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# Puzzles

## Find them all

Find all continuous positive functions, $$f$$ on $$[0,1]$$ such that:
$$\int_0^1 f(x) dx=1\\ \mathrm{and }\int_0^1 xf(x) dx=\alpha\\ \mathrm{and }\int_0^1 x^2f(x) dx=\alpha^2$$

## Odd and even outputs

Let $$g:\mathbb{N}\times\mathbb{N}\rightarrow\mathbb{N}$$ be a function.
This means that $$g$$ takes two natural number inputs and gives one natural number output. For example if $$g$$ is defined by $$g(n,m)=n+m$$ then $$g(3,4)=7$$ and $$g(10,2)=12$$.
The function $$g(n,m)=n+m$$ will give an even output if $$n$$ and $$m$$ are both odd or both even and an odd output if one is odd and the other is even. This could be summarised in the following table:
 $$n$$ odd even $$m$$ odd even odd e odd even
Using only $$+$$ and $$\times$$, can you construct functions $$g(n,m)$$ which give the following output tables:
 $$n$$ odd even $$m$$ odd odd odd e odd odd
 $$n$$ odd even $$m$$ odd odd odd e odd even
 $$n$$ odd even $$m$$ odd odd odd e even odd
 $$n$$ odd even $$m$$ odd odd odd e even even
 $$n$$ odd even $$m$$ odd odd even e odd odd
 $$n$$ odd even $$m$$ odd odd even e odd even
 $$n$$ odd even $$m$$ odd odd even e even odd
 $$n$$ odd even $$m$$ odd odd even e even even
 $$n$$ odd even $$m$$ odd even odd e odd odd
 $$n$$ odd even $$m$$ odd even odd e odd even
 $$n$$ odd even $$m$$ odd even odd e even odd
 $$n$$ odd even $$m$$ odd even odd e even even
 $$n$$ odd even $$m$$ odd even even e odd odd
 $$n$$ odd even $$m$$ odd even even e odd even
 $$n$$ odd even $$m$$ odd even even e even odd
 $$n$$ odd even $$m$$ odd even even e even even
Tags: functions

## Bézier curve

A Bézier curve is created as follows:
1) A set of points $$P_0$$, ..., $$P_n$$ are chosen (in the example $$n=4$$).
2) A set of points $$Q_0$$, ..., $$Q_{n-1}$$ are defined by $$Q_i=t P_{i+1}+(1-t) P_i$$ (shown in green).
3) A set of points $$R_0$$, ..., $$R_{n-2}$$ are defined by $$R_i=t Q_{i+1}+(1-t) Q_i$$ (shown in blue).
.
.
.
$$n$$) After repeating the process $$n$$ times, there will be one point. The Bézier curve is the path traced by this point at $$t$$ varies between 0 and 1.

What is the Cartesian equation of the curve formed when:
$$P_0=\left(0,1\right)$$ $$P_1=\left(0,0\right)$$ $$P_2=\left(1,0\right)$$

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