Ritangle 2019 took place 28 September - 9 December 2016.

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

King Edward VI School, Stratford-upon-Avon

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In the below \(r, i, t, a, n, g, l\) and \(e\) are non-zero real numbers.

A sequence is defined as follows:

\(u1 = r, u2 = i, u3 = t, u4 = a, u5 = n, u6 = g, u7 = l\) and \(u8 = e\).

Subsequent terms are defined as \(\frac{1}{\textrm{product of previous eight terms}}\)

What is \(u_100\) ? This is the first part of a key to unlock a clue for the main competition.

Take the four numbers from the bag and put them into the circles in some order (no repeats!).

How many different equations can you make? What are the solutions?

Write down the possible positive integer solutions in descending order.

This is the second part of a key to unlock a clue for the main competition.

A triangle has angles in degrees that are all integers.

One is a square, another is a cube and the third is a fourth power.

Write down the sizes of the three angles in descending order.

This is the third part of a key to unlock a clue for the main competition.

Given a positive integer \(n\), we say \(s(n)\) is the sum of all the factors of \(n\) not including \(n\) itself.

Thus \(s(6) = 1 + 2 + 3 = 6; s(7) = 1; s(8) = 1 + 2 + 4 = 7; s(9) = 1 + 3 = 4.\)

It is easy to find even numbers n so that \(s(n) > n\), for example, \(s(12) = 1 + 2 + 3 + 4 + 6 = 16\).

It's harder to find odd numbers n where \(s(n) > n\), but it is possible; for example, \(s(1575) = 1649 > 1575.\)

There is one odd number \(n\) smaller than 1575 so that \(s(n) > n\).

This number is the fourth part of a key to unlock a clue for the main competition.

Replace the question marks with the whole numbers from 5 to 29 inclusive (no repeats!).

? and ? are numbers with first digit 2, that add to 50.

3 and ? are the prime factors of ?

? is the square of ?

? and ? are twin primes.

The number of Archimedean solids is ?, which is half ?

? is both an odd number and a cube.

? > ? are each one more than a Fibonacci number, and one less than a triangular number.

? and ? and ? and ? multiply to 73370.

? and ? have an HCF that is one less than ?

? + ? = 20, and their LCM is ?

? and ? multiply to 1 less than an odd square.

Write down the three red question-mark numbers in descending order.

The equation of the perpendicular bisector of the line \(AB\), where \(A = (2, 5)\) and \(B = (6, 3)\) is what?

Take a positive integer \(a\), cube all its digits and add the numbers you get together to get a positive integer \(b\). Now do the same to \(b\), to get a positive integer \(c\).

If \(a = 1\), then \(c = a\). What is the next value of a so that \(c = a\)?

Note: this value is less than \(1000\).

A right-angled triangle has sides of length \(x, y\) and \(z\) where \(x, y\) and \(z\) are integers and \(x < y < z\).

Adding the three side lengths gives \(810\), while multiplying the three side lengths gives \(13284\) times this.

What is the area of the triangle?

The line \(y = mx + k\) touches the parabola \(y = ax^2 + bx + c\) (where \(a ≠ 0\)) at the point (\(p, q\)).

If \(m = 8a^2 + 4ab + 12ac + b\), what is \(p\) (in terms of \(a; b\) and \(c\))?

An arithmetic progression has third term \(32j + 19k\) and tenth term \(18j + 12k\).

What, in terms of \(j\) and \(k\), is the sixteenth term?

Consider the following equation (in radians):

\(sin(10^9x) = 0.1\)

Let \(n\) be the number of roots this equation has in the interval \(0 ≤ x ≤ 325\).

What is the value of \(n\) rounded to three significant figures?

If \(\large\frac{x^5}{y^2}\)\(=98304\) and \(\large\frac{y^5}{x^2}\)\(=\frac{6561}{64}\) then what is \(x\) ?

Multiply this answer by one million.

