Ritangle 2022

\(a\) to \(j\) are the digits from 0 to 9 in some order.
1 divides the one-digit number \(a\),
2 divides the two-digit number \(ab\),
3 divides the three-digit number \(abc\),
\(\cdots\)
10 divides the ten-digit number \(abcdefghij\).

What is the ten-digit number \(abcdefghij\) ?

 You may choose to write a computer program to tackle this question. You can learn to do this HERE; check the 'Using technology' link.
 
 To get your final answer, multiply this by \(1.7 \times 10^{-7}\) and take the integer part.

The missing values \(a\), \(b\), \(c\), \(d\), \(e\) and \(f\) are the digits \(1\), \(2\), \(3\), \(4\), \(5\) and \(6\) in some order.

The numbers

\[
\large \overset{a}{\square}\overset{b}{\square},\overset{c}{\square}5\overset{d}{\square},\overset{e}{\square}4\overset{f}{\square}
\]
are in arithmetic progression.

What is the maximum possible product for the three numbers? 

To get your final answer, multiply this by  \(3.8 \times 10^{-6}\) and take the integer part. 

The lines \(y=x, y=ax\) and \(y=b x+c\)
where \( b \lt a \lt 0 \) and \(c\gt 0\)
enclose an equilateral triangle of area \(1 \).
What is the sum (to 3 s.f.) of the \(y\)-coordinates of the three vertices of the triangle?

To get your final answer, multiply this by \(476\) and take the integer part.

The diagram shows a regular octagon and a circle that each have area \(1 \).
The circle passes through two vertices of the octagon as shown.
What (to 3 s.f.) is the area of the part of the circle inside the octagon?

To get your final answer, multiply this by \(29388\) and take the integer part.
 

The polynomial \(f(x)\) is cubic, while the polynomial \(g(x)\) is quadratic.

\(y=f(x)+g(x)\) while \(z=f(x)-g(x) \).

\(y^{\prime \prime} \ldots {}^{\prime \prime} (a)\), where there are \(n\) dashes, means
'the \(n^{th}\) derivative of \(y\) with respect to \(x\) evaluated at \(x=a\).'

You are given that \(4=y^{\prime \prime \prime}(3)=y^{\prime \prime}(2)=y^{\prime}(1)=y(0)\), 
and in addition \(16=z^{\prime \prime}(2)=z^{\prime}(1)=z(0)\).

What (to 3 s.f.) is \(y(1)\) ?

To get your final answer, multiply this by \(0.7\) and take the integer part.

If \(a^{b}=c^{12}\), \(c^{d}=e^{7}\) and \(e^{f}=a^{5}\) what is \(b \times d \times f\)?

To get your final answer, multiply this by 0.172 and take the integer part

In the diagram, \(\mathrm{CD}\) is a diameter of the circle, \(\mathrm{O}\) is the centre, and \(\mathrm{DB}=2 \mathrm{AD}\).

If angle \(\mathrm{AOB}\) is \(100^{\circ}\), what (in degrees to 3s.f.) is the angle \(\mathrm{ACD}\) ?

To get your final answer, multiply this by \(2.23\) and take the integer part.

\(P\) is the sum of an infinite series as follows.

\[
P=1+\frac{1}{2}-\frac{1}{4}-\frac{1}{8}+\frac{1}{16}-\frac{1}{32}-\frac{1}{64}+\frac{1}{128}-\frac{1}{256}-\frac{1}{512}+\frac{1}{1024}-\ldots
\]

Now we swap the plus and minus signs to get \(Q\) (the \(1\) at the front does not change sign).

\[
Q=1-\frac{1}{2}+\frac{1}{4}+\frac{1}{8}-\frac{1}{16}+\frac{1}{32}+\frac{1}{64}-\frac{1}{128}+\frac{1}{256}+\frac{1}{512}-\frac{1}{1024}+\ldots
\]

What (to 3 s.f.) is \( \frac{P}{Q} \)?

To get your final answer, multiply this by \(61\) and take the integer part. 

\(abc, def\) and \(ghi\) are three three-digit numbers, where \(a\) to \(i\) are the nine digits from \(1\) to \(9\) in some order.

\(T\) (if it exists) is the triangle with sides length \(abc, def\) and \(ghi\). 

If \(P=\) maximum possible area for \(T\), 
and \(Q=\) minimum possible non-zero area for \(T\), 
then what, to the nearest whole number, is \( \large \frac{P}{Q}\) ?

