[Problem M1—M5; F1—F4]

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MATEMATIKE SOFTWARE

M1. In the country of Anchuria, where President Miraflores* rules, the time for new presidential elections has approached. There are approximately 20 million voters in the country, of which only one percent (the regular army of Anchuria) supports Miraflores. Miraflores naturally wants to be elected, but on the other hand he wants the elections to appear democratic. What Miraflores calls “democratic voting” is this: all voters are divided into several equal groups, then each of these groups is again divided into a number of equal groups, then these last groups are again divided into equal groups, etc.; in the smallest groups, a representative of the group is chosen - an elector, then an elector is chosen by a representative to vote in an even larger group, etc.; finally, representatives of the largest groups choose a representative. Miraflores divides voters into groups as he wishes and instructs his parties on how to vote. Will he be able to organize “democratic elections” in such a way that he will be elected president? (If the votes are equal, the opposition wins.)

XXXII Moscow Mathematical Olympiad

*) See O’Henry, Kings and Cabbage. Selected works in two volumes, GIHL, 1955, vol. 1, p. 7.

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Below are several problems in mathematics and physics, starting from Problem Book «Kvant». Many of them are quite difficult. The best solutions to these problems proposed by readers will be published in the journal.

M2. Given a sphere of radius 1. Equal circles $\gamma_0, \gamma_1, \dots, \gamma_n$ of radius $r (n \geqslant 3)$ are located on it. Circle $\gamma_0$ touches all circles $\gamma_1, \dots, \gamma_n$; in addition, circles $\gamma_1$ and $\gamma_2$ touch each other; $\gamma_2$ and $\gamma_3$; …; $\gamma_n$ and $\gamma_1$. At what $n$ is this possible? Calculate the corresponding radius $r$.

XVIII Mathematical Olympiad
Czechoslovaks

Fig. 1.

M3. On figure 1 the plane is covered with squares of fives and flowers. The centers of squares of the same color are located at the tops of the square grid. With what number of colors is a similar filling of the plane possible?

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On figure 2 the plane is covered with hexagons of seven colors so that the centers of hexagons of the same color form the vertices of a lattice of identical regular triangles. With what number of colors is a similar construction possible?

Notes. In the first problem, the number of colors can be one (all squares are the same color) or two (like on a chessboard). In the second problem you can easily find solutions with one color and with three colors. It is advisable to give a complete solution to the problems, i.e., describe all colorings that satisfy the specified conditions.

Fig. 2.

loviyam. However, please send incomplete solutions if they seem interesting to you. Think, for example, is there a solution to the second problem with thirteen colors?

A. N. Kolmodorov

M4. Given a segment AB. Find on the plane a set of points C such that in triangle ABC the median
drawn from vertex A is equal to the height drawn from vertex B.

XXXII Moscow Mathematical Olympiad

M5. In a set E consisting of n elements, m different subsets (different from E itself) are identified so that for every two elements of E there is exactly one of the given subsets that contains both of these elements.

Prove that m≥n. In what cases is equality possible?

N. Bourbaki

IN PHYSICS

F1. Three communicating vessels with water, the centers of which are at the same distance a from each other, are covered with pistons of the same thickness, made of the same materials (Fig. 3). Vertical identical rods are attached to the pistons, which are pivotally connected to rod AB. At what point on the rod can a load be attached to it so that in the equilibrium position the rod remains horizontal, if the masses of the rod and rods are negligible compared to the masses of the pistons and the load. Vessel diameters are indicated at figure.

F2. Two balls with masses m₁ and m₂ lie on a horizontal plane, fastened together