1.
The picture shows three flying arrows and nine fixedballoons. When an arrow hits a balloon, it bursts, and the arrow flies further in the same direction.How many balloons will be hit by the arrows?
3.
Diana played darts throwing arrows toward a target with three sections. First she got 14 points with twoarrows on the target. The second time she got 16 points. How many points did she get the third time?
4.
A garden is divided into identical squares. A fast snail and a slow snail move along the perimeter of thegarden starting simultaneously from the corner S but in different directions. The slow snail moves at thespeed of 1 metre per hour (1 m/h) and the fast one at 2 metres per hour (2 m/h).At what point will the two snails meet?
5.
In which of the four squares is the fraction of the black area the largest?
6.
A star is made out of four equilateral triangles and a square. The perimeter of the square is 36 cm. What is the perimeter of the star?
7.
From the list 3, 5, 2, 6, 1, 4, 7 Masha chose 3 different numbers whose sum is 8. From the same list Dasha chose 3 different numbers whose sum is 7. How many common numbers have been chosen by both girls?
9.
There are 3 squares in the figure. The side length of the smallest square is 6 cm. What is the side length of the biggest square?
10.
In the following figure, the circles are light bulbs connected to some other light bulbs. Initially, all light bulbs are off. When you touch a light bulb, this light bulb and all its neighbours (e.g., the light bulbs connected to it) are lit. At least how many light bulbs do you have to touch to turn on all the light bulbs?
11.
Part B: Each correct answer is worth 4 points
Each square contains one of the numbers 1, 2, 3, 4, or 5, so that both of the calculations following the arrows are correct. A number may be used more than once. What number goes into the box with the question mark?
14.
In a box there are many one-euro, two-euro and five- euro coins. A dispenser draws coins out of the box – one at a time, and stops when three identical coins are taken out. What is the largest possible amount that can be withdrawn?
15.
Two girls, Eva and Olga and three boys, Adam, Isaac and Urban play with a ball. When a girl has the ball, she throws it to the other girl or to a boy. When a boy has the ball, he throws it to another boy but never to the boy from whom he just received it. Eva starts by throwing the ball to Adam. Who will do the fifth throw?
16.
Emily wants to enter a number into each cell of the triangular table. The sum of the numbers in any two cells with a common edge must be the same. She has already entered two numbers. What is the sum of all the numbers in the table?
17.
John coded a correct addition calculation naming the digits A, B, C and D .Which digit is represented by B?
18.
On Monday Alexandra shares a picture with 5 friends. For several days, everybody who receives the picture, sends it once on the next day to two friends. On which day does the number of people who have seen the picture (including Alexandra) become greater than 75, if it is known that no one receives the picture more than once?
19.
The sum of the ages of Kate and her mother is 36, and the sum of the ages of her mother and her grandmother is 81. How old was the grandmother when Kate was born?
20.
Annie replaced the letters with numbers in the word KANGAROO (identical letters with the same digits, different letters with different digits) so that she got the largest possible 8-digit number, which is not a multiple of 4. What is the sum of the last three digits replacing the word ROO?
21.
Part C: Each correct answer is worth 5 points.
Captain Hook has plundered a safe that contains 2520 gold coins. During the night, each of his pirates secretly took out some coins just for themselves. The first one took out 1/2 of the coins, the second one 1/3 of the remaining coins, the third one 1/4 of the remaining coins and so on. When Captain Hook opened the safe in the morning, he found only 252 coins inside. How many pirates are commanded by Captain Hook?
22.
n the figure on the right, the five balls A, B, C, D and E weigh 30, 50, 50, 50 and 80 grams, but not necessarily in this order. Which ball weighs 30 grams?
23.
If A, B, C are distinct digits, which of the following numbers cannot be the largest possible 6-digit number written using three digits A, two digits B, and one digit C?
24.
In the World of Numbers, there are many number-machines, which work in the following way: the machine adds the two beginning digits of the number and replaces them by their sum. For example, beginning with the number 87312 and using six such machines we obtain:
How many such machines should be used in order to get the number
from the number
25.
Nick wants to arrange the numbers 2, 3, 4, ..., 10 into several groups such that the sum of the numbers in each group is the same. What is the largest number of groups he can get?
26.
Peter cut an 8-cm wide wooden plank with a saw into 9 parts across the width of the plank.One piece was a square, the other were rectangles. Then he arranged all the pieces together as shown in the picture. What was the length of the plank?
27.
Write 0 or 1 in each cell of the 5×5 table so that each 2×2 square of the 5×5 table containsexactly 3 equal numbers. What is the largest possible sum of all the numbers in the table?
28.
14 people are seated at a round table. Each person is either a liar or tells the truth. Everybody says: "Both myneighbours are liars". What is themaximum number of liars at the table?
29.
There are eight domino tiles on the table (pic 1). One half of one tile is covered. The 8 tiles can be arrangedinto a 4×4 square (pic 2), so that the number of dots in each row and column is the same.
How many dots are on the covered part?
