2022 AMC 10A P12&P13
P12
On
Halloween 
children walked into
the principal's office asking for candy. They can be classified into three
types: Some always lie; some always tell the truth; and some alternately lie
and tell the truth. The alternaters arbitrarily choose their first response,
either a lie or the truth, but each subsequent statement has the opposite truth
value from its predecessor. The principal asked everyone the same three
questions in this order.
"Are
you a truth-teller?" The principal gave a piece of candy to each of
the 
children who answered
yes.
"Are
you an alternater?" The principal gave a piece of candy to each of
the 
children who answered
yes.
"Are
you a liar?" The principal gave a piece of candy to each of the 
children who answered
yes.
How
many pieces of candy in all did the principal give to the children who always
tell the truth?
P13
Let 
be a
scalene triangle. Point 
lies on 
so
that 
bisects 
The line
through 
perpendicular to intersects
the line through 
parallel to at
point 
Suppose 
and 
What is 
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