New exercises and problems in Mathematics
November 2003

Please read The Conditions of the Problem Solving Competition.

New exercises

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

C. 735. The points A₁, B₁, C₁, D₁ lie on the sides AB, BC, CD, DA of a unit square ABCD. Given that

\(\displaystyle AA_1=BB_1=CC_1=DD_1=\frac{1}{5}, \)

find the area of the quadrilateral A₁B₁C₁D₁.

C. 736. While downloading a 1.5-MB file from the internet, our computer continually displays the estimated time needed till the task is complete. The estimation is based on the average speed of the downloading completed so far. Looking at the screen, we saw that the downloading was exactly half complete and the estimated time remaining was 2 minutes. Then we observed that, irrespective of the time t elapsed, the time display kept reading 2 minutes, owing to the net being busy. Express the size of the downloaded part of the file as a function of the time t. (Suggested by L. Koncz, Budapest)

C. 737. A company sells sweets in rectangular boxes. They also make 10-packs of the boxes held together by a thin wrapper. The manager would like the 10-pack to be geometrically similar to a single box. Is that possible?

C. 738. Find the maximum possible area of a triangle, given that its sides are at most 2 units long.

C. 739. A kite is made of sticks 3 cm and 4 cm long, flexible at the joints to make the angles variable. How long are the diagonals of the kite when its area is half the maximum possible area?

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. 3672. Prove that by cutting off two triangles every triangle can be made into a pentagon, such that there is an appropriate point of the pentagon where each side of it subtends an acute angle. (3 points) (J. Surányi)

B. 3673. Solve the following simultaneous equations on the set of positive real numbers: x²+y²+xy=7, x²+z²+xz=13, y²+z²+yz=19. (4 points)

B. 3674. For what positive integers k may the product of the first k primes be the sum of two positive cube numbers? (4 points)

B. 3675. The medians of a triangle divide it into six triangles. Is it possible for one of these triangles to be similar to the original one? (3 points) B. 3676. In the football pool of Quasiland, one bets on four games. Charlie Yarborough wants to show off to his friends by demonstrating that he lives up to his name. If he is clever enough, how many tickets does he need to fill out in order to be certain to have one that contains no right guesses at all? (5 points) (S. Dobos)

B. 3677. Find all pairs (\(\displaystyle \alpha\),\(\displaystyle \beta\)) of positive integers for which the set of positive integers can be divided into two disjoint sets A and B, such that {\(\displaystyle \alpha\)a | a\(\displaystyle \in\)A}= {\(\displaystyle \beta\)b | b\(\displaystyle \in\)B}. (4 points) (IMC10, Cluj-Napoca, Romania, 2003)

B. 3678. The line drawn through the vertex A of a quadrilateral ABCD parallel to the diagonal BD and the line drawn through vertex B parallel to diagonal AC intersect at the point E. Prove that the line EC divides the diagonal BD in the same ratio as line ED divides the diagonal AC. (3 points) (J. Surányi)

B. 3679. Prove that

\(\displaystyle \frac{1}{a}+\frac{1}{b}+\frac{4}{c}+\frac{16}{d}\ge\frac{64}{a+b+c+d} \)

for all positive numbers a, b, c, d. (4 points)

B. 3680. Consider the triangle PQR formed by the tangents drawn to the circumscribed circle of the triangle ABC at the vertices A, B and C, such that C lies on side PQ, B lies on side PR and A on side QR. Let C₁ denote the foot of the altitude of triangle ABC drawn from vertex C on the side AB. Prove that CC₁ bisects the angle QC₁P. (5 points) (Gillis-Turán competition)

B. 3681. Five of the six edges of a tetrahedron are known to be at most 2 units long. Prove that its volume is at most 1. (4 points)

New advanced problems

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

A. 329. A circle k₁ of the plane lies in the interior of a circle k₂. Point P lies inside k₁, and point Q lies outside k₂ in the plane. Given the circles and the points, draw an arbitrary line e through P that does not pass through Q. Let e intersect k₁ at A and B. Let the circumscribed circle of ABQ intersect k₂ at C and D. Prove that all line segments CD obtained in this way are concurrent.

A. 330. The sequence of Lucas numbers is defined by the following recurrence: L₀=2, L₁=1, L_n+1=L_n+L_n-1. Show that L_(p+1)/2^p-1 is divisible by L_p for all primes p>3.

A. 331. According to the records of the Bureau of Statistics in Quasiland, any pair of citizens who know each other have exactly one common acquaintance, while any two people who do not know each other have at least ten common acquaintances. Is it possible that the records are accurate?

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or by e-mail to:

solutions@komal.elte.hu

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