In a basket, there are 30 red and green apples. Among any 12 apples there is at least one red one, and among any 20 apples there is at least one green one. How many red apples and how many green apples are there in the basket?
Does there exist a flat quadrilateral in which the tangents of all interior angles are equal?
On an island there are 1,234 residents, each of whom is either a knight (who always tells the truth) or a liar (who always lies). One day, all of the inhabitants of the island were broken up into pairs, and each one said: “He is a knight!" or “He is a liar!" about his partner. Could it eventually turn out to be that the number of “He is a knight!" and “He is a liar!" phrases is the same?
Solve the inequality: \(|x + 2000| <|x - 2001|\).
Solving the problem: “What is the solution of the expression \(x^{2000} + x^{1999} + x^{1998} + 1000x^{1000} + 1000x^{999} + 1000x^{998} + 2000x^3 + 2000x^2 + 2000x + 3000\) (\(x\) is a real number) if \(x^2 + x + 1 = 0\)?”, Vasya got the answer of 3000. Is Vasya right?
Let \(M\) be the point of intersection of the medians of the triangle \(ABC\), and \(O\) an arbitrary point on a plane. Prove that \[OM^2 = 1/3 (OA^2 + OB^2 + OC^2) - 1/9 (AB^2 + BC^2 + AC^2).\]
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?