H. Cube

Codeforces

IDCF10185H

Time1000ms

Memory256MB

Difficulty—

English · Original

Formal · Original

In geometry, a cube is a three-dimensional solid object bounded by six square faces, with three meeting at each vertex. The image bellow is an example of a cube. You are given the surface area of a cube, and your task is to find the length of that cube's edge. Can you? The first line contains an integer T (1 ≤ T ≤ 1000), in which T is the number of test cases. Each test case consists of a line containing an integer a (1 ≤ a ≤ 6 × 106), giving the surface area of a cube. For each test case, print a single line containing the length of the given cube's edge. *It is guaranteed that all answers are integer numbers. Do not print any floating-point values.* ## Input The first line contains an integer T (1 ≤ T ≤ 1000), in which T is the number of test cases.Each test case consists of a line containing an integer a (1 ≤ a ≤ 6 × 106), giving the surface area of a cube. ## Output For each test case, print a single line containing the length of the given cube's edge.*It is guaranteed that all answers are integer numbers. Do not print any floating-point values.* [samples]

**Definitions** Let $ n \in \mathbb{Z}_{\geq 0} $ be the starting floor. Let $ a \in \mathbb{Z}_{\geq 0} $ be the number of battles. Let $ p \in [0, 100] $ be the win probability percentage; define $ q = \frac{p}{100} \in [0, 1] $ as the probability of winning a single battle. **Constraints** $ 0 \leq n, a \leq 10^9 $, $ 0 \leq p \leq 100 $. **Objective** Let $ X $ be the random variable denoting the final floor after $ a $ battles. Each win increases floor by 1, each loss decreases floor by 1. Let $ W \sim \text{Binomial}(a, q) $ be the number of wins. Then the number of losses is $ a - W $, and: $$ X = n + W - (a - W) = n + 2W - a $$ The expected final floor is: $$ \mathbb{E}[X] = n + 2\mathbb{E}[W] - a = n + 2aq - a = n + a(2q - 1) $$

API Response (JSON)

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