Prove that there are at most three primes between $10$ and $10^{10}$ all of whose decimal digits are $1$.
1997 Israel Grosman Mathematical Olympiad
Is there a planar polygon whose vertices have integer coordinates and whose area is $1/2$, such that this polygon is (a) a triangle with at least two sides longer than $1000$? (b) a triangle whose sides are all longer than $1000$? (c) a quadrangle?
Find all real solutions of $\sqrt[4]{13+x}+ \sqrt[4]{14-x} = 3$.
Prove that if two altitudes of a tetrahedron intersect, then so do the other two altitudes.
Consider partitions of an $n \times n$ square (composed of $n^2$ unit squares) into rectangles with one integer side and the other side equal to $1$. What is the largest possible number of such partitions among which no two have an identical rectangle at the same place?
In the plane are given $n^2 + 1$ points, no three of which lie on a line. Each line segment connecting a pair of these points is colored by either red or blue. A path of length $k$ is a sequence of $k$ segments where the end of each segment (except for the last one) is the beginning of the next one. A path is simple if it does not intersect itself. Prove that there exists a monochromatic simple path of length $n$.