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Brilliant Polynomial Roots Contest - Season 1

Welcome to the first ever Brilliant Polynomial Roots Contest. This is inspired by many other contests in Brilliant. The aim is to improve the skills of Brilliant users in olympiad problems that ask you to find some functions involving the roots of a polynomial by vieta's, newton sums or other methods. The rules are:

  1. I will post the first problem. If someone solves it, he or she can post a solution and then must post a new problem.

  2. A solution must be posted below the thread of the problem. Then, the solver must post a new problem as a separate thread.

  3. Please make a substantial comment.

  4. Make sure you know how to solve your own problem before posting it, in case no one else is able to solve it within 36 hours. Then, you must post the solution and you have the right to post a new problem.

  5. If the one who solves the last problem does not post a new problem in 24 hours, the creator of the previous problem has the right to post another problem.

  6. No restriction in techniques you can use! use of calculus and roots of unity is allowed. use of cyclotomic polynomials and möbius inversion is also allowed.

  7. It is NOT compulsory to post original problems. But make sure it has not been posted on Brilliant.

  8. Your question must have a polynomial. it can be like \(f(x)=\dfrac{x^2-1}{x-1}\) or \(f(x)=x+1\). Both are allowed as long as the simplified form is a polynomial.

Format your proofs as

SOLUTION TO PROBLEM n

proof here

Question as

PROBLEM n

ask question relevant question here

To answer the latest question just shift to the "newest" mode


EDIT: for external discussion go to the discussion board

Note by Aareyan Manzoor
11 months, 1 week ago

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PROBLEM 7

If \(a, b, c\) are the roots of the equation

\[x^3 - 2x^2 - 3x - 4 = 0\]

Show that \(\frac{a^{2016} - b^{2016}}{a - b} + \frac{b^{2016} - c^{2016}}{b - c} + \frac{c^{2016} - a^{2016}}{c - a}\) is an integer.

This problem has been solved by Aareyan Manzoor but after time was finished, so Dev Sharma can Post the next problem Dev Sharma · 11 months, 1 week ago

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@Dev Sharma i will outline a proof. by GP we have \[\sum_{n=0}^{2016}(\sum_{sym}a^nb^{2016-n})\] write thi inside as \[\sum a^n(p_{2016-n}-a^{2016-n})=p_np_{2016-n}-p_{2016}\] which is always an integer. this is merely an outline as the time was up i cannot post the next problem. it is up to you @Dev Sharma Aareyan Manzoor · 11 months, 1 week ago

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@Dev Sharma Simple induction man, it not only holds for 2016 but for all natural no. n in the powers. Reetun Maiti · 10 months, 3 weeks ago

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PROBLEM 1 given the roots of \[y^8-y^7+y^6-1=0\] are \(y_1,y_2,...,y_7,y_8\) find the value of \[\dfrac{1}{y_1^9}+\dfrac{1}{y_2^9}+...+\dfrac{1}{y_7^9}+\dfrac{1}{y_8^9}\] This problem has been solved by dev sharma Aareyan Manzoor · 11 months, 1 week ago

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@Aareyan Manzoor SOLUTION TO PROBLEM 1

\[y^8 - y^7 + y^6 - 1 = 0\]

Now we have to form a polynomial whose roots are \(\frac{1}{y_{1}}, .....\)

Now, let \(x = \frac{1}{y}\)

then \(y = \frac{1}{x}\)

Putting it in the above equation, we get

\[x^8 + x^2 - x + 1 = 0\]

\[x^8 = x - x^2 - 1\]

\[x^9 = x^2 - x^3 - x ...(1)\]

Putting \(x = \frac{1}{y_{1}},..., \frac{1}{y_{8}}\) repeatedly in the equation and add them all and using bit Newton Sum (\(S_{1} = S_{2} = S_{3} = 0\))

And answer is 0. Dev Sharma · 11 months, 1 week ago

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PROBLEM 6

Let the roots of \(p(x)=x^3-2x-2\) be \(x_1,x_2,x_3\). Evaluate: \[\sum_{i=1}^{3} \dfrac{x_i+1}{x_i^2+x_i+1}\]

This problem has been solved by Dev Sharma Alan Enrique Ontiveros Salazar · 11 months, 1 week ago

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@Alan Enrique Ontiveros Salazar SOLUTION TO PROBLEM 6

\[x^3 - 2x - 2 = 0\]

\[x^3 - 1 = 2x + 1 ...(1)\]

Now, lets see what we need to find,

\(\frac{x + 1}{x^2 + x + 1}\)

Now multiplying \(x - 1\) in numerator and denominator.

