Ritangle 2019 took place 7 October - 17 December 2019.

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

King Edward VI School, Stratford-upon-Avon

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The numbers \(1\), \(2\), \(3\), \(4\), \(5\) and \(6\) are placed into the squares below (no repeats!) in some order in a way that makes a truthful sentence.

The line segment \(AB\),

where \(A = ( \Box ,\Box )\) and \(B = ( \Box ,\Box )\),

has midpoint \(( \Box ,\Box)\).

What is the largest that the length \(AB\) can be?

Bella and Carissa need to travel the same one kilometre on level ground.

They possess a (rather rusty) bicycle. Bella cycles at 6 km/hr, and walks at 4 km/hr, while Carissa cycles at 5 km/hr and walks at 3 km/hr. They set off together; Bella cycles for x km, leaves the bike and walks the rest of the way. Carissa starts off walking, before picking up the bike and cycling the rest of the way.

At time \(t\) hours, both Bella and Carissa have finished. What is the smallest that \(t\) can be?

Luke wrote down this equation, in variables \(x\) and \(z\), on a piece of paper;

$$\frac{x-I}{9}-\frac{6}{z}=I$$He then turned his piece of paper through 180\(^\circ\), revealing a second equation in variables \(x\) and \(z\). He decided to solve the two equations together as a pair of simultaneous equations.

If, after doing this, he multiplies all the possible values of \(x\) together, what does he get?

A right-angled triangle has hypotenuse length \(1\), which happens to be equal to the sum of the length of one side and the height of the triangle if taking the hypotenuse as the base.

Find the perimeter of the triangle to 3 sig. figs.

Technology may be needed here to find a suitably accurate approximation to the solution of an equation.

The diagram shows an equilateral triangle with some lengths labelled.

Find \(x\)

A tree was planted on April 1^{st} \(abcd\) (where \(a\), \(b\), \(c\) and \(d\) are distinct non-zero digits; see ‘Notation’ below).

The tree will be cut down on April 1st \(dabc\), when its age two years later would have been \(dcc\) (using the same ‘digit notation’ as for years).

What is \(a+b+c+d\)?

**Notation:** \(pqrs\) is the year \(1234\) if and only if \(p=1\), \(q=2\), \(r=3\) and \(s=4\).

The diagram, which is not to scale, shows triangles \(ABC\) and \(BED\).

The length of \(BC\) is \(6\), the length of \(AB\) is \(8\) and the length of \(AC\) is \(10\).

The length of \(BE\) is \(13\), the length of \(BD\) is \(12\) and the length of \(ED\) is \(5\).

What is the shaded area?

Three ants find themselves at \(A\), \(B\) and \(C\) where \(ABC\) is an equilateral triangle of side \(10cm\).

At time \(t=0\), they set off walking along the sides of the triangle; ant A travels at \(1cm/s\), ant B travels at \(2cm/s\), while ant C travels at \(3cm/s\).

Ant A chooses whether to travel towards \(B\) or \(C\) at random, and similarly the other two ants choose randomly from the two directions available to them. When an ant gets to a vertex it continues to travel around the perimeter of the triangle in the direction it is going.

To 3 significant figures, what will be the average (expected) time in seconds that elapses before the first collision between two ants?

You are given a circle with a rectangle that touches the circle on three sides.

You are also given one measurement as in the diagram.

What is the area of the rectangle?

Planet Zog has exactly 60 seconds in a minute, 60 minutes in an hour, 24 hours in a day and 365 days in a year. It uses the same system of months, days and years that we do in a non-leap year (leap years don’t exist on planet Zog). Given a number \(x\) with at least 10 digits, \(x\)-time on Zog is given from the first 10 digits of \(x\) (ignoring any decimal places) as follows:

Day Month Year Year Year Year Sec Sec Min Hour

For example, since \(\pi=3.1415926536\)…., \(\pi\)-time is the 3^{rd} day of January, 4159, 26 seconds and five minutes past 3am (so no rounding).

For some numbers \(x\) with at least 10 digits, \(x\)-time will not exist, e.g. for \(9.012345678\) (there is no month 0).

There are times which are not \(x\)-time for any value of \(x\) (e.g. any time in October, or any time in the afternoon).

Find, in seconds, the time elapsed on Zog between \(\varphi\)-time and \(e\)-time, where \(\varphi\) is the golden ratio.

Two rectangles \(P\) and \(Q\) are said to be friendly if the size of the area of \(P\) equals the size of the perimeter of \(Q\), and also the size of the perimeter of \(P\) equals the size of the area of \(Q\). If \(P\) and \(Q\) are friendly, we say \(P*Q\); if \(P*P\) we say \(P\) is self-friendly.

