Min-max

Algebra Level 5

$\Large \sum_{\stackrel{(x,y) \in \mathbb N^2}{\max(x,y) = 70}} \text{min}(x,y) = \, ?$

Clarification : $\mathbb N$ refers to the set of all positive integers.

The answer is 4900.

2 solutions

Arjen Vreugdenhil
Mar 26, 2016

The sum is over the ordered pairs $(1, 70)\ \text{through}\ (69,70);\ \ (70,1)\ \text{through}\ (70,69);\ \ (70, 70).$ Thus we must add $2\cdot(1 + 2 + \cdots + 69) + 70 = 2\cdot \tfrac12\cdot 69\cdot 70 + 70 = (69+1)\cdot 70 = 70^2 = \boxed{4900}.$

Why is it level 5? It is so easy.

Saarthak Marathe - 5 years, 1 month ago

@Arjen Vreugdenhil Can you please help me out???

In the question it stated that (x,y) belongs to $N^{2}$ and also maximum (x,y) is 70 so doesn't it mean that (x,y) belong to (1 , 4 , 9 , 16 , 25 , 36 , 49 , 64)

Ankit Kumar Jain - 5 years, 2 months ago

By ${\mathbb N}^2$ mathematicians typically means $\mathbb N \times \mathbb N$ , the Cartesian product of two sets, which consists of all tuples $(x,y)$ with $x,y \in \mathbb N$ .

Arjen Vreugdenhil - 5 years, 2 months ago

I understood now... Thanks...

Ankit Kumar Jain - 5 years, 2 months ago

Adrian Castro
Mar 27, 2016

$\sum _{(x,y)\epsilon\mathbb{N}^2, max(x,y)=70}min(x,y)=\sum_{i=1}^{70}min(i,70)+\sum_{i=1}^{70}min(70,i)-min(70,70)$

$=2\sum_{i=1}^{70}i-70=70(71)-70=70^2=\boxed{4900}$

Min-max

The answer is 4900.

2 solutions

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