Solution Problem 20
A binary operation has the properties that that
for all nonzero real numbers
and
and (Here the dot represents the usual
can be
multiplication operation.) The solution to the equation written as
where and are relatively prime positive integers. What is
Solution Problem 21
A quadrilateral is inscribed in a circle of radius have length
What is the length of its fourth side?
Solution Problem 22
How many ordered triples satisfy
of positive integers
and
Solution Problem 23
Three numbers in the interval
?
Three of the sides of this quadrilateral
are chosen independently and at random. What is the
probability that the chosen numbers are the side lengths of a triangle with positive area?
Solution Problem 24
There is a smallest positive real number such that there exists a positive real number such that all the roots of the polynomial are real. In fact, for this value of the value of is unique. What is the value of
Solution Problem 25
Let be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows: Bernardo starts by writing the smallest perfect square with digits. Every time Bernardo writes a number, Silvia erases the last digits of it. Bernardo then writes the next perfect square, Silvia erases the last digits of it, and this process continues until the last two numbers that remain on the board differ by at least 2. Let
be the smallest
positive integer not written on the board. For example, if , then the numbers that Bernardo writes are , and the numbers showing on the board after Silvia erases are of
and , and thus
?
. What is the sum of the digits
2016 AMC 12A Answer Key
1 B 2 C 3 B 4 D 5 E 6 D 7 D 8 D 9 E 10 B 11 E 12 C 13 A 14 C 15 D 16 D 17 B 18 D 19 B 20 A 21 E 22 A 23 C 24 B 25 E