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The footnote is 'correct', but for totally the wrong reason. The only reason why it is 'correct' in this case is because we are considering three functions that have a uniformly equidistant from each other. This distance happens to be less than 1 and therefore the conclusion is made. No no no...

Have you done anything to do with expected value?

So what is your definition of a 'good trial'?

Loose leaf ruled.

This.

\\ $Find the last three digits of the number $ 7^{9999}

\\ $Let $ a,b,c $ denote three distinct integers, and let $ P(x) $ be a polynomial with integer coefficients.$ \\\\ $Show that there does NOT exist a polynomial, where $ P(a)=b, P(b)=c $ and $ P(c)=a.

N points are given on the circumference of a circle, and all possible chords are constructed. The points are chosen such that no three chords are concurrent. How many triangles are there with all of its vertices lying inside the circle?

\\ $Define $ f_n(x)=f_0(f_{n-1}(x)) $ and let $ f_0(x)=\frac{1}{1-x}. \\\\ $Evaluate $ f_{2012}(2012)

Both correct^^. RealiseNothing, your solution is very intuitive.

I like what you did there^^

My apologies John. I meant for k to be a perfect cube. I had my mind set on the word square due to the nature of the configuration.

Correct.

Learning how to draw a nice and smooth curve is a far more valuable skill because it allows for modifications according to restrictions or features.

\\ $Find all integer triplets $ (x,y,z) $ that satisfy $ x^3+y^3+z^3=(x+y+z)^3

\\ $Suppose $ n \in \mathbb{N}. $ Find the sum of the digits appearing in the integers$\\\\ 1,2,3,...,10^n-2,10^n-1

The square given below has the unique property, that the products of the rows, columns, and the two diagonals, yield the same value k. A B C D E F G H I So ABC=DEF=GHI=ADG=BEH=CFI=AEI=CEG=k. Prove that if A,B,..,I are all integers, then k must be a perfect cube.

Any gel-pens or felt-tip pens tend to be inky. Just go to Officeworks and look around. Personally, I enjoy shopping for pens.

I always use a compass and a set-square.

Why don't you just buy an 'inky pen', then?