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=...


![](chall.png)

## 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):

![](Condition1.PNG)

![](Condition2.PNG)

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.

![](table1.PNG)

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}**

![](Flag.PNG)

Original writeup (https://github.com/ManhNDd/CTFwriteup/tree/master/INShAck-2018/Math%20killer%20-%20easy).