\(\frac{x^2-1}{x^3 -1}\)

Now using (1)

\(\frac{x^2 - 1}{2x + 1}\)

Now, using vieta

\(x_{1} + x_{2} + x_{3} = 0\)

\(x_{1}x_{2} + x_{2}x_{3} + x_{3}x_{1} = -2\)

\(x_{1}x_{2}x_{3} = 2\)

Now putting \(x = x_{1}, x_{2}, x_{3}\) in the expression we want to find out and simplifying it, we get

\(\frac{x_{1}^2}{2x_{1} + 1} + \frac{x_{2}^2}{2x_{2} +1} + \frac{x_{3}^2}{2x_{3} + 1} -(\frac{1}{2x_{1} + 1} + \frac{1}{2x_{2} + 1} + \frac{1}{2x_{3} + 1})\)

Now using those vieta relations, we can get answer \(\frac{-1}{3}\) Dev Sharma · 11 months, 1 week ago

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@Dev Sharma We can simplify our last expression like this -

\(\frac{x_{1}^2}{2x_{1} + 1} + \frac{x_{2}^2}{2x_{2} +1} + \frac{x_{3}^2}{2x_{3} + 1} -(\frac{1}{2x_{1} + 1} + \frac{1}{2x_{2} + 1} + \frac{1}{2x_{3} + 1})\)

= \(\frac{x_{1}^2(4x_{2}x_{3} + 2x_{2} + 2x_{3} + 1) + x_{2}^2(4x_{1}x_{3} + 2x_{1} + 2x_{3} + 1) + x_{3}^2(4x_{1}x_{2} + 2x_{1} + 2x_{2} + 1)}{(2x_{1} + 1)(2x_{2} + 1)(2x_{3} + 1)}\) - \(\frac{(4x_{2}x_{3} + 2x_{2} + 2x_{3} + 1) + (4x_{1}x_{3} + 2x_{1} + 2x_{3} + 1) + (4x_{1}x_{2} + 2x_{1} + 2x_{2} + 1)}{(2x_{1} + 1)(2x_{2} + 1)(2x_{3} + 1)}\)

Now, simplify it and get the answer!! Dev Sharma · 11 months, 1 week ago

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@Dev Sharma Natural approach, there's another that is easier but it still good, well done. Now let's see your problem! Alan Enrique Ontiveros Salazar · 11 months, 1 week ago

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Let \(w_k\) denote the 2015th primitive roots of unity. Find

\[\sum _{n=1}^{\phi \left(2015\right)}w_n\prod _{k=1}^{\phi \left(2015\right)}w_k\]

This problem has been solved by Aareyan Manzoor Julian Poon · 11 months, 1 week ago

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@Julian Poon SOLUTION TO PROBLEM 4 you forgot to format your question properly! the proof anyways:

claim: the product of all the primitive roots of unity are 1.

proof: we know that each primitive root of unity can be written as \(e^{\dfrac{2k\pi i}{n}}\).for k coprime to n. when we multiply all of these we find: \[exp(\dfrac{2\pi i}{n}\sum_{1≤x≤2015,gcd(x,2015)=1} x)\] the formula for the sum of all co-primes less then n is \[\dfrac{n\phi(n)}{2}\] view this. put that there to get \[exp(\phi(n)\pi i)\] \(\phi(n)\) is always even apart from n=1,2. so this will always be 1 for n>2. at 2015 it is one then. the rest is easy i guess. we put 1 to find \[\sum w_n=\mu(2015)=\boxed{-1}\] Aareyan Manzoor · 11 months, 1 week ago

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PROBLEM 20:

If \(a,b,c\) are roots of the polynomial \(x^3-3x^2+2x+3=0\) , find the value of \(\dfrac{1}{a^3+b^3} + \dfrac{1}{b^3+c^3} + \dfrac{1}{a^3+c^3} \)

Easy , because its created by me ;)

This problem is solved by Aareyan Manzoor Nihar Mahajan · 10 months, 3 weeks ago

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@Nihar Mahajan By newtons sum \(P_3=0\). so \[\sum \dfrac{1}{a^3+b^3}=\sum \dfrac{1}{-c^3}\] pu x=1/x to have \[x^3+\dfrac{2}{3} x^2-x+\dfrac{1}{3}=0\] \[-\sum x^3= -((\sum(x))((\sum x)^2-3(\sum x_1x_2))+3\prod x)\] the result follows from vietas. Aareyan Manzoor · 10 months, 3 weeks ago

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@Nihar Mahajan Yo @Aareyan Manzoor damn you wrote the solution first , Chinmay Sangawadekar · 10 months, 3 weeks ago

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Problem 16

Among the polynomials \(P(x), Q(x), R(x)\) with real coefficients at least one has degree two and one has degree three. If \(P(x)^2 + Q(x)^2 = R(x)^2\) prove that one of the polynomials of degree three has three real roots.

Solved by Khang Nguyen Thanh Xuming Liang · 10 months, 4 weeks ago

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@Xuming Liang SOLUTION OF PROBLEM 16: This is a problem of All-Russian MO 2002.

Define \(\deg(P)\) to mean the degree of \(P(x)\), and likewise \(\deg(Q)\) and \(\deg(R)\).

Observe that \(\max\{\deg(P), \deg(Q)\} = \deg(R)\).

This is because \(\deg(R^2) = \deg(P^2 + Q^2) = \max \{\deg(P^2), \deg(Q^2)\}\) (leading coefficients of \(P^2\) and \(Q^2\) are positive and do not cancel), and \(2\deg(R) = \deg(R^2) = \max \{\deg(P^2), \deg(Q^2)\} = 2 \max \{\deg(P), \deg(Q)\}\).

