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Mathematics Olympiad Samples

Mathematics Olympiad Samples MATHEMATICS SOME OF MIPT Olympiad SAMPLES'1999

Find the positive integer n, such that the sum of all positive integers composed of n digits each ends with 1998 zeros.

A line that is tangent to the parabola y = 2x2 touches this parabola at point A and intersects the parabola y = x2 + x - 2.5 at B and C. It's known that A is the midpoint of the segment BC. Find the tangent of the angle between the lines OA and BC, where O is the point with coordinates (0 ; 0).

Let ABC is a triangle with BC=9 and AC=1. Angle ACB is four times bigger than BAC. Find AB.

There are N points on a plane. The straight lines are drawn through every pair of these N points. It turns out that we have drawn 20 different lines in total. Find all N for which this is possible. Find minimal m and maximal M values of N. Explain why this situation is impossible for Nm and N>M.(вставить знак меньше)

SABC is a pyramid with AB=BC=CA=22 and SA=SB=SC, SO is the altitude of this pyramid. Point P belongs to this altitude and PO=5. It turns out that the radiuses of circumspheres of the pyramids SPAB and PABC are equal. Find volume of SABC.

Find all triplets of real numbers (x; y; z) satisfying the system of equations:

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