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# Puzzles

## 10 December

For all values of $$x$$, the function $$f(x)=ax+b$$ satisfies
$$8x-8-x^2\leqslant f(x)\leqslant x^2.$$
What is $$f(65)$$?
Edit: The left-hand quadratic originally said $$8-8x-x^2$$. This was a typo and has now been corrected.

## 6 December

$$p(x)$$ is a quadratic with real coefficients. For all real numbers $$x$$,
$$x^2+4x+14\leq p(x)\leq 2x^2+8x+18$$
$$p(2)=34$$. What is $$p(6)$$?

## Two tangents

Source: Reddit
Find a line which is tangent to the curve $$y=x^4-4x^3$$ at 2 points.

Source: Luciano Rila (@DrTrapezio)
$$p(x)$$ is a quadratic polynomial with real coefficients. For all real numbers $$x$$,
$$x^2-2x+2\leq p(x)\leq 2x^2-4x+3$$
$$p(11)=181$$. Find $$p(16)$$.

## 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:
$$P_0=\left(0,1\right)$$ $$P_1=\left(0,0\right)$$ $$P_2=\left(1,0\right)$$

## Parabola

On a graph of $$y=x^2$$, two lines are drawn at $$x=a$$ and $$x=-b$$ (for $$a,b>0$$. The points where these lines intersect the parabola are connected.
What is the y-coordinate of the point where this line intersects the y-axis?

## Two lines

Let A and B be two straight lines such that the gradient of A is the y-intercept of B and the y-intercept of A is the gradient of B (the gradient and y-intercept of A are not the same). What are the co-ordinates of the point where the lines meet?

## Archive

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