Thus one of \(P, Q\) (without loss of generality, \(Q\)) must have degree 2, and both \(P\) and \(R\) must have degree 3.

Now we write \(P^2 = (R + Q).(R−Q)\). We know that both factors have degree 3.

Since \(P\) has either 1 or 3 real roots, \(R+Q\) and \(R-Q\) either both have one real root or three real roots accordingly.

Assume that \(P, R+Q\) and \(R-Q\) each have one real root; let the root of \(P\) be \(r_1\).

Since \(r_1^2|P^2\), this means that \(r_1\) is a root of both \(R+Q\) and \(R-Q\), and is therefore a root of both \(R\) and \(Q\).

Now let \(P_0, Q_0\), and \(R_0\) be the three polynomials \(P, Q\), and \(R\) with the common root divided out.

We have \(P_0^2 +Q_0^2 = R_0^2\).

Then all three polynomials have real coefficients, \(P_0\) and \(R_0\) have degree 2, and \(Q_0\) has degree 1.

We also know that \(P_0\) has 0 real roots, \(Q_0\) has 1, and \(R_0\) has 0 or 2.

Since the latter case would imply that \(R\) has 3 real roots, we instead assume that \(R_0\) has 0 real roots.

Writing \(Q_0^2 = (R_0 + P_0).(R_0 − P_0)\), we see that one of the factors must have degree 2 and the other degree 0.

If the first factor has degree 2, we may write \(P_0 = ax^2 + bx + c, R_0 = ax^2 + bx + d\) for real \(a, b, c, d\).

Then \(Q_0 = (d − c)(2ax^2 + 2bx + c + d)\). Since \(P_0\) and \(R_0\) have no real roots, we know that \(b^2 − 4ac < 0\) and \(b^2 − 4ad <0\).

Adding the two inequalities and multiplying by 2, this means \(4b^2 − 8a(c + d) < 0\).

But then \(Q_0\) has no real roots, a contradiction.

Similarly, if \(R_0 + P_0\) has degree 0, \(R_0 − P_0\) has degree 2, we write \(P_0 = ax^2 + bx + c, R_0 = −ax^2 − bx + d\).

Then \(Q_0 = −(c + d)(2ax^2 + 2b + (c − d))\).

Again, \(P_0\) and \(R_0\) have no real roots, so \(b^2 − 4ac < 0\) and \(b^2 + 4ad < 0\).

Adding and multiplying by 2 gives \(4b^2 − 8a(c − d) < 0\), which again implies that \(Q_0\) has no real roots.

Thus our original assumption is false, and either \(P\) or \(R\) must be a third-degree polynomial with 3 real roots. Khang Nguyen Thanh · 10 months, 3 weeks ago

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@Xuming Liang Clearly, \(R(x)\) has degree \(3\) and \(P(x)\) and \(Q(x)\) have degree \(2\) and \(3\) (W.L.O.G.)

Also,

\(R(x) = Q(x) \pm i P(x)\)

From the above equation, it is clear that if \(R(x)\) has a real root, it is also a root of \(P(x)\) and \(Q(x)\). So, \(R(x)\) has at most \(2\) real roots (counted with multiciplity). (Since \(P(x)\) has degree \(2\) and non real roots occur in conjugate pairs)

If \(R(x)\) has \(2\) real roots (one with multiplicity), then we are done. If not, then let \(\alpha\) be the common real root to all three polynomials.

Let \(\beta_{1}\) and \(\beta_{2}\) be the other two roots of \(Q(x)\),

Since,

\(Q^2 (x) = (R(x) + P(x))(R(x) - P(x))\)

and the two polynomials in the brackets are not identical, the set of roots must be \(\alpha\), \(\beta_{1}\), \(\beta_{1}\) and \(\alpha, \beta_{2}, \beta_{2}\). Again, since non real roots occur in conjugate pair, we must have \(\beta_{1}, \beta_{2} \in \mathbb R\). Thus \(Q(x)\) has three real roots. Ishan Singh · 10 months, 3 weeks ago

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Problem 15

Prove that there doesn't exist polynomials with integer coefficients \(f_k(x)\)(k=1,2,3,4), such that \[ 9x+4=f_1(x)^3+f_2(x)^3+f_3(x)^3+f_4(x)^3\]

solved by Xuming Liang Xuming Liang · 11 months ago

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@Xuming Liang Solution: Consider the third root of unity \(w\ne 1\). Since \(w^2=-1-w\), we can express \(f_i(w)=aw+b\) for some integers \(a,b\). Thus \(f_i(w)^3=(aw+b)^3=3ab(b-a)w+a^3+b^3-3a^2b\). Note that the coefficient of \(w\) is always even, therefore the coefficient of \(w\) on the LHS of the equality is even. However the same coefficient on the RHS is odd, thus these polynomials cannot exist. Xuming Liang · 10 months, 4 weeks ago

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@Xuming Liang dont you think you need to prove thet \(f_i\) is linear? Aareyan Manzoor · 10 months, 4 weeks ago

