TUYMAADA 1995

Give a geometric proof of the statement that the fold line on a sheet of paper is straight.

Let $x_1=a, x_2=a^{x_1}, ..., x_n=a^{x_{n-1}}$ where $a>1$ . What is the maximum value of $a$ for which lim exists $\lim_{n\to \infty} x_n$ and what is this limit?

Prove that the equation $(\sqrt5 +1)^{2x}+ (\sqrt5 -1)^{2x}=2^x(y^2+2)$ has an infinite number of solutions in natural numbers.

It is known that the merchant's $n$ clients live in locations laid along the ring road. Of these, $k$ customers have debts to the merchant for $a_1,a_2,...,a_k$ rubles, and the merchant owes the remaining $n-k$ clients, whose debts are $b_1,b_2,...,b_{n-k}$ rubles, moreover, $a_1+a_2+...+a_k=b_1+b_2+...+b_{n-k}$ . Prove that a merchant who has no money can pay all his debts and have paid all the customer debts, by starting a customer walk along the road from one of points and not missing any of their customers.

A set consisting of $n$ points of a plane is called an isosceles $n$ -point if any three of its points are located in vertices of an isosceles triangle. Find all natural numbers for which there exist isosceles $n$ -points.

Given a circle of radius $r= 1995$ . Show that around it you can describe exactly $16$ primitive Pythagorean triangles. The primitive Pythagorean triangle is a right-angled triangle, the lengths of the sides of which are expressed by coprime integers.

Find a continuous function $f(x)$ satisfying the identity $f(x)-f(ax)=x^n-x^m$ , where $n,m\in N , 0<a<1$

Inside the triangle $ABC$ a point $M$ is given . Find the points $P,Q$ and $R$ lying on the sides $AB,BC$ and $AC$ respectively and such so that the sum $MP+PQ+QR+RM$ is the smallest.