IMSO 2008 Mathematics - International Short Answer

The diagram shows a way to factorize 330. Find the largest possible value of p+q.

A rectangular tank is half full of water. Its height is 30 cm and its base is 20 cm× 40 cm. If 4000 cm³ of water is added, what is the water depth in cm?

During the first 48 days of a year, what is the greatest possible number of Mondays?

The picture shows 12 drinking cans. Each row has one fewer can than the row directly below it. Anto arranges 39 cans in a similar formation. What is the maximum height, in rows?

In 2007, 33% of olympiad participants were male. In 2008, the total number increases by 12.5%, while female participants decrease by 10%. What percentage of the 2008 participants is male?

Which two labelled points must be connected to A to divide region ABCD into three equal areas?

Using each of the digits 1, 2, 3, 4 once gives 24 four-digit numbers. Sort them from smallest to largest. What is the rank of 4213?

Rectangle PQRS is inscribed in a semicircle of radius 10 cm. P, Q lie on the diameter and R, S lie on the arc. If PQ=2 QR, find the area inside the semicircle but outside the rectangle.

Deny’s and Ali’s ages total 29; Banu’s and Carli’s ages total 35; Ali’s and Banu’s ages total 31. Banu is younger than Ali. Who is oldest?

Two distinct straight lines meet at most once; three distinct lines meet at most three times. What is the greatest number of intersection points of eight distinct straight lines?

Reversing the digits of a two-digit natural number makes a number exactly 20% larger. Find the original number.

The figure is a cube net with faces numbered 1 to 6. Find the product of the numbers on all faces adjacent to face 1.

Boxes are labelled 1, 2, 3, … consecutively. The digit 2 appears 50 times in all labels. How many boxes are there?

A 4 m× 5 m× 6 m wooden block is painted on its outside and cut into 1 m unit cubes. N cubes have exactly two painted faces; M cubes have three painted faces. Find N: M.

A pair of positive whole numbers containing no 0 is called commensurable if their digit sums are equal. For example 14 is commensurable with 5, 14, 32, 1121, 11111. How many numbers are commensurable with 10?

A rectangle intersects a circle of radius 5 as shown. BC=6 and DE=7. Find AC.

The ratio of black to white balls is 3: 4. Every white ball is marked A or B. The number of A balls is 2/3 of the number of B balls; their difference is 52. How many black balls are there?

Fatimah has 100 ml of 20% orange juice. How many ml of 80% orange juice must she add to obtain 40% orange juice?

Seventeen unit cubes are arranged to form a solid. What is the minimum possible surface area, in square units?

M is the centre of the circle. F=AC∩BD and E=CM∩BD; CM is perpendicular to BD. In the diagram A, M, B are collinear. If angle MBE=32°, find angle CFD.