We can say \(cos x + sin x\) is one of these, while \(cos x + sin(πx)\) is not one of these, but \(cos^2 x + sin^2 x\) is one of these, although \(cos(x^2) + sin(x^2)\) is not one of these, however, \(cos x sin x\) is one of these…

The polynomial \(ax^3 + bx^2+ cx – 68000\) gives a remainder of \(6000\) when divided by \(x – 1\), a remainder of \(5000\) when divided by \(x – 2\) , and a remainder of \(4000\) when divided by \(x – 3\). What's the remainder when we divide it by \(x – 4\)?

*Make a hat charm? I may, I may not* (anagram). *When did I die?*

The Indian mathematician Ramanujan famously pointed out that the number \(1729\) was special, since \(1729 = 13 + 123 = 93 + 103\).

The value \(1729\) is in fact the smallest that can be written as the sum of two positive cubes in two different ways.

What's the smallest number that can be written as the sum of a positive cube and a fourth power in two different ways?

The answer is \(4097 = 13+84 = 163+14\). *We can debate as to how different these ways actually are!*

What's the next smallest such number?

What’s \(\large\frac{(6a^2+9ab+3b^2)(6a^2-8ab+2b^2)}{(a^2-b^2)(18a-6b)}\) when simplified?

In a triangle \(ABC, A, P, Q\) and \(B\) are collinear. \(A\) is the point \((1, 2)\).

\(P\) is the point \((4, –0.25)\) and is the foot of the altitude from \(C\) to \(AB\).

\(Q\) is the point \((5, –1)\) and is the foot of the median from \(C\) to \(AB\).

The length \(PC\) is the same as the length \(AB\).

Find the coordinates of \(C\), multiply them together and then multiply this by \(150\).

A triangle has two sides of length \(\sqrt{380}\) and \(2+\sqrt{95}\)

The angle between them is \(60^\circ\).

What's the length of the third side?

What’s number 47 in this sequence?

- Given a line segment \(AB\), it’s possible to construct an equilateral triangle with \(AB\) as one of the sides.
- Give a line segment \(AB\) and a point \(C\), it’s possible to construct a line segment \(CD\) so that the lengths of \(AB\) and \(CD\) are equal.
- ...

The expression

\(\Bigg(\Big((3x^3+7x+1)^2+(1-x)^5\Big)^5+\Big((3x^2-12)^3+(9x^3-x)^5\Big)^2\Bigg)^2\)

is a what?

If \(A\) is the point

\(\large(1-\frac{1}{4}+\frac{1}{16}-\frac{1}{64}+...,1+\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+...)\)

and \(B\) is the point

\(\large(1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+...,1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...)\)

then \(–5\) represents the what?

Over the interval \(0 < x < 2\pi\) what do these curves all have?

\(y=(x-\pi)^4-(x-\pi)^2\) \(y=\frac{3x}{2}+sin(\frac{3x}{2})\) \(y=cosx\)

As x varies, what is the minimum value of

\(y=2x^2-12ax-16bx+18a^2+48ab+2a+32b^2-3b\)?

Did he use his mathematical these to solve an age-old puzzle?

The numbers \(a, b\) and \(c\) are positive

In the figure, \(a=arccos(\frac{11}{13}), b=arccos(-\frac{1}{7})\) and \(y=2arcsin(\frac{1}{7})\)

what is the length of \(AB\)? What is length of \(AC\)? What is the length of \(AD\)? The length of \(AD\) is two digits, what are they?

What are \(x_1 < x_2 < x_3 < x_4\) if

\( \begin{array}{cc} 4\\\large\sum\\r=1\end{array}x_r=41, \begin{array}{cc} 4\\\large\sum\\r=1\end{array}x^{2}_r=579, \begin{array}{cc} 4\\\large\sum\\r=1\end{array}x^{3}_r=10241, \begin{array}{cc} 4\\\large\sum\\r=1\end{array}x^{4}_r=201603\)

Both \(x\) and \(y\), where \(x\) and \(y\) are integers greater than \(1\) and \(x < y\), are less than \(20,000\).

The proper factors of \(x\) add to \(y\) and the proper factors of \(y\) add to \(x\).

Here, the proper factors of an integer \(n > 1\) include \(1\) but do not include \(n\).

Which pair of such numbers \(x\) and \(y\) am I thinking of?