+1}F_n+6F_n^2)\\ &= 3F_{n+2}(2F_{n+1}+F_n)(F_{n+1}+2F_n)\\ &= 3F_{n+2}F_{n+3}(F_{n+1}+2F_n) \end{align*} \] which is clearly divisible by \(F_{n+3}\).

Note that \(a^3=x^2(y+z)\). Multiplying the three cyclic variants of this equation gives \(a^3b^3c^3=x^2y^2z^2(x+y)(y+z)(z+x)\). However, we also know \(abc=xyz\) so plugging in and simplifying gives \(xyz=(x+y)(y+z)(z+x)\implies xyz+\sum\limits_{sym}x^2y=0\). However, note that \(a^3+b^3+c^3+abc=xyz+\sum\limits_{cyc}x^2(y+z)=xyz+\sum\limits_{sym}x^2y=0\) so we're done.

\(\displaystyle\sum_{cyc}\dfrac{a^4}{(b-1)^2}=\sum_{cyc}\dfrac{(a-1+1)^4}{(b-1)^2}\ge \sum_{cyc}\dfrac{(2\sqrt{a-1})^4}{(b-1)^2}=\sum_{cyc}\dfrac{16(a-1)^2}{(b-1)^2}\ge 48\) where both inequalities are by AM-GM.

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n = 1.25 is a solution for question 1

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An easier way to solve question number 2. Is by writing in form of \[=8F^{3}_{n+2}-F^{3}_{n}-(F^{3}_{n+1}+F^{3}_{n+2})\] And then expand we get in one bracket \((2F_{n+2}-F_{n})=(F_{n+2}+F_{n+2}-F_{n})=(F_{n+2}+F_{n+1})=(F_{n+3})\) and other bracket as \((F_{n+1}+F_{n+2})=(F_{n+3})\) and hence we take \(F_{n+3}\) as overall common and the result gets proved .

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I could only solve the first two. My solution for the first question is about the same.

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In question 4th X^2 +4>= 4x thus x^4>=16(x-1)^2 Dividing by (y-1)^2 and adding for all x,y,z Then apply AM.GM and get the result

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In 3 question put values of a^3 b^3 c^3 and abc from the question and it can be easily proved.

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q1) write as one radical \[\sqrt{2n+2\sqrt{n^2-1}}\]. let this be rational number k. then \[2n+2\sqrt{n^2-1}=k^2\] the LHS is rational, so RHS must be rational.this implies n^2-1 is a square of a rational number. as n is an integer, that rational number Must also be an integer. this implies \[n^2-1=a\Longrightarrow (n-a)(n+a)=1=1*1=-1*-1\] \[n=1\quad or\quad -1(n/a)\] at n=1, the expression is not a perfect square, so no positive integral solution.

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@Nihar Mahajan @Surya Prakash

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n = 1.25 is a solution for question 1

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When will be the result of rmo 2015 will be declared for rajasthan region??

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For question 1 Since the sum of two square roots is rational implies both are rational. But n is integer which forces each term to be an integer We can just say that both n-1 and n+1 are perfect squares and the difference between these two perfect squares is 2 which does not yield any solution.

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"the sum of two square roots is rational implies both are rational", this is not justified, marks would not have been given. justify it please.

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I don't think any justification is required for such a trivial statement. Still if u want it is as follows . The two square roots could be (rational,rational),(rational,irrational),(irrational,irrational) rational plus irrational can't give rational. irrational plus rational can give rational only if both cancel each other which means one of the square roots is negative which is again ruled out So u r left with the only possibility of both being rational.

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Another example where the terms are explicitly square roots (just not of integers) is:

\[ \sqrt{ 3 + 2 \sqrt{2} } + \sqrt{ 3 - 2 \sqrt{2} } = 2 \]

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These are questions from AMTI final 2014 for junior level isn't it ?

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@Daniel Liu

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Q1 there are no such numbers

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Prove it

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Why does the question ask for real numbers? Every positive integer n will make the expression a real number.

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If p,q,r are positive and in AP.Then show that quadratic equation px^2+qx+r=0 are real for |p/r-7|greater than or equal to 4√3

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