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PROBLEM 5 given the polynomial \[x^4+x^3-2x+11=0\] a,b,c,d are its roots. evaluate \[\sum_{cyc}\dfrac{a^3(a+1)}{a^4+11}\]

This problem has been solved by Alan Enrique Ontiveros Salazar Aareyan Manzoor · 11 months, 1 week ago

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@Aareyan Manzoor SOLUTION TO PROBLEM 5

Surely there's an easier way, but this is a new way:

\(x^4=-x^3+2x-11 \\ x^5=x^3+2x^2-13x+11 \\ x^6=x^3-13x^2+13x-11\)

So \(\dfrac{x^3(x+1)}{x^4+11}=\dfrac{2x-11}{2x-x^3}\)

Now we say that \(2x-11=(2x-x^3)(Ax^3+Bx^2+Cx+D)\). Expanding and substituting the first three relations we get:

\(2x-11=(-3A+B+C-D)x^3+(13A-2B+2C)x^2+(-9A+13B-2C+2D)x-11A-11B+11C\)

Matching the coefficients and solving the system we get \(A=\dfrac{2}{15}\), \(B=\dfrac{4}{15}\), \(C=-\dfrac{3}{5}\) and \(D=-\dfrac{11}{15}\).

Hence, \(\dfrac{2x-11}{2x-x^3}=\dfrac{2}{15}x^3+\dfrac{4}{15}x^2-\dfrac{3}{5}x-\dfrac{11}{15}\)

So we want to find \(\displaystyle \dfrac{2}{15} \sum x_i^3+\dfrac{4}{15}\sum x_i^2-\dfrac{3}{5} \sum x_i - \dfrac{11}{15}(4)\)

Using Newton's Sums we get \(S_1=-1,S_2=0,S_3=2,S_4=11\) and \(P_1=S_1=-1\), \(P_2=S_1P_1-2S_2=1\), \(P_3=S_1P_2-S_2P_1+3S_3=5\).

So the sum is \(\dfrac{2}{15}(5)+\dfrac{4}{15}(1)-\dfrac{3}{5}(-1)-\dfrac{11}{15}(4)=\boxed{-\dfrac{7}{5}}\). Alan Enrique Ontiveros Salazar · 11 months, 1 week ago

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PROBLEM 3 find \[\sum \ln(1-\omega^3)\] where \(\omega\) are 2015th roots of unity≠1.

This problem has been solved by Julian Poon Aareyan Manzoor · 11 months, 1 week ago

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@Aareyan Manzoor \[\sum _{ }^{ }\ln \left(1-w^3\right)=\ln\prod _{ }^{ }\left(1-w^3\right)\]

Let

\[f\left(x\right)=\prod _{ }^{ }\left(x-w\right)=\frac{x^{2015}-1}{x-1}=\sum _{n=0}^{2014}x^n\]

\[\prod _{ }^{ }\left(1-w^3\right)=\prod _{ }^{ }\left(1-w\right)\left(e^{\frac{2i\pi }{3}}-w\right)\left(e^{\frac{4i\pi }{3}}-w\right)=f\left(1\right)f\left(e^{\frac{2i\pi }{3}}\right)f\left(e^{\frac{4i\pi }{3}}\right)=2015\]

Hence, the sum is \(\ln{(2015)}\) Julian Poon · 11 months, 1 week ago

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PROBLEM 9

The polynomial \(ax^3+bx^2+cx+d\) has integral coefficients \(a,b,c,d\) with \(ad\) odd and \(bc\) even. Show that at least one zero of the polynomial is irrational.

Nobody was able to solve this,so Svatejas Shivakumar can post the next problem Svatejas Shivakumar · 11 months, 1 week ago

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@Svatejas Shivakumar SOLUTION OF PROBLEM 9

Let \(x_i,i=1,2,3\) be the rational roots of the given.polynomial. Then \((ax)^3+b(ax)^2+ac(ax)+a^2d=0\).

Setting \(y=ax\),we get \(y^3+by^2+acy+a^2d=0\).

\(y_i\) are the three rational zeros of the above equation,i.e. they must be integers. Also, since they are divisors of \(a^2d\), they must be odd. Since \(y_1+y_2+y_3=-b\) and \(y_1y_2+y_2y_3+y_3y_1=ac\), both \(b\) and \(ac\) must be odd,i.e., \(b\) and \(c\) are odd hence \(bc\) is odd. This contradicts the assumption that \(bc\) is even. Hence at least one zeros of the polynomial is irrational. Svatejas Shivakumar · 11 months ago

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PROBLEM 8

Let \(p(x)\) be a polynomial with integer coefficients. Assume that \(p(a) = p(b) = p(c) = -1\), where \(a, b, c\) are three different integers. Prove that \(p(x)\) has no integral zeroes.

This problem has been solved by Svatejas Shivakumar Dev Sharma · 11 months, 1 week ago

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@Dev Sharma SOLUTION TO PROBLEM 8

Lemma: If \(f(x)\) is a polynomial with integeral coefficients and \(a\) is an integeral root of \(f(x)\) and \(m\) is any integer different from \(a\), then \(a-m\) divides \(f(x)\).

