Difference between revisions of "2021 AIME II Problems/Problem 7"
Arnigam2007 (talk | contribs) (→Solution 2) |
Arnigam2007 (talk | contribs) (→Solution 2) |
||
Line 23: | Line 23: | ||
Solving this quadratic yields that <math>d \in {-5, \frac{3}{2}}</math> | Solving this quadratic yields that <math>d \in {-5, \frac{3}{2}}</math> | ||
+ | |||
+ | Now we just try these 2 cases. | ||
+ | |||
+ | |||
+ | For <math>d = \frac{3}{2}</math> substituting in Equation 1 gives a quadratic in <math>c</math> which has roots <math>c \in \frac{10}{3}, -2</math> | ||
+ | |||
+ | - Arnav Nigam | ||
==See also== | ==See also== | ||
{{AIME box|year=2021|n=II|num-b=6|num-a=8}} | {{AIME box|year=2021|n=II|num-b=6|num-a=8}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 00:02, 23 March 2021
Contents
[hide]Problem
Let and
be real numbers that satisfy the system of equations
There exist relatively prime positive integers
and
such that
Find
.
Solution 1
From the fourth equation we get substitute this into the third equation and you get
. Hence
. Solving we get
or
. From the first and second equation we get
, if
, substituting we get
. If you try solving this you see that this does not have real solutions in
, so
must be
. So
. Since
,
or
. If
, then the system
and
does not give you real solutions. So
. From here you already know
and
, so you can solve for
and
pretty easily and see that
. So the answer is
.
~ math31415926535
Solution 2
can be rewritten as
.
Hence,
Rewriting , we get
.
Substitute
and solving, we get,
call this Equation 1
gives
.
So,
, which implies
or
call this equation 2.
Substituting Eq 2 in Eq 1 gives,
Solving this quadratic yields that
Now we just try these 2 cases.
For substituting in Equation 1 gives a quadratic in
which has roots
- Arnav Nigam
See also
2021 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.