Problems

Josie and Kevin are each thinking of a two digit positive integer. Josie’s number is twice as big as Kevin’s. One digit of Kevin’s number is equal to the sum of digits of Josie’s number. The other digit of Kevin’s number is equal to the difference between the digits of Josie’s number. What is the sum of Kevin and Josie’s numbers?

A rectangular sheet of paper is folded so that one corner lies on top of the corner diagonally opposite. The resulting shape is a pentagon whose area is $20\mathrm{\%}$ one-sheet-thick, and $80\mathrm{\%}$ two-sheets-thick. Determine the ratio of the two sides of the original sheet of paper.

A shop sells golf balls, golf clubs and golf hats. Golf balls can be purchased at a rate of $25$ pennies for two balls. Golf hats cost $\text{mathsterling}1$ each. Golf clubs cost $\text{mathsterling}10$ each. At this shop, Ross purchased $100$ items for a total cost of exactly $\text{mathsterling}100$ (Ross purchased at least one of each type of item). How many golf hats did Ross purchase?

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 $8\otimes 5$.

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 $A\times E\times T$?