To get your final answer, multiply this by \(0.081\) and take the integer part. 

Luke expands \((a+b x)^{5}\) for some values of \(a\) and \(b\) and notices
- the coefficients of \(x^{3}\) and \(x^{4}\) are equal
- the coefficients of \(x\) and \(x^{2}\) add to 1 . 

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

To get your final answer, multiply this by \(237\) and take the integer part.

The missing values \(a\), \(b\), \(c\), \(d\), \(e\) and \(f\) are the digits \(1\), \(2\), \(3\), \(4\), \(5\) and \(6\) in some order.
\[
\frac{8 x^{2}+\overset{a}{\square} x+2 \overset{b}{\square}}{\left(x^{2}+4\right)(\overset{c}{\square}  x+1)}\equiv \begin{aligned}
& \frac{x+\overset{d}{\square} }{x^{2}+\overset{e}{\square} }+\frac{\overset{f}{\square}}{2 x+1}
\end{aligned}
\]

Note, \(2 \overset{b}{\square}\) means \( '20+b'\).

What is the six-digit number \(abcdef\)?

To get your final answer, multiply this by \(4.6 \times 10^{-5}\) and take the integer part. 

If the solutions to the equation
\[
a x+b=\lvert x^{2}+c x+d \rvert
\]
are \(x=1,4,9\) and \(16\) , 

what is \(a^{2}+b^{2}+c^{2}+d^{2}\) ?

To get your final answer, multiply this by \(0.054\) and take the integer part.

If \( \mathrm{AB}=\sin \left(2 x^{\circ}\right), \mathrm{CD}=\cos x^{\circ}\), and \(\mathrm{EF}=\sin x^{\circ}\), and \(0 \lt x \lt 90\), 

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

To get your final answer, multiply this by \(1.98\) and take the integer part.

Define the pointiness of an angle \(x^{\circ} \) for \( 0 \leq x \leq 180\)   by

\[
P\left(x^{\circ}\right)=1-\frac{x}{180}
\]

Thus \( P\left(90^{\circ}\right)=0.5, P\left(1^{\circ}\right)= \frac{179}{180}, P\left(180^{\circ}\right)=0 .\)

In the diagram the angle \(\mathrm{OAB}\) has pointiness \(0.9\),
while \(\mathrm{A B C}\) has pointiness \(0.8, \mathrm{B C D}\) has pointiness \(0.7\) and so on.

The points \(\mathrm{O}, \mathrm{B}, \mathrm{D}, \mathrm{F}\) and \(\mathrm{H}\) lie in a straight horizontal line,
and the same is true for \(\mathrm{A, C, E}\) and \(\mathrm{G}\). 

\(\mathrm{O A}\) is vertical and is of length \(1 \mathrm{~cm}\).

An ant walks from \(\mathrm{O}\) to \(\mathrm{A}\) to \(\mathrm{B}\)... to \(\mathrm{H}\) along the line segments above.

What distance (in \(\mathrm{cm}\), to 3s.f.) does the ant travel?

To get your final answer, multiply this by \(0.74\) and take the integer part.

The points \(A=(a, a)\) and \(B=(b, b)\), where \(a \lt 0\) and \(b \gt 0\), lie on the curve \(y=p x^{2}+q x+4\).

The gradient of the curve at \(A\) is \(b\), and the gradient of the curve at \(B\) is \(a\).

What is \(a^{2}+b^{2} ?\)

To get your final answer, multiply this by \(1.21\) and take the integer part.

The triangle with sides \(\sqrt{a}, \sqrt{b}, \sqrt{c}\) exists and has area \(A\).

The point \((a, b, c)\) lies on a sphere, centre the origin, with radius \(\sqrt{899}\).

The cuboid with sides \(a,b\) and \(c\) has surface area \(1702 .\)

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

To get your final answer, multiply this by \(1.8\) and take the integer part.

A robot is given a pile of \(1024 \left(=2^{10}\right)\) cards numbered from \(1\) to \(1024\) in some random order.

It sorts them into a pile that is in strict numerical order.

It does this by dividing them into \(2^m\) equal piles \((0 \lt m \leq  10) \), sorting each pile, and then by collating pairs of equally-sized piles repeatedly until it has a single ordered pile.  

