IKMC 2012 - Grades 11-12 - Paper 02 - Self-study

A clock has three hands of different lengths, for hours, minutes and seconds. We do not know which length belongs to which hand, but the clock runs correctly. At 12:55:30 the hands were in the positions shown. Which picture shows this clock at 8:10:00?

The water level in a port city rises and falls on a certain day as shown in the figure. For how many hours was the water level above 30 cm on that day?

How many different rectangles have area 60 and whole-number side lengths?

The positive integers are coloured in order red, blue, green: 1 red, 2 blue, 3 green, 4 red, 5 blue, 6 green, and so on. Which colour could the sum of a red number and a blue number be?

The number $\sqrt[3]{2\sqrt 2}$ is equal to:

In a list of five numbers, the first number is 2 and the last is 12. The product of the first three numbers is 30, the product of the middle three is 90, and the product of the last three is 360. Which number is in the centre?

A rectangular piece of paper ABCD measuring 4 cm × 16 cm is folded along MN so that vertex C coincides with vertex A, as shown. What is the area of quadrilateral ANMD′?

The sum of the digits of a nine-digit integer is 8. What is the product of these digits?

What is the largest natural number $n$ for which $n^{200}<5^{300}$?

Quintus’s age is a two-digit power of five, and Sekundus’s age is a two-digit power of two. The sum of all the digits of their ages is odd. What is the product of those digits?

What is the angle α marked at a point of the regular five-pointed star?

A real number $x$ satisfies $x^3<64<x^2$. Which statement is correct?

A travel agency organizes four different trips for a group. Each trip has a participation rate of 80%. What is the minimum percentage of the group that took part in all four trips?

For a ski race, consecutive starting numbers are handed out. One number was accidentally handed out twice. The sum of all numbers handed out is 857. Which number was handed out twice?

In one class, a test had an average mark of exactly 4. The boys averaged 3.6, while the girls averaged 4.2. Which statement is correct?

The picture is the plan of a rose bed. White roses grow in the two equal squares, red roses in the third square, and yellow roses in the right-angled triangle. The bed has both length and width 16 m. What is its total area?

A right-angled triangle has side lengths a = 8, b = 15 and c = 17. What is the radius r of the inscribed semicircle shown?

Square ABCD has side length 2. E and F are the midpoints of AB and AD respectively. G lies on CF with 3CG = 2GF. What is the area of triangle BEG?

The clock shown has a rectangular dial, but each hand moves at a constant rate like a normal clock. The distance between 8 and 10 is 12 cm, and that between 1 and 2 is x cm. What is x?

Renate wants to glue ordinary dice into a straight “dicebar” as shown. Opposite faces of each die sum to 7 dots. She glues together only faces with equal numbers of dots. The total on the unglued faces must be 2012. How many dice must she use?