Three non-coplanar vectors are given. Is it possible to find a fourth vector perpendicular to the three vectors given?
Find the volume of an inclined triangular prism whose base is an equilateral triangle with sides equal to a if the side edge of the prism is equal to the side of the base and is inclined to the plane of the base at an angle of \(60^{\circ}\).
The grandad is twice as strong as the grandma, the grandma is three times stronger than the granddaughter, the granddaughter is four times stronger than the dog, the dog is five times stronger than the cat and the cat is six times stronger than the mouse. The grandad, the grandma, the granddaughter, the dog and the cat together with the mouse can pull out the pumpkin from the ground, which they cannot do without the mouse. How many mice should be summoned so that they can pull out the pumpkin themselves?
Is it possible to cut out such a hole in a sheet of paper through which a person could climb through?
There are two hourglasses – one for 7 minutes and another for 11 minutes. An egg is boiled for 15 minutes. How can this time be measured with the help of the available hourglasses?
How can you divide a pancake with three straight sections into 4, 5, 6, 7 parts?
Is it possible to bake a cake that can be divided by one straight cut into 4 pieces?
What is the maximum number of pieces that a round pancake can be divided into with three straight cuts?
The best student in the class, Katie, and the second-best, Mike, tried to find the minimum 5-digit number which consists of different even numbers. Katie found her number correctly, but Mike was mistaken. However, it turned out that the difference between Katie and Mike’s numbers was less than 100. What are Katie and Mike’s numbers?
The best student in the class, Katie, made up a huge number, writing out in a row all of the natural numbers from 1 to 500: \[123 \dots 10111213 \dots 499500.\] The second-best student, Tom, erased the first 500 digits of this number. What do you think, what number does the remaining number begin with?