Ritangle 2017 took place 2 October - 7 December 2017.

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 the page for this year's competition.

Congratulations to the winning team from

Thomas Hardye School, Dorchester

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The number 3211000 is called **self-descriptive** since it contains three 0s, two 1s, one 2, one 3, zero 4s, zero 5s and zero 6s.

Find the two smallest self-descriptive numbers and add them together.

You are given nine rods of lengths 6, 7, 8, 10, 15, 17, 24, 25 and 26.

You pick three at random.

\(p\) is the probability that you can form a triangle with your rods.

The choice (6, 7, 26) is a fail, and so is (7, 10, 17).

In addition, \(q\) is the probability that your three rods make a right-angled triangle.

What is \(\large\frac{q}{p}\)?

Two competing shops have a suit for sale, and both are asking for the same price.

Both shops have a sale; the first shop drops the price of the suit by £18, the second drops it by 18%.

The following week, the first shop drops the price of the suit by a further 21%, while the second shop takes off a further £21.

After this second round of deductions, the two shops are again offering the suit at the same price.

What was the original price of the suit in pounds?

A triangle \(ABC\) has a perimeter of \(P\) cm and an area of \(Q\) cm\(^2\), where \(P = 2Q\).

Triangle \(DEF\) is similar to \(ABC\).

The sum of the perimeters of the two triangles in cm is equal numerically to the sum of their areas in cm\(^2\).

\(DEF\) has an area \(k\) times larger than \(ABC\). What is \(k\)?

A circle of radius 1 rolls along the \(x\)-axis towards the origin until it is stopped by the line \(y = x\).

What is the \(x\)-coordinate of its centre now?

In the diagram above, you are permitted to journey horizontally and vertically from any start point.

You are also allowed to retrace your steps.

Thus possible journeys include: **1, 12, 214, 123654, 2541, 12321**.

The journeys **233** and **126** are impossible.

Now consider the diagram below. It contains the journeys for the first thirteen numbers from a simple sequence where all the terms are different.

What is the fourteenth number?

What is \(u_{101}\) in this sequence?

\(u_1\) = thousand, \(u_2\) = million, \(u_3\) = billion, ….

A triangle \(ABC\) is isosceles, with the length of \(AC\) and the length of \(BC\) both 1.

The point \(D\) lies on \(BC\) produced so that the length of \(AD\) is 2..

Angle \(ABC\) = \(α\) and angle \(ADC\) = \(β\).

\(a+β=\large\frac{\pi}{6}\)

To two decimal places, what is the length of \(CD\)?

Two ants walk together along the x-axis from the origin to (1,0).

At (1,0) they part company:

- the first ant goes north a distance 0.9, then west (0.9)\(^2\), south (0.9)\(^3\), east (0.9)\(^4\), north (0.9)\(^5\)…
- the second ant travels south 0.8, west (0.8)\(^2\), north (0.8)\(^3\), east (0.8)\(^4\), south(0.8)\(^5\)...

The first ant ends up at point \(A\), and the second ant at point \(B\).

If \(m\) is the gradient of \(AB\), what is |\(m\)|?

Luke has four tiles, each with a different shape, size and colour, and each bearing a different number.

The tiles are circular, square, triangular and hexagonal, and they are blue, yellow, red and green in some order.

The sizes are tiny, small, large and huge, and the four numbers are 1000, 2000, 3000 and 4000.

You are given these facts:

- The yellow tile is circular and bears the number 3000.
- The tiny tile bears either the number 1000 or the number 4000.
- The red tile is not square.
- One of the huge tile and the triangular tile is green, while the other bears the number 2000.
- The tile bearing the number 2000 is either small or large.
- The small tile's number is 1000 less than the red tile's number.

Now work out the hexagonal tile's number times the blue tile's number as your answer.

You may find this grid helpful.

A rectangle has sides with lengths 12 and 8.

A square with side length \(c\) is drawn in one corner, creating the rectangular areas \(P, Q, R\) and \(S\) as in the diagram.

What is the minimum value that \(\large\frac{\textrm{area of } Q + \textrm{ area of } R}{\textrm{area of } P + \textrm{ area of } S}\) can take?

We define \(\lfloor{x}\rfloor\) to be the integer part of \(x\), so \(\lfloor45\rfloor = 45\), \(\lfloor56.8\rfloor = 56\).

