How Cheenta works to ensure student success?
Explore the Back-Story

Largest Possible Value | PRMO-2019 | Problem 17

Try this beautiful problem from PRMO, 2019 based on Largest Possible Value.

Largest Possible Value | PRMO | Problem-17


Let a, b, c be distinct positive integers such that b + c - a,c + a - b and a + b - c are all perfect squares.
What is the largest possible value of a + b + c smaller than 100?

  • 20
  • 91
  • 13

Key Concepts


Number theory

Perfect square

Integer

Check the Answer


Answer:91

PRMO-2019, Problem 17

Pre College Mathematics

Try with Hints


Let b + c - a = x^2 … (i)
c + a - b = y^2 … (ii)
a + b - c = z^2 … (iii)

Now since a,b, c are distinct positive integers,
Therefore, x, y, z will also be positive integers,
add (i), (ii) and (iii)
a + b + c = x^2 + y^2 + z^2
Now, we need to find largest value of a + b + c or x^2 + y^2 + z^2 less than 100
Now, to get a, b, c all integers x,y, z all must be of same parity, i.e. either all three are even or all three
are odd.

Can you now finish the problem ..........

Let us maximizex^2 + y^2 + z^2, for both cases.
If x, y, zare all even.
Therefore,

b + c - a = 8^2 = 64
c + a - b = 42 = 16
a + b - c = 22 = 4
Which on solving, givea = 10,b = 34, c = 40 and a + b + c = 84
If x, y, z are all odd
\Rightarrow  b + c - a = 92 = 81
c + a - b = 32 = 9
a + b - c = 12 = 1
Which on solving, give a = 5 ,b = 41, c = 45 anda + b + c = 91

Can you finish the problem........

Therefore Maximum value of a + b + c < 100 = 91

Subscribe to Cheenta at Youtube


Try this beautiful problem from PRMO, 2019 based on Largest Possible Value.

Largest Possible Value | PRMO | Problem-17


Let a, b, c be distinct positive integers such that b + c - a,c + a - b and a + b - c are all perfect squares.
What is the largest possible value of a + b + c smaller than 100?

  • 20
  • 91
  • 13

Key Concepts


Number theory

Perfect square

Integer

Check the Answer


Answer:91

PRMO-2019, Problem 17

Pre College Mathematics

Try with Hints


Let b + c - a = x^2 … (i)
c + a - b = y^2 … (ii)
a + b - c = z^2 … (iii)

Now since a,b, c are distinct positive integers,
Therefore, x, y, z will also be positive integers,
add (i), (ii) and (iii)
a + b + c = x^2 + y^2 + z^2
Now, we need to find largest value of a + b + c or x^2 + y^2 + z^2 less than 100
Now, to get a, b, c all integers x,y, z all must be of same parity, i.e. either all three are even or all three
are odd.

Can you now finish the problem ..........

Let us maximizex^2 + y^2 + z^2, for both cases.
If x, y, zare all even.
Therefore,

b + c - a = 8^2 = 64
c + a - b = 42 = 16
a + b - c = 22 = 4
Which on solving, givea = 10,b = 34, c = 40 and a + b + c = 84
If x, y, z are all odd
\Rightarrow  b + c - a = 92 = 81
c + a - b = 32 = 9
a + b - c = 12 = 1
Which on solving, give a = 5 ,b = 41, c = 45 anda + b + c = 91

Can you finish the problem........

Therefore Maximum value of a + b + c < 100 = 91

Subscribe to Cheenta at Youtube


Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy
Cheenta

Cheenta Academy

Aditya Birla Education Academy

Aditya Birla Education Academy

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com
Menu
Trial
Whatsapp
Math Olympiad Program
magic-wandrockethighlight