IKMC 2012 - Grades 11-12 - Paper 01 - Self-study

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?

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 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?

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 integer $n$ for which $n^{200}<5^{300}$?

Which function satisfies the equation $f(1/x)=1/f(x)$ (where the expressions are defined)?

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

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

My age is a two-digit integer that is a power of 5, and my cousin’s age is a two-digit integer that is a power of 2. The sum of all the digits of our ages is odd. What is the product of all those digits?

A travel agency organized four optional tours of Sicily for a group of tourists. Each tour had a participation rate of 80%. What is the smallest possible percentage who took part in all four tours?

What is the solution set of $|x|+|x-3|>3$?

School marks in Slovakia run from 1 (best) to 5. In a Slovak school, a test in the 4th class had an average mark of 4. Boys averaged 3.6 and girls 4.2. Which statement about the class 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?

All tickets for the first row of a cinema were sold. Seats are numbered consecutively starting with 1. By mistake, an extra ticket was sold for one seat. The sum of the seat numbers on all sold tickets is 857. Which seat had two tickets sold?

A right-angled triangle has side lengths a, b and c, with legs a and b as shown. What is the radius r of the inscribed semicircle in the figure?

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?

A kangaroo wants to build a row of standard dice; opposite faces of each die sum to 7 dots. Two faces can be glued only if their numbers of dots are equal. The total number of dots on the outer, unglued faces is to be 2012. How many dice are needed?