Find all friendly pairs of rectangles and all self-friendly rectangles in which all sides are positive integers. List all the side lengths you have found. You may use the fact that all these values are less than 100.

Remove repeats of any numbers in your list and add up the list of distinct numbers you have left.

You may find technology helpful for working on this problem.

The sequence of fractions below tends towards the number \(e\).

\(a_0=2\),

\(a_1=2+\cfrac{1}{1}\),

\(a_2=2+\cfrac{1}{1+\cfrac{1}{2}}\),

\(a_3=2+\cfrac{1}{\cfrac{1}{2+\cfrac{2}{3}}}\),

\(a_4=2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{2}{3+\cfrac{3}{4}}}}\)

Calculate the value of \(a_6\).

To three significant figures, what is the size of the percentage error in this estimate of \(e\)?

- Find \(z\) in terms of \(x\) and \(y\). This is just for interest!
- Annie takes 7 hours to complete a painting job, while Leroy takes 8 hours to do the same job. How long in minutes should they take working together?
- What is the connection between parts a and b? This is just for interest!

Find \(\displaystyle \int_1^{10}\cfrac{\{x\}}{[x]}dx \)

Notation: considering the case when \(x\) is positive,

- \([x]\) is the integer part of \(x\), so \([3.2]=3\), while \([3]=3\)
- \(\{x\}\) is the 'fractional part' of \(x\), so \(\{3.2\}=0.2\), while \(\{3\}=0\)
- you can see that \(x-[x]=\{x\}\), so \( \cfrac{22}{7} - \left[\cfrac{22}{7}\right] = \cfrac{22}{7}-3 = \cfrac{1}{7} = \Big\{\cfrac{22}{7}\Big\} \)

In 2020, four English football teams make it to the last eight of the Champions League.

The draw for the next round is made at random. ‘Home’ and ‘Away’ can be ignored.

What is the probability that exactly one pairing features two English teams?

What is the maximum possible area of a quadrilateral with these sides?

You may find technology helpful for working on this problem.

Not drawn to scale

With reference to triangle in the diagram, to three significant figures, what is \(x\)?

You may find technology helpful for working on this problem.

Each rectangle contains a number; some numbers are hidden (i.e. they are not shown).

The sum of the numbers in the four rectangles around a dot is the same for all twenty dots.

What is the value of \(xy\)?

Add together all the positive integrer values of \(x\) that satisfy this equation:

\( (x^2-199x+9899)^{(x^2-192x+9215)}=1\)

You pick three distinct numbers \(a\), \(b\) and \(c\). The diagram now gives you six equations to solve for \(x\).

For example,

\(ax+b=cx+a\) and \(bx+a=ax+c\)

The solution to one of the equations is \(x=3\).

What is the sum of the squares of the solutions to the six equations?

A sequence starts with the terms \(10\), \(20\).

Rule 1: to get the next term, divide the previous term by the one before that.With this rule the third term in the sequence would be \(\cfrac{20}{10}=2\).

Rule 2: to get the next term, add 1 to the previous term and then divide by the term before that. With this rule the third term would be \(\cfrac{20+1}{10}=2.1\).

Applying Rule 1 repeatedly gives you one sequence, $$u_1=10, u_2=20, u_3=2,u_4=...$$

while applying Rule 2 repeatedly gives you another, $$v_1=10, v_2=20, v_3=2.1, v_4=...$$

What is the first value of \(n\) greater than \(1\) such that \(u_n=v_n\) and \(u_{n+1}=v_{n+1}\)?

What is the sum of the infinite series:

$$a+2ar+3ar^2+4ar^3+...$$when \(a=r=\cfrac{1}{8}\)?

A \(900\times2250\) rectangle is divided into a grid of \(2,025,000\) squares, each of side \(1\).

How many of the \(1\times1\) squares does a diagonal of the rectangle pass through?

Note: going through the corner of a square does not count as passing through the square.

The area enclosed by the curves

$$y=3x^2+2x+a$$and

$$y=ax^2+2x+3$$is \(1\)

What is the sum of all possible values for \(a\)?

In the expression

$$\left(ax+\cfrac{b}{x}\right)^c+\left(bx+\cfrac{c}{x}\right)^a+\left(cx+\cfrac{a}{x}\right)^b$$\(a\), \(b\) and \(c\) are consecutive positive integers with \(a<b<c\).

The value of the constant term when the three brackets are expanded and all the resultant terms are added is \(1102707270\).

What is \(a\)?

Final task - part A

You are given two biased dice, each bearing the numbers from 1 to 6.

For the first dice; the probability of scoring x when the dice is rolled is proportional to x.

For the second dice; the probability of scoring x when the dice is rolled is inversely proportional to x.

The dice are rolled together; what (to 3 s.f.) is the probability that they both show a prime number?

