1

Part of 2002 Germany Team Selection Test

Problems(2)

Divisible by 121

Source: VAIMO 1, German Pre-TST 2002

Determine the number of all numbers which are represented as

x^2+y^2

x, y \in \{1, 2, 3, \ldots, 1000\}

and which are divisible by 121.

quadraticsnumber theory unsolvednumber theory

Two-variable function

Source: VAIMO 4, German Pre-TST 2002

P

denote the set of all ordered pairs

\left(p,q\right)

of nonnegative integers. Find all functions

f: P \rightarrow \mathbb{R}

satisfying f(p,q) \equal{} \begin{cases} 0 & \text{if} \; pq \equal{} 0, \\ 1 \plus{} \frac{1}{2} f(p+1,q-1) \plus{} \frac{1}{2} f(p-1,q+1) & \text{otherwise} \end{cases}
Compare IMO shortlist problem 2001, algebra A1 for the three-variable case.

functionquadraticsIMO Shortlistalgebra unsolvedalgebra