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Re: HSC 2014 4U Marathon $The BoS problem-solving seminar consists of n students sitting at desks. Let $u_k$ be the number of ways of rearranging the seating so that exactly $k$ students stay in the same spot.\\ \\ Prove that: \\ \\ $\sum_{k=0}^n ku_k=n!.$ $

Re: HSC 2014 4U Marathon $Prove that not all roots of the polynomial:\\ \\ $P(x):=x^n-5x^{n-1}+12x^{n-2}-15x^{n-3}+a_{n-4}x^{n-4}+\ldots+a_0\\ \\$are positive real numbers. $

Re: HSC 2014 4U Marathon If you enforce A,B,C,D > 0, then this statement is only (technically) true because the implicitly defined curve is NEVER an ellipse. Your inequality in fact just gives a necessary (but not sufficient) condition for at least one point in the plane to satisfy your...

Re: HSC 2014 4U Marathon One that I've asked a couple of times before: Prove that any real polynomial is non-negative if and only if it is the sum of the squares of two real polynomials.

Re: HSC 2014 4U Marathon It's a shame more polynomial questions of this sort aren't asked in the HSC.

Re: HSC 2014 4U Marathon This is not true as stated. It is true iff w is a PRIMITIVE n-th root of unity.

Re: HSC 2014 4U Marathon I don't think cases are necessary but that looks about right. My solution: $Since both sequences are decreasing:\\ \\ $(x_i-x_j)(y_i-y_j)\geq 0.$\\ \\ Summing, we get:\\ \\ $\sum_{i=1}^n\sum_{j=1}^n (x_i-x_j)(y_i-y_j)=2n\sum_{i=1}^n x_i y_i-2\left(\sum_{i=1}^n...

Re: HSC 2014 4U Marathon Yeah, proving this is a bit easier than proving the rearrangement inequality.

Re: HSC 2014 4U Marathon $Let $(x_k)$ and $(y_k)$ be two decreasing real-valued sequences. \\ \\ Prove that:\\ \\ $n\sum_{k=1}^n x_ky_k \geq \left(\sum_{k=1}^n x_k\right)\left(\sum_{k=1}^n y_k\right).$ $

Re: HSC 2014 4U Marathon $Let $a,b,c>0$ with $abc=8$. Prove that:\\ \\ $\frac{a^2}{\sqrt{(1+a^3)(1+b^3)}}+\frac{b^2}{ \sqrt{(1+b^3)(1+c^3)} }+\frac{c^2}{\sqrt{(1+c^3)(1+a^3)}}\geq \frac{4}{3}.$ $

5' 11", 78kg.

Re: HSC 2014 4U Marathon Here is my solution based on calculus: $Consider: \\$f(x,y,z):=\frac{1}{1+x^3}+\frac{1}{1+y^3}+\frac{1}{1+z^3}-\frac{3}{1+xyz}$\\ for $x,y,z>1.$ \\ \\ So $\frac{\partial f}{\partial x}=3(g(yz)-g(x^2))$\\ \\ where \\ \\ $g(t):=\frac{t}{(1+xt)^2}.$\\ \\ Calculus...

Here is an outline that is high school level: Suppose the graphs of two polynomials p and q of degree < n pass through the same n points (no two of which lie on the same vertical line). Then p-q is a poly of degree < n which has at least n roots. The only such polynomial is the zero...

What makes a "legitimate genius" in your view?

Re: HSC 2014 4U Marathon $For $a,b,c >1$, show that:\\ \\ $\frac{3}{1+abc}\leq \frac{1}{1+a^3}+\frac{1}{1+b^3}+\frac{1}{1+c^3}.$ $

Yep. I prefer the units and tech tree of bw. The only obvious advantage of sc2 are the mechanics requiring less apm, but as I don't play at a high level this isn't a large factor for me.

Re: HSC 2014 4U Marathon $Prove that:\\ \\ $(1+m)^{-1/n}+(1+n)^{-1/m}>1$\\ \\ for positive integers $m,n>1.$ $

Re: HSC 2014 4U Marathon Well yeah, that's what I did above. Really doesn't change the time it takes to do it.

Re: HSC 2014 4U Marathon Lol, would hardly call that an alternate solution :P. It's like exactly the same difficulty.

Re: HSC 2014 4U Marathon How is that a solution though? That is just noting that the LHS expression is homogenous, which is what allowed me to make the assumption the variables have sum 1. Sy did the same thing explicitly, by dividing by x+y+z. Do you have an easy solution to the inequality...