Ritangle 2020 took place 5 October - 15 December 2020.
Ritangle is a competition designed for teams of students aged 16-18 who are currently studying either:
For more information about Ritangle, please visit the page for this year's competition.
Congratulations to the winning team from
The Cherwell School, Oxford
The team from King Edward VI School, Stratford-Upon-Avon were runners up. They also produced some excellent work in stage 3 of the final question.
Teams from the following schools also reached stage 3 of the final question. Less than 5% of the teams that entered achieved this in a very challenging Ritangle 2020. Congratulations to all the schools and colleges below:
Released: 0900 GMT, 10th December 2020
In the long string of numbers formed from the answers to questions 1 to 24 you should see lots of 1 and 2 digit primes. Now think how that might be a code.
Released: 0900 GMT, 11th December 2020
From questions A to H you have another string of digits with lots more 1 and 2 digit primes in there. But this time the code is harder. Look back at all those names in questions 1-24 and A-H. They were chosen for a reason.
Kanesha is examining her gas bill. She sees that a unit of gas sells for \(£a\). This price is then increased by \(b\)%.
A month later, Kanesha notices that the new price is reduced by \((b-1)\)%, and that takes the unit price back to \(£a\).
What, to 3 significant figures, is \(b\)?
Evan rolls a fair dice 10 times.
What is \(\large\frac{P(\textrm{Evan gets exactly four fours})}{P(\textrm{Evan gets exactly five fives})}\)?
Yu Yan is watching two cricket teams play a 20 overs a side game. The team chasing have been set a target of \(T\) runs, but Yu Yan can’t see what \(T\) is on the scoreboard.?
After 10 overs they have scored 78 runs, and after 11 overs they have scored 88 runs. The run-rate (average runs already scored per over already bowled) and the required run-rate (required average runs per over remaining to be bowled) are calculated after the 10th and 11th overs, and Yu Yan notices that on the scoreboard they both increase by the same amount.
What is \(T\)?
Wendy defines the graph of a periodic function as follows for positive x;
and so on, all with radius \(n\).
Wendy discovers that if all the curves \(y-f_n(x)\) are drawn for all positive integers \(n\), then the line \(y=-mx\) (where \(m\) is finite and as large as possible) is a tangent to every curve. What is the value of \(m\) to 3 significant figures??
Olivia draws a quadrilateral with four interior angles that are in geometric progression.
The common ratio \(r\) is an integer.
The smallest of the four angles is \(a\) degrees, and \(10a\) is an integer. ?
If Olivia has drawn the quadrilateral where \(r\) takes the largest possible value, what is \(a\)?
Ramesh constructs an acute angle of size \(x\)°.
He notices that sin \(x\)°, cos \(x\)° and tan \(x\)° in that order are consecutive terms from an arithmetic progression.
What is x to 3 significant figures?
Davina is examining the world record times for the marathon in various years.
She writes down two such times as follows;
Belayneh Densamo (Ethiopia) 2:06:50 Rotterdam, 1988
Eliud Kipchoge (Kenya) 2:01:39 Berlin, 2018
Davina assumes the graph of world record time against year is linear, and on the basis of these two times calculates the expected world record time for the marathon in 2050 to the nearest second. What is this time?
Assume that all these marathons take place at the same time of year.
Give your answer as HMMSS.
Sita puts 26 tiles bearing the letters A to Z into a bag and draws three tiles one by one at random without replacing them as she goes.
What’s the probability, to 3 significant figures, that she draws the three tiles in alphabetical order?
Donald draws ten mathematical objects that can be labelled from \(3\) to \(12\).
Donald puts them into a natural sequence as follows;
\(10-12-11-7-6-a-b-c-d-e\)
where \(a, b, c, d\) and \(e\) are the digits \(3, 4, 5, 8\) and \(9\) in some order.
What is the five-digit number \(abcde\)?
Ian constructs a large cube in layers from unit cubes as follows.
The first layer is one cube scoring \(100\). This layer scores \(100\).
The second layer is \(26\) cubes surrounding the first layer, creating a solid \(3×3×3\) cube, with each of these new cubes scoring \(99\). This new layer scores \(26×99 = 2574\).
The third layer is \(98\) cubes surrounding the second layer, creating a solid \(5×5×5\) cube, with each of these cubes scoring \(98\). This layer scores \(98×98 = 9604\). ?
This continues until Ian adds the \(100th\) layer, with each of these cubes scoring \(1\).
What is the score for the highest scoring layer?
Oscar draws a pair of circles.
The circumferences of the two circles add to \(πa\), while their areas add to \(πb\).
