H. Annuity Payment Scheme

Codeforces

IDCF10050H

Time1000ms

Memory64MB

Difficulty—

English · Original

Formal · Original

At the peak of the Global Economic Crisis BerBank offered an unprecedented credit program. The offering was so attractive that Vitaly decided to try it. He took a loan of s burles for m months with the interest rate of p percent. Vitaly has to follow the scheme of annuity payments, meaning that he should make fixed monthly payments — x burles per month. Obviously, at the end of the period he will pay m·x burles to the bank in total. Each of the monthly payments is divided by BerBank into two parts as follows: For example, if s = 100, m = 2, p = 50 then x = 90. For the first month a1 = s'·p / 100 = s·p / 100 = 50 and b1 = 90 - 50 = 40. For the second month a2 = (100 - 40)·50 / 100 = 30, so b2 = 90 - 30 = 60 and the debt is paid off completely. Your task is to help Vitaly and write a program that computes x given the values of s, m and p. The single line of the input contains three integers s, m and p (1 ≤ s ≤ 106, 1 ≤ m ≤ 120, 0 ≤ p ≤ 100). Output the single value of monthly payment x in burles. An absolute error of up to 10 - 5 is allowed. ## Input The single line of the input contains three integers s, m and p (1 ≤ s ≤ 106, 1 ≤ m ≤ 120, 0 ≤ p ≤ 100). ## Output Output the single value of monthly payment x in burles. An absolute error of up to 10 - 5 is allowed. [samples]

**Definitions** Let $ s \in \mathbb{R}^+ $ be the initial loan amount (in burles). Let $ m \in \mathbb{Z}^+ $ be the number of months. Let $ p \in \mathbb{R} $ be the monthly interest rate (in percent). Let $ x \in \mathbb{R}^+ $ be the fixed monthly payment. Let $ d_0 = s $, and for $ i = 1, \dots, m $, define the debt after $ i $ payments recursively: $$ d_i = d_{i-1} \left(1 + \frac{p}{100}\right) - x $$ **Constraints** 1. $ 1 \leq s \leq 10^6 $ 2. $ 1 \leq m \leq 120 $ 3. $ 0 \leq p \leq 100 $ **Objective** Find $ x $ such that $ d_m = 0 $. That is, solve for $ x $ in: $$ s \left(1 + \frac{p}{100}\right)^m = x \sum_{k=0}^{m-1} \left(1 + \frac{p}{100}\right)^k $$ Equivalently: $$ x = s \cdot \frac{r (1 + r)^m}{(1 + r)^m - 1}, \quad \text{where } r = \frac{p}{100} $$

API Response (JSON)

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