On the questioners’ planet (where everyone can only ask questions. Cricks can only ask questions to which the answer is yes, and Goops can only ask questions to which the answer is no), you meet 4 alien mathematicians.
They’re called Alexander Grothendieck, Bernhard Riemann, Claire Voisin and Daniel Kan (you may like to shorten their names to \(A\), \(B\), \(C\) and \(D\)).
Alexander asks the following question “Am I the kind who could ask whether Bernhard could ask whether Claire could ask whether Daniel is a Goop?"
Amongst the final three (that is, Bernhard, Claire and Daniel), are there an even or an odd number of Goops?
Four different digits are given. We use each of them exactly once to construct the largest possible four-digit number. We also use each of them exactly once to construct the smallest possible four-digit number which does not start with \(0\). If the sum of these two numbers is \(10477\), what are the given digits?
On the questioners’ planet, there are two types of aliens, Cricks and Goops. These aliens can only ask questions. Cricks can only ask questions to which the answer is yes, Goops can only ask questions to which the answer is no.
There are 19 aliens standing in a circle. Each of them asks the following question “Do I have a Crick standing next to me on both sides?" Then one of them asks you in private “Is 57 a prime number?" How many Cricks were actually in the circle?
Suppose that \(n\) is a natural number and \(p\) is a prime number. How many numbers are there less than \(p^n\) that are relatively prime to \(p^n\)?
Prove that one can tile the whole plane without spaces and overlaps, using any non self-intersecting quadrilaterals of the same shape. Note: quadrilaterals might not be convex.
It is impossible to completely tile the plane using only regular pentagons. However, can you identify at least three different types of pentagons (each with at least two different corresponding sides AND angles) that can be used to tile the plane in three distinct ways? Here essentially different means the tilings have different patterns.
Draw how to tile the whole plane with figures, made from squares \(1\times 1\), \(2\times 2\), \(3\times 3\), where squares are used the same number of times in the design of the figure.
Consider a line segment of length \(3m\). Jack chose \(4\) random points on the segment and measured all the distances between those \(4\) points. Prove that at least one of the distances is less than or equal to \(1m\).
The kingdom of Triangland is an equilateral triangle of side \(10\) km. There are \(5\) cities in this kingdom. Show that some two of them are closer than \(5\) km apart.
Margaret marked three points with integer coordinates on a number line with a red crayon. Meanwhile Angelina marked the midpoint of each pair of red points with a blue crayon. Prove that at least one of the blue points has an integer coordinate.