Mathematical Olympiad - practice problems - page 5 of 11
Number of problems found: 210
- Digit puzzle
Four cards each showed a different digit, one of which was zero. Victor formed the largest four-digit number from the cards, and Martin formed the smallest four-digit number. Adam wrote the difference between Victor's and Martin's numbers on the board. Th - Card expression Tomas received nine cards with the following numbers and math symbols for math olympiad results. 18, 19, 20, 20, +, -, x, (,) Note 4 numbers and operators plus, minus, times, left parenthesis, right parenthesis. He stored the cards so that there were neve
- Year 2018
The product of the three positive numbers is 2018. What are the numbers? - Three-digit number
Find all three-digit numbers n with three different non-zero digits divisible by the sum of all three two-digit numbers we get when we delete one digit in the original number. - Triangle Geometry Proof
On side AB of triangle ABC, points D and E are given such that |AD| = |DE| = |EB|. Points A and B are the midpoints of segments CF and CG. Line CD intersects line FB at point I, and line CE intersects line AG at point J. Prove that the intersection of lin - Z5 – I – 2 MO 2018
Tereza received four identical right-angled triangles with sides of lengths 3 cm, 4 cm, and 5 cm. From these triangles (not necessarily all four) she tried to put together new shapes. She gradually managed to put together quadrilaterals with perimeters of - Quadrilaterals II
In the ABCDEFGHIJKL, the two adjacent sides are perpendicular to each other, and all sides except the AL and GF sides are identical. The AL and GF sides are twice as long as the other sides. The lines BG and EL intersect at point M and divide the dodecago - Perpendicular sides In the ABCDEFGHIJKL, the two adjacent sides are perpendicular to each other, and all sides except the AL and GF sides are identical. The AL and GF sides are twice as long as the other sides. The lines BG and EL intersect at point M. The quadrilateral ABMJ
- Z7–I–5 MO 2018
At the Rose garden centre, one shop ordered a total of 120 roses in red and yellow, the second shop a total of 105 roses in red and white, and the third shop a total of 45 roses in yellow and white. The garden centre fulfilled the order, in such a way tha - Point construction
Given an isosceles right triangle ABS with base AB. On a circle centered at point S and passing through points A and B, point C lies such that triangle ABC is isosceles. Determine how many points C satisfy the given conditions and construct all such point - Dwarf heights
Mr. Ticháček had three gypsum dwarfs in the garden: the largest was called Maško, the middle Jarko, and the smallest Franko. Since he liked to play with them, he discovered that when he put Fan on Jerry, they were as tall as Maško. On the other hand, when - Pear sharing
Ivan and Miranda shared pears in the mission. Ivan always takes two pears, and Miranda takes half of what remains in the mission. Thus, Ivan, Miranda, Ivan, Miranda, and finally Ivan, who took the last two pears, took them away. Definitely. Who had more p - Z9 – I – 5 MO 2018
Peter and Ivan created decorations from mutually identical white circles. Peter used four circles, which he placed so that each touched two other circles. Between them he then inserted another circle, which touched all four white circles, and he coloured - MO C–I–1 2018 An unknown number is divisible by just four numbers from the set {6, 15, 20, 21, 70}. Determine which ones.
- Olympiad Participants Set Problem Out of 50 pupils, 44 solved at least one of the Olympiads - MO Mathematical Olympiad and BO Biology Olympiad. Twenty pupils still need to solve the MO. Of those who dealt with both Olympiads, 1/3 of those who dealt with just one were. How many pupils solv
- Bedrich and Adam
When Bedrich is as old as Adam today, Adam will be 14 years old. When Adam was as old as Bedrich, Bedrich was two years old today. How old are Adam and Bedrich today? - MO Z8-I-1 2018 Frank and David meet daily in the elevator. One morning, they found that if they multiply their current age, they get 238. If they did the same after four years, this product would be 378. Determine the sum of the current ages of Frank and David.
- MO Z9-I-3 2018
In our town there are three cinemas, which are named according to the compass points. About their opening hours it is known that: • every day at least one cinema is open, • if the southern cinema is open, then the northern cinema is not open, • the northe - Crayon count
Michelle has five crayons. Victor has fewer of them than Michelle. Wendell has as many as Michelle and Walt have together. All three have seven times more crayons than Victor. How many crayons does Wendell have? - Z8-I-2 MO 2018
A new pupil came into the class, about whom it was known that apart from English he still knew excellently one foreign language. Three classmates were arguing about which language it is. The first judged: "It is not French." The second guessed: "It is Spa
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