Try this beautiful problem from Algebra based on Integer
Integers a, b, c satisfy \(a + b – c = 1\) and \(a^2 + b^2 – c^2 = –1\). What is the sum of all possible values of
\(a^2 + b^2 + c^2\) ?
Algebra
quadratic equation
Factorization
Answer:$18$
PRMO-2018, Problem 6
Pre College Mathematics
Use \((a + b – 1 )= c\) this relation
Can you now finish the problem ..........
\((a + b – 1 )^2= c^2\) (squaring both sides.......)
Can you finish the problem........
Given that a,b,c are integer satisfy \(a + b – c = 1\) and \(a^2 + b^2 – c^2 = –1\).
Now
\((a + b – c )= 1\)
\(\Rightarrow (a + b – 1 )= c\)
\(\Rightarrow (a + b – 1 )^2= c^2\) (squarring both sides.......)
\(\Rightarrow (a^2 + b^2 +1^2+2ab-2a-2b)= c^2\)
\(\Rightarrow (a^2 + b^2 –c^2+1+2ab)= 2(a+b)\)
\(\Rightarrow (-1)+1+2ab= 2(a+b)\) ( as \(a^2 + b^2 – c^2 = –1\).
\(\Rightarrow ab=a+b\)
\(\Rightarrow (a-1)(b-1)=1\)
Therefore the possibile cases are \((a-1)=\pm 1\) and \((b-1)=\pm 1\)
Therefore \( a=1\),\(b=1\) or \(a=0\),\(b=0\)
From the equation \((a + b – c )= 1\) ,C =3 (as a=b=2) and C=-1(as a=b=0)
Therefore \((a^2 +b^2 +c^2)=4+4+9=17\) and \((a^2 +b^2 +c^2)=0+0+1=1\)
sum of all possible values of \(a^2 + b^2 + c^2=17+1=18\)
AMC - AIME Boot Camp... for brilliant students.
Use our exclusive one-on-one plus group class system to prepare for Math Olympiad
Publishing original scientific research in a peer-reviewed journal is an achievement many scholars work toward for years. Achieving this feat early in one's academic journey requires exceptional intellectual rigor, creativity, and persistence. Aaditya Punatar and Muskan Sadana have reached this remarkable milestone, successfully publishing their collaborative research paper in the prestigious Journal of Emerging Investigators […]
Her project on quantum-accelerated Steiner trees for 15-minute cities earned a place among the finalists at the world's largest pre-college STEM competition. Prishaa Shrimali of Marymount School of New York, was selected as a Grand Finalist at the 2026 Regeneron International Science and Engineering Fair (ISEF) in Phoenix, Arizona. Her project was Optimizing Urban Accessibility: […]
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's M.Stat Entrance. They ranked within the first 50 in the entire country in these entrances. I.S.I. M.Stat Entrance
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's B.Stat Entrance and Chennai Mathematical Institute's B.Sc. Math Entrance. They ranked within the first 200 in the entire country in these entrances. Most of these students attended the problem solving workshops regularly, which happen 5 days every week. CMI B.Sc. Math Entrance […]