C. 985. A two-digit number is multiplied by 4, and the original two-digit number is written behind the number obtained. The resulting number has exactly six divisors. What may be the original two-digit number?

C. 987. The lengths of the sides of a triangle cut out of paper are 8 cm, 10 cm and 12 cm. The triangle is folded along a line through the common vertex so that the shortest side overlaps with the longest side. A double-layer part and a single-layer part are obtained. Prove that the single-layer part is an isosceles triangle.

C. 988. Given that 4 passengers in a metro train of 6 carriages have colds, what is the probability that there are at most two carriages in which there is a passenger who has a cold?

C. 989. A cubical playing die is made out of a spherical body by cutting off six identical spherical caps. Each of the circular sections is tangent to the four adjacent ones. What percentage is the total area of the six circles of the whole surface area of the resulting die?

B. 4175. Let A, B, C, D be any points in the plane. Prove that if the circles ABC and ABD intersect each other at right angles then the circles ACD and BCD also intersect each other at right angles.

B. 4177. The tangents drawn to the circumscribed circle of triangle ABC at the points B and C intersect each other at point M. The line drawn through M parallel to AB intersects line AC at N. Prove that AN=BN.

B. 4179. The vertex C of a parabola is the centre of a circle that passes through the focus F of the parabola. Let the intersections of the parabola and the circle be A and B, let AB and CF intersect at E, and let D denote the point of the circle that lies opposite to F. Show that DE is the geometric mean of FE and the diameter of the circle.

A. 480. Let p(z) be a complex polynomial of degree n, and suppose that all (complex) roots of p(z) are of unit modulus. For every real number c0, show that the roots of the polynomial