The query that identifies the rows might look like this:
select t.*
from table t
where not exists (select 1
from table t2
where t2.ParticipantGroup = t.ParticipantGroup and
t2.StageNo = t.StageNo and
t2.time < t.time
);
The question is then how you turn this into an update. For MySQL, you would do:
update table StageResults sr join
(select t.*
from table t
where not exists (select 1
from table t2
where t2.ParticipantGroup = t.ParticipantGroup and
t2.StageNo = t.StageNo and
t2.time < t.time
)
) toupdate
on toupdate.ParticpantNumber = sr.ParticpantNumber
set sr.points = sr.points + 1000;
The syntax for SQL Server would be a bit different, but your question is tagged MySQL.
EDIT:
For SQL Server:
with toupdate as (select t.*
from table t
where not exists (select 1
from table t2
where t2.ParticipantGroup = t.ParticipantGroup and
t2.StageNo = t.StageNo and
t2.time < t.time
)
)
update toupdate
set points = points + 1000;