Understand the problem

 For \(n\ge3 \), determine all real solutions of the system of \(n\) equations :

                                               \(x_1+x_2+\cdots+x_{n-1}=\frac{1}{x_n}\)

                                                                 ………………………….

                      \(x_1+x_2+\cdots+x_{i-1}+x_{i+1}+\cdots+x_{n-1}+x_n=\frac{1}{x_i}\)

                                                                 …………………………..

                                               \(x_2+x_3+\cdots+x_{n-1}+x_n=\frac{1}{x_1}\).

Source of the problem

I.S.I. (Indian Statistical Institute) B.Math.(Hons.) Entrance Examination 2008. Subjective Problem no. 9.

Topic

System of \(n\) Equations

Difficulty Level

9 out of 10

Suggested Book

Mathematical Olympiad Challenges by Titu Andreescu & Razvan Gelca.

Start with hints

Do you really need a hint? Try it first!

 Let \( s = x_1+x_2+\cdots+x_i+\cdots +x_{n-1}+x_n\).

Then the system of \(n\) equations are equivalent to \(x_i^2-sx_i+1=0\) for \(i=1,2,….,n\).

It follows that \(x_1,x_2,…..,x_{n-1},x_n\) are solutions to the quadratic equation: \(u^2-su+1=0\).                                                   \(\cdots\cdots\cdots\cdots (i)\)

Now we have two possible cases.

Case – I :  Two roots of the equation (i) are equal, then , all \(x_i\) are equal. 

i.e. , \(x_1=x_2=\cdots=x_{n-1}=x_n=u\).

\(\Rightarrow s=nu\).

Putting this value of \(s\) in equation (i) we get, \((n-1)u^2=1\).

\(\Rightarrow u=\frac{1}{\pm \sqrt{n-1}}\).

Case – II : Two roots of equation (i) are not equal. Then let these two roots are \(u_1 \) and \(u_2\). Also let among {\(x_1,x_2,……,x_{n-1},x_n\)}   \( k\)  of them equal to \(u_1\) and \((n-k) \) of them equal to \(u_2\), where \(0<k<n\).

      In this case, we have , (a) \(u_1+u_2=s\).    [Sum of roots of equation (i)]

                                              (b) \(u_1\cdot u_2=1\).    [ Product of roots of equation (i)].

Now,.   \(s=x_1+x_2+\cdots+x_{n-1}+x_n=k\cdot u_1+(n-k)\cdot u_2\)

        \(\Rightarrow s=(u_1+u_2)+(k-1)\cdot u_1+(n-k-1)\cdot u_2\) 

        \(\Rightarrow (k-1)\cdot u_1+(n-k-1)\cdot u_2=0\)   [using (a)]

        \(\Rightarrow  (k-1)\cdot u_1^2=(k+1-n)\cdot u_1\cdot u_2=k+1-n\le 0\) [using (b) and since ,\( k<n\Rightarrow k+1\le n\)].

      \(\Rightarrow u_1^2\le 0\)

But \(u_1\neq 0\) as the coefficient of \(u^0\) is 1(\(\neq 0\)) in equation (i).

        \(\Rightarrow u_1^2<0\)

\(\Rightarrow u_1\) is not real. So we have no real solution to the system of \(n\) equations in this case.

 

 

 

Thus the total no. of real solutions to the system of \(n\) equations is two and these two solutions are:

                        \(x_1=x_2=\cdots=x_{n-1}=x_n=\frac{1}{\sqrt{n-1}}\)      and

                        \(x_1=x_2=\cdots=x_{n-1}=x_n=-\frac{1}{\sqrt{n-1}}\).(Ans.)

Connected Program at Cheenta

I.S.I. & C.M.I. Entrance Program

Indian Statistical Institute and Chennai Mathematical Institute offer challenging bachelor’s program for gifted students. These courses are B.Stat and B.Math program in I.S.I., B.Sc. Math in C.M.I.

The entrances to these programs are far more challenging than usual engineering entrances. Cheenta offers an intense, problem-driven program for these two entrances.

Similar Problem

The Product of Digits, ISI Entrance 2017, Subjective Solution to problem – 5.

Understand the problem  Let \(g : \mathbb{N} \to \mathbb{N} \) with \( g(n) \) being the product of digits of \(n\).        (a) Prove that \( g(n)\le n\) for all \( n \in \mathbb{N} \) .        (b) Find all \(n \in \mathbb{N} \) , for which \( n^2-12n+36=g(n) \)....

Three Primes, ISI Entrance 2017, Subjective solution to Problem 6.

Understand the problemLet \(p_1,p_2,p_3\)  be primes with \(p_2\neq p_3\), such that \(4+p_1p_2\) and \(4+p_1p_3\) are perfect squares. Find all possible values of \(p_1,p_2,p_3\).   Start with hintsDo you really need a hint? Try it first!Let \(4+p_1p_2=m^2\) and...

System of n equations, ISI Entrance 2008, Solution to Subjective Problem No. 9.

Understand the problem For \(n\ge3 \), determine all real solutions of the system of \(n\) equations :                                                \(x_1+x_2+\cdots+x_{n-1}=\frac{1}{x_n}\)                                                                ...

C.M.I. 2019 Entrance – Answer Key, Sequential Hints

CMI (Chennai Mathematical Institute) Entrance 2019, Sequential hints, answer key, solutions.

A Trigonometric Substitution, ISI Entrance 2019, Subjective Solution to Problem – 6 .

Understand the problem For all natural numbers\(n\), let          \(A_n=\sqrt{2-\sqrt{2+\sqrt{2+\cdots +\sqrt{2}}}}\)           (\( n\) many radicals) (a) Show that for \(n\ge 2,  A_n=2\sin \frac{π}{2^{n+1}}\). (b) Hence, or otherwise, evaluate the limit              ...

Two Similar Triangles, ISI B.Math/B.Stat Entrance 2018, Subjective Solution of Problem No. 2

Understand the problemSuppose that \(PQ\) and \(RS\) are two chords of a circle intersecting at a point \(O\) , It is given that \(PO=3\) cm and \( SO=4\) cm . Moreover, the area of the triangle \(POR\) is \(7 cm^2 \) . Find the are of the triangle \(QOS\) .   I.S.I....

Pythagorean Triple, ISI B.Math/B.Stat Entrance 2018, Subjective Solution of Problem No. 7

Understand the problem Let \(a,b,c \in \mathbb{N}\) be such that \(a^2+b^2=c^2\) and \(c-b=1\).Prove that (i) \(a\) is odd,(ii) \(b\) is divisible by 4,(iii) \(a^b+b^a\) is divisible by \(c\).   I.S.I. (Indian Statistical Institute) B.Stat/B.Math Entrance...

Powers of 2 – I.S.I. B.Math/B.Stat Entrance 2019 Subjective Solution Problem 1

Understand the problem Prove that the positive integers \(n\) that cannot be written as a sum of \(r\) consecutive positive integers, with \(r>1\) ,are of the form \(n=2^l\) for some \(l\ge 0\).   I.S.I. (Indian Statistical Institute) B.Stat/B.Math Entrance...

Clocky Rotato Arithmetic

Do you know that CLOCKS add numbers in a different way than we do? Do you know that ROTATIONS can also behave as numbers and they have their own arithmetic? Well, this post is about how clocks add numbers and rotations behave like numbers. Consider the clock on earth....

Shortest distance between curves – I.S.I. Entrance 2019 Subjective Solution Problem 8

Shortest path between two smooth curves is along the common normal. We use this fact to solve 8th problem of I.S.I. Entrance 2019 (UG Subjective)