Select Page

Understand the problem

Prove that the number of ordered triples $(x, y, z)$  in the set of residues of $p$ such that $(x+y+z)^2 \equiv axyz \mod{p}$, where $gcd(a, p) = 1$ and $p$ is prime is $p^2 + 1$.
Number Theory
Medium
Suggested Book
Elementary Number Theory by David Burton

Do you really need a hint? Try it first!

First, assume that at least one of $x,y,z$ is zero. Show that there are $3p-2$ solutions in this case. Next we need to consider the case where none of them is zero.

Let us denote the residue class of $p$ by $\mathbb{Z}_p$. Show that, there exist non-zero $b,c$ in $\mathbb{Z}_p$ such that $y\equiv bx\;\text{mod}\; p$ and $z\equiv cy\;\text{mod}\; p$.

From hint 2, show that $(1+b+bc)^2\equiv ab^2cx\;\text{mod}\;p$. This means that $x\equiv a^{-1}c^{-1}(1+b^{-1}+c)^2\;\text{mod}\;p$ (you need to convince yourself that the inverses exist).  Now it becomes a matter of simply choosing $b,c$.

Note that, $1+b^{-1}+c$ cannot be zero. Given any $b\neq p-1$, there exists exactly one non-zero $c$ such that $1+b^{-1}+c$ is 0 modulo $p$. Hence, in this case there are $(p-2)^2$ choices. For $b=p-1$, this special $c$ is actually 0. Hence in this case there are $p-1$ choices. Thus, the total number of choices is $(p-2)^2+(p-1)=p^2-4p+4+(p-1)=p^2-3p+3$. Adding to this the $3p-1$ cases considered in hint 1, we get $p^2+1$ as the answer.

Connected Program at Cheenta

Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year. Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.

Similar Problems

Coin Toss Problem | AMC 10A, 2017| Problem No 18

Try this beautiful Problem on Probability from AMC 10A, 2017. Problem-18, You may use sequential hints to solve the problem.

GCF & Rectangle | AMC 10A, 2016| Problem No 19

Try this beautiful Problem on Geometry on Rectangle from AMC 10A, 2010. Problem-19. You may use sequential hints to solve the problem.

Fly trapped inside cubical box | AMC 10A, 2010| Problem No 20

Try this beautiful Problem on Geometry on cube from AMC 10A, 2010. Problem-20. You may use sequential hints to solve the problem.

Measure of angle | AMC 10A, 2019| Problem No 13

Try this beautiful Problem on Geometry from AMC 10A, 2019.Problem-13. You may use sequential hints to solve the problem.

Sum of Sides of Triangle | PRMO-2018 | Problem No-17

Try this beautiful Problem on Geometry from PRMO -2018.You may use sequential hints to solve the problem.

Recursion Problem | AMC 10A, 2019| Problem No 15

Try this beautiful Problem on Algebra from AMC 10A, 2019. Problem-15, You may use sequential hints to solve the problem.

Roots of Polynomial | AMC 10A, 2019| Problem No 24

Try this beautiful Problem on Algebra from AMC 10A, 2019. Problem-24, You may use sequential hints to solve the problem.

Set of Fractions | AMC 10A, 2015| Problem No 15

Try this beautiful Problem on Algebra from AMC 10A, 2015. Problem-15. You may use sequential hints to solve the problem.

Indian Olympiad Qualifier in Mathematics – IOQM

Due to COVID 19 Pandemic, the Maths Olympiad stages in India has changed. Here is the announcement published by HBCSE: Important Announcement [Updated:14-Sept-2020]The national Olympiad programme in mathematics culminating in the International Mathematical Olympiad...

Positive Integers and Quadrilateral | AMC 10A 2015 | Sum 24

Try this beautiful Problem on Rectangle and triangle from AMC 10A, 2015. Problem-24. You may use sequential hints to solve the problem.