Advent calendar 2022

It's nearly Christmas and something terrible has happened: an evil Christmas-hater has set three drones loose above Santa's stables. As long as the drones are flying around, Santa is unable to take off to deliver presents to children all over the world. You need to help Santa by destroying the drones so that he can deliver presents before Christmas is ruined for everyone.

Each of the three drones was programmed with four integers between 1 and 20 (inclusive): the first two of these are the drone's starting position; the last two give the drone's daily speed. The drones have divided the sky above Santa's stables into a 20 by 20 grid. On 1 December, the drones will be at their starting position. Each day, every drone will add the first number in their daily speed to their horizontal position, and the second number to their vertical position. If the drone's position in either direction becomes greater than 20, the drone will subtract 20 from their position in that direction. Midnight in Santa's special Advent timezone is at 5am GMT, and so the day will change and the drones will all move at 5am GMT. For example, if a drone's starting position was (1, 12) and its movement was (5, 7), then:

You need to calculate each drone's starting position and daily speed, then work out where the drone currently is so you can shoot it down.

You can attempt to shoot down the drones here.

The expression \((3x-1)^2\) can be expanded to give \(9x^2-6x+1\). The sum of the coefficients in this expansion is \(9-6+1=4\).

What is the sum of the coefficients in the expansion of \((3x-1)^7\)?

How many numbers are there between 100 and 1000 that contain no 0, 1, 2, 3, or 4?

Ivy makes a sequence by starting with the number 35, then repeatedly making the next term by reversing the digits of the current number and adding 6. The first few terms of this sequence are:

What is the first number in Ivy's sequence that is smaller than the previous term?

In the annual tournament of Christmas puzzles, each player must play one puzzle match against each other player. Last year there were four entrants into the tournament (A, B, C, and D), and so 6 matches were played: A vs B, C vs D, A vs D, A vs C, D vs B, and finally B vs C.

This year, the tournament has grown in popularity and 22 players have entered. How many matches will be played this year?

The diagram to the right shows (two copies of) quadrilateral ABCD.

The sum of the angles ABC and BCD (green and blue in quadrilateral on the left) is 180°. The sum of the angles ABC and DAB (green and orange in quadrilateral on the left) is also 180°. In the diagram on the right, a point inside the quadrilateral has been used to draw two triangles.

The area of the quadrilateral is 850. What is the smallest that the total area of the two triangles could be?

120 is the smallest number with exactly 16 factors (including 1 and 120 itself).

What is the second smallest number with exactly 16 factors (including 1 and the number itself)?

Noel writes the integers from 1 to 1000 in a large triangle like this:

The number 12 is directly below the number 6. Which number is directly below the number 133?

Put the digits 1 to 9 (using each digit exactly once) in the boxes so that the sums are correct. The sums should be read left to right and top to bottom ignoring the usual order of operations. For example, 4+3×2 is 14, not 10. Today's number is the product of the numbers in the red boxes.

Noel writes the integers from 1 to 1000 in a large triangle like this:

The rightmost number in the row containing the number 6 is 9. What is the rightmost number in the row containing the number 300?

There are 3 even numbers between 3 and 9.

What is the only odd number \(n\) such that there are \(n\) even numbers between \(n\) and 729?

Holly draws a line of connected regular pentagons like this:

She continues the pattern until she has drawn 204 pentagons. The perimeter of each pentagon is 5. What is the perimeter of her line of pentagons?

Today's number is given in this crossnumber. The across clues are given as normal, but the down clues are given in a random order: you must work out which clue goes with each down entry and solve the crossnumber to find today's number. No number in the completed grid starts with 0.

The determinant of the 2 by 2 matrix \(\begin{pmatrix}a&b\\c&d\end{pmatrix}\) is \(ad-bc\).

If a 2 by 2 matrix's entries are all in the set \(\{1, 2, 3\}\), the largest possible deteminant of this matrix is 8.

What is the largest possible determinant of a 2 by 2 matrix whose entries are all in the set \(\{1, 2, 3, ..., 12\}\)?

There are five 3-digit numbers whose digits are all either 1 or 2 and who do not contain two 2s in a row: 111, 112, 121, 211, and 212.

How many 14-digit numbers are there whose digits are all either 1 or 2 and who do not contain two 2s in a row?

A line is tangent to a curve if the line touches the curve at exactly one point.

The line \(y=-160\,000\) is tangent to the parabola \(y=x^2-ax\). What is \(a\)?

Put the digits 1 to 9 (using each digit exactly once) in the boxes so that the sums are correct. The sums should be read left to right and top to bottom ignoring the usual order of operations. For example, 4+3×2 is 14, not 10. Today's number is the largest number you can make with the digits in the red boxes.

The equation \(x^5 - 7x^4 - 27x^3 + 175x^2 + 218x = 840\) has five real solutions. What is the product of all these solutions?

What is the area of the largest triangle that fits inside a regular hexagon with area 952?

There are 21 three-digit integers whose digits are all non-zero and whose digits add up to 8.

How many positive integers are there whose digits are all non-zero and whose digits add up to 8?

Put the digits 1 to 9 (using each digit exactly once) in the boxes so that the sums are correct. The sums should be read left to right and top to bottom ignoring the usual order of operations. For example, 4+3×2 is 14, not 10. Today's number is the product of the numbers in the red boxes.

The last three digits of \(5^5\) are 125.

What are the last three digits of \(5^{2,022,000,000}\)?

Write the numbers 1 to 81 in a grid like this:

Pick 9 numbers so that you have exactly one number in each row and one number in each column, and find their sum. What is the largest value you can get?

What is the smallest number that is a multiple of 1, 2, 3, 4, 5, 6, 7, and 8?

One of the vertices of a rectangle is at the point \((9, 0)\). The \(x\)-axis and \(y\)-axis are both lines of symmetry of the rectangle.

What is the area of the rectangle?