Mathematical Olympiad - practice problems - page 8 of 11
Number of problems found: 210
- Four-digit number
Find the smallest four-digit number abcd such that the difference (ab)²− (cd)² is a three-digit number written in three identical digits. - Alarm clock
The old watchmaker has a unique digital alarm in its collection that rings whenever the sum of the alarm's digits equals 21. Find out when the alarm clock will ring. What is their number? List all options. - Chocolate 6
An international gang of chocolate smugglers led by the famous Jack Crooked Nose, who has 7 collaborators, singled out the Bratislava airport as a crossroads for their business. An aircraft from Bratislava to Stockholm flies every third day. An aircraft f - Shuttle transport
A river 777 km long flows through forests and meadows, and here and there a port flickers on its bank. Between 2 of 5 ports on the river, there is a permanent shuttle transport. It is provided by the 77-year-old ship Quendolína with captain Bluebeard, who - Bicycles
You're the owner of the transport's learning playground. Buy bicycles of two colors, but you've got to spend accurately 120000 CZK. The Blue bike costs 3600 CZK and the red bicycle 3200 CZK. - MO Z6-6-1
Write integers greater than 1 to the blanks in the following figure so that each darker box is a product of the numbers in the neighboring lighter boxes. What number is in the middlebox? - 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. - Desserts cost
Annie has 50 CZK, Anežka has 46 CZK, and they want to use all the money to buy desserts for a family celebration. They decide between cakes and pinwheels. A pinwheel is CZK 4 more expensive than a cake, and for all the money, you could buy a third more ca - Largest number n
Find the largest natural number d that has that property for any natural number n; the number V(n) is the value of the expression: V (n) = n ^ 4 + 11n²−12 is divisible by d. - Polynomial coefficients
Find all triplets P (x) = a * x² + b * x + c with the integer coefficients a, b, and c to which it applies P (1) - Inequality triangle
The heel of height from the vertex C in the triangle ABC divides the side AB in the ratio 1:2. Prove that in the usual notation of the lengths of the sides of the triangle ABC, the inequality 3 | a-b | < c. - Christmas trees
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 - MO-I-Z6
A square with a side of 4 cm is divided into squares with a side of 1 cm as shown in the figure. Divide the square along the marked lines into two figures each with a perimeter of 16 cm. Find at least three different solutions (i.e. three solutions such t - Isosceles - isosceles
It is given a triangle ABC with sides /AB/ = 3 cm /BC/ = 10 cm, and the angle ABC = 120°. Draw all points X such that the BCX triangle is an isosceles and triangle ABX is an isosceles with the base AB. - 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 Margerith left only the same type of cake. How many coconuts, Meringue Cookies an - Four families
Four families were on a joint trip. The first family had three siblings: Alice, Betty, and Cyril. In the second family were four siblings: David, Eric, Philip, and Gabby. In the third family, there were two siblings, Hugo and Ivy. Three siblings in the fo - 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 Mira
Centipede Mira 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 a sock on her left foot from the end of the third article and then in every other third article. Simil
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