E. Meetings

Codeforces

IDCF10050E

Time1000ms

Memory64MB

Difficulty—

English · Original

Formal · Original

Two cities A and B are connected by a straight road that is exactly l meters long. At the initial moment of time a cyclist starts moving from city A to city B at a speed v1 meters/second, and a pedestrian starts moving from city B to city A at a speed v2 meters/second. When one of them reaches a city, the road ends, so the person has to turn around and start moving in the opposite direction by the same road, keeping the original speed. As a result, the cyclist and the pedestrian are traveling between cities A and B indefinitely. Your task is to calculate the number of times they will meet during the first t seconds. If they meet in exactly t seconds after the initial moment of time, this meeting should also be counted. The only line of input contains four integer numbers: l, v1, v2 and t. All numbers are between 1 and 109, inclusively. Print a single integer — the number of times the cyclist and the pedestrian will meet during the first t seconds. ## Input The only line of input contains four integer numbers: l, v1, v2 and t. All numbers are between 1 and 109, inclusively. ## Output Print a single integer — the number of times the cyclist and the pedestrian will meet during the first t seconds. [samples]

**Definitions** Let $ l, v_1, v_2, t \in \mathbb{Z}^+ $ denote the road length, cyclist speed, pedestrian speed, and total time, respectively. **Constraints** $ 1 \leq l, v_1, v_2, t \leq 10^9 $ **Objective** Compute the number of meetings between the cyclist and pedestrian in the time interval $ [0, t] $, where: - The cyclist starts at position $ 0 $ (city A) moving toward $ l $ (city B) at speed $ v_1 $. - The pedestrian starts at position $ l $ (city B) moving toward $ 0 $ (city A) at speed $ v_2 $. - Upon reaching either end, each reverses direction instantaneously without loss of speed. Model their positions as periodic functions with period $ \frac{2l}{v_1} $ and $ \frac{2l}{v_2} $, respectively. The relative motion is equivalent to a single particle moving with relative speed $ v_1 + v_2 $ on a circle of circumference $ 2l $, where each meeting corresponds to covering a distance of $ 2l $ relative to each other. The number of meetings in time $ t $ is: $$ \left\lfloor \frac{(v_1 + v_2) t}{2l} \right\rfloor $$

API Response (JSON)

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      "content": "Two cities A and B are connected by a straight road that is exactly l meters long. At the initial moment of time a cyclist starts moving from city A to city B at a speed v1 meters/second, and a pedest",
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      "content": "Two cities A and B are connected by a straight road that is exactly l meters long. At the initial moment of time a cyclist starts moving from city A to city B at a speed v1 meters/second, and a pedest...",
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