Difference between revisions of "2023 IOQM/Problem 4"
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Find the maximum possible value of <math>x + y</math>. | Find the maximum possible value of <math>x + y</math>. | ||
==Solution1(Diophantine)== | ==Solution1(Diophantine)== | ||
| − | <math>x^{4}=(x-1)(y^{3}-23)-1</math> | + | <math>x^{4}=(x-1)(y^{3}-23)-1</math>, subtracting 1 on both sides we get <math>x^{4}- 1^{4}=(x-1)(y^{3}-23)-2</math> [[factorizing]] the LHS we get |
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| + | <math>(x^{2}+ 1)(x-1)(x+1) =(x-1)(y^{3}-23)-2</math>. Now [[divide]] the [[equation]] by <math>x-1</math> to get <cmath>(x^{2}+ 1)(x+1) = (y^{3}-23)-\frac{2}{x-1}</cmath> Since <math>x</math> and <math>y</math> are [[integers]], this implies <math>x-1</math> [[divides]] 2, so possible values <math>x-1</math> are -1, -2, 1, 2 | ||
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| + | This means <math>x</math>= 0, -1(Rejected as <math>x</math> is a [[positive integer]]), 2, 3. so <math>x</math> =2 or 3. Now checking for each value, we find that when <math>x</math>=2, there is no [[integral]] value of <math>y</math>. When <math>x</math>= 3 <math>y</math>=4 which is the only possible [[integral]] solution. | ||
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| + | So, <math>x+y</math>= 3+ 4 = <math>\boxed{7}</math> | ||
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| + | ~SANSGANKRSNGUPTA(inspired by PJ AND AM sir) | ||
Revision as of 12:46, 24 October 2023
Problem
Let 
be positive integers such that
![\[x^{4}=(x-1)(y^{3}-23)-1\]](http://latex.artofproblemsolving.com/5/0/9/509f87eace1256402c53d3e8ceac7a97e2c01ef0.png)
Find the maximum possible value of 
.
Solution1(Diophantine)
, subtracting 1 on both sides we get 
factorizing the LHS we get
. Now divide the equation by 
to get ![\[(x^{2}+ 1)(x+1) = (y^{3}-23)-\frac{2}{x-1}\]](http://latex.artofproblemsolving.com/6/b/b/6bb107f77bda0217ae41a5237a8a7fce673607b0.png)
Since 
and 
are integers, this implies divides 2, so possible values are -1, -2, 1, 2
This means = 0, -1(Rejected as is a positive integer), 2, 3. so =2 or 3. Now checking for each value, we find that when =2, there is no integral value of . When = 3 =4 which is the only possible integral solution.
So, 
= 3+ 4 = 
~SANSGANKRSNGUPTA(inspired by PJ AND AM sir)
