When $a_0, a_1, \dots , a_{1000}$ denote digits, can the sum of the $1001$-digit numbers $a_0a_1\cdots a_{1000}$ and $a_{1000}a_{999}\cdots a_0$ have odd digits only?

In a triangle $ABC$ assume $AB = AC$, and let $D$ and $E$ be points on the extension of segment $BA$ beyond $A$ and on the segment $BC$, respectively, such that the lines $CD$ and $AE$ are parallel. Prove $CD \ge \frac{4h}{BC}CE$, where $h$ is the height from $A$ in triangle $ABC$. When does equality hold?

Find all functions $f$ such that \[f(f(x) + y) = f(x^2-y) + 4yf(x)\] for all real numbers $x$ and $y$.

Show that for any integer $n \ge 2$ the sum of the fractions $\frac{1}{ab}$, where $a$ and $b$ are relatively prime positive integers such that $a < b \le n$ and $a+b > n$, equals $\frac{1}{2}$. (Integers $a$ and $b$ are called relatively prime if the greatest common divisor of $a$ and $b$ is $1$.)