K. 405.a) Find all sets of three integers such that their product is a positive prime and, if they are listed in increasing order, the differences of consecutive numbers are the same?

b) Find all sets of three integers such that their product is the double of a positive prime and, if they are listed in increasing order, the differences of consecutive numbers are the same?

K. 406. We say that an integer is a ``mountain number'' if it consists of digits distinct from one another and from 0, and its digits are increasing from the first one to the ``mountain top'', then decreasing from the top to the last digit. The mountain top cannot be the first or the last digit.

a) Determine the largest and the smallest mountain numbers.

b) How many mountain numbers with 4 digits are there?

C. 1205. Find all right-angled triangles in which the measures of the sides are two-digit integers, the length of the hypotenuse is obtained by interchanging the digits of one leg, and the three two-digit numbers consist of exactly three kinds of digits, each occurring twice.

C. 1207.E, F and G are points on each side of triangle ABC and n is a natural number such that , , and . Show that if n5 then the area of triangle EFG is greater than the half of the area of triangle ABC.

C. 1208. The vertices of the parallelograms PQRL and LSTA in the plane are labelled in the same sense around the clock. The parallelograms do not have a point in common, except L. Prove that there exists a pentagon ABCDE (degenerated cases are allowed) in the plane such that the midpoints of the sides are P, Q, R, S, T, in this order.

C. 1209. The tangents drawn from a point C lying outside a circle touch the circle at points A és B. M is a point on the shorter arc AB. Let MN, ME and MD be the line segments drawn from M, perpendicular to the line segments AB, BC and CA, respectively. Given that MN=4, MD=2 and , find the area of triangle MNE.

B. 4592. What may be the number of people in a company in which every member has exactly three acquaintances, and any two people have an acquaintance in common exactly if the two of them do not know each other?

B. 4593. An ant starts crawling at constant speed from the left end of a 4-metre-long rubber rope towards the right end. It covers exactly one metre per minute. The rope is fixed at the left end, in a horizontal position. At the end of each minute, the rope is stretched uniformly by one metre. In which minute will the ant reach the right end of the rope? The ant is considered pointlike, the act of stretching takes negligible time, and the rope can be stretched to any length without breaking it.

B. 4594. Consider a sequence of reflections in the lines of all four sides of a square in some order. How many different transformations are obtained by the composition of four such reflections?

B. 4597. The radii of the escribed circles of a triangle are r_{a}, r_{b} and r_{c}, the radius of the circumscribed circle is R. Determine the angles of the triangle if r_{a}+r_{b}=3R and r_{b}+r_{c}=2R.

B. 4598. The diagonals of a cyclic quadrilateral ABCD intersect at E, the midpoints of sides AB and CD are K and M, and the perpendicular projections of point E on the sides BC and AD are L and N, respectively. Prove that the lines KM and LN are perpendicular to each other.

B. 4601. One face of a tetrahedron is a regular triangle of unit side. The length of the other three edges is a. What is the maximum possible area of the perpendicular projection of the tetrahedron on a plane?

A. 607. The circles k_{1}, k_{2} and k_{3} are pairwise externally tangent to each other; the point of tangency between k_{1} and k_{2} is T. One of the external common tangents of k_{1} and k_{2} meets k_{3} at points P and Q. Prove that the internal common tangent of k_{1} and k_{2} bisects the arc PQ of k_{3} which is closer to T.