NIKKEI Programming Contest 2019-2 Task E. Non-triangular Triplets

$ equire{enclose}$

# 必要条件 一方面 $sum\_{i=1}^{N}(a_i + b_i) le sum\_{i=1}^{N} c_i implies 2sum\_{i=1}^{N} c_i ge sum\_{i=1}^{N}(a_i + b_i + c_i) = sum\_{i=K}^{K+3N-1} i = frac{3N(2K+3N-1)}{2}$ 另一方面 $sum\_{i=1}^{N} c_i le sum_{i=K+2N}^{K+3N-1} i = frac{N(2K+5N-1)}{2}$ $N(2K+5N-1) le frac{3N(2K+3N-1)}{2} implies 2K - 1le N$

此必要条件也可用另一种方法推导出来：
由于 $sum_{i = 1}^{N} (a_i + b_i) ge sum_{i=K}^{K+2N-1} i $ 且 $sum_{i=1}^{N} c_i le sum_{i = K+2N}^{K+3N-1} i$，因此 $sum_{i = 1}^{N} (a_i + b_i) le sum_{i=1}^{N} c_i implies sum_{i=K}^{K+2N-1} i le sum_{i = K+2N}^{K+3N-1} i implies 2K - 1le N$。

构造

the pattern is $(x, y)$, $(x+2, y -1)$, ...

例子
$K = 2, N = 6$
egin{matrix}
3 & 5 & enclose{right}{7} & 2 & 4 & 6 \
10 & 9 & enclose{right}{8} & 13 &12 & 11 \
hline
14 & 15 & 16 & 17 & 18 & 19
end{matrix}
$K = 2, N = 7$
egin{matrix}
2 & 4 & 6 & 8 & 3 & 5 & 7 \
15 & 14 & 13 & 12 & 11 & 10 & 9 \
hline
19 & 20 & 21 & 22 & 16 & 17 & 18
end{matrix}