Tags: misc math
Rating:
# Category: Misc/Math
## Problem:
**Description:**
```
95% of ppl can't solve this equation!!!
If ur smart enough, send ur solution to https://math-killer.ctf.insecurity-insa.fr/solve?a=...&b=...&c=...
```
## Solution:
According to the image, the equation to be solved:
```
a/(b+c) + b/(a+c) + c/(a+b) = 6
```
By submit some input like a=1&b=2&c=3.0, I figure out that a, b, c must be integers (both positives and negatives accepted):
Let's solve the equation. I tried to set a = 0:
```
b/c + c/b = 6
<=> b = (3+2*sqrt(2))*c
or b = (3-2*sqrt(2))*c
```
=> Cannot get integer roots.
I also tried to solve the origin equation with a unknown and b, c coefficients, and got a very complex cubic equation :( => failed
Let's try to transform the equation. Let x = b+c, y = a+c, z = a+b => x, y, z are non-zero integers.
```
=> a = (y+z-x)/2, b = (x+z-y)/2, c = (x+y-z)/2 (=> x, y, z must be all odd or even)
a/(b+c) + b/(a+c) + c/(a+b) = 6
<=> (y+z-x)/(2*x) + (x+z-y)/(2*y) + (x+y-z)/(2*z) = 6 (*)
```
Let z unknown and x, y coefficients, we transform (*) into quaratic equation, and get:
```
Delta = (x^2+y^2-15*x*y)^2 - 4*(x+y)*(b^2*c+c^2*b)
z = (15*x*y - x^2 - y^2 + sqrt(Delta))/2/(x+y)
or z = (15*x*y - x^2 - y^2 - sqrt(Delta))/2/(x+y)
```
Then, to find x, y such that z is integer, I used excel to get a table of z values in some x, y ranges. To learn how to get the table, watch https://www.youtube.com/watch?v=jUoo_7KQfO0.
From the table, we get some integer roots. With (x, y, z) = (6, 4, 30), we get (a, b, c) = (14, 16, -10)
Submit and we get the flag:
**INSA{try_positive_solutions_now}**
