Ritangle 2021

Brian is thinking about:

• his son Alan’s age, \(A\),
• his own age, \(B\), and
• his grandfather Colin’s age, \(C\).

He realises that in some order, \(A\), \(B\) and \(C\) are:

• a triangle number,
• a square number and
• a Fibonacci number.

He then realises that in exactly nine years' time, assuming everyone is still around, the same will be true again. How old is Alan?

Take the Age of Consent to be 16; this is respected in this family at all times! Assume that no one is older than 100 here.  

How do we decide if a number is divisible by \(11\)?

Write down the digit sum but alternating \(+\) and \(-\) signs.

The result is divisible by \(11\) if and only if the original number is. 

So given \(12345\), \(1 - 2 + 3 - 4 + 5 = 3\), and since \(11\) does not go into \(3\), \(11\) does not go into \(12345\).

On the other hand, given \(92345\), \(9 - 2 + 3 - 4 + 5 = 11\), since \(11\) does go into \(11\), \(11\) goes into \(92345\).

What’s the first multiple of \(11\) to contain all 10 digits?

Note; the number cannot begin with zero.

The positive integers \(a\), \(b\) and \(c\) are the smallest that satisfy the equations:

\(a^5 = 3b^4 = 2c^3\)

If \(log_{10}a + log_{10}b + log_{10}c = plog_{10}2 + qlog_{10}3\), where \(p\) and \(q\) are positive integers, then what is \(p + q\)?

\(A\) is the point \((0, 1)\) while \(B\) is the point \((2, 0)\).

\(C\) is a point \((c, 2c)\) where \(c > 0\), and \(D\) is the point \((d, -\cfrac{d}{2}\)) where \(d > 0\).

The triangles \(ABC\) and \(ABD\) are congruent.

How long is \(CD\)?

This is a partly completed 4 by 4 magic square, where all rows, columns and both diagonals add to the same amount.

What is \(x\)?

Can you work out what is happening here?

The letters \(a\) to \(h\) are all positive integers.

What is the number \(a + b + c + d + e + f + g + h\)?

You are given that the infinite series

\(cos^2 x - sin^2 x + cos^4 x - sin^4 x +cos^6x - sin^6x +\)...

adds to \(1\) for some value of \(x\).

You are also given that \(0 \,{<}\, x \,{<}\, \pi/2\)

What to 3 s.f. is \(x\)?

To get your final answer, multiply this by 35489 and take the integer part.

The sequence \(u_1, u_2, u_3...\) can be written as \((u_n)\).

You are given that

• \((u_n)\) is an arithmetic progression with first term \(a\) and common difference \(b\)
• \((v_n)\) is an arithmetic progression with first term \(c\) and common difference \(d\)
• \((u_n + v_n)\) is an arithmetic progression with first term \(5\) and common difference \(8\)
• \((u_nv_n)\) is \((w_n + k(n-1)^2)\) where \((w_n)\) is an arithmetic progression with first term \(6\) and common difference \(7\), and where \(k\) is a constant.

Find \(|a\times b\times c\times d|\)

In a cyclic quadrilateral \(ABCD\), the lengths of the sides \(AB, BC, CD\) and \(DA\) are in arithmetic progression with common difference \(d > 0\).

If angle \(DAB\) is \(60^{\circ}\), what (to 3s.f.) is \(\cfrac{AB}{d}\)?

You are given the cubic polynomial \(y = x^3 + ax^2 + bx + c\).

Let \(\cfrac{d^ny}{dx^n}(k)\) denote the \(nth\) derivative of \(y\) with respect to \(x\) when evaluated at \(x = k\).

You are given that \(\cfrac{d^ny}{dx^n}(m)=\cfrac{d^my}{dx^m}(n)\) for all integers \(m\), \(n\), where \(0 \leq m \leq 2,\) \(0 \leq n \leq 2\).

What is \(a\times b\times c\)?

Fit the digits \(0, 1, 2, 3, 4, 5, 6, 7, 8, 9\) into the squares in a way that makes the equations truthful.

where \(fg\) is a two digit number, {{\(1 < a < b < c < d < e\)}} and {{\(1 < h < i < j\)}}.

What is \(x\)?

