IMSO 2009 Mathematics Contest in Taiwan – Short Answer

Find the largest possible divisor of the three numbers 480608, 508811 and 723217 so that the remainder is the same in each case.

In a small group of people all of the following relationships were present: father, mother, son, daughter, brother, sister, cousin, nephew, niece, uncle and aunt. What is the smallest group of people for which this is possible?

Seven cubes are glued together face to face as shown in the diagram. The volume of the solid formed in this way is $189\text{ cm}^3$. Find its surface area.

Jack said to Jim: “If I give you 6 pigs for one horse, then you will own twice as many animals as I own.” Dan said to Jack: “If I give you 14 sheep for one horse, then you will own three times as many animals as I own.” Jim said to Dan: “If I give you 4 cows for one horse, then you will own six times as many animals as I own.” How many animals in total do Jack, Jim and Dan own?

By adding brackets in various ways to the expression $1\div3\div5\div7\div11\div13$, what is the maximum number of different values the expression can have?

Replace the asterisks with digits so that the multiplication below is correct. What is the product?

Tom has a contract to dig out some foundations, and it must be done in 30 days. His own machine, which he wishes to use as much as possible, would take 48 days to do all the work. He can hire a bigger machine which would do the complete job in 21 days, but it costs 300 dollars a day. There is only enough room for one machine at a time. What is the least number of days for which he will have to hire the larger machine?

Four different right-angled triangles all have sides of integral length and their perimeters are the same length. Find the smallest perimeter for which this is possible.

The diagram shows an irregular pentagon with all five diagonals drawn. How many distinct triangles (not necessarily different in shape or size) can be found using only the lines, or parts of lines, shown?

I have a rectangular picture whose edges are each an exact number of centimetres in length. At a quick glance it could be mistaken for a square, but it is not a square. It is placed inside a black border which is 3 cm wide all the way around. The area of the border is exactly equal to the area of the picture. What is the area, in $\text{cm}^2$, of the picture alone?

A combination lock on a safe needs a six-letter sequence to open it, made from A, B, C, D, E, F with none used twice. Here are three guesses:

| Guess | 1 | 2 | 3 | 4 | 5 | 6 |
| --- | --- | --- | --- | --- | --- | --- |
| FIRST | C | B | A | D | F | E |
| SECOND | A | E | D | C | B | F |
| THIRD | E | D | F | A | C | B |

In the FIRST guess only ONE letter is in its correct place. In the SECOND guess only TWO letters are correctly placed, and those two positions are not next to each other. In the THIRD guess only THREE letters are correctly placed. Each of the six letters is in its correct place once across the guesses. What is the correct combination?

Given that $ABCD$ is a square and $EA:EB:EC=1:2:3$, determine the size of angle $AEB$, in degrees.