So for \(m = 4\), it divides the cards into \(16\) piles of \(64\) each, and sorts each of the \(64\)-card piles. It then collates the \(16\) piles into \(8\) piles of \(128\) each, then into \(4\) piles of \(256\) each, then into \(2\) piles of \(512\) each, then finally into one pile of \(1024\). 

- Dividing the pack into \(2^m\) equal piles takes \(1000\) centiseconds.
- To order the \(k^{\text {th }}\) card within a pile takes \(k\) centiseconds.
- To collate two piles of cards takes \(1\) centisecond per card, plus \(1\) centisecond.

What is the smallest time, to the nearest second, in which the robot can sort the pack of cards?

To get your final answer, multiply this by \(0.017\) and take the integer part. 

On Planet Zog a clock has two hands,
one to show the hours (of which there are \(7\) in a Zog day)
and another to show the minutes (and there are \(100\) minutes in a Zog hour).
The hour hand goes round the clock once a day.
The minute hand goes round the clock seven times a day.
The clock is started with both hands vertical at time 0 hours, 0 minutes.
The hands are pointing to opposite sides of the clock for the first time in the day after \(a\) minutes, and for the last time in the day after \(b\) minutes.
What to the nearest integer is \(b - a\) ?

To get your final answer, multiply this by \(0.247\) and take the integer part.

You are given three statements about a positive integer \(a\).
- \(\mathrm{A}_{1}\) says \(a\) is even,
- \(\mathrm{A}_{2}\) says \(a^2\) is even,
- \(\mathrm{A}_{3}\) says \(a^2+a\) is even.
If \(\mathrm{A}_{i}\) (where \(i=1\) or \(2\) or \(3\) ) implies \(\mathrm{A}_{j}\) (where \(j=1\) or \(2\) or \(3\) ) then \({b}_{{ij}}=i+j\), otherwise \({b}_{ij}=0\).
If \(B\) is the \(3 \times 3\) matrix whose elements are given by the numbers \(b_{i j}\), what is the absolute value of the determinant of \(B\) ?

To get your final answer, multiply this by \(0.51\) and take the integer part.

An \(a\)-pointer prime \(p\) is one where \(p\) plus the sum of the digits of  \(p\) is the next prime after \(p\).

The first \(a\)-pointer primes are \( 11, 13, 101, 103, 181,293, 631,701,811, 1153.\)

An \(m\)-pointer prime \(q\) is one where \(q\) plus the product of the digits of  \(q\) is the next prime after \(q\).

The first \(m\)-pointer primes are \( 23, 61, 1123, 1231, 1321, 2111, 2131, 11261.\)

Define an \(am\)-pointer prime \(r\) as one where \(r\) plus the sum of the digits of \(r\) plus the product of the digits of \(r\) is the next prime after \(r\). Notice that all \(a\)-pointer primes containing the digit zero are \(am\)-pointer primes.

Find the first \(am\)-pointer prime that is not an \(a\)-pointer prime.

Clue

We say the integer \(m \geq 1\), where \(m\) is in base \(10\) , is a repunit base \(n\), where \(n\) is in base \(10\) and \(n>1\), if the base \(n\) representation of \(m\) is all \(1 s\)
(so \(63\) is a repunit base \(2\) ).

\(31\) is the smallest number \(m\) to be a repunit base \(n\) for three different positive integers \(n\).

(\(31\) is \(11\) in base \(30\),  \(111\) in base \(5\) , and \(11111\) in base \(2\)). 

What is the next smallest integer so that this is true? You are given that this number is less than \(10000\).

Clue

A triangle of area \(A\) has three altitudes of length \(9 \mathrm{~cm}\), \(10 \mathrm{~cm}\) and \(11 \mathrm{~cm}\). What (to 3 s.f.) is  \(A\)?

Clue

A sequence is defined as follows;
\[
u_{1}=a, u_{2}=b, u_{n+1}=\left|u_{n}\right|-u_{n-1} \text {. }
\]
If \(m\) is the smallest positive integer so that \(u_{m}=u_{1}\) for all values of \(a\) and \(b\), what is \(m\) ?

Clue

Given two positive integers \(a\) and \(b\) with \(a>b\), 
form the integer \(c\) that is the number \(a-b\) concatenated with the number \(a+b\).
So \(a=9\) and \(b=8\) would lead to \(117\) .
Neither \(a+b\) nor \(a-b\)  can begin with \(0\) .
You are told that for a certain choice of \(a\) and \(b\), \( c\) is prime with five digits. What is the smallest \(c\) can be?

