NZ-NZMO 2024

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?

Prove the following inequality

$$\frac{6}{2024^{3}} < \left(1 - \frac{3}{4}\right)\left(1 - \frac{3}{5}\right)\left(1 - \frac{3}{6}\right)\left(1 - \frac{3}{7}\right)\dots \left(1 - \frac{3}{2025}\right).$$

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 $20\%$ one-sheet-thick, and $80\%$ two-sheets-thick. Determine the ratio of the two sides of the original sheet of paper.

A dot-trapezium consists of several rows of dots such that each row contains one more dot than the row immediately above (apart from the top row). For example here is a dot-trapezium consisting of 15 dots, having 3 rows and 4 dot in the top row.

A positive integer $n$ is called a trapezium- number if there exists a dot- trapezium consisting of exactly $n$ dots, with at least two rows and at least two dots in the top row. How many trapezium- numbers are there less than 100?

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

Let $\omega$ be the incircle of scalene triangle $ABC$ . Let $\omega$ be tangent to $AB$ and $AC$ at points $X$ and $Y$ . Construct points $X'$ and $Y'$ on line segments $AB$ and $AC$ respectively such that $AX' = XB$ and $AY' = YC$ . Let line $CX'$ intersects $\omega$ at points $P, Q$ such that $P$ is closer to $C$ than $Q$ . Also let $R$ be the intersection of lines $CX'$ and $BY'$ . Prove that $CP = RX'$ .

Some of the 80960 lattice points in a $40 \times 2024$ lattice are coloured red. It is known that no four red lattice points are vertices of a rectangle with sides parallel to the axes of the lattice. What is the maximum possible number of red points in the lattice?

Let $a$ , $b$ and $c$ be any positive real numbers. Prove that

$${\frac{a^{2}+b^{2}}{2c}}+{\frac{a^{2}+c^{2}}{2b}}+{\frac{b^{2}+c^{2}}{2a}}\geqslant a+b+c.$$