$21$ distinct numbers are chosen from the set $\{1,2,3,\ldots,2046\}.$ Prove that we can choose three distinct numbers $a,b,c$ among those $21$ numbers such that \[bc<2a^2<4bc\]

The points $D,E$ and $F$ are chosen on the sides $BC,AC$ and $AB$ of triangle $ABC$, respectively. Prove that triangles $ABC$ and $DEF$ have the same centroid if and only if \[\frac{BD}{DC} = \frac{CE}{EA}=\frac{AF}{FB}\]

Let $M=\{1,2,3,\ldots, 10000\}.$ Prove that there are $16$ subsets of $M$ such that for every $a \in M,$ there exist $8$ of those subsets that intersection of the sets is exactly $\{a\}.$

Find all positive integers $n$ such that we can divide the set $\{1,2,3,\ldots,n\}$ into three sets with the same sum of members.

In a tetrahedron we know that sum of angles of all vertices is $180^\circ.$ (e.g. for vertex $A$, we have $\angle BAC + \angle CAD + \angle DAB=180^\circ.$) Prove that faces of this tetrahedron are four congruent triangles.

Super number is a sequence of numbers $0,1,2,\ldots,9$ such that it has infinitely many digits at left. For example $\ldots 3030304$ is a super number. Note that all of positive integers are super numbers, which have zeros before they're original digits (for example we can represent the number $4$ as $\ldots, 00004$). Like positive integers, we can add up and multiply super numbers. For example: \[ \begin{array}{cc}& \ \ \ \ldots 3030304 \\ &+ \ldots4571378\\ &\overline{\qquad \qquad \qquad }\\ & \ \ \ \ldots 7601682 \end{array} \] And \[ \begin{array}{cl}& \ \ \ \ldots 3030304 \\ &\times \ldots4571378\\ &\overline{\qquad \qquad \qquad }\\ & \ \ \ \ldots 4242432 \\ & \ \ \ \ldots 212128 \\ & \ \ \ \ldots 90912 \\ & \ \ \ \ldots 0304 \\ & \ \ \ \ldots 128 \\ & \ \ \ \ldots 20 \\ & \ \ \ \ldots 6 \\ &\overline{\qquad \qquad \qquad } \\ & \ \ \ \ldots 5038912 \end{array}\] a) Suppose that $A$ is a super number. Prove that there exists a super number $B$ such that $A+B=\stackrel{\leftarrow}{0}$ (Note: $\stackrel{\leftarrow}{0}$ means a super number that all of its digits are zero). b) Find all super numbers $A$ for which there exists a super number $B$ such that $A \times B=\stackrel{\leftarrow}{0}1$ (Note: $\stackrel{\leftarrow}{0}1$ means the super number $\ldots 00001$). c) Is this true that if $A \times B= \stackrel{\leftarrow}{0}$, then $A=\stackrel{\leftarrow}{0}$ or $B=\stackrel{\leftarrow}{0}$? Justify your answer.