New exercises and problems in Mathematics October
2002
 New exercises Maximum
score for each exercise (sign "C") is 5 points. 
C. 685. The surface area of a cylinder is 1000 m^{2} and its height is 1000 km. Find the volume of the cylinder in litres.
C. 686. Anna was bored in class, and to kill time she made a list of integers. Starting with a certain number, she obtained the next number by either adding or multiplying the digits of the previous number on the list. She continued writing down numbers hence obtained, and observed that all the numbers were odd. With how many initial values at most six digits is that possible?
C. 687. The coordinates of the vertices of a quadrilateral are A(0;0), B(16;0), C(8;8), D(0,8). Find the equation of the line parallel to AC that halves the area of the quadrilateral.
C. 688. Solve the equation [x/2]+[x/4]=x. ([x] denotes the greatest integer not greater than the number x.)
C. 689. Solve the equation
\(\displaystyle
x^{\log_2(16x^2)}4x^{\log_2(4x)+1}16x^{\log_2(4x)+2}+64
x^3=0.
\)
(Suggested by A. Mosóczi, Budapest)
 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. 3572. Solve the equation [x/2]+[x/4]=[x]. ([x] denotes the greatest integer not greater than the number x.) (3 points)
B. 3573. Given the line segment AB its midpoint F and the point P. With the help of a single straight edge, construct a line parallel to AB through the point P. (4 points)
B. 3574. The circle k_{1} touches the circle k_{2} internally at the point P. The chord AB of the larger circle touches the smaller circle at C, AP and BP intersect the smaller circle at the points D and E, respectively. Given that AB=84, PD=11, and PE=10. Find the length of the line segment AC. (4 points)
B. 3575. Let X denote the set of positive integers that contain different digits in decimal notation. If n\(\displaystyle \in\)X, let A_{n} denote the set of numbers obtained by the permutations of the digits of n, and let d_{n} be the greatest common factor of the elements of A_{n}. What is the maximum value of d_{n}? (4 points)
B. 3576. A random sequence is produced using the digits 0, 1, 2. How long should the sequence be so that the probability that all three digits occur in the sequence is at least 0.61? (4 points) (Suggested by E. Fried, Budapest)
B. 3577. Solve the equation sin 3x + 3 cos x = 2 sin 2x (sin x+cos x). (4 points) (Suggested by A. Mosóczi, Budapest)
B. 3578. The centre of the face ABCD of the cube in the Figure is the point M. Find the points P and Q on the lines AB and EM, respectively, such that the distance PQ is equal to the distance of the lines AB and EM. (3 points)
B. 3579. Solve the equation
\(\displaystyle
x=\sqrt{3+4\sqrt{3+4\sqrt{3+4x}}}.
\)
(5 points)
B. 3580. Out of all obtuse triangles of integer side lengths with the obtuse angle equal to the double of one acute angle, which one has the shortest perimeter? (4 points)
B. 3581. Find the minimum value of the function
f(x)=1001+1000x+999x^{2}+^{...}+2x^{999}+x^{1000}.
(5 points)
 New advanced problems Maximum score
for each advanced problem (sign "A") is 5 points. 
A. 299. P and Q are interior points of the square ABCD such that PAQ\(\displaystyle \angle\)= PCQ\(\displaystyle \angle\)= 45^{o}. Determine the length PQ in terms of the lengths BP and DQ.
A. 300. Find all pairs (a, b) such that a and b are whole numbers and a^{2} + ab + b^{2}is a multiple of 7^{5}.
A. 301. Let a_{0},a_{1},... a sequence of non negative numbers such that for every k, m \(\displaystyle \ge\)0, a_{k+m} \(\displaystyle \le\)a_{k+m+1} + a_{k} a_{m}. Assume, additionally, that na_{n} < 0.2499 holds for sufficiently large n. Prove that there exists a number q for which 0<q<1 and a_{n}<q^{n} if n is large enough.
Send your solutions to the following address:
KöMaL Szerkesztőség (KöMaL feladatok), Budapest 112,
Pf. 32. 1518, Hungary or by email to:
Deadline: 15 November 2002
