Ax + By < C

AtCoder

IDabc372_g

Time3000ms

Memory256MB

Difficulty—

You are given three length-$N$ sequences of positive integers: $A=(A_1,A_2,\ldots,A_N)$, $B=(B_1,B_2,\ldots,B_N)$, and $C=(C_1,C_2,\ldots,C_N)$. Find the number of pairs of positive integers $(x, y)$ that satisfy the following condition: * $A_i \times x + B_i \times y < C_i$ for all $1 \leq i \leq N$. It can be proved that the number of such pairs of positive integers satisfying the condition is finite. You are given $T$ test cases, each of which should be solved. ## Constraints * $1 \leq T \leq 2 \times 10^5$ * $1 \leq N \leq 2 \times 10^5$ * $1 \leq A_i, B_i, C_i \leq 10^9$ * The sum of $N$ over all test cases is at most $2 \times 10^5$. * All input values are integers. ## Input The input is given from Standard Input in the following format. Here, $\mathrm{case}_i$ refers to the $i$\-th test case. $T$ $\mathrm{case}_1$ $\mathrm{case}_2$ $\vdots$ $\mathrm{case}_T$ Each test case is given in the following format: $N$ $A_1$ $B_1$ $C_1$ $A_2$ $B_2$ $C_2$ $\vdots$ $A_N$ $B_N$ $C_N$ [samples]

Samples

Input #1

Output #1

2
0

In the first test case, there are two valid pairs of integers: $(x, y) = (1, 1), (2,1)$. Thus, the first line should contain $2$.
In the second test case, there are no valid pairs of integers. Thus, the second line should contain $0$.

Input #2

3
7
138 16011 918976
5478 7748 499926
5234 17727 748589
1157 10511 643136
31200 3005 721285
28839 14469 798851
1933 5378 864127
9
17775 1665 386430
37001 863 922418
9756 4182 746671
12379 9106 807578
3984 4049 640539
25333 9869 780810
20372 7000 688738
16107 11974 827227
10779 10531 770510
5
4916 14132 460944
11856 45422 610561
56014 18216 825793
10363 6220 945356
37418 33866 851593

Output #2

660
995
140

API Response (JSON)

{
  "problem": {
    "name": "Ax + By < C",
    "description": {
      "content": "You are given three length-$N$ sequences of positive integers: $A=(A_1,A_2,\\ldots,A_N)$, $B=(B_1,B_2,\\ldots,B_N)$, and $C=(C_1,C_2,\\ldots,C_N)$. Find the number of pairs of positive integers $(x, y)$ ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 3000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc372_g"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given three length-$N$ sequences of positive integers: $A=(A_1,A_2,\\ldots,A_N)$, $B=(B_1,B_2,\\ldots,B_N)$, and $C=(C_1,C_2,\\ldots,C_N)$.\nFind the number of pairs of positive integers $(x, y)$ ...",
      "is_translate": false,
      "language": "English"
    }
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}

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