Ritangle 2018 took place 1 October - 12 December 2018.

Ritangle is a competition designed for teams of students aged 16-18 who are currently studying either:

- A level Mathematics
- the International Baccalaureate
- Scottish Highers
- qualifications with equivalent content.

For more information about Ritangle, please visit our information page.

Congratulations to the winning team from

Thomas Hardye School, Dorchester

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How many 8 digit numbers are there which are both

- divisible by \(18\) and
- such that every digit is either a \(1\) or a \(2\) or a \(3\)?

In this question \(a\) > \(0\).

The line \(y = 3ax\) and the curve \(y = x^2 + 2a^2\) enclose an area of size \(a\).

What is \(a\)?

Let \(f(x) =10x^2 + 100x + 10\).

Suppose \(f(a) = b\) and \(f(b) = a\).

Given that \(a ≠ b\), what is \(f(a + b)\)?

In this question \(a\) and \(b\) are positive. A quadrilateral is formed by the points \(A, B, C\) and \(D\) where \(A\) is \((a, 0)\), \(B\) is \((0, b)\), \(C\) is \(\left(-\frac{1}{b},0\right)\) and \(D\) is \(\left(0,-\frac{1}{a}\right)\).

\(ABCD\) is always a trapezium.

If \(a = 11\), what value of \(b\) minimises the area of trapezium \(ABCD\)?

**A spreadsheet may help you with this question.**

In this question angles are in radians. An infinite sequence \(x_0, x_1, x_2, x_3,\)… is defined as follows:

\(x_0 = 1\), \(x_{2n+1} = cos(x_{2n})\), \(x_{2n+2}\) = arctan\((x_{2n+1})\) for all integers \(n ≥ 0\).

Find the limit to which the sequence \(y_n = x_{2n+1} – x_{2n+2}\) \((n ≥ 0)\) converges.

What percentage of the regular octagon shown is shaded?

A biased six-sided dice showing the faces 1, 2, 3, 4, 5, 6 is rolled 21 times, giving 21 results. One face shows once, another twice, a third three times, a fourth four times, a fifth five times and the sixth six times.

The median result is 3. The IQR is 4. The sum of the results, \(\sum x\) , is 80.

What is \(\sum x^2\)?

For a real number \(x\), the floor function,\(\lfloor{x}\rfloor\) , is defined as the largest integer less than or equal to \(x\), while the ceiling function,\(\lceil{x}\rceil\) , is defined as the smallest integer greater than or equal to \(x\). Thus \(\lfloor{3}\rfloor\)=\(\lceil{3}\rceil\)=\(3\) and \(\lfloor{5.1}\rfloor\)=\(\lceil{4.9}\rceil\)=\(5\).

Define a sequence \(u_n\)=\(\lfloor(\frac{n}{10}^2)\rfloor\)+\(\lceil(\frac{n}{10}^2)\rceil\) for positive integers \(n\).

What is the smallest value of \(n\) so that \(u_{n+1} = u_n+4?\)

Two numbers \(x\) and \(y\) are such that \(0 < x < 1\) and \(0 < y < 1\).

The sum to infinity of the geometric series with first term \(x\) and common ratio \(y\) is 2. The sum to infinity of the geometric series with first term \(y\) and common ratio \(x\) is 3.

What is \(xy\)?

A palindromic number is one that reads the same forwards as backwards. For example 121 is a palindromic number but 122 is not.

It’s recently been shown that every positive integer is the sum of three positive palindromic numbers. For example 2587876 = 2534352 + 18981 + 34543.

You are given that 652641310 = 1_5_1 + 34_6_43 + 649_4_946 where the three numbers on the right are palindromic.

Find the six digits that fill the gaps. What is the product of these six digits?

What number do you get if the number of distinct arrangements of the letters in the string HUBBAHUBBA is divided by the number of distinct arrangements of the letters in the string HUBBA?

Given any triangle \(ABC\),

- the line perpendicular to \(BC\) which passes through \(A\)
- the line perpendicular to \(AC\) which passes through \(B\)
- the line perpendicular to \(AB\) which passes through \(C\)

will always meet at a point called the *orthocentre*.

A triangle has its orthocentre at the origin. One of its sides is part of the line \(3y = x + 2\), while another side is part of the line \(x = 1\). Find the perimeter of the triangle.

What is the value of the term in the expansion of

\(\large\left(6x^3-\frac{5}{x^2}\right)^{10}\)

that is independent of \(x\)?

Define

\(f(x)=30sin(40x+72)=40cos(72x+30)+72tan(30x+40)\)

(the input into each trigonometric function is in degrees).

What is the period of \(f\) ?

You are given that \(a^bc^d = 46656\)

where \(a ≤ c\) and \(a, b, c\) and \(d\) are all integers with \(a, b, c, d ≥ 2\).

How many possibilities for \((a, b, c, d)\) are there?

Part of the Franciscan church in Nice looks like this. Suppose a nearby church includes this in its architecture, as in the diagram on the right. The grid is comprised of sixteen 1 by 1 squares.

The curves \(AB\) and \(AF\) are arcs from circles centred at \(G\) and \(H\) respectively. The curves \(BC, CD, DE\) and \(EF\) are all arcs from circles with their centres on the straight line that includes \(C\) and \(E\). What is the shaded area?

You are given that \(f(x)=ax^3+bx^2+cx+d\).

You are also told that \(f(0)=0, f'(1)=1, f''(2)=2, f'''(3)=3\).

What is \((c+d)^{a+b}\)?

An equilateral triangle \(ABC\) with side length 1 is divided into three triangles \(ABM, MCN\) and \(CAN\) each with the same area. This is shown in the diagram.

What is the length \(x\)?

$$\text{Find} \begin{array}{cc} 100\\ \large\sum \\ r=1\end{array}\left(r^2\begin{array}{cc} \sqrt[r]{r+1} \\ {\displaystyle\int} \\ \sqrt[r]{r}\end{array}x^{r-1}dx\right)$$

A square with side length \(x\) has perimeter \(P\) and area \(A\).

A rectangle with sides of lengths \(x\) and \(y\), where \(y ≠ x\), has perimeter \(P´\) and area \(A´\).

The numerical values \(P´, P, A´, A\) are four consecutive terms from an arithmetic sequence.

What is the numerical value of \(P + A + P´ + A´\) ?

You are given that \(\large\frac{a+\sqrt{b}}{5-3\sqrt{b}}\)\(=c+d\sqrt{b}\).

Where \(a, b, c, d\) are integers so that \(0 < a, b, c, d < 7\) and \(b\) is not a square.

Find \(abcd\).

The area inside the first quadrant \((x ≥ 0, y ≥ 0)\) enclosed between the curves \(y=x^k\) and \(y=x^{\frac{1}{k}}\), where \(k > 1\), is \(\frac{1}{100}\).

What is \(k\)?

The sizes of the three angles of a triangle, measured in degrees, are three consecutive terms from a geometric sequence.

The same three values (the sizes of the three angles, measured in degrees) multiply together to give 20.

What, in degrees, is the smallest angle?

The point \(A\) is on the parabola \(y = x^2 + 2\).

The point \(B\) is on the parabola \(x = y^2 + 2\).

What is the smallest that the distance \(AB\) can be?