Problems

Is it possible to place a positive integer in every cell of a $10\times 10$ array in such a way that both the following conditions are satisfied?

1. Each number (not in the bottom row) is a proper divisor of the number immediately below.

2. The numbers in each row, rearrange if necessary, form a sequence of 10 consecutive numbers.

A round-robin tournament is one where each team plays every other team exactly once. Five teams take part in such a tournament getting: $3$ points for a win, $1$ point for a draw and $0$ points for a loss. At the end of the tournament the teams are ranked from first to last according to the number of points.
Is it possible that at the end of the tournament, each team has a different number of points, and each team except for the team ranked last has exactly two more points than the next-ranked team?

$ABCD$ is a rectangle with side lengths $AB=CD=1$ and $BC=DA=2$. Let $M$ be the midpoint of $AD$. Point $P$ lies on the opposite side of line $MB$ to $A$, such that triangle $MBP$ is equilateral. Find the value of $\mathrm{\angle }PCB$.

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?