2023 AIME I Problems/Problem 15
Problem 15
Find the largest prime number for which there exists a complex number
satisfying
- the real and imaginary parts of
are integers;
, and
- there exists a triangle with side lengths
, the real part of
, and the imaginary part of
.
Answer: 349
Suppose ; notice that
, so by De Moivre’s theorem
and
. Now just try pairs
going down from
, writing down the value of
on the right; and eventually we arrive at
the first time
is prime. Therefore,
.
Solution
Denote . Thus,
.
Thus,
Because ,
,
are three sides of a triangle, we have
and
.
Thus,
Because ,
,
are three sides of a triangle, we have the following triangle inequalities:
We notice that , and
,
, and
form a right triangle. Thus,
.
Because
,
.
Therefore, (3) holds.
Conditions (4) and (5) can be written in the joint form as
We have
and
.
Thus, (5) can be written as
Therefore, we need to jointly solve (1), (2), (6).
From (1) and (2), we have either , or
.
In (6), by symmetry, without loss of generality, we assume
.
Thus, (1) and (2) are reduced to
Let . Plugging this into (6), we get
Because is a prime,
and
are relatively prime.
Therefore, we can use (7), (8), , and
and
are relatively prime to solve the problem.
To facilitate efficient search, we apply the following criteria:
\begin{enumerate}
\item To satisfy (7) and , we have
.
In the outer layer, we search for
in a decreasing order.
In the inner layer, for each given
, we search for
.
\item Given
, we search for
in the range
.
\item We can prove that for
, there is no feasible
.
The proof is as follows.
For , to satisfy
, we have
.
Thus,
.
Thus, the R.H.S. of (8) has the following upper bound
Hence, to satisfy (8), a necessary condition is
However, this cannot be satisfied for .
Therefore, there is no feasible solution for
.
Therefore, we only need to consider
.
\item We eliminate that are not relatively prime to
.
\item We use the following criteria to quickly eliminate that make
a composite number.
\begin{enumerate}
\item For
, we eliminate
satisfying
.
\item For
(resp.
), we eliminate
satisfying
(resp.
).
\end{enumerate}
\item For the remaining , check whether (8) and the condition that
is prime are both satisfied.
The first feasible solution is and
.
Thus,
.
\item For the remaining search, given , we only search for
.
\end{enumerate}
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Following the above search criteria, we find the final answer as and
.
Thus, the largest prime
is
.
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)