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2010 AMC 12A Problems and Solution
Problem 1
What is
?
Solution
.
Problem 2
A ferry boat shuttles tourists to an island every hour starting at 10 AM until its last trip, which starts at 3 PM. One day the boat captain notes that on the 10 AM trip there were 100 tourists on the ferry boat, and that on each successive trip, the number of tourists was 1 fewer than on the previous trip. How many tourists did the ferry take to the island that day?
Solution
It is easy to see that the ferry boat takes trips total. The total number of people taken to the island is
Problem 3
Rectangle Square
, pictured below, shares shares
of its area with square
. What is
?
.
of its area with rectangle
Solution Solution 1
Let
, let
, and let
.
Solution 2
The answer does not change if we shift horizontal lines to divide
to coincide with
, and add new
into five equal parts:
This helps us to see that
.
and
, where
. Hence
Problem 4
If
, then which of the following must be positive?
Solution
is negative, so we can just place a negative value into each expression and find the one that is positive. Suppose we use
.
Obviously only
is positive.
Problem 5
Halfway through a 100-shot archery tournament, Chelsea leads by 50 points. For each shot a bullseye scores 10 points, with other possible scores being 8, 4, 2, and 0 points. Chelsea always