MATHEMATICS WITHOUT BORDERS
AGE GROUP 6
AUTUMN 2018
INSTRUCTIONS
1. Please DO NOT open the test papers before receiving the proctor’s
permission.
2. The test contains 20 problems with open answers.
3. You must write down your answers in the ANSWER SHEET.
4. You will get 2 points for each correct answer, 1 point for an incomplete
answer, and 0 points for a wrong or missing answer.
5. Using calculators, phones or other electronic devices, as well as books or
formula sheets is NOT ALLOWED.
6. You have 60 minutes to complete the test. In the case of two students having
the same number of points, the student who completed the test quicker will get a
higher ranking place.
7. Taking the test papers and any other notes out of the room is NOT
ALLOWED.
8. Receiving any help from a proctor or anyone else during the competition is
NOT ALLOWED. The organisers insist on honesty and fair play on the part of all
participants in the tournament.
GOOD LUCK!
1
Arithmetics
Problem 1. Calculate
1+2×2+3×3+4×4
2×2+4×4+6×6+8×8
Problem 2. How many integers are there from 0 to 10 that have an odd number of positive
divisors?
Problem 3. Calculate the sum of 0.0025 ÷ 50 and 1.99995.
Problem 4. The least common multiple of two natural numbers is 60, and the greatest common
divisor of the same numbers is 6. Calculate the sum of these numbers.
Problem 5. Ivan started writing the odd numbers from 1 onwards in a row on a whiteboard. He
stopped when two digits 9 appeared next to each other on the board. How many digits did he
write before the two digits 9?
Logical Thinking
Problem 6. What is the smallest three-digit number that, when divided by 4, 5 and 6, leaves a
remainder of 3?
Problem 7. A salesman bought some items from the wholesale market and determined a price at
which to sell the items in his own store in order to make 20% profit. Later on he lowered the
price by 10% and sold the items at the new price. What % profit did he get in the end?
Problem 8. Ivan wrote down all natural numbers from 1 to 201 (including those two numbers).
Then he removed all numbers that are divisible by 3 and by 5. How many numbers are left?
2
Problem 9. At least how many symbols „+“ should we place to the left of the following
expression in order for the numerical equality to be correct?
222. . .2 = 2018
37 digits 2
Problem 10. Find the number of irreducible proper fractions with 18 as their denominator.
Geometry
Problem 11. What is the greatest possible number of rectangles that can be formed using 10
straight lines?
Problem 12. The diagonals АС and BD of the quadrilateral ABCD are mutually perpendicular and
have lengths of 10 cm and 8 cm, respectively. If the diagonal АС bisects the diagonal BD,
calculate the area of the quadrilateral ABCD in square centimeters.
Problem 13. The faces of a cube with an edge of 6 cm were painted and then the cube was
divided into smaller cubes with edges of 1 cm. How many of the smaller cubes have at least one
painted face?
Problem 14. The points А, В and С lie on a straight line in such a way that:
The distance from point А to point В is 6 cm;
The distance from point С to point А is 2 cm;
The distance between the midpoints of the segments АВ and АС is 2 cm.
Find the length of the segment BC in centimeters.
Problem 15. There are three isosceles triangles with lengths expressed in integer centimeters and
a perimeter of 16 cm. Find the smallest side of these triangles in centimeters.
(Hint: The sum of any two side lengths of a triangle is always greater than the third side.)
3
Combinatorics
Problem 16. We are given three-digit numbers. If we only remove the first digit of each of those
numbers, we would get a perfect square. If we only remove the last digit of each of those
numbers, we would get a prime. How many such three-digit numbers are there?
Problem 17. How many digits are required to write down the smallest natural number that only
contains the digits 0 and 1 and is divisible by 72?
Problem 18. In how many ways can we go from point A to point B if we are only allowed to
move along the arrows?
Problem 19. The points A, B, C and D lie on a straight line. Another straight line passes through
point D, along which lie another 3 points: X, Y and Z. How many triangles have their vertices
among the points?
Problem 20. Which number should we place in □, so that the following equality would be
correct?
11 + □ 3
=
41 + □ 8