If \(u_n=\lfloor(n+1)^\frac{3}{2}\rfloor+1-3n\) for \(n ≥ 1\), and \(k\) is the first positive integer so that \(u_k\) is positive, what is \(k\)?

The polynomial \(ax^3 + bx^2 + cx + 1\) gives a remainder of 21 when divided by \(x – 2\).

The polynomial \(cx^3 +ax^2 +bx+1\) gives a remainder of 25 when divided by \(x – 2\).

The polynomial \(bx^3+cx^2+ax+1\) gives a remainder of –1 when divided by \(x – 2\).

Find \(a^{(b^c)}\)

You are given that \(a\) = 18530, \(b\) = 38114, \(c\) = 45986.

Confirm that \(a + b, b + c\) and \(c + a\) are all perfect squares.

There is a fourth number \(d\) so that \(a + d = p^2\), \(b + d = q^2\) and \(c + d = r^2\), where \(d, p,q\) and \(r\) are all positive integers.

Find \(d\).

A rectangle \(R_1\) has sides \(j\) and \(k\).

The next rectangle in the sequence \(R_2\) has sides \(\frac{j}{2}\) and \(2_k\), while \(R_3\) has sides \(\frac{j}{4}\) and \(3_k\), and \(R_4\) has sides \(\frac{j}{8}\) and \(4_k\), and so on.

What, in terms of \( j\) and \(k\), is the sum of the areas of all the rectangles in the sequence?

Taken in one order the integers \(y < z < a\) are consecutive terms from an arithmetic sequence, and taken in another they are three consecutive terms from a geometric sequence.

What is \(y^2 + z^2 + a^2\) in terms of \(z\)?

If \(a\) and \(b\) are the smallest positive integers so that \(5a^7 = 7b^5\), what is \(ab\)?

The two parabolas \(y = x^2 + 5x + 2\) and \(x = y^2 + 5y + 2\) intersect in four points, where two of them, \(A\) and \(B\), are on the line \(y = x\).

What is the length of \(AB\)?

For what value of \(a\) do the curves \(y = a^x\) and \(y =\) log\(_a\) \(x\) touch?

The point A is \(\large(\frac{1}{5}, \frac{1}{7})\)and is on the same axes as the line \(L\) which is \(2x + 3y + q = 0\), where \(q\) is positive.

Initially \(A\) is not on \(L\). But

- if we change the \(x\)-coefficient to 2 – \(a\), then the revised line \(L\) goes though \(A\).
- if we instead change the \(y\)-coefficient to 3 – \(b\), then the revised line \(L\) goes though \(A\).
- if we instead change the constant term to \(q - c\), then this revised line \(L\) goes though \(A\).

If \(a+b+c = \large\frac{377}{35}\)+ \(d\) then what is \(d\) in terms of \(q\)?

Fred the policeman sees the man he wants, Roger the burglar, in a car down the straight road ahead.

Fred is cycling along at a steady speed of 3m/s.

As he passes a lamp-post Roger spots him and starts to drive away from rest with a steady acceleration of

0.1m/s\(^2\).

Fred's front wheel just grazes Roger's back bumper before Roger disappears into the distance.

How far was Roger's car initially in metres from the lamp-post?

A square contains the largest possible regular hexagon.

What is \(\large\frac{\textrm{area of hexagon}}{\textrm{area of square}}\)?

**nearest integer** multiple of \(p\).

Luke is working with logarithms to base 10.

He makes three mistakes in a row; he says that:

- \(log(6) + log(a) = log\)\((\)\(6 + a)\)
- \(log(b) log(\)6\() = log(b – \)6\()\)
- \(log(c^6) = (log(c))^6\)

Strangely, however, \(a, b\) and \(c\) are all numbers bigger than 1 such that the equations he's written down do in fact hold.

What is the integer part of \(abc\)?

A particular kind of arithmagon works like this:

Find **?** in this arithmagon:

The function of four variables \(C(a, b, c, d) = \large\frac{(a-b)(c-d)}{(a-c)(b-d)}\) is called the cross-ratio function.

What is the maximum number of different values for \(C(a, b, c, d)\) that we can find if we pick four distinct numbers and assign them to \(a, b, c\) and \(d\) in all possible ways, without repeats?