A convex polygon $\mathcal P$ in the plane is dissected into smaller convex polygons by drawing all of its diagonals. The lengths of all sides and all diagonals of $\mathcal P$ are rational numbers. Prove that the lengths of all sides of all polygons in the dissection are also rational numbers.
Let $n$ be a positive integer. For every integer sequence $A=(a_0,a_1,\ldots,a_n)$ satisfying $0\le a_i\le i$, define $t(A)=(t(a_0),t(a_1),\ldots,t(a_n))$ by letting $t(a_i)$ be the number of terms preceding $a_i$ in $A$ that are different from $a_i$. Show that fewer than $n$ applications of $t$ lead from any such $A$ to a sequence $B$ with $t(B)=B$.
Let $ABC$ be a triangle. A circle passing through $A$ and $B$ intersects segments $AC$ and $BC$ at $D$ and $E$, respectively. Lines $AB$ and $DE$ intersect at $F$, while lines $BD$ and $CF$ intersect at $M$. Prove that $MF=MC$ if and only if $MB\cdot MD=MC^2$.
Proposed by Titu Andreescu and Zuming Feng (as credited in the supplied source).
Let $a,b,c$ be positive real numbers. Prove that
$$\frac{(2a+b+c)^2}{2a^2+(b+c)^2}+\frac{(2b+c+a)^2}{2b^2+(c+a)^2}+\frac{(2c+a+b)^2}{2c^2+(a+b)^2}\le8.$$
Proposed by Zuming Feng and Titu Andreescu (as credited in the supplied source).
At the vertices of a regular hexagon are written six nonnegative integers whose sum is $2003^{2003}$. Bert may pick a vertex and replace the number there by the absolute value of the difference between the numbers at its two neighboring vertices. Prove that Bert can make a sequence of moves after which all six vertices contain $0$.