Ritangle 2019

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.

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 \(s=4\).

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

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?

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

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

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 \(a\)?

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.

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.

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

If we write the triangle as

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

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 \(ABC\)?

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?

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.

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.

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).

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.)