MathDB

8.1 Find all primes

p,q

and

r

such that

pqr= 5(p + q + r).

8.2 Prove that if

\overline{ab}/\overline{bc} = a/c

, then

\overline{abb...bb}/\overline{bb...bbc} = a/c

(each number has

n

digits).
8.3 / 9.1 Construct a triangle with perimeter, altitude and angle at the base.
8.4. / 9.4 Prove that the square of the sum of N distinct non-zero squares of integers is also the sum of

N

squares of non-zero integers.
8.5. In the quadrilateral

ABCD

the diagonals

AC

and

BD

are drawn. Prove that if the circles inscribed in

ABC

and

ADC

touch each other each other, then the circles inscribed in

BAD

and in

BCD

also touch each other.
8.6 / 9.6 If the numbers

A

and

n

are coprime, then there are integers

X

and

Y

such that

|X| <\sqrt{n}

|Y| <\sqrt{n}

and

AX-Y

is divided by

n

. Prove it.
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983461_1964_leningrad_math_olympiad]here.

Problem Statement