New exercises and problems in Mathematics
September 1998

New exercises for practice in September 1998

C. 509. An escalator has 125 stairs. Walking upwards while the escalator is going up at a constant speed, once we reached the upper end of the escalator after making 45 steps. Next time we were able to make 55 steps before reaching the top of the escalator. How much faster were we walking the second time?

Fried Ervin (Budapest)

C. 510. Let A be a ten-thousand-million-digit number divisible by 9. Let B denote the sum of the digits of A and let C denote the sum of the digits of B. Determine the sum of the digits of C.

C. 511. A certain parallelogram has 4 cm and 7 cm long sides. The lengths of its diagonals differ by 2 cm. How long are the diagonals of the parallelogram?

C. 512. After the guests had left the party, a piece of a cake, in the shape of a quarter of a circle of radius 15 cm, was still lying on the table. We cut two 1.5 cm wide strips off the cake, parallel to each straight edge of the piece, and ate them up. What fraction of the piece of the cake did we consume?

New exercieses in September 1998

Gy. 3214. At a party, one is called `timid' if he or she knows at most 3 other people. Prove that if everybody at the party has at least 3 timid acquaintances, then everybody is timid. How many participants may be there at the party in this case?

Gy. 3215. For any integer n, let \(\displaystyle sigma" ALIGN=ABSCENTE\)(n) denote the sum of its positive divisors. (For example, \(\displaystyle sigma" ALIGN=ABSCENTE\)(3)=1+3=4, \(\displaystyle sigma" ALIGN=ABSCENTE\)(6)=1+2+3+6=12.) Prove or disprove: if \(\displaystyle sigma" ALIGN=ABSCENTE\)(n)>2n, then \(\displaystyle sigma" ALIGN=ABSCENTE\)(m)>2m for every multiple m of n.

Gy. 3216. Find those pairs a, b for which equations

and

have a common solution.

Gy. 3217. Find those prime numbers p for which the system

admits an integral solution.

German competition problem

Gy. 3218. Let A and B denote two points of a circle centered at a point O, and let P be a point inside the circle, different from O. Construct points C and D on the circle such that triangles PAB and PCD are similar to each other and have the same orientation.

Based on the submission of Bohner Géza (Cegléd)

Gy. 3219. Each side of a convex quadrilateral is tangent to a circle of radius r whose centre is denoted by O. Show that at least one of the circles, circumscribed to triangles OAB and OCD, respectively, has a radius larger than or equal to r.

Gy. 3220. A cyclic trapezium has bases a and b, a<b. The angle at one vertex of the longer base of the trapezium, subtended by the shorter base, measures 45^o. Determine the area of the trapezium.

Gy. 3221. Prove that six congruent regular octahedra and eight regular tetrahedra, all of the same edge-length can be arranged to form a regular octahedron whose edges are twice as large as those of the original polyhedra.

New problems in September 1998

F. 3238. Prove that inequality

holds for arbitrary real numbers a, b, c.

F. 3239. Find those natural numbers x, y which satisfy equality x²(x²+y²)=y^100.

F. 3240. Show that there exist infinitely many pairs a, b of positive integers such that a divides b²+b+1 and b divides a²+a+1.

F. 3241. In a triangle ABC, points A₁, B₁ and C₁ are incident to sides CB, AC and AB, respectively, such that

Let r₁, r₂, r₃, r₄ and r denote the radii of the inscribed circles of triangles AC₁B₁, BC₁A₁, CA₁B₁, A₁B₁C₁ and ABC, respectively. Determine those values of \(\displaystyle lambda" ALIGN=ABSCENTE\) for which equality

holds.

Bíróo Bálint (Eger)

F. 3242. The distance between a (closed) region K and a point P is defined as the minimum of all distances PQ, where \(\displaystyle Q eleme K-nak" ALIGN=ABSCENTE\). Given are in the plane a square and a circle outside each other such that the centre of the circle lies on the extension of one of the sides of the square. Determine the locus of points which lie in the same distance from the circle and the square.

F. 3243. Let a, b, c, d be pairwise skew lines in three dimensional Euclidean space such that the normal transversals of a, b and c, d are orthogonal to each other, and so are the normal transversals of a, c and b, d, too. Prove that the normal transversals of a, d and b, c are also orthogonal to each other.

New advanced problems in September 1998

N. 179. Given 3 collinear points in the plane, show that, for every positive integer n, it is possible to draw n distinct concentric circles around each point such that the number of points which are incident to 3 circles is at least cn², where c is a positive constant independent of the position of the given points.

N. 180. In a trapezium ABCD, the intersection of diagonals AC and BD is denoted by F. Point E is incident to leg AD such that EF is parallel to AB. There are given two parabolas whose axes are parallel to the bases of the trapezium. One of them passes through points B, C, E, the other touches lines AD, BD, AC. Prove that the parabolas touch each other.

N. 181. Given are a few balls in the three dimensional Euclidean space whose interiors have a point in common. These balls are rearranged in such a way that the distance between the centres of any two of them decreases. Prove that in their new position, the balls still have a common interior point.

N. 182. Find all pairs a, b of positive integers for which a divides b²+b+1 and b divides a²+a+1.