Mathematical Olympiad - practice problems - page 7 of 11
Number of problems found: 210
- Card sum puzzle
On the table lay eight cards with the numbers 2, 3, 5, 7, 11, 13, 17, and 19. Frank chose three cards. He added the numbers on them and found that their sum was 1 more than the sum of the numbers on the remaining cards. Which cards could have been left on - Divisible number
Six cards with digits 1, 2, 3, 4, 5, and 6 are on the table. Agnes made a six-digit number from these cards, divisible by six. Then she gradually removed the cards from the right. A five-digit number divisible by five remained on the table when she remove - Circumscribed by triangle
Inside the rectangle ABCD, the points E and F lie so that the line segments EA, ED, EF, FB, and FC are congruent. Side AB is 22 cm long, and the circle circumscribed by triangle AFD has a radius of 10 cm. Determine the length of side BC. - Z9–I–4 MO 2017
Numbers 1, 2, 3, 4, 5, 6, 7, 8, and 9 were prepared for a train journey with three wagons. They wanted to sit out so that three numbers were seated in each carriage, and the largest of the three was equal to the sum of the remaining two. The conductor sai - Candies - coloured
There were red and green candies in a tin. Charlie ate 2/5 of all the red candies, and Susan ate 3/5 of all the green candies. Now the red candies make up 3/8 of all the candies in the tin. How many candies were originally in the tin? - MO Z6 I-3 2017 jars
Jano had 100 identical preserving jars, from which he built triangular pyramids. The highest floor of the pyramid always has one jar, the second floor from the top represents an equilateral triangle, whose side consists of two jars, etc. An example of the - Area of triangle
Two pairs of parallel lines, AB to CD and AC to BD, are given. Point E lies on the line BD, point F is the midpoint of the segment BD, point G is the midpoint of the segment CD, and the area of the triangle ACE is 20 cm². Determine the area of triangle DF - MO Z7–I–3 2017
A zoological garden offered school groups an advantageous admission: every fifth pupil gets a ticket free of charge. The teacher of 6.A calculated that if he buys admission for the children from his class, he will save for four tickets and will pay 19.95 - MO Z8–I–4 2017
Robots Robert and Hubert assemble and disassemble coffee grinders. Each of them, however, assembles a grinder four times faster than the other one disassembles it. When they came to the workshop in the morning, several grinders were already assembled ther - Unadjusted clock
Matej was finding out how precisely the tower clock measures time. He came to the conclusion that if no one adjusted it on an ongoing basis, it would show completely exact time once every 200 days. a) Calculate by how many seconds the time measured by the - Intersection of the altitudes
In the acute triangle KLM, the angle KLM is 68°. Point V is the intersection of the altitudes, and P is the foot of the altitude on the side LM. The angle P V M axis is parallel to the side KM. Compare the sizes of angles MKL and LMK. - Zero insertion
Annie and Blanka each wrote one double-digit number, which started with a seven. The girls chose different numbers. Then, each inserted a zero between the two digits, giving them a three-digit number. Everyone subtracted their original two-digit number fr - MO Z6-I-2 2017
Erica wanted to offer chocolate to her three friends. When she took it out of her backpack, she found that it was broken, as shown in the picture. (The marked squares are identical.) The girls agreed not to break the chocolate any further and drew lots to - Adela number
Adela had two numbers written on the paper. When she added their greatest common divisor and least common multiple, she was given four different numbers less than 100. She was amazed that if she divided the largest of these four numbers by the least, she - MO Z9–I–3 - 2017 Robots Robert and Hubert assemble and disassemble coffee grinders. Each of them assembles a grinder four times faster than he disassembles one himself. When they came to the workshop in the morning, several grinders were already assembled there. At 7:00 H
- Average age
The average age of all people at the celebration was equal to the number of people present. After the departure of one person who was 29 years old, the average age was again equal to the number present. How many people were originally at the celebration? - Cake and cone cost
Johnny got pocket money and wants to buy something good for it. If he purchased four cakes, it would increase by 0.50 euros. If he wanted to buy five cakes, he would miss 0.60 euros. He would spend all his pockets on the rest if he bought two cakes and th - MO8-Z8-I-5 2017
Identical rectangles ABCD and EFGH are positioned such that their sides are parallel to the same. The points I, J, K, L, M, and N are the intersections of the extended sides, as shown. The area of the BNHM rectangle is 12 cm², the rectangle MBCK area is 6 - One million
Write the million number (1000000) using only nine numbers and algebraic operations plus, minus, times, divided, powers, and squares. Find at least three different solutions. - Competition participants In the mathematical competition, its participants solved two tasks. Everyone solved at least one problem, while 80% of the participants solved the first problem, and 50% solved the second problem. Sixty participants solved both tasks. How many participant
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