Today, I came across an equation in practice mock-test of my coaching institute, aiming for engineering entrance examination (The course for the test wasn't topic-specific, it was a test of complete high school mathematics). It was having four variables and only one equation. While analyzing my test paper, this is the only problem I (and my friends too) couldn't figure out even after giving this problem several hours. So I came here for some help.
Question : Solve for $a,b,c,d \in \Bbb R$, given that $$a^2+b^2+c^2+d^2-ab-bc-cd-d+\frac 25 =0$$
Since only one equation is given, there must be involvement of making of perfect squares, such that they all add up to $0$. Thus, resulting in few more equations. But how to?
I tried a lot of things, such as making $(a-b)^2 $ by adding the missing terms and subtracting again, but got no success.
Thanks!
Best Answer
Let $F(a,b,c,d) = a^2+b^2+c^2+d^2-ab-bc-cd-d+\frac25$.
With help of a CAS, one can verify
$$\begin{align} F\left(\frac15+p,\frac25+q,\frac35+r,\frac45+s\right) &= p^2 - pq + q^2 - qr + r^2 -rs + s^2\\ &= \frac12\left(p^2 + (p-q)^2 + (q-r)^2 + (r-s)^2 + s^2\right)\end{align}$$
If one set $(a,b,c,d)$ to $\left(\frac15+p,\frac25+q,\frac35+r,\frac45+s\right)$, one find $$\begin{align} F(a,b,c,d) = 0 &\iff p = p-q = q-r = r-s = s = 0\\ &\iff p = q = r = s = 0 \end{align} $$ This implies the equation at hand has a unique solution: $$(a,b,c,d) = \left(\frac15,\frac25,\frac35,\frac45\right)$$
Update
About the question how I come up with this. I first write $F(a,b,c,d)$ as
$$\begin{align} F(a,b,c,d) &= a^2 + b^2 + c^2 + d^2 - ab - bc - cd - da + d(a-1) + \frac25\\ &= \frac12((a-b)^2+(b-c)^2+(c-d)^2+(d-a)^2) + d(a-1) + \frac25 \end{align}\tag{*1} $$ To simplify the term $d(a-1)$, I introduce $\lambda, \mu$ such that
$$\begin{cases} d &= \frac12 + \lambda + \mu\\ a &= \frac12 + \lambda - \mu \end{cases} \quad\implies\quad d(a-1) = \lambda^2 - \left(\frac12+\mu\right)^2 $$ Now $d-a = 2\mu$ and $(a-b)^2 + (b-c)^2 + (c-d)^2 \ge 3\left(\frac{d-a}{3}\right)^2 = \frac43 \mu^2$.
If one substitute this back into $(*1)$, one find
$$F(a,b,c,d) \ge \frac83\mu^2 + \lambda^2 - (\frac12 + \mu)^2 + \frac25 = \lambda^2 + \frac53\left(\mu - \frac{3}{10}\right)^2$$
In order for $F(a,b,c,d) = 0$, we need
$$\lambda = 0,\quad\mu = \frac{3}{10} \quad\text{ and }\quad(a-b)^2 + (b-c)^2 + (c-d)^2 = \frac13(d-a)^2$$ The last condition forces $d-c = c-b = b-a = \frac13(d-a)$ and leads to the solution $(a,b,c,d) = \left(\frac15,\frac25,\frac35,\frac45\right)$. This is a little bit sloppy to describe, so I look at expansion of $F(a,b,c,d)$ near the solution and obtain a simpler description of $F$ in terms of $p,q,r,s$.