# Puzzles

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time geometry 2d shapes 3d shapes numbers spheres trigonometry complex numbers algebra lines graphs coordinates odd numbers fractions differentiation calculus folding tube maps ellipses triangle numbers money bases triangles squares area square numbers chess probability circles averages speed sport multiples dates factors parabolas functions logic cards games people maths shape prime numbers irreducible numbers probabilty angles proportion dice integration sum to infinity dodecagons hexagons multiplication factorials coins shapes regular shapes colouring grids floors integers rugby crosswords percentages digits sums christmas rectangles clocks menace routes taxicab geometry remainders chalkdust crossnumber palindromes sequences means unit fractions division square roots surds doubling quadratics indices planes volume number partitions ave pascal's triangle mean advent symmetry arrows addition cube numbers star numbers perfect numbers## 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 \(p(6)\)?

## Between quadratics

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 \(p(16)\).

## Balanced sets

A set of points in the plane is called 'balanced' if for any two points \(A\) and \(B\) in the set, there is another point \(C\) in the set such that \(AC=BC\) (here \(AC\) is the distance between \(A\) and \(C\)).

For all \(n\geq3\), find a balanced set of \(n\) points.

## 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).

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\(n\)) 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

Source:

*Alex Through the Looking-Glass: How Life Reflects Numbers and Numbers Reflect Life*by Alex BellosOn 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?

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