Given a triangle \(T\), with sides \(a\), \(b\) and \(c\) and perimeter \(P\), where \(a,b,c > \cfrac{P}{6}\) define \(D(T)\), the dual of \(T\), to be the triangle with sides \(\cfrac{2}{3}P-a,\cfrac{2}{3}P - b, \cfrac{2}{3}P - c\).

Notice that \(T\) and \(D(T)\) have the same perimeter, and that \(D(D(T)) = T\).

(Why is the condition \(a, b, c > \cfrac{P}{6}\) important?)

If \(a = 7\), \(b = 11\), then what is the smallest possible value for \(c\) if the areas of \(T\) and \(D(T)\) are equal?

Note; you may find Heron’s formula for the area of a triangle useful.

The diagram shows three congruent equilateral triangles beneath two congruent right-angled triangles creating an irregular pentagon of area \(A\).

If the areas of the five triangles are all equal, and the pentagon has perimeter \(P\), find \(\displaystyle\left\lvert\cfrac{P^2}{A}\right\rvert\) to 3s.f.

The top \(n + 1\) rows of Pascal’s Triangle are shown here.

You are told that the average of all these numbers is \(1533\).

What is \(n\)?

You could find Excel useful here.

Define a rectangle to be DAPPER if its diagonal, area and perimeter are all positive integer values.

If a rectangle has perimeter \(48\) and a diagonal of length \(20\), show that it is dapper, and find its area.

How many four-digit numbers (in base 10) are there so that

• all the digits are different
• two of the digits add to the sum of the other two?

A number cannot begin with \(0\).

The points \(A, B, C\) and \(D\) form a cyclic quadrilateral.

The circle \(ABCD\) is shown in the diagram.

The angles at \(A\) and \(B\) are acute, the angles at \(C\) and \(D\) are obtuse.

The digits \(1\) to \(9\) are put into the squares to make a truthful picture.

How many different ways is it possible to do this?

Define \(\textrm{f}(x) = \int_0^x (\int (\int y - k\textrm{d}y) \textrm{d}y) \textrm{d}y\) where \(k\) is a constant.

If \(\textrm{f}(1) = 1\), \(\textrm{f}(2) = 2\), and \(\textrm{f}(3) = 3\), what is \(k\)?

Define \(L(S)\) to be the length of the longest straight line segment lying wholly inside the shape S.

So if \(T\) is a triangle, \(L(T)\) is the length of the longest side, and if \(R\) is a rectangle, \(L( R)\) is the length of the diagonal.

A convex pentagon \(P = ABCDE\) has right angles at \(A\) and \(B\), and \(AB = x\), \(BC = 1\), \(CD = 1\), \(DE = 1\), and \(EA = 1\), where {{\(0 < x < 2\)}}.

For what value of \(x\) (to 3.s.f.) is \(L(P)\) a minimum?

A sequence is defined as follows for \(n \geq 3\);

\(u_1 = a, u_2 = b, u_n = \dfrac{u_{n-1} + u_{n-2} }{2}\)

So the next term is the average of the previous two. You are told that as \(n\) tends to infinity, \(u_n\) tends to \(12\).

For the sequence with the same rule that starts \(u_1 = b, u_2 = c\) as \(n\) tends to infinity, \(u_n\) tends to \(15\).

For the sequence with the same rule that starts \(u_1 = c, u_2 = a\) as \(n\) tends to infinity, \(u_n\) tends to \(18\).

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

A Christmas tree consists of a green equaliteral triangle and a brown square inside a circle of radius \(1\) as shown.

What is the area of the tree?

Give your answer to 3s.f

The diagram shows the part of the ellipse \(x^2 + 4y^2 = 1\) that lies in the first quadrant.

The area between the ellipse, the \(x\)-axis and the \(y\)-axis in the diagram is \(E\).

\(A\), \(B\) and \(C\) are the first three squares in an infinite sequence of squares.

Let \(S\) be the sum of the areas of the squares \(A\), \(B\), \(C\)…  to infinity.

What (to 4 s.f.) is \(\cfrac {S}{E}\)?

Rishi needs £1 trn (one trillion pounds = one million million pounds) to repair his house.

