Reason + Mathematical Olympiad - practice problems - page 5 of 6
Number of problems found: 112
- Z7-I-4 stars 4949
Write instead of stars digits, so the next write of the product of the two numbers is valid: ∗ ∗ ∗ · ∗ ∗ ∗ ∗ ∗ ∗ ∗ 4 9 4 9 ∗ ∗ ∗ ∗ ∗ ∗ 4 ∗ ∗ - Pyramid Z8–I–6
Each brick of the pyramid contains one number. Whenever possible, the number in each brick is the lowest common multiple of two numbers of bricks lying directly above it. May that number be in the lowest brick? Determine all possibilities. - Coefficients 4445
Find all triplets P (x) = a * x² + b * x + c with the integer coefficients a, b, and c to which it applies P (1) - Shopkeeper 4433
The seller of Christmas trees sold spruces for 220 CZK, pines for 250 CZK, and hemlocks for 330 CZK. In the morning he had an equal number of spruces, hemlocks, and pines. In the evening, he had sold all the trees and received a total of 36,000 CZK for th
- Trapezoid MO-5-Z8
ABCD is a trapezoid in that lime segment CE is divided into a triangle and parallelogram. Point F is the midpoint of CE, the DF line passes through the center of the segment BE, and the area of the triangle CDE is 3 cm². Determine the area of the trapezoi - Cakes Z8-I-5
Mom brought ten cakes of three types: coconut was less than Meringue Cookies, and most were caramel cubes. John chose two different kinds of cakes. Stephan did the same and for Margerith left only the cakes of the same type. How many coconut, Meringue Coo - Four families
Four families were on a joint trip. In the first family, there were three siblings: Alica, Betka, and Cyril. In the second family were four siblings: David, Erik, Filip, and Gabika. In the third family, there were two siblings, Hugo and Iveta. Three sibli - Star equation
Write digits instead of stars so that the sum of the written digits is odd and is true equality: 42 · ∗8 = 2 ∗∗∗ - Centipede 4257
Centipede Mirka consists of a head and several articles. Each pair has one pair of legs. When it got cold, she decided to get dressed. Therefore, she put on a sock on her left foot in the third article from the end and then on every other third article. S
- MO-Z5-3-66 tiles
The picture shows square tiles with a side of 10 dm, composed of four identical small rectangles and squares. The circumference of a small square is five times smaller than the circumference of the entire tile. Determine the dimensions of the rectangle. - Candy - MO
Gretel deploys different numbers to the vertex of a regular octagon, from one to eight candy. Peter can then choose which three piles of candy to give Gretel others retain. The only requirement is that the three piles lie at the vertices of an isosceles t - Octahedron - sum
On each wall of a regular octahedron is written one of the numbers 1, 2, 3, 4, 5, 6, 7, and 8, wherein on different sides are different numbers. John makes the sum of the numbers written on three adjacent walls for each wall. Thus got eight sums, which al - Z9–I–1
In all nine fields of given shape to be filled with natural numbers so that: • each of the numbers 2, 4, 6, and 8 is used at least once, • four of the inner square boxes containing the products of the numbers of adjacent cells of the outer square, • in th - Abbreviation 4148
From point A to point C, an educational trail passes through point B and a red tourist sign; see the picture. In addition, an undrawn abbreviation 1500 meters long, starting at A and ending on the nature trail, can be used. Vojtech found that • the trip f
- Skiing meeting
Four friends came to the skiing meeting from 4 world directions and led the next interview. Charles: "I did not come from the north or from the south." Mojmir "But I came from the south." Joseph: "I came from the north." Zdeno: "I come from the south." We - Z9-I-4
Kate thought of a five-digit integer. She wrote the sum of this number and its half in the first line of the workbook. On the second line, write a total of this number, and its one fifth. She wrote a sum of this number and its one nines on the third row. - Tunnels
Mice had built an underground house consisting of chambers and tunnels: • each tunnel leading from the chamber to the chamber (none is blind) • from each chamber lead just three tunnels into three distinct chambers, • from each chamber, mice can get to an - Fluid
We have vessels containing 7 liters, 5 liters, and 2 liters. The largest container is filled with fluid, the others empty. Can you only get 5 liters and two 1 liter of fluid by pouring? How much pouring is needed? - Pet store
They sold fish from one aquarium from the breeding product (Zverimex). Ondrej wanted half of all the fish, but to avoid cutting any fish, he got half the fish more than he wanted. Matej wanted half of the remaining fish, but like Ondrej, he got half the f
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