Problems
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).
The letters \(A\), \(E\) and \(T\) each represent different digits from \(0\) to \(9\) inclusive. We are told that \[ATE\times EAT\times TEA=36239651.\] What is \(A\times E\times T\)?
Starting at one of the vertices, an ant wishes to walk each of the \(12\) edges of a sugar cube exactly once. Prove that this is impossible.
Noah has \(10\) dogs, who he wishes to group into \(5\) pairs for \(5\) families, each of whom want two dogs. However, the dogs are quite picky, and can’t be paired with most of the other dogs. None in the first group will go with each other: an alsatian, a border collie, a chihuahua, a dachshund and an English bulldog. None in the second group will go with each other: a foxhound, a greyhound, a harrier, an Irish setter and a Jack Russell. Furthermore,
The alsatian is the least picky and can be paired with any in the second group.
The border collie won’t go with the foxhound, but will go with any other dog in the second group.
The chihuahua and the dachshund will only go with the Irish setter and the Jack Russell.
Additionally, none of the foxhound, greyhound and harrier will go with the English bulldog.
Is it possible to pair up the \(10\) dogs?
Multiply an odd number by the two numbers either side of it. Prove that the final product is divisible by \(24\).