# New exercises and problems in Mathematics

January
2004

## Please read The Conditions of the Problem Solving Competition.

## New exercisesMaximum score for each exercise (sign "C") is 5 points. |

**C. 745.** Are there 2004 positive integers whose sum is equal to their product?

**C. 746.** How many numbers of the form \(\displaystyle \overline{ababab}\) are divisible by *a*) 217; *b*) 218?

**C. 747.** The base of an isosceles triangle is unity and the lengths of its legs is *b*. How long is the base of the isosceles triangle whose legs enclose an angle equal to the angles lying on the base of the first triangle and whose legs are one unit long?

**C. 748.** Solve the following equation on the set of integers: \(\displaystyle \sin\left(\frac{\pi}{3}\,
\big(x-\sqrt{x^2-3x-12}\,\big)\right)=0\).

**C. 749.** The edges of the cube in the *Figure* (see page 39) are 6 units long. The points *K* and *L* trisect the edge *AE*. The cube is cut into pieces by the planes *LHG* and *KFG*. Find the volume of the part containing vertex *B*.

## 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. 3692.** We have two boxes of pebbles. One box contains *p* pebbles and the other one contains *q*. We are allowed to take either one pebble out of each box or multiply the number of pebbles in one box by three. Is it possible to empty both boxes by repeating these two moves if *a*) *p*=100, *q*=200; *b*) *p*=101, *q*=200? (*4 points*)

**B. 3693.** When a positive integer *k* divided by the prime number *p*, the remainder is 6. If the number 1000-*k* is divided by *p*, the remainder is the same, and, furthermore, 10 000-*k* is divisible by *p*. What is the prime number *p*? (*3 points*) (Suggested by *G. Bakonyi,* Budapest)

**B. 3694.** Given the convex quadrilateral *ABCD*, construct the four lines passing through the vertex *A* that divide the area of the quadrilateral into five equal parts. (*3 points*)

**B. 3695.** Let *n* be a positive integer. Prove that 5^{n}-8*n*^{2}+4*n*-1 is divisible by 64. (*3 points*)

**B. 3696.** The endpoints of a number of open rays lying on a line divide the line into parts. Prove that there are at most *q*+1 parts that are covered by exactly *q* rays. (*4 points*)

**B. 3697.** Solve the following equation:

\(\displaystyle x^2+\left(\frac{5x}{x-5}\right)^{\!\!2}=11. \)

(*4 points*)

**B. 3698.** Given the third vertices of the equilateral triangles drawn over the sides of a quadrilateral ``outside'' construct the triangle. No discussion of the solutions is expected this time. (*5 points*)

**B. 3699.** The point *P* lies on the diameter *AB* of a unit circle. A chord through *P* intersects the circle at the points *C* and *D*. What is the maximum area of the quadrilateral *ABCD*? (*5 points*)

**B. 3700.** Prove that if *x*_{1},*x*_{2},...,*x*_{n} are positive numbers then

\(\displaystyle \frac{x_1^3}{x_1^2+x_1x_2+x_2^2}+ \frac{x_2^3}{x_2^2+x_2x_3+x_3^2}+\ldots+ \frac{x_n^3}{x_n^2+x_nx_1+x_1^2}\ge\frac{1}{3}(x_1+x_2+\ldots+ x_n). \)

(*5 points*) (*Gillis-Turán Competition*)

**B. 3701.** Let *P* denote an arbitrary point of a cube. Prove that there are at least six tetrahedra that contain *P* and the vertices of which are vertices of the cube. (*4 points*)

## New advanced problemsMaximum score for each advanced problem (sign "A") is 5 points. |

**A. 335.** By only using identical integers, the four basic operations and brackets, Steve is preparing further positive integers. For example:

\(\displaystyle \frac{100\cdot100}{\frac{100}{100}- \frac{100}{100+100}}=20\;000 \)

Playing around, he maneged to produce a positive integer less than \(\displaystyle \frac{|n|}{2^k}\) using the number *n* exactly *k* times. Prove that the expression found by Steve was an identity, that is, he would have obtained the same result by plugging any other number for *n* in the expression (unless the expression is undefined). (Based on the idea of *L. Szobonya*)

**A. 336.** Let *a* denote an odd number and let *b* denote an even number, such that *a*^{2}+*b*^{2}=*p* is a prime number. Prove that the equation *x*^{2}-*py*^{2}=*a* has an integer solution.

**A. 337.** The lines bounding a number of open half planes in the plane divide the plane into convex domains. Find a quadratic expression *C*(*q*) such that it is true for any integer *q*\(\displaystyle \ge\)1 that if the half planes cover each point of the plane at least *q* times then the set of points covered exactly *q* times is the union of at most *C*(*q*) domains. (*Schweitzer Competition, 2003*)

### Send your solutions to the following address:

- KöMaL Szerkesztőség (KöMaL feladatok),

Budapest 112, Pf. 32. 1518, Hungary