Proof: On dividing \(f(x)\) by \(x-m\) we get \(f(x)=(x-m)q(x)+f(m)\), where \(q(x)\) is a polynomial with integral coefficients. For \(x=a\), we get \(f(a)=0=(a-m)q(a)+f(m)\) or \(f(m)=-(a-m)q(a)\). Hence \(a-m\) divides \(f(m)\).

Suppose \(d\) is an integeral root of \(p(x)\), then by the lemma, \(d-a,d-b,d-c\) divides \(-1\) . But \(-1\) has only \(2\) factors namely \(-1,1\). Hence at least two of \(a,b,c\) are the same but \(a,b,c\) are different. Hence, \(p(x)\) has no integral zeroes. Svatejas Shivakumar · 11 months, 1 week ago

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PROBLEM 2

If \(1, x_{1}, x_{2}, ....., x_{48}\) are 49th roots of unity, then find

\[\displaystyle\sum_{i=1}^{48} \frac{1}{(1 - x_{i})^3}\] This problem has been solved by Aareyan Manzoor Dev Sharma · 11 months, 1 week ago

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@Dev Sharma SOLUTION TO PROBLEM 2 the polynomial with roots \(x_i\) is \[x^{48}+x^{47}+...+x+1=\dfrac{x^{49}-1}{x-1}=0\] let \(y=\dfrac{1}{1-x}\) or \(x=\dfrac{y-1}{y}\).then we substitute: \[(\dfrac{y-1}{y})^{49}-1=0,y\neq 0\] \[y^{49}=(y-1)^{49}\] \[y^{48}-24y^{47}+376y^{46}-4324y^{45}+.....+\dfrac{1}{49}=0\] using newtons sum \[\begin{array} a p_1&=&s_1&=&24\\ p_2&=&s_1p_1-2s_2&=&-176\\ p_3&=&s_1p_2-s_2p_1+3s_3&=&\boxed{-276}\end{array}\] we are done. Aareyan Manzoor · 11 months, 1 week ago

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Problem 21 given a poly \[F(x)=x^4+x+1\] if its roots are a,b,c,d find \[\sum_{sym}\dfrac{1}{a^{10}+b^{9}}\] Aareyan Manzoor · 10 months, 3 weeks ago

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@Aareyan Manzoor Quite a messy problem. Do you have an easier way of doing it? If so, could you please post it since the time is up? Samuel Jones · 10 months, 2 weeks ago

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@Samuel Jones :p forgot about the contest.

I am not feeling to write the solution..... i will try to later. You can post the next problem(as you reminded me of the contest). Aareyan Manzoor · 10 months, 2 weeks ago

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@Aareyan Manzoor Is the solution tedious or elegant? Harsh Shrivastava · 10 months, 2 weeks ago

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@Harsh Shrivastava The former. It is too big! Aareyan Manzoor · 10 months, 2 weeks ago

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@Aareyan Manzoor Please post the solution.It's been over 2 weeks. Abdur Rehman Zahid · 10 months ago

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@Abdur Rehman Zahid :p the contest ended Aareyan Manzoor · 10 months ago

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PROBLEM 19

If \(x_1,x_2,x_3\) are the roots of \(P(x)=x^3-9x^2+14x-1\), find all the possible values of \(\dfrac{x_1}{x_2}+\dfrac{x_2}{x_3}+\dfrac{x_3}{x_1}\).

This problem is solved by both , Xuming and Nihar. Alan Enrique Ontiveros Salazar · 10 months, 3 weeks ago

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@Alan Enrique Ontiveros Salazar SOLUTION TO PROBLEM 19:

First I let \(a=\dfrac{x_1}{x_2} , b= \dfrac{x_2}{x_3} , c= \dfrac{x_3}{x_1}\).

Now using \(x_1x_2x_3=1\) (By Vieta's) , we can easily obtain that \(ab+bc+ac=\dfrac{1}{a} + \dfrac{1}{b}+\dfrac{1}{c}\) .

Again by using \(x_1x_2x_3=1\) we obtain :

\[\dfrac{b}{a}+ \dfrac{c}{b}+ \dfrac{a}{c} = x_1^3+x_2^3+ x_3^3 \\ =(x_1+x_2+x_3)^3-3(x_1+x_2+x_3)(x_1x_2+x_2x_3+x_1x_3)+3x_1x_2x_3 \\ = 9^3-3(9)(14)+3 \\ =\boxed{354}\]

Again by using \(x_1x_2x_3=1\) we obtain : \(\dfrac{a}{b}+ \dfrac{b}{c}+ \dfrac{c}{a}=(x_1x_2)^3+(x_2x_3)^3+(x_1x_3)^3\). Now we let \(k=x_1x_2 \ , \ m=x_2x_3 \ , \ n=x_1x_3\) So we have:

\[k^3+m^3+n^3=(k+m+n)^3-3(k+m+n)(km+mn+kn) + 3(k^2m^2n^2)\]

Again by using \(x_1x_2x_3=1\) we obtain : \(km+mn+kn=x_1+x_2+x_3\) and \(k^2m^2n^2=1\). Thus by using Vieta's again ,

\[k^3+m^3+n^3=(14)^3-3(14)(9)+3 = \boxed{2370}\]

