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2018-11-26 Advanced

         

As we can see in involving involutions, there are a number of functions \(f:\mathbb{R}\to\mathbb{R}\) that satisfy the relation

\[{\color{purple}f\big(f(x)\big)=}{\color{forestgreen}\, x} \ \ \forall x\in\mathbb{R}.\]

Is there any function \(f:\mathbb{R}\to\mathbb{R}\) with the property \({\color{purple}f\big(f(x)\big)=}{\color{red} -x} \ \ \forall x\in\mathbb{R}?\)

We form a number triangle by first placing the positive integers along a down-and-right diagonal of an infinite square grid. Then, all other spaces in the triangle are filled by summing the numbers directly up and to the right of that space.

The sequence \(1,3,8,20,48,112,256,576,...\) occurs in the leftmost column of this triangle.

If \(S(x)=x+3{ x }^{ 2 }+{ 8 }x^{ 3 }+{ 20x }^{ 4 }+48{ x }^{ 5 }+112{ x }^{ 6 }+256{ x }^{ 7 }+576{ x }^{ 8 }+\cdots,\) what is \(S\big(\frac13\big)?\)


Bonus: Generalize \(S(n).\)

The above curve-stitching is formed by joining \( (i,0) \) to \( (0,10-i) \) for \( i = 1, 2, \ldots, 9 \).

If \(i\) is all the numbers between 1 and 10, not just integers, the outer curve will be smooth rather than kinked.

Now, after a rotation and a translation shown below, this smooth curve can be expressed as \( y = \frac{\sqrt{a}}{{b}}x^2\), where \(a\) and \(b\) are positive integers with \(a\) square-free.

What is the value of \( \frac{b^2}{a}?\)

If a cubic equation has real zeroes \(\pm p\) and 0, and its two horizontal tangent points and two non-zero \(x\)-intercepts can be joined together to form a rectangle (as pictured), then the ratio of the rectangle’s larger side to its shorter side can be expressed as \(\frac{\sqrt{a} + \sqrt{b}}{c}\), where \(a\), \(b\), and \(c\) are square-free integers.

What is \(a + b + c?\)

  • \(\sqrt[3]{a} + \sqrt[3]{b}\) is a root of \(10x^3 - 30x - 29\).
  • \(\sqrt[5]{a} + \sqrt[5]{b}\) is a root of \(10x^5 - 50x^3 + 50x - 29\).
  • \(\sqrt[7]{a} + \sqrt[7]{b}\) is a root of \(n_1x^7 + n_2x^5 + n_3x^3 + n_4x + n_5\).

If \(\gcd(n_1, n_2, n_3, n_4, n_5) = 1\) with \(n_1 > 0 \), what is the value of \(|n_1|+|n_2|+|n_3|+|n_4|+|n_5|?\)

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