MathDB

4

Part of 1990 China Team Selection Test

Problems(2)

double difference square and summed up

Source: China TST 1990, problem 4

6/27/2005
Number aa is such that ∀a1,a2,a3,a4∈R\forall a_1, a_2, a_3, a_4 \in \mathbb{R}, there are integers k1,k2,k3,k4k_1, k_2, k_3, k_4 such that ∑1≤i<j≤4((ai−ki)−(aj−kj))2≤a\sum_{1 \leq i < j \leq 4} ((a_i - k_i) - (a_j - k_j))^2 \leq a. Find the minimum of aa.
functionalgebra unsolvedalgebra
maximum number of circles that can be drawn

Source: China TST 1990, problem 8

6/27/2005
There are arbitrary 7 points in the plane. Circles are drawn through every 4 possible concyclic points. Find the maximum number of circles that can be drawn.
set theorycombinatorics unsolvedcombinatorics