We also have \((a+b+c)\left(\dfrac{1}{a} + \dfrac{1}{b}+\dfrac{1}{c}\right) =3 + \dfrac{a}{b}+ \dfrac{b}{c}+ \dfrac{c}{a}+ \dfrac{b}{a}+ \dfrac{c}{b}+ \dfrac{a}{c}\) that is :

\[(a+b+c)(ab+bc+ac)=3 + 2370+354 = \boxed{2727}\]

We also have \( \left(\dfrac{a}{b}+ \dfrac{b}{c}+ \dfrac{c}{a}\right)\left( \dfrac{b}{a}+ \dfrac{c}{b}+ \dfrac{a}{c}\right) = a^3+b^3+c^3+a^3b^3+b^3c^3+a^3c^3+3\)

Now we let \(p=ab \ , \ q=bc \ , r = ac\) , and we have \(p^3+q^3+r^3=(p+q+r)^3-3(p+q+r)(pq+qr+pr) +3p^2q^2r^2\) . Now using \(abc=1\) , we have \(pq+qr+pr=ab+bc+ac\) ,

\[p^3+q^3+r^3=(p+q+r)^3-3(p+q+r)(pq+qr+pr) +3p^2q^2r^2 \\ = (ab+bc+ac)^3-3(ab+bc+ac)(a+b+c)+3\]

and we also have

\[a^3+b^3+c^3=(a+b+c)^3 - 3(a+b+c)(ab+bc+ac)+3\]

Adding above both equations ,

\[p^3+q^3+r^3+a^3+b^3+c^3 = (ab+bc+ac)^3+ (a+b+c)^3-6(a+b+c)(ab+bc+ac)+6 \\ \Rightarrow \left(\dfrac{a}{b}+ \dfrac{b}{c}+ \dfrac{c}{a}\right)\left( \dfrac{b}{a}+ \dfrac{c}{b}+ \dfrac{a}{c}\right) = (ab+bc+ac)^3+ (a+b+c)^3-6(2727)+6 \\ \Rightarrow (ab+bc+ac)^3+ (a+b+c)^3 = 855336\]

Let \(a+b+c=f \ , \ ab+bc+ac = g\) , we have \(f^3+g^3=855336\) and \(fg = 2727\) and substituting \(g=\dfrac{2727}{f}\) in the former equation and solving the obtained quadratic , we get \(f=a+b+c = 29 , 94\).

HUSH!

Nihar Mahajan · 10 months, 3 weeks ago

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@Nihar Mahajan Well done! :D Alan Enrique Ontiveros Salazar · 10 months, 3 weeks ago

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@Nihar Mahajan Yeah for sure problem has long calculations ;) Chinmay Sangawadekar · 10 months, 3 weeks ago

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@Alan Enrique Ontiveros Salazar The answers are indeed \(94,29\). The key to finding the answer is constructing equations using symmetric sums. Note that our desired sum is equal to \(\displaystyle \sum_{cyc}x_1^2x_3\), which we denote \(A\). Let \(B\) denote its symmetric counterpart: \(\displaystyle \sum_{cyc} x_1^2x_2\).

Note that \(A+B=\displaystyle \sum_{sym} x_1^2x_2=9\cdot 14-3=123\).

\(A\cdot B=\displaystyle \sum_{sym} x_1^3+\displaystyle \sum_{sym} (\dfrac {1}{x_1})^3+3(x_1x_2x_3)^2=2726\). Here I omit the routine calculation for the two sums.

Solving for \(A,B\) gives us \(94,29\). Xuming Liang · 10 months, 3 weeks ago

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@Xuming Liang Wow! I haven't learnt this method...

So who must post the next question? I got the correct answer first but you wrote the solution first... Nihar Mahajan · 10 months, 3 weeks ago

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@Nihar Mahajan I nominate you :) Xuming Liang · 10 months, 3 weeks ago

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@Xuming Liang Thank you! :) Nihar Mahajan · 10 months, 3 weeks ago

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@Xuming Liang Yes, that is correct, you used the same method than me. Alan Enrique Ontiveros Salazar · 10 months, 3 weeks ago

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@Alan Enrique Ontiveros Salazar After a very long process , tedious calculations , I got the possible values as approximately \(94 , 29\) (It may be wrong too) . Please verify @Alan Enrique Ontiveros Salazar . Thanks. Nihar Mahajan · 10 months, 3 weeks ago

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Problem 17

Consider the sequence of polynomials \(\{P_n(x)\}_{n=1,2,3,...}\) such that \(P_n(2\cos x)=2^n\cos nx, \forall x\in\mathbb{R},\forall n\in\mathbb{N}^*\).

