These are some (but not all) academically competitive Public High Schools that you might want to consider. They are listed by district, because district correlates to general location, which is usually an important factor.
district
code
school
01
M539
New Explorations into
Science, Technology and Math High School
01
M696
Bard High School Early
College
02
M408
Professional Performing Arts High School
02
M411
Baruch College Campus High School
02
M412
N.Y.C. Lab School for Collaborative Studies
02
M414
N.Y.C. Museum School
02
M416
Eleanor Roosevelt High School
02
M418
Millennium High School
02
M519
Talent Unlimited High School
02
M545
High School for Dual Language and Asian Studies
03
M479
Beacon High School
03
M485
Fiorello H. LaGuardia High School of Music & Art and
Performing Arts
03
M541
Manhattan / Hunter Science High School
04
M435
Manhattan Center for Science and Mathematics
05
M362
Columbia Secondary School
05
M692
High School for Mathematics, Science and Engineering at City
College
10
X696
High School of American Studies at Lehman College
13
K595
Bedford Academy High School
13
K670
Benjamin Banneker Academy
14
K449
Brooklyn Latin School, The
15
K684
Millennium Brooklyn HS
17
K543
Science, Technology and Research Early College High School at
Erasmus
17
K590
Medgar Evers College Preparatory School
22
K405
Midwood High School
22
K535
Leon M. Goldstein High School for the Sciences
22
K555
Brooklyn College Academy
24
Q299
Bard High School Early College Queens
24
Q610
Aviation Career & Technical Education High School 3.18
25
Q252
Queens School of Inquiry, The
27
Q323
Scholars' Academy
27
Q650
High School for Construction Trades, Engineering and
Architecture
28
Q284
York Early College Academy 3.20
28
Q680
Queens Gateway to Health Sciences Secondary School
.