New exercises and problems in Mathematics
October 2002

Please read The Conditions of the Problem Solving Competition.

New exercises

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

C. 685. The surface area of a cylinder is 1000 m² 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 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. 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 k₁ touches the circle k₂ 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+999x²+^...+2x⁹⁹⁹+x¹⁰⁰⁰|.

(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 a² + ab + b²is a multiple of 7⁵.

A. 301. Let a₀,a₁,... 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<qⁿ if n is large enough.

Send your solutions to the following address:

or by e-mail to:

solutions@komal.elte.hu

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