Clue

The missing values \(a\), \(b\), \(c\), \(d\), \(e\) and \(f\) are the digits \(1\), \(2\), \(3\), \(4\), \(5\) and \(6\) in some order (no repeats!)

\[
\large\int_{\overset{a}{\square}}^{2}  
\overset{b}{\square}x^{\overset{c}{\square}}+2\overset{e}{\square}xdx=\frac{\overset{d}{\square}(144)}{\overset{f}{\square}}
\]

Given that the numerator and denominator of the fraction have no common factors, what is the six-digit number \(abcdef\)?

Clue

The turning points on the curve \(y=x^{4}-2 a^{2} x^{2}+a^{4}\), where \(a\) is positive, form an equilateral triangle. 

What (to 4 s.f.) is \(a\)?

Clue

The functions \(3 \sin x^{\circ}, 2 \sin \left(2 x^{\circ}\right)\) and \(\sin \left(3 x^{\circ}\right)\)
are evaluated at \(x=a\),
and the results, in some order, are in arithmetic progression.
(You are given that \(0 \lt a \lt 180)\).
If \(\cos \left(a^{\circ}\right)=\frac{\sqrt{k}-1}{2}\), where \(k\) is an integer, what (to 3 s.f. ) is \(\sin (a^{\circ})\) ?

Clue

Welcome to the fantasy island of Volcania!

Volcania is roughly the north-east quadrant of a circle. Its national grid, measured in kilometres, has \(x=0\) along its western edge and \(y=0\) on its southern edge. These two shorelines are mostly high cliffs. The NE shoreline is roughly circular. The sea reaches every whole-kilometre grid point that is 28 km or more from the SW corner, and no grid points that are closer than 28 km to the corner.

Volcania takes its name from its magical conical mountains. The mountains have their summits on the kilometre grid points \((a, b)\) and have height \(0.1875 c \) km above sea level, where \(a, b\) and \(c\) are each of the possible sets of positive integers that satisfy \( a^2+b^2+c^2=734\) . Each mountain is steep, with a constant gradient; lava always has the same viscosity and thermal properties, so all slopes make an equal angle \( \alpha\) with the horizontal.

There is a flat coastal plain, at zero altitude, between the NE shoreline and the mountains.

Sabrina is a runner who wishes to visit each of the summits in the shortest possible time (no peak can be visited twice). She arrives by boat and lands at one of the grid points on the NE shore. She then visits each summit in turn before returning to a grid point on the NE shore (possibly the starting point but not necessarily). The magic of the island means that once she has picked her next destination, the other mountains (apart from one she is standing on, if any) all dematerialise. Her route will generally be directly down until she meets the slope up to the next target mountain, then directly up that mountain to its summit, where she chooses her next target. Nowhere on the island is below sea level; if Sabrina reaches flat ground at any point, she runs straight across it heading for her next chosen mountain.

If Sabrina chooses a ‘next mountain’ that is so high and so close to her that she is below ground level when it materialises, she will be buried alive. She would like to avoid this fate. Similarly, she cannot choose as her next target a mountain for which the summit is below the surface of her current mountain.

Sabrina’s speed across the map (i.e. the horizontal component of her speed over the ground surface) is \(u=2(\tan \beta (1+\cos 2 \theta )+\sin 2\theta)ms^{-1}.\) 
The angle \( \theta\)  is in Sabrina’s control (it depends on her running style). She picks different values for \( \theta\) for running uphill, downhill and on the flat, so as to maximise her speed across the map at every stage. 

The angle \( \beta\) is the angle to the horizontal at which she is running (so this is either \( 0, \alpha \)  or \(-\alpha\)).

It transpires that Sabrina’s downhill speed (constant) is exactly four times her uphill speed (also constant).

Your task is to advise Sabrina at which grid point on the shoreline she should start, which order to tackle the mountains, and at which grid point she should finish. To support your advice you will need to calculate the total time taken for your best route in seconds.

Submit your answer in comma-separated format as follows:

Time taken to complete the route (in seconds, to 2 decimal places)

\((x, y)\) coordinates of start point (whole kilometres)

\((x, y)\) coordinates of each mountain visited (whole kilometres, in the order that Sabrina runs them)

\((x, y)\) coordinates of finish point (whole kilometres)

For example:
123456.78
28,0
1,2
2,3
…
0,28