Show that given any nine points on a sphere, there is a closed hemisphere that contains six of them. A closed hemiphere is one that contains the equator with respect to the division.
I’m thinking of a positive number less than \(100\). This number has remainder \(1\) when divided by \(3\), it has remainder \(2\) when divided by \(4\), and finally, it leaves remainder \(3\) when divided by \(5\). What number am I thinking of?
I’m thinking of two prime numbers. The first prime number squared is thirty-six more than the second prime number. What’s the second prime number?
We have a very large chessboard, consisting of white and black squares. We would like to place a stain of a specific shape on this chessboard and we know that the area of this stain is less than the area of one square of the chessboard. Show that it is always possible to place the stain in such a way that it does not cover a vertex of any square.
Does the equation \(9^n+9^n+9^n=3^{2025}\) have any integer solutions?
Mark one card with a \(1\), two cards with a \(2\), ..., fifty cards with a \(50\). Put these \(1+2+...+50=1275\) cards into a box and shuffle them. How many cards do you need to take from the box to be certain that you will have taken at least \(10\) cards with the same mark?
For every pair of integers \(a\),
\(b\), we define an operator \(a\otimes b\) with the following three
properties.
1. \(a\otimes a=a+2\);
2. \(a\otimes b = b\otimes a\);
3. \(\frac{a\otimes(a+b)}{a\otimes
b}=\frac{a+b}{b}.\)
Calculate \(8\otimes5\).
During a tournament with six players, each player plays a match against each other player. At each match there is a winner; ties do not occur. A journalist asks five of the six players how many matches each of them has won. The answers given are \(4\), \(3\), \(2\), \(2\) and \(2\). How many matches have been won by the sixth player?
Let \(n\) be an integer (positive or negative). Find all values of \(n\), for which \(n\) is \(4^{\frac{n-1}{n+1}}\) an integer.
Klein tosses \(n\) fair coins and Möbius tosses \(n+1\) fair coins. What’s the probability that Möbius gets more heads than Klein? (Note that a fair coin is one that comes up heads half the time, and comes up tails the other half of the time).