Note: 1 is not a prime number.

Final task - part B

You are given that for three positive numbers x, y and z,

\(z^{(y^x)}=(z^x)^{\frac{1}{y}}\)

What is the largest that \(y\) can be?

A graphing program may be needed here to find a maximum point.

Final task - part C

Pasquale says that Pasquale's Triangle is created as below:

1

2

2

3

4

3

4

7

7

4

5

11

14

11

5

...

...

...

...

...

...

If we write the triangle as

\(^0Q_0\)

\(^1Q_0\)

\(^1Q_1\)

\(^2Q_0\)

\(^2Q_1\)

\(^2Q_2\)

\(^3Q_0\)

\(^3Q_1\)

\(^3Q_2\)

\(^3Q_3\)

\(^4Q_0\)

\(^4Q_1\)

\(^4Q_2\)

\(^4Q_3\)

\(^4Q_4\)

...

...

...

...

...

...

Then what is \(^{100}Q_{99}\)?

Final task - part D

The number \(a\), \(b\), \(c\) are consecutive positive whole numbers.

The lines \(y=bx+c\) and \(y=bx+c\) meet at \(B\).

The lines \(y=bx+c\) and \(y=cx+a\) meet at \(C\).

The lines \(y=cx+a\) and \(y=ax+b\) meet at \(A\).

What is the area of triangle \(ABC\)?

Final task - part E

A rook starts from the bottom left corner of a chess board, and journeys to the top right corner. It can only travel north or east, and it cannot cross the given thick line.

How many possible journeys are there?

Final task - part F

There are two values of \(x\) for which the tangents to the cubic curves

$$y=f(x)=2x^3+x^2+3x+4$$and

$$y=g(x)=x^3+7x^2-6x+1$$are parallel.

These four tangents create a parallelogram; what (to 3 s.f.) is the x-coordinate of its centre?

Final task - part G

The function \(|x|\) means 'the positive size of \(x\)', so \(|5|=5\), \(|-5|=5\).

Explicitly,

$$|x|=\left\{ \begin{array}{lr} x\text{ for } x\geq 0\\ -x \text{ for } x<0\end{array}\right.$$The curve \(y=2a|x|+a^2x\) consists of two infinite half-lines that meet at the origin. If \(a\) is positive, and the two half-lines are perpendicular, what is \(a\)?

Add together all possible values for \(a\) and round to 3s.f.

Final task - part G

The last time I stayed in a hotel, I was given the room number \(316\).

‘That’s an interesting number,’ I thought, ‘since \(316 =100 + 216 = 10^2 + 6^3\).’

Now \(1 = 1^2 + 0^3 = 1^3 + 0^2\).

So \(1\) can be written as the sum of a square and a cube in more than one way.

What is the sum of all the numbers from \(0\) to \(99\) that can be written as the sum of a square and a cube in more than one way?

Note; both the cube and the square need to be non-negative.

You may find Excel useful to you in this problem.

You are given many copies of two kinds of tile, \(5\times5\) and \(7\times7\).

The tiles are used to cover a rectangular \(n \times m\) floor without gaps like so; tiles of type A fill the bottom left rectangle as shown, while tiles of type B cover the remainder of the \(m \times n\) rectangle. The tiles type A rectangle is \(a \times b\), where \(0 \leq a \leq m\), \(0 \leq b \leq n\). If \(m = 98\) and \(n = 126\), what is the largest total number of tiles that could be used to cover the floor completely?

Final task - part J

A triangle has two angles \(x\) and \(2x\) (measured in degrees).

The side opposite the angle size \(x\) is of length \(x + 20 cm\), the side opposite the angle size \(2x\) is of length \(2x + 20 cm\).

Find the length of the third side in \(cm\) (to 3 sig figs).

Tiebreaker

A computer prints out the days of the year in the 21st century in the form day-month-year without any leading zeroes, so that the tenth of October 2054 is \(10 \text{-} 10 \text{-} 54\) and the first of July 2002 is \(1 \text{-} 7 \text{-} 2\).

One day the computer is asked to print out all the days from \(1 \text{-} 1 \text{-} 1\) to \(31 \text{-} 12 \text{-} 99\), but due to a glitch, it omits the hyphens. For some dates, this does not really matter; \(111\) can only be \(1 \text{-} 1 \text{-} 1\), and \(311299\) must be \(31 \text{-} 12 \text{-} 99\) (such dates are called CLEAR). But some dates are ambiguous; \(11211\) could be \(1 \text{-} 12 \text{-} 11\) or \(11 \text{-} 2 \text{-} 11\) (such dates are called UNCLEAR).

How many UNCLEAR dates are printed by the computer?

(The number \(11211\), for example, will appear twice, and will be counted twice in your count.)