A number theory problem by Ankit Kumar Jain

Number Theory Level 3

If $(x , y) \in \mathbb Z^+$ such that $(x + y)^2 + 3x + y = 1994$ , evaluate $y$ .

This is part of the set My Problems and THRILLER

The answer is 37.

3 solutions

Ankit Kumar Jain
Feb 22, 2017

$(x + y)^2 + 3x + y = 1994$

$\Rightarrow (x + y)(x + y + 1) + 2x = 1994$

$x + y > 44 \Rightarrow (x + y)(x + y + 1) + 2x > 1994$ .

$x + y < 44 \Rightarrow (x + y)(x + y + 1) \leq 1892 \Rightarrow 2x > 104 \Rightarrow x > 52 \Rightarrow (x + y) > 52$ CONTRADICTION!!

$\therefore x + y = 44 \Rightarrow (x , y) = (7 , 37)$

Abhijit Dixit
Feb 22, 2017

If both $x,y$ are positive then $3x + y > x + y ........(1)$

Thus problem revolves around expressing $1994$ as sum of a perfect square + positive no $. .........(2)$

this can be done in many ways satisfying both $1$ and $2$

$44^2 + 58 = 1994$

$43^2+ 145=1994$

$42^2+230=1994$

and so on.....

but except $44^2 + 58 = 1994$ all lead to negative values of y.

thus the only possibility is

$3x+y = 58$

$x+ y = 44$

Nice intuition!!!

A Former Brilliant Member - 4 years, 3 months ago

Thanks :) your way was more general.

ABHIJIT DIXIT - 4 years, 3 months ago

@ABHIJIT DIXIT I guess you also need to add the same thing in your solution as @A E .See comments under his solution.

Ankit Kumar Jain - 4 years, 3 months ago

my mistake I took that thing for granted... really nice problem @Ankit Kumar Jain !

ABHIJIT DIXIT - 4 years, 3 months ago

Thanks @ABHIJIT DIXIT :)

Ankit Kumar Jain - 4 years, 3 months ago

@Ankit Kumar Jain – @ABHIJIT DIXIT You can try my other problems too.

I have given a link to my set in the question itself and reshare if you like it

And there is another set named THRILLER which you can find in my prfile.

Ankit Kumar Jain - 4 years, 3 months ago

A Former Brilliant Member
Feb 27, 2017

Simplify the given expression and solve it as a quadratic in y.Now make discriminant a perfect square to obtain x=7.Now put this value the the given expression to get,y= -52& 37.As x&y are positive ,therefore (x,y)=(7,37). Easy question :p

@A E By making discriminant a perfect square , you get $-8x + 7977 = k^2$ . How do you solve this to get $7$ ?

Ankit Kumar Jain - 4 years, 3 months ago

Nearest square number to 7977 is 7921 ie (89)^2. Therefore ,if k=89 ,x comes out as 7.

A Former Brilliant Member - 4 years, 3 months ago

What about $x = 233 , k = 88$ . This is correct though it will contradict the conditions of the main equation. So I think you should mention that in your solution that " For $k \leq88 \Rightarrow x \geq 233$ . But $x + y \leq 44 \Rightarrow x \leq 43$ . This leads to contradictory results and hence $\boxed{k = 89}$ is the only possibility."

Ankit Kumar Jain - 4 years, 3 months ago

@Ankit Kumar Jain – Ya...thanks

A Former Brilliant Member - 4 years, 3 months ago

A number theory problem by Ankit Kumar Jain

The answer is 37.

3 solutions

0 pending reports