Problems

a) What is the minimum number of pieces of wire needed in order to weld a cube’s frame?

b) What is the maximum length of a piece of wire that can be cut from this frame? (The length of the edge of the cube is 1 cm).

Prove that in a bipartite planar graph $E\ge 2F$, if $E\ge 2$ ($E$ is the number of edges, $F$ is the number of regions).

Find the last digit of the number $1\times 2+2\times 3+\cdots +999\times 1000$.

Is the number 12345678926 square?

Solve the equation in integers $2x+5y=xy-1$.

Prove there are no integer solutions for the equation $x^2+1990=y^2$.

Prove there are no integer solutions for the equation $4^k-4^l={10}^n$.

There are 100 notes of two types: $a$ and $b$ pounds, and $a\ne b(\mathrm{mod}101)$. Prove that you can select several bills so that the amount received (in pounds) is divisible by 101.

Recall that a natural number $x$ is called prime if $x$ has no divisors except $1$ and itself. Solve the equation with prime numbers $pqr=7(p+q+r)$.

Solve the equation with natural numbers $1+x+x^2+x^3=2^y$.