He has the opportunity to invest 1p with the Leprechaun Unlimited Corporation (LUC).

The investment has the following features:

• with a probability of \(0.98\), the money doubles overnight,
• with a probability of \(0.01\) the money is returned unchanged the next day, and
• with a probability of \(0.01\) everything is lost, and Rishi cannot restart.

Unless the money is all lost, Rishi plans to continue to entrust the accumulated sum to LUC on each subsequent night, with the same probabilities.

What is the probability of Rishi achieving his target of £1 trn within three years?

Express your answer as a decimal, to three significant figures.

If ‘^’ means ‘raise to the power of’, so that \(5\)^{^}\(7\) means \(5^7\), what are the last five digits of \(5\)^{^}\((7\)^{^}\((5\)^{^}\(7))\)? 

Take these final five digits as your final answer.

\(0 \leq x \leq 430, 0 \leq y \leq 2020\).

If \(43x = 1\) (mod \(431\)), find \(x\).

If \(201y = 1\) (mod \(2011\)), find \(y\).

What is \(x + y\)?

Potentially useful facts: if \(a\) and \(b\) are integers then \(a = b\) (mod \(n\)) if \(a\) and \(b\) have the same remainder when divided by \(n\).

\(a = b\) (mod \(n\)) \(\Rightarrow ka = kb\) (mod \(n\)), where \(k\) is any integer.

Roger defines a function \(R\) that acts on a pair of values \((x,y)\), turning them into \(R(x,y) = (\cfrac{\sqrt3}{2}x - \cfrac{1}{2}y, \cfrac{1}{2}x + \cfrac{\sqrt3}{2}y)\)

So for example, \(R (0,0) = (0,0)\) and \(R (1,0) = (\cfrac{\sqrt3}{2}, \cfrac{1}{2})\)

He defines \(R^2 (x,y)\) as the result of applying the function \(R\) twice, i.e. \(R^2(x,y) = R(R(x,y)), R^3(x,y)\) as the result of applying \(R\) three times, and so on.

He defines the pair of values \((a,b)\) as equal to \(R^{2021}(1,0)\)

Work out the value of \(|a + b|\) to three significant figures.

Sanvitha defines a function \(S\) that reverses the action of Roger’s function \(R\), so that \(S(R(x,y))=(x,y)\). For example, \(S(0,0)= (0,0)\) and \(S (\cfrac{\sqrt3}{2},\cfrac{1}{2})= (1,0)\).

We can say that \(S\) ‘undoes’ \(R\), and that \(S\) is the inverse of \(R\).

Potentially useful fact: if \(R\) is represented by the matrix \(M\), then \(S\) is represented by the inverse of \(M = M^{-1}\).

\(M=\begin{pmatrix}a & b\\c & d\end{pmatrix}\)\(\Rightarrow M^{-1}=\)\(\begin{pmatrix}a & b\\c & d\end{pmatrix}^{-1}\)\(= \cfrac{1}{ad-bc}\)\(\begin{pmatrix}d & -b\\-c & a\end{pmatrix}\)

\(ad - bc\) is called the determinant of \(M\).

How do we multiply two matrices?

\(\begin{pmatrix}a & b\\c & d\end{pmatrix}\)\(\begin{pmatrix}p & q\\r & s\end{pmatrix}\) \(=\begin{pmatrix}ap + br & aq + bs\\cp + dr & cq + ds\end{pmatrix}\)

She defines the pair of values \((c,d)\) as equal to \(S(0,2)\).

Work out the value of \(|c + d|\) to three significant figures.

Potentially useful fact: \(\cfrac{1}{p}=q\)\((mod\) \(n)\) \(\Leftrightarrow 1=pq\) \((mod\) \(n\))

Find all pairs of natural numbers \((x, y)\) where \(0 \,{<} x \,{<}\, 26, 0 \,{<}\, y \,{<}\, 26\) for which \(xy = 1\) (mod \(26\)).

Take \((y, x)\) to be the same pair as \((x,y\)), and order each pair so that in \((x,y)\), \(x \leq y\).

Order the pairs in ascending order of the first number in each pair.

Concatenate all the numbers into one long number.

The first pair is \((1, 1)\) so the long number begins '\(11\)'.