New exercises and problems in Mathematics April
2004
 New exercises Maximum
score for each exercise (sign "C") is 5 points. 
C. 760. We paid with a 1000forint (HUF) note in the grocery shop. The bill showed the amount to be payed and the amount we got back. We observed that these two numbers had the same digits but in different orders. What was the sum of the digits?
C. 761. Given the lengths of two sides of a triangle, and given that the corresponding medians are perpendicular, calculate the length of the third side.
C. 762. The surface area of the circumscribed sphere of a cube K_{1} is twice the surface area of the inscribed sphere of another cube K_{2}. Let V_{1} denote the volume of the inscribed sphere of the cube K_{1} and let V_{2} denote the volume of the circumscribed sphere of the cube K_{2}. Determine the ratio \(\displaystyle \frac{V_1}{V_2}\).
C. 763. There are three 30 cm x40 cm shelves on a stand in the corner of a room. The distances between them are equal. There were three spiders sitting at the point where the middle shelf and both walls meet. One of them crawled diagonally up one wall to the corner of the upper shelf. Another spider crawled diagonally down the other wall to the corner of the lower shelf. The third one remained where it was and observed that he could see his fellows at an angular distance of 120^{o} from each other. Find the (equal) separation between the shelves.
C. 764. Given the real number s, solve the inequality \(\displaystyle \log_{\frac{1}{s}}\log_s\)\log_s
\log_sx">.
 New problemsThe maximum
scores for problems (sign "B") depend on the difficulty. It is allowed
to send solutions for any number of problems, but your score will be
computed from the 6 largest score in each month. 
B. 3722. The secant passing through the intersections of a circle of 4cm radius centred at O_{1} and a circle of 6cm radius centred at O_{2} intersects the line segment O_{1}O_{2} at the point T. The length of O_{1}O_{2} is not smaller than 6 cm. The larger circle intersects the line segment O_{1}O_{2} at A, the smaller circle intersects it at B, and AT:BT =1:2. Find the length of the line segment O_{1}O_{2}. (3 points) (Suggested by Sz. Békéssy, Budapest)
B. 3723. We have prepared a convex solid out of pentagons and hexagons. Three faces meet at every vertex. Each pentagon is attached to five hexagons along its edges, and each hexagon has common edges with three pentagons. How many faces does the solid have? (4 points)
B. 3724. Find all polynomials p(x) for which the polynomials p(x)^{.}p(x+1) and p(x+p(x)) are identically equal. (4 points)
B. 3725. Prove that if a and b are positive numbers then \(\displaystyle 2\sqrt a+3\sqrt[3]{b}\ge5\sqrt[5]{ab}\). (4 points)
B. 3726. What is the digit preceding the decimal point in the number \(\displaystyle \big(3+\sqrt{7}\,\big)^{2004}\)? (4 points)
B. 3727. In a convex quadrilateral, the square of the distance between the midpoints of each pair of opposite sides is equal to one half the sum of the squares of those two sides. Prove that the quadrilateral is a rhombus. (4 points)
B. 3728. Consider the sequence with a_{0}=1 where a_{2n+1}=a_{n} and a_{2n+2}=a_{n}+a_{n+1} for all integers n\(\displaystyle \ge\)0. Prove that every positive rational number occurs among the elements of the set
\(\displaystyle
\left\{\frac{a_{n+1}}{a_n}\colon n\ge1\right\}=\left\{\frac{1}{1},\frac{1}{2},
\frac{2}{1},\frac{1}{3},\frac{3}{2},\dots\right\}.
\)
(5 points)
B. 3729. The triangle ABC has unit area. E, F and G are points on the sides BC, CA and AB, respectively, such that AE bisects BF at the point R, BF bisects CG at the point S, and finally CG bisects AE at the point T. Find the area of the triangle RST. (5 points)
B. 3730. The common centre of a circle k_{a} of radius a and a circle k_{b} of radius b (a>b) is O. One of the tangents drawn from a point A lying on k_{a} to the circle k_{b} touches k_{b} at E. The perpendicular drawn to the radius OE at an arbitrary point P of it intersects k_{b} at the points Q and R, and intersects the line AO at the point T. The points S_{1} and S_{2} are obtained by marking off those points of the perpendicular drawn to AO at T whose distance from T are equal to PQ. What is the locus of the points S_{i} as P moves along the radius OE? (4 points) (Suggested by M. Kárpáti, Bük)
B. 3731. How many positive integers \(\displaystyle \overline{a_1a_2\dots a_{2n}}\) are there, such that none of the digits are 0 and the sum a_{1}a_{2}+a_{3}a_{4}+...+a_{2n1}a_{2n} is even? (5 points)
 New advanced problems Maximum score
for each advanced problem (sign "A") is 5 points. 
A. 344. Does there exist such a lattice rectangle which can be decomposed into lattice pentagons congruent to the one shown in the Figure?
A. 345. Let r and s be arbitrary positive integers and let t be the minimum of \(\displaystyle 2^k\big(\lceil r/2^k\rceil+\lceil s/2^k\rceil1\big)\) on the nonnegative integer values of k. Prove that (a) \(\displaystyle \binom{t}{j}\) is even whenever ts<j<r; (b) t is the smallest integer with this property. (\(\displaystyle \lceil x\rceil\) denotes the smallest integer not smaller than x.)
A. 346. Determine all functions \(\displaystyle f\:\mathbb{Q}\to\mathbb{R}\) satisfying f(xy)=f(x)f(y) and f(x+y)\(\displaystyle \le\)max (f(x),f(y)) for all x, y real numbers.
Send your solutions to the following address:
KöMaL Szerkesztőség (KöMaL feladatok), Budapest 112,
Pf. 32. 1518, Hungary or by email to:
Deadline: 15 May 2004