Prove that: \(\large 1\le\dfrac{\sqrt[n]{P_n(x)}-2}{x-2}\le n, \forall x>2, \forall n\in\mathbb{N}^*\) Khang Nguyen Thanh · 10 months, 3 weeks ago

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@Khang Nguyen Thanh solution to Problem 17 \[P_n(2x)=2^n\cos(n\arccos(x))=2^nT_n(x)\] chebyshev polynomials used here(of the first kind). we know:\(T_2(x)=2x^2-1 ,T_3(x)=4x^3-3x,T_{n+1}(x)=2xT_n(x)-T_{n-1}(x)\) we see hat cases 2,3 are allways satisfied.so \[2x-2≤2\sqrt[n+1]{T_{n+1}(2x)}-2≤(n+1)(2x-2)\\ 2x≤2\sqrt[n+1]{2xT_n(2x)-T_{n-1}(2x)}≤(n+1)(2x-2)+2\\ x^{n+1}≤4xP_n(2x)-4P_{n-1}(2x)≤((n+1)(2x-2)+2)^{n+1}\] we had \[(2x)^n≤P_n(2x)≤(n(2x-2)+2)^n\\-((n-1)(2x-2)+2)^n≤-P_{n-1}(x)≤-(2x)^{n-1}\] multiplying and adding \[4(2x)^{n+1}-4((n-1)(2x-2)+2)^n≤4xP_n(2x)-4P_{n-1}(2x)≤4x(n(2x-2)+2)^n-4(2x)^{n-1}\] now notice that \[4(2x)^{n+1}-4((n-1)(2x-2)+2)^n≥x^{n+1}\\4x(n(x-2)+2)^n-4(2x)^{n-1}≤((n+1)(2x-2)+2)^{n+1}\] Hence proved by induction. Aareyan Manzoor · 10 months, 3 weeks ago

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@Aditya Agarwal here is a problem!


Problem 14

Let \(P(x)\) be a polynomial of degree \(n>1\) with integer coefficients, and let \(k\) be a positive integer. Consider the polynomial \(Q(x) = P( P ( \ldots P(P(x)) \ldots ))\), where \(P\) occurs \(k\) times. Prove that there are at most \(n\) integers \(t\) such that \(Q(t)=t\).

Dan Shwarz,Romania

solved by Xuming Liang Sualeh Asif · 11 months ago

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@Sualeh Asif This is the fifth problem of IMO 2006, I recognized it from a documentary about the USA IMO team. I will outline the basic idea of the proof:

The first idea shows why the case \(k=2\) is important(sufficient):

If \(Q(s)=s\) but \(P(s)\ne s\), then \(P(P(s))=s\). In other words, if \(s\) is a fixed point of \(Q\) but not of \(P\), then it is a fixed point of \(P(P(x))\). So it suffices to consider(count) the fixed points of \(k=2\).

We now prove the case for \(k=2\):

If all fixed points of \(P(P(x))\) are fixed points of \(P(x)\), then the result holds because \(P\) has at most \(n\) fixed points.

If not, then there exist \(a\ne b\) such that \(P(a)=b, P(b)=a\). The next observation is that all pairs \((a',b')\) that satisfy the previous equations(\(a',b'\) need not to be distinct) have the same sum, i.e. \(a+b=a'+b'=c\) for some constant \(c\). Thus all the numbers in these pairs(fixed points of \(P(P(x))\)) are the roots to the polynomial \(P(x)+x-c\). Since this has the same degree as \(P\)(\(n>1\)), we are done.


I know this isn't the full solution, so feel free to fill in the steps. Xuming Liang · 11 months ago

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@Sualeh Asif This has totally stumped me! I tried proving it for k=2. But even when I did that, I was unsure of what to do? Aditya Agarwal · 11 months ago

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@Aditya Agarwal Keep trying though!
The \(k=2\) is a very crucial step! Sualeh Asif · 11 months ago

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PROBLEM 11:

Prove: For any polynomial \(g(x)\), \(\text{deg}(g(x))>1\), another polynomial \(k(x)\) can be substituted for \(x\), in such a way that \(g(k(x))\) can be expressed as a product of non-constant polynomials. (All the polynomials have integral coefficients)

Nobody posted the solution, Aditya Agarwal posted the solution Aditya Agarwal · 11 months ago

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@Aditya Agarwal Solution for Problem 11:

It is evident that, \(g(a)-g(x)\) is divisible by \(a-x\). Now lets us take \(a\), such that \(a-x\), is divisible by \(g(x)\), for example, \(a=k(x)=x+p(x)\). So, \(g(k(x))\) is divisible by \(g(x)\). Now because \(\text{deg}(g(k(x)))\) is greater than \(\text{deg}(g(x))\), the second factor is not constant. Aditya Agarwal · 11 months ago

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@Aditya Agarwal Please post the solution and the next problem as well since no one has solved it within the time limit. Svatejas Shivakumar · 11 months ago

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PROBLEM 10

The polynomial \(P(x)=x^{n}+a_{1}x^{n-1}+\ldots+a_{n-1}x+1\) with nonnegative coefficients \(a_{1},\ldots,a_{n-1}\) has \(n\) real roots. Prove that \(P(2) \ge 3^{n}\).

This problem has been solved by Aditya Agarwal Svatejas Shivakumar · 11 months ago

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@Svatejas Shivakumar Because of the given condition (non-negative coefficients), the roots would be non-positive.