If \(a\) is 24, what is
(maximum possible value of \(b\)) – (minimum possible value of \(b\))?
A point can be seen as a circle of zero radius.
Paula draws the lines \(y=ax+1\) and \(y=2ax+3\), which meet at A. ?
She then draws the lines \(y=3ax+6\) and \(y=4ax+10\), which meet at B.
The value \(a\) is a positive integer. ?
If AB \(\large=\frac{\sqrt{629}}{5}\), what is \(a\)?
Haris is making models of a regular tetrahedron, \(T\), a regular octahedron, \(O\), and a cube, \(C\).
He is using sticks of various lengths to make the models. Each of the edges of the cube \(C\) is a 10cm long stick.
Haris wants the combined surface areas of \(T\) and \(O\) to add to the total surface area of \(C\). He also wants the combined total stick-length for \(T\) and \(O\) to add to the total stick-length for \(C\).
Find the length of one of the sticks needed to build \(T\) in cm to 3 significant figures.
Aisha is trying to save \(£x\).
After a week, she’s saved a total of \(£p\), which is \(q\)% of the total required.
After a further week, she’s saved a total of \(£q\), which is \((p+7.5)\)% of the total required.
Aisha notices that \(p+q=\large\frac{x}{10}\). What is \(x\)?
Nigel is initially baffled by this cryptic clue;
Neither impossible nor easy,
this elegant examination needs terrific insight
(made eventually straightforward, thankfully)
when everything looks very enigmatic…
But eventually he solves it to get a three digit number. What is this number?
Tom constructs the diagram shown, which consists of four squares and two equilateral triangles, all of side 1 unit, that surround a hexagon.
The six angles labelled \(a\)° are all equal.
Find \(a\).
Una is examining the polynomial \(f(x)=ax^3+bx^2+cx+d\), where \(a, b, c\) and \(d\) are all integers.
She notices that \(\large\frac{d^2}{dx^2}\)\((f(x)+x^3)=12\sqrt[3]{f(x)+3x^2}\) for all values of \(x\).
What is \(a\) x \(b\) x \(c\) x \(d\)?
Susanna cracks the code
1968119625 256144441361 368136400625-36181576
to get a two digit number. What is this number?
Francois is making models of a cube and a dodecahedron using sticks for the edges.
He wants each solid to have the same surface area.
If the cube has surface area 100cm\(^2\), then what, in cm to 3 significant figures, is the total stick length required to make the pair of models?
Eleanor is reading about mathematics on the internet.
She discovers that \(17^2+19^2+23^2+29^2=2020\), so \(2020\) is the sum of the squares of four consecutive primes. The diagram shows this fact, via five ?squares surrounding three right-angled triangles.
The sides \(a, b, c, d\) are \(17, 19, 23, 29\) in some order.
In the case when \(a<b\), Eleanor wonders what order would bring the marked angle PQR as close as possible to 180° without exceeding it.
Give your answer as the values for \(a, b, c\), and \(d\) concatenated
Roger draws three circles \(X, Y\) and \(Z\) that touch at A, B and C as in the diagram.
ABC is a diameter of \(X\), AB is a diameter of \(Y\) and BC is a diameter of \(Z\).
Circle \(X\) has area \(A_X\) and radius \(x\), circle \(Y\) has area \(A_Y\) and radius \(y\), and circle \(Z\) has area \(A_Z\) and radius \(z\).
If , \(A_X+2A_Y+2A_Z=\pi\) what is the largest (to 3 significant figures) that \(x\) can be?
Mollie sings in a group with five other people.
In a performance, she has to stand next to Tevin.
In addition, she cannot stand next to Becki.
Jonny has to stand on one end of the group of six.
As the audience views things, in how many different arrangements can the group stand?
Alan is faced with a code, for which he is given a keyword clue.
Flippant and comic, even tongue-in-cheek?
Investigate our useful starters.
523418525-5237 561749 4612135
Alan cracks the code to give a three-digit number; what is this?
Troppo the clown puts on his shoes and socks using ‘a routine’.
A routine involves choosing a foot, putting a shoe or sock onto that foot, and then repeating this three further times.
At the end of a routine, he is always wearing two shoes and two socks. A shoe is always over a sock.
Troppo has four pairs of socks (yellow, orange, purple and white).
He also has three pairs of shoes (blue, red, and green).
His shoes must match, but his socks must not match.
The socks are (unusually) labelled left and right, and his shoes (as usual) are left and right; Troppo always obeys these instructions.
How many possible different routines (where different colours for shoes or socks count as different routines) does? Troppo have to choose from?
