Weekly Challenges Are Here!

To celebrate the launch of the weekly challenges we asked the Ask NRICH community to suggest their favourite short problems in mathematics. There was a great response from many of our most enthusiastic problem solvers.
Here are their suggestions for you to consider at your leisure, presented in the order in which they were suggested. Some are harder than others, some lead into more involved questions and some will be much harder than the typical weekly challenge. We will leave it to you to decide which are which!

 
1. Let $ABC$ be an isosceles triangle with a right angle at $B$. Construct $D$ inside $ABC$ such that $AB = AD$ and angle $BAD = 30$ degrees. Prove that $BD = DC$.


2. What is the algebraic expression in terms of $x, y$ and $z$ for the area of the triangle with vertices $(x, 0, 0)$, $(0, y, 0)$, and $(0, 0, z)$?

3. Let four equilateral triangles be sides of a square-based pyramid: find the ratio of the volume of this pyramid to a tetrahedron made of the same four triangles.

4. Prove that the square of any prime number greater than $3$ is one more than a multiple of $24$. 
 
5. Prove that among $n+1$ numbers from the set ${1,2, \dots ,2n}$ there are always two such that one divides the other.

6. Prove that $n!$ is a divisor of the product of any $n$ consecutive natural numbers. 

7. Simplify $$\sqrt{n+\sqrt{n+\sqrt{n+\sqrt{n+\sqrt{n+\sqrt{n+\sqrt{n}...}}}}}}$$
 
8.  In which ways could you represent a list of numbers using only three symbols $0$, $1$, and $,$? In which ways could you represent a list of numbers using only the two symbols $1$ and $,$? In which ways could you represent a list of numbers using only one symbol: $1$? 
 
9. Make as many numbers as you can using only 4 fours. For example:
$$1 = (4/4)\times (4/4) \quad  2 = (4/4) + (4/4)\quad 3 = 4 - \sqrt4 + 4/4$$

10) Draw ten points inside a square of side length $3$. Show that you can always find two points less than $\sqrt{2}$ apart. (Note: This is not the 'best' theoretical bound. As a more difficult extension you might wish to explore ways in which it might be improved upon.) 

11.1) Six people are in a room and each two can either be friends or enemies with each other. Show that there are always three of them who are all friends or all enemies with each other.

11.2) Following on from (11.1), this time you have $17$ people in the room and each two can be friends, enemies or neutral towards each other. Show there are always three of them who all share the same view about each other.

12.1) You have nine pool balls and an ordinary set of scales. One ball is heavier than the others, and your task is to find out which one. You're only allowed two readings from the scales (You can play with an interactivity for this problem)

12.2) Following on from (12.1), you have twelve balls this time and one is a different weight from the others. You have three weighings this time to determine which ball is different, and whether it's heavier or lighter than the rest. (You can also play with an interactivity  for this problem)
 
13)  A hole is bored diametrically through a solid sphere. If the cylindrical wall of the hole is six inches long, what is the remaining volume of the sphere? (You might also enjoy reading about the intriguingly named  Mouhefanggai)
 
14) There is a corridor of infinite length which is lined with shut doors numbered $1, 2, 3, \dots $ in order. Someone then walks along the corridor changing the state of every other door starting with door $2$, someone else following behind changes the state of every third door starting at door $3$, someone else follows behind changing the state of every fourth door starting at door 4, and so on. Assuming the above process comes to an end, which doors will be open? 
By "changing the state of" we mean opening a closed door or shutting an open door.

15) Imagine that a wire passes round the equator of the earth, which we can think of as a circle of radius $6000$km. An engineer wishes to fabricate a similar piece of wire going round the equator at a height of $2$ metres above the ground. How much more wire will she need?

16) Two people are going to compete by drawing balls from a bowl containing 10 balls. Either A or B goes first. Each person must draw 1 or 2 balls from the bowl. The winner (or loser) is the person who takes the last balls. What is A's strategy to ensure he wins? The same game is now played where 1, 2 or 3 balls can be removed. The loser is the person who takes the last ball. What is the strategy now? (You can play this with an  interactivity  for this problem)

17) In base ten the number twenty-three is represented as $23=2\times 10^1+3\times 10^0$. In base $10$, the digits '23' represent a prime number. Do the digits '23' represent a prime number in base 9? Do they represent a prime number in any other base? Construct representations of prime numbers in different bases. Is there any link between the primes in different bases? If so, why? If not, why not?
 
18) We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.
 
19) An experiment is repeated many times: two real numbers $A$ and $B$ are randomly, uniformly and independently chosen between $-1$ and $1$ and the smaller number taken. What is the average value of this smaller number? (NB: by 'smaller' we mean here that $A$ is smaller than $B$ if and only if $A< B$)
 
20) Find three different numbers such that $z^3=1$