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Prove that in any triangle ABC sin1/2(B-C)=((b-c)/a)cos1/2A.
In a circle, if the measure of the major arc is 40 times more than the measure of the minor arc, then find the measure of minor arc.
An airline tracks data on its flight arrivals. Over the past six months, 65 flights on one route arrived early, 273 arrived on time, 218 were late, and 44 were cancelled. What is the probability that a flight is early? On time? Late? Cancelled? Are these outcomes mutually exclusive? what is the probability that a flight is either early or on time?
On the plane there are several points, all the pairwise distance between them are different. Each of these points connect with the nearest. Can this set be a closed broken line?
N ≥ 3 points lie on a plane, but not all of them lie on a straight line. Prove that there is a circle passing through three of these points and containing none of the remaining points.
Prove that if a polygon has several axes of symmetry, they all intersect in one point.
Prove that if one convex polygon lies inside another one, then the perimeter of the inner polygon does not exceed the perimeter of the outer polygon.
Sum of four unit vectors equals zero. Prove that they can be divided into two pairs of opposite vectors.
Polygon has the center of symmetry O. Prove that the sum of the distances to its vertices is minimal for point O.
The lengths of two sides of the triangle are 3.14 and 0.67. Find the length of the third side if known that it is an integer.