So the polynomial can be expressed as \[P(x)=(x+b_1)...(x+b_n)\] where \(b_i=-x_i\>0), and \(x_i\) are the roots.

Now, by the AM-GM inequality, we get, \[2+b_i\geq3{b_i}^{\frac13}\]

Now by Vieta's Formulas, we have, that the product of the roots is \(1\).

So \[P(2)=(2+b_1)...(2+b_n)\geq3^n\]


Those who are saying that the product \(b_1b_2...b_n=-1\), it won't be. Because \(b_i>0\), and thus the product has to be greater than \(-1\), so the product would definitely by \(1\). (The confusion arised because of the substitution, \(b_i=-x_i\)) Aditya Agarwal · 11 months ago

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PROBLEM 18 consider \[P(x)=x^4-3x+1=0\] has roots \(x_i\forall i=1,2,3,4\). find \[\sum_{cyc} \dfrac{x^{7}}{(x^3-3)^2} \] Aareyan Manzoor · 10 months, 3 weeks ago

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@Aareyan Manzoor Solution to problem 18

\(x^4=3x-1\)

\(x^5=3x^2-x\)

\(x^7=3x^4-x^3=3(3x-1)-x^3=-x^3+9x-3\)

\(x^4-3x=-1 \implies x^3-3=-\dfrac{1}{x}\)

Then, the sum is \(\displaystyle \sum_{cyc} \dfrac{-x^3+9x-3}{\left(-\dfrac{1}{x}\right)^2}=\sum_{cyc}(-x^5+9x^3-3x^2)=\sum_{cyc}=(-(3x^2-x)+9x^3-3x^2)=\sum_{cyc}(9x^3-6x^2+x)\)

We know that \(S_1=0,S_2=0,S_3=3,S_4=1\) and, by Newton's Sums, \(P_1=0,P_2=0,P_3=9\). So the sum is \(9(9)-6(0)+0=\boxed{81}\). Alan Enrique Ontiveros Salazar · 10 months, 3 weeks ago

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@Alan Enrique Ontiveros Salazar Correct! Post next problem! Aareyan Manzoor · 10 months, 3 weeks ago

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This is a very nice problem I came across

PROBLEM 13

The polynomial \(f(x)=x^{n}+a_{1}x^{n-1}+\ldots+a_{n-1}x+a_{n}=0\) with integral non-zero coefficients, has \(n\) real roots. Prove that if the roots are pairwise coprime, then \(a_{n-1}\) and \(a_{n}\) are coprime.

Solved by Aditya Agarwal Svatejas Shivakumar · 11 months ago

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@Svatejas Shivakumar SOLUTION OF PROBLEM 13

Proof by contradiction:

Lets assume that the last two coefficients are not coprime, i.e, \(\gcd(a_{n-1},a_n)\neq1\).

By Vieta's formulas we have that the product of roots, let them be \(\{z_i\}_1^{n}\) , is \((-1)^na_n\). So this means that one of the roots, let it be \(z_1\) is divisible by the number, let it be \(k\), by which \(a_{n-1}\) and \(a_n\) are divisible. Also, by Vieta's Formulas, we have \[z_1...z_{n-1}+z_1z_3...z_n+...+z_2...z_n=(-1)^{n-1}a_{n-1}\] Now we know that \(a_{n-1},z_1\) are divisible by \(k\). So this means that the last term, namely the term which doesn't contain \(r_1\), is also divisible by \(k\). But this contradicts the fact that all the roots are coprime. Hence, the last two coefficients should also be coprime.


P.S.: I have not got any good problem at the moment, and also, it doesn't seem that many users are online today. So I would resist posting a problem today, if anyone wants to do it, he can, in place of mine. Aditya Agarwal · 11 months ago

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@Aditya Agarwal Nice solution! I too don't have any good problem. I searched a lot to find this problem. Svatejas Shivakumar · 11 months ago

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@Svatejas Shivakumar I am weary now! I will post one tomorrow if no one posts one today. Aditya Agarwal · 11 months ago

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PROBLEM 12

Let \(f(x)\) be a monic polynomial with integral coefficients. If there are four different integers \(a,b,c,d\) such that \(f(a)=f(b)=f(c)=f(d)=5\), then show that there is no integer \(k\), so that \(f(k)=8\).

*Solved Svatejas Shivakumar * Aditya Agarwal · 11 months ago

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@Aditya Agarwal SOLUTION OF PROBLEM 12

\(f(x)=g(x)(x-a)(x-b)(x-c)(x-d)+5\) where \(g(x)\) is a polynomial with integral coefficients

If \(f(k)=8,\) then \(g(k)(k-a)(k-b)(k-c)(k-d)=3\).

\(g(k)(k-a)(k-b)(k-c)(k-d)\) is a product of five integers of which at least four are distinct(since a,b,c,d are distinct) but 3 can be expressed as a product of at most three distinct factors i.e \(1×(-1)×(-3)\).

Hence there is no integer such that \(f(k)=8\). Svatejas Shivakumar · 11 months ago

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@Svatejas Shivakumar Nice solution! Just, edit it to "SOLUTION OF PROBLEM 12". And post problem 13. Aditya Agarwal · 11 months ago

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