g New exercises and problems in Mathematics - February 2004
Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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# New exercises and problems in MathematicsFebruary 2004

## New exercises

Maximum score for each exercise (sign "C") is 5 points.

C. 750. A train left the station on schedule. Having covered 8 kilometres, the engineer looked at his watch to see that the short and long hands of the watch were exactly in the same position. The average speed of the train over the first 8 kilometres was 33 kilometres per hour. When did the train leave the station?

C. 751. The sides of a kite are a and b (a$\displaystyle \ne$b). The sides of different lengths enclose a right angle. Find the radius of the circle touching each line obtained by extending the sides.

C. 752. Prove that if the positive numbers a, b, c are consecutive terms of a geometric progression then a+b+c, $\displaystyle \sqrt{3(ab+bc+ca)}$ and $\displaystyle \sqrt[3]{27abc}$ are also consecutive terms of a geometric progression.

C. 753. Some 1.5-litre mineral water bottles have a narrower part at the middle to provide a more comfortable grip. The normal perimeter of a bottle is 27.5 cm but only 21.6 cm at the narrow part. The two cylindrical surfaces are connected by 2-cm-high conical surfaces both above and below the narrow part. How much taller are such bottles than those of normal perimeter without a narrower part?

C. 754. Solve the equation $\displaystyle \frac{2003x}{2004}=2003^{\log_x2004}$.

## New problems

The 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. 3702. Two players play the following game: They take turns taking at least one but at most nine out of 110 counters. No player is allowed to repeat the previous move. The player unable to move is losing the game. Which player has a winning strategy? (4 points)

B. 3703. In a sequence an, a1=1337 and a2n+1=a2n=n-an for all positive integers n. Find the value of a2004. (3 points)

B. 3704. Construct a triangle, given the lines of the perpendicular bisectors of the sides and a point lying on the line of a side. (4 points)

B. 3705. The positive integers a and b are relative primes but the greatest common factor of the numbers A=8a+3b and B=3a+2b is not 1. What is the greatest common factor of A and B? (3 points)

B. 3706. Consider the twelve planes containing one edge of a unit cube, not intersecting the cube and enclosing 45o angles with the faces meeting at that edge. What is the volume of the convex solid bounded by these planes? (3 points)

B. 3707. The edges of the tetrahedron ABCD are AB=c, BC=a, CA=b, DA=a1, DB=b1, and finally DC=c1. Let h denote the length of the line segment connecting the vertex D of the tetrahedron to the centroid of the opposite face. Prove that $\displaystyle h^2=\frac{1}{3} \big(a_1^2+b_1^2+c_1^2\big)-\frac{1}{9}\big(a^2+b^2+c^2\big)$. (4 points)

B. 3708. A line through a given point P is cutting a circle k at points A and B such that PA=AB=1. The tangents drawn from P to k touch the circle at the points C and D. M is the intersection of AB and CD. Determine the distance PM. (4 points)

B. 3709. Given is the quadratic equation ax2+bx+c=0. Its coefficients a, b, c satisfy 2a+3b+6c=0. Prove that the equation has a root x such that 0<x<1. (4 points)

B. 3710. Ti is the foot of the altitude drawn from the vertex Ai of the acute-angled non-isosceles triangle A1A2A3. Let Bi denote the intersection of the lines AjAk and TjTk (where i, j and k are all different). Show that the points B1, B2, B3 are collinear. (5 points)

B. 3711. The sum of the non-negative real numbers s1, s2,..., s2004 is 2 and s1s2+s2s3+ ...+ s2003 s2004+ s2004s1=1. Find the largest and smallest possible values of S=s12+s22+...+s20042. (5 points)

Maximum score for each advanced problem (sign "A") is 5 points.

A. 338. For any positive integer n denote the closest integer to $\displaystyle \sqrt{n}$ by f(n). Calculate the value of the sum

$\displaystyle \sum_{n=1}^\infty\frac{2^{f(n)}+2^{-f(n)}}{2^n}.$

A. 339. We want to select 4 tuples from a 28-element set with the following properties: a) Any two 4-tuple has at most two common elements; b) for any element x and 4-tuple A that is not containing x there exists at least one 4-tuple B that contains x-et and it has exactly two common elements with A. Is it possible to select such a system of 4-tuples?

A. 340. Is it possible that the length of the intersection of a circular disc of unit radius and a parabola is greater than 4 units?