MathDB

p1. In the correct identity

(x^2 - 1)(x + ...) = (x + 3)(x- 1)(x +...)

two numbers were replaced with dots. What were these numbers?
p2. A merchant is carrying money from point A to point B. There are robbers on the roads who rob travelers: on one road the robbers take

10\%

of the amount currently available, on the other -

20\%

, etc. . How should the merchant travel to bring as much of the money as possible to B? What part of the original amount will he bring to B? https://cdn.artofproblemsolving.com/attachments/f/5/ab62ce8fce3d482bc52b89463c953f4271b45e.png
p3. Find the angle between the hour and minute hands at

7

hours

38

minutes.
p4. The lottery game is played as follows. A random number from

1

1000

is selected. If it is divisible by

2

, they pay a ruble, if it is divisible by

10

- two rubles, by

12

- four rubles, by

20

- eight, if it is divisible by several of these numbers, then they pay the sum. How much can you win (at one time) in such a game? List all options.
p5.The sum of the digits of a positive integer

x

is equal to

n

. Prove that between

x

and

10x

you can find an integer whose sum of digits is

n + 5

.
p6.

9

people took part in the campaign, which lasted

12

days. There were

3

people on duty every day. At the same time, the duty officers quarreled with each other and no two of them wanted to be on duty together ever again. Nevertheless, the participants of the campaign claim that for all

12

days they were able to appoint three people on duty, taking into account this requirement. Could this be so?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.

Problem Statement