MathDB

2

Part of 1993 China Team Selection Test

Problems(2)

a/b + c/d for fixed n

Source: China TST 1993, problem 2

6/27/2005
Let n≥2,n∈Nn \geq 2, n \in \mathbb{N}, a,b,c,d∈Na,b,c,d \in \mathbb{N}, ab+cd<1\frac{a}{b} + \frac{c}{d} < 1 and a+c≤n,a + c \leq n, find the maximum value of ab+cd\frac{a}{b} + \frac{c}{d} for fixed n.n.
quadraticsfunctionalgebra unsolvedalgebrainequalities
Subset without squares

Source: China TST 1993, problem 5

6/27/2005
Let S={(x,y)∣x=1,2,…,1993,y=1,2,3,4}S = \{(x,y) | x = 1, 2, \ldots, 1993, y = 1, 2, 3, 4\}. If T⊂ST \subset S and there aren't any squares in T.T. Find the maximum possible value of ∣T∣.|T|. The squares in T use points in S as vertices.
algebra unsolvedalgebra