TUYMAADA 1998

Write the number $\frac{1997}{1998}$ as a sum of different numbers, inverse to naturals.

Solve the equation $(x^3-1000)^{1/2}=(x^2+100)^{1/3}$

The segment of length $\ell$ with the ends on the border of a triangle divides the area of that triangle in half. Prove that $\ell >r\sqrt2$ , where $r$ is the radius of the inscribed circle of the triangle.

Given the tetrahedron $ABCD$ , whose opposite edges are equal, that is, $AB=CD, AC=BD$ and $BC=AD$ . Prove that exist exactly $6$ planes intersecting the triangular angles of the tetrahedron and dividing the total surface and volume of this tetrahedron in half.

A right triangle is inscribed in parabola $y=x^2$ . Prove that it's hypotenuse is not less than $2$ .

Prove that the sequence of the first digits of the numbers in the form $2^n+3^n$ is nonperiodic.

All possible sequences of numbers $-1$ and $+1$ of length $100$ are considered. For each of them, the square of the sum of the terms is calculated. Find the arithmetic average of the resulting values.

Given the pyramid $ABCD$ . Let $O$ be the midpoint of edge $AC$ . Given that $DO$ is the height of the pyramid, $AB=BC=2DO$ and the angle $ABC$ is right. Cut this pyramid into $8$ equal and similar to it pyramids.