IMSO 2009 — International Mathematics — Essay

Bob bought a coat and a shirt. The normal prices of both items are the same, but when Bob bought them, the shirt was discounted by $50\%$ and the coat by $25\%$. If he bought them for 130 dollars, what was the normal price of the shirt?

The area of a rhombus is $36\text{ cm}^2$. One of its diagonals has length twice the other diagonal. What is the length of the smaller diagonal?

I plan to travel by car from City A to City B. I travel the first half of the distance at $30\text{ km/h}$. To get an average traveling speed of $40\text{ km/h}$, what speed should I use for the second half of the distance?

The mean mathematics mark of nine students is 70. When Lambert’s mark is added, the mean becomes 69. When John’s mark is added, the mean becomes 72. What is the mean if both Lambert’s and John’s marks are added?

Andy bought three packages costing 35, 30, and 40 dollars. The first contains 2 books, 1 pencil, and 1 eraser. The second contains 1 book, 1 pencil, and 2 erasers. The third contains 3 books and 2 erasers. What is the price of a fourth package containing 2 books, 1 pencil, and 3 erasers?

The figure shows a regular hexagon. On each side there is an isosceles right triangle, with one side forming its right angle being a side of the hexagon. Determine the marked angle $x$.

Bob and Ivan mow grass in their yard. One day they both work. After that, Bob works every 8 days and Ivan every 6 days. Each time Bob earns 15 dollars and Ivan earns 17.5 dollars. When they work again on the same day, how much money will each have earned by the end of that day?

Mum’s kitchen scale is set incorrectly, but otherwise works fine. A bag of sugar reads 1.5 kg, a bag of flour reads 1.3 kg, and the two together read 2.5 kg. What will the scale show for a piece of butter whose true weight is 0.3 kg?

Alan, Billy, Candy, and David occur in that alphabetical order in a queue of 48 people. Alan is seventh from the front; David is ninth from the back. There are equally many people between Alan and Billy as between Candy and David, and six people between Billy and Candy. How many people are between Alan and Candy?

A rectangle has vertical and horizontal axes of symmetry. Folding along the vertical axis produces a rectangle of perimeter 40 cm. Folding the original rectangle along the horizontal axis produces a rectangle of perimeter 50 cm. What is the perimeter of the original rectangle?

Brad and Jake travel 9 km from A to B. Jake first cycles at $8\text{ km/h}$, leaves the bicycle at some point, then walks at $5\text{ km/h}$ to B. Brad first walks at $4\text{ km/h}$, picks up Jake’s bicycle, then cycles at $10\text{ km/h}$ to B. They start together and arrive at B together. For how many minutes is the bicycle left waiting before Brad uses it?

In the figure, $BC=25\text{ cm}$, $BE=8\text{ cm}$, and $AD=4\text{ cm}$. What is the area of triangle $CDF$?

Tom must finish digging some foundations in 30 days. His own machine would take 50 days to do the whole job, and he wants to use it as much as possible. A larger machine that he can hire would finish the job in 21 days. Only one machine can work at a time. What is the least number of days for which he must hire the larger machine?