MathDB

8.1 Four circles are placed on planes so that each one touches the other two externally. Prove that the points of tangency lie on one circle. https://cdn.artofproblemsolving.com/attachments/9/8/883a82fb568954b09a4499a955372e2492dbb8.png
8.2. Let the integers

a

and

b

be represented as

x^2-5y^2

, where

x

and

y

are integer numbers. Prove that the number

ab

can also be presented in this form.
8.3 Solve the equation

x(x + d)(x + 2d)(x + 3d) = a

.
8.4 / 9.1 Let

a+b+c=1

m+n+p=1

. Prove that

-1 \le am + bn + cp \le 1

8.5 Inscribe a triangle with the largest area in a semicircle.
8.6 Three circles of the same radius intersect at one point. Prove that the other three points intersections lie on a circle of the same radius. https://cdn.artofproblemsolving.com/attachments/4/7/014952f2dcf0349d54b07230e45a42c242a49d.png
8.7 Find the circle of smallest radius that contains a given triangle.
8.8 / 9.2 Given a polynomial

x^{2n} +a_1x^{2n-2} + a_2x^{2n-4} + ... + a_{n-1}x^2 + a_n,

which is divisible by

x-1

. Prove that it is divisible by

x^2-1

.
8.9 Prove that for any prime number

p

other than

2

and from

5

, there is a natural number

k

such that only ones are involved in the decimal notation of the number

pk

..
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983459_1962_leningrad_math_olympiad]here.

Problem Statement