The numbers \(a\) and \(b\) are such that the first equation of the system \[\begin{aligned} \cos x &= ax + b \\ \sin x + a &= 0 \end{aligned}\] has exactly two solutions. Prove that the system has at least one solution.
Determine all natural numbers \(m\) and \(n\) such as \(m! + 12 = n^2\).
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 \(60^{\circ}\).
Prove that the following facts are true for any graph:
a) The sum of degrees of all vertices is equal to twice the number of edges (and therefore it is even);
b) The number of vertices of odd degree is even.
48 blacksmiths must shoe 60 horses. Each blacksmith spends 5 minutes on one horseshoe. What is the shortest time they should spend on the work? (Note that a horse can not stand on two legs.)
Is the sum of the numbers \(1 + 2 + 3 + \dots + 1999\) divisible by 1999?
A schoolboy told his friend Bob:
“We have thirty-five people in the class. And imagine, each of them is friends with exactly eleven classmates...”
“It cannot be,” Bob, the winner of the mathematical Olympiad, answered immediately. Why did he decide this?
This problem is from Ancient Rome.
A rich senator died, leaving his wife pregnant. After the senator’s death it was found out that he left a property of 210 talents (an Ancient Roman currency) in his will as follows: “In the case of the birth of a son, give the boy two thirds of my property (i.e. 140 talents) and the other third (i.e. 70 talents) to the mother. In the case of the birth of a daughter, give the girl one third of my property (i.e. 70 talents) and the other two thirds (i.e. 140 talents) to the mother.”
The senator’s widow gave birth to twins: one boy and one girl. This possibility was not foreseen by the late senator. How can the property be divided between three inheritors so that it is as close as possible to the instructions of the will?