2021 CMO 第 1 题

Let $n$ be positive integer such that there are exactly 36 different prime numbers that divides $n.$ For $k=1,2,3,4,5,$ $c_n$ be the number of integers that are mutually prime numbers to $n$ in the interval $[\frac{(k-1)n}{5},\frac{kn}{5}] .$ $c_1,c_2,c_3,c_4,c_5$ is not exactly the same.Prove that $$\sum_{1\le i<j\le 5}(c_i-c_j)^2\geq 2^{36}.$$

设 $n$ 为正整数，使得正好有 36 个不同的素数整除 $n。$ 对于 $k=1,2,3,4,5，$ $c_n$ 是区间 $[\frac{(k-1)n}{5},\frac{kn}{5}] 中与$n$互质的整数的数量。$ $c_1,c_2,c_3,c_4,c_5$ 不完全是相同。证明 $$\sum_{1\le i<j\le 5}(c_i-c_j)^2\geq 2^{36}。$$