IMSO 2007 Mathematics - International Short Answer

If 672 is divided into three parts proportional to 3, 4 and 5, what is the largest number?

Anelka’s weight is a whole number. If Anelka gains 6 kilograms, his weight will be less than 36 kilograms. If he gains 8 kilograms, his weight will be more than 36 kilograms. What is Anelka’s weight?

In the figure, a circle is inscribed in the larger square. The four vertices of the smaller square are on the circle. The area of the larger square is 49 square centimetres. What is the area of the smaller square?

Initially, two elevators were at the same floor. The first elevator went up 4 floors and then went down 7 floors. The second elevator went up 11 floors and then went down 9 floors. After the movements, how many floors apart were the two elevators?

Use each of the digits 1, 2, 3, 4, 5 and 6 exactly once to form two 3-digit numbers. What is the largest possible product of the two 3-digit numbers?

In the expression $\frac ab+\frac cd+\frac ef$, each letter is replaced by a different digit among 1, 2, 3, 4, 5 and 6. What is the smallest possible value of this expression?

Some cats and some pigeons are in the pigeon loft. The total number of their heads is 34 and the total number of their legs is 80. How many cats are there?

A 10 cm × 10 cm × 10 cm cube is cut parallel to its three faces as shown in the figure. After cutting, eight smaller solids are formed. What is the total surface area of these smaller solids?

What is the largest 2-digit prime number that can be obtained by adding two prime numbers?

In the figure, two equal circles are placed in a rectangle of width 14 and height y. The distance between their centres is $\frac 34 y$. What is y?

Sixteen matchsticks form Figure 1, thirty-six form Figure 2, and so on with the pattern shown. The middle row of Figure n contains 2 n+1 squares. How many matchsticks are needed for Figure 10?

In the figure, circles a, b and c touch at P. The centre of c is on b, and the centre of b is on a. What is the ratio of the shaded area to the unshaded area of circle c?

In the following grid, how many shortest paths are there from A to B?

Mary has six stamps worth 0.1 dollars, five worth 0.4 dollars, four worth 0.5 dollars, three worth 5 dollars and two worth 10 dollars. She must pay exactly 38.1 dollars in postage. At most how many stamps can remain?

Tony draws triangles whose side lengths are whole numbers and whose perimeter is 7 units. At most how many different sizes of triangles can he draw?

The average of the squares of seven consecutive integers is 629. What is the smallest integer?

Some people are waiting in a queue. Abel is exactly in the middle. Sarah is number 14 and somewhere behind Abel. Bart is number 20. At most how many people are in the queue?

In the figure below, what is x in degrees?

Given $a\mathbin{\Delta}b=(a-b)(a+b)$ and $c\otimes d=d^2-\frac 1{30}c$, find $(10\mathbin{\Delta}5)\otimes 3$.

The diagonal of a 4 × 4 square passes through 5 corner points of the small squares; the diagonal of a 4 × 2 rectangle passes through 3. How many such corner points does a diagonal of a 45 × 60 rectangle pass through?