1936 Moscow Mathematical Olympiad

Part of Moscow Mathematical Olympiad

Subcontests

(10)

1

MMO 031 Moscow MO 1936 3 planes and a ball in space, ways

Given three planes and a ball in space. In space, find the number of different ways of placing another ball so that it would be tangent the three given planes and the given ball. It is assumed that the balls can only touch externally.

1

MMO 030 Moscow MO 1936 10^6 as product of three factors

How many ways are there to represent

10^6

as the product of three factors? Factorizations which only differ in the order of the factors are considered to be distinct.

1

MMO 029 Moscow MO 1936 rectangle with integer sides and diagonals

The lengths of a rectangle’s sides and of its diagonal are integers. Prove that the area of the rectangle is an integer multiple of

12

1

MMO 028 Moscow MO 1936 line construction, given perimeter

Given an angle less than

180^o

M

outside the angle. Draw a line through

M

so that the triangle, whose vertices are the vertex of the angle and the intersection points of its legs with the line drawn, has a given perimeter.

1

MMO 027 Moscow MO 1936 system x+y=a, x^5 +y^5 = b^5

Solve the system

\begin{cases} x+y=a \\ x^5 +y^5 = b^5 \end{cases}

1

MMO 026 Moscow MO 1936 4 consecutive with product is 1680

4

consecutive positive integers whose product is

1680

1

MMO 025 Moscow MO 1936 diameter construction

Consider a circle and a point

P

outside the circle. The angle of given measure with vertex at

P

subtends a diameter of the circle. Construct the circle’s diameter with ruler and compass.

1

MMO 024 Moscow MO 1936 P. Dirac&rsquo;s problem, representation with 3 2s

Represent an arbitrary positive integer as an expression involving only

3

twos and any mathematical signs. (P. Dirac)

1

MMO 023 Moscow MO 1936 isosceles triangle construction

All rectangles that can be inscribed in an isosceles triangle with two of their vertices on the triangle’s base have the same perimeter. Construct the triangle.

1

MMO 022 Moscow MO 1936 4-digit perfect square

Find a four-digit perfect square whose first digit is the same as the second, and the third is the same as the fourth.