MathDB

p1. Given two irreducible fractions. The denominator of the first fraction is

4

, the denominator of the second fraction is

6

. What can the denominator of the product of these fractions be equal to if the product is represented as an irreducible fraction?
p2. Three horses compete in the race. The player can bet a certain amount of money on each horse. Bets on the first horse are accepted in the ratio

1: 4

. This means that if the first horse wins, then the player gets back the money bet on this horse, and four more times the same amount. Bets on the second horse are accepted in the ratio

1:3

, on the third -

1:1

. Money bet on a losing horse is not returned. Is it possible to bet in such a way as to win whatever the outcome of the race?
p3. A quadrilateral is inscribed in a circle, such that the center of the circle, point

O

, is lies inside it. Let

K

L

M

N

be the midpoints of the sides of the quadrilateral, following in this order. Prove that the bisectors of angles

\angle KOM

and

\angle LOC

are perpendicular (Fig.). https://cdn.artofproblemsolving.com/attachments/b/8/ea4380698eba7f4cc2639ce20e3057e0294a7c.png
p4. Prove that the number

\underbrace{33...33}_{1999 \,\,\,3s}1

is not divisible by

7

.
p5. In triangle

ABC

, the median drawn from vertex

A

to side

BC

is four times smaller than side

AB

and forms an angle of

60^o

with it. Find the greatest angle of this triangle.
p6. Given a

7\times 8

rectangle made up of 1x1 cells. Cut it into figures consisting of

1\times 1

cells, so that each figure consists of no more than

5

cells and the total length of the cuts is minimal (give an example and prove that this cannot be done with a smaller total length of the cuts). You can only cut along the boundaries of the cells.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.

Problem Statement