Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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# New exercises and problems in MathematicsMay 2003

## New exercises

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

C. 720. There have been three school trips in this academic year. In the first trip, 75% of the participants were boys. The second time, 60% were boys. Then exactly those students went on the third trip who had taken part in at least one of the first two. Show that there were at least as many boys as girls on the third trip, too.

C. 721. a, b, c are non-zero real numbers. How many different values may the expression

$\displaystyle \frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{abc}{|abc|}$

have?

C. 722. For every real number x, let f(x) denote the minimum of 4x+1, x+2, and -2x+4. What is the largest possible value of f(x)?

C. 723. A first-generation automatic elf crawler can only proceed along a straight line. In order to change direction, it needs to be stopped, turned in the new direction, and started again. At least how many times must the crawler be stopped along a circular corridor 1 m wide with an inner diameter of 30 metres, so that it finally returns to its starting point? (The size of the automatic elf crawler is negligible.)

C. 724. The base of a right prism is a right-angled triangle. The length of one leg of the triangle is equal to the height of the prism. The total length of the other leg and the hypotenuse is 8 cm. What is the maximum possible volume of the prism?

## 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. 3642. Seven-digit numbers are prepared by using each of the digits 1,2, 3, 4, 5, 6, 7. Is it possible that one such number divides another? (4 points)

B. 3643. Divide a cube into six congruent tetrahedra. (3 points)

B. 3644. There is a circle drawn about each vertex of an equilateral triangle. The circles do not intersect each other, and they do not intersect the lines of the opposite sides either. How to choose the radii of the circles in order to cover the greatest possible part of the triangle? (4 points)

B. 3645. Find all possible real functions f defined on non-zero integers, such that

$\displaystyle f\left(\frac{x+y}{3}\right)=\frac{f(x)+f(y)}{2}$

for all real x and y. (3 points)

B. 3646. The vertices of the triangle escribed'' to a side of a given triangle are the points where the escribed circle touches the lines of the sides. The sides of a triangle are a, b, c. Find the ratio of the areas of the triangles escribed to the sides a and b. (4 points) (S. Kiss, Szatmárnémeti)

B. 3647. X is an interior point of side AB of the triangle ABC. Y is an interior point of side BC. AY and CX intersect at Z. Prove that if AY=YC and AB=ZC, then the points B, X, Y, Z lie on a circle. (4 points)

B. 3648. Solve the following simultaneous equations:

2y+x-x2-y2=0, z-x+y-y(x+z)=0, -2y+z-y2-z2=0.

(4 points)

B. 3649. Let a0=5 and an+1=2an+1. Prove that for every natural number n there exists a k different from n, such that an|ak. (4 points)

B. 3650. $\displaystyle \alpha$, $\displaystyle \beta$, $\displaystyle \gamma$ are the angles of a triangle, and

cos 3$\displaystyle \alpha$+ cos 3$\displaystyle \beta$+ cos 3$\displaystyle \gamma$=1.

Prove that the triangle has a 120o angle. (5 points)

B. 3651. A positive integer n is said to be undivided if 1<k<n and (k,n)=1 imply that k is a prime. How many undivided numbers greater than 2 are there? (5 points)

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

A. 320. Jerry is swimming in a square pool. He wants to get away from Tom waiting for him at the edge of the pool. Tom cannot swim, cannot run as fast as Jerry, but runs p times faster than Jerry swims. For what values of the number p can Jerry escape, whatever strategy Tom may use?

A. 321. Given 2n-1 irrational numbers, prove that it is possible to select n numbers x1,...,xn, such that if a1,...,an are arbitrary non-negative rational numbers that are not all 0 then a1x1+...+anxn is irrational. (Bulgarian competition problem)

A. 322. We would like to warm up one litre of cold water of 0 oC temperature with the help of one litre of 100 oC temperature water. To do so we can divide both the cold and the warm water into not necessarily equal parts. After this we can stepwise arrange a cold and a warm part without mixing them in order to exchange heat until their temperature equalizes. After a sufficient number of steps the components of the originally cold water are mixed into a single container. What is the highest temperature of the originally cold water that can be achieved this way?