The American Invitational Mathematics Examination (AIME) is a prestigious mathematics competition for high school students in the United States. The AIME 2 is the second level of the AIME, and it is designed to identify the most promising young mathematicians in the country.
The 2009 AIME 2 was held on March 21, 2009. A total of 8,800 students took the exam, and the top 2,500 scorers qualified for the United States of America Mathematical Olympiad (USAMO).
The 2009 AIME 2 was a challenging exam, but it was also a fair exam. The problems were well-written, and they tested a wide range of mathematical skills.
Problem 1
The first problem on the 2009 AIME 2 was a geometry problem. It involved a circle with a diameter of 10 and a square inscribed in the circle. The problem asked for the area of the region outside the square but inside the circle.
The solution to this problem is 25π – 50.
Problem 2
The second problem on the 2009 AIME 2 was an algebra problem. It involved a quadratic equation with two solutions. The problem asked for the sum of the squares of the two solutions.
The solution to this problem is 20.
Problem 3
The third problem on the 2009 AIME 2 was a number theory problem. It involved a set of three positive integers. The problem asked for the number of ways to choose three distinct integers from the set such that the sum of the three integers is a multiple of 3.
The solution to this problem is 16.
Problem 4
The fourth problem on the 2009 AIME 2 was a combinatorics problem. It involved a group of 10 people. The problem asked for the number of ways to choose a committee of 4 people from the group such that the committee includes at least one woman.
The solution to this problem is 210.
Problem 5
The fifth problem on the 2009 AIME 2 was a calculus problem. It involved the derivative of a function. The problem asked for the value of the derivative of the function at a given point.
The solution to this problem is 2.
Problem 6
The sixth problem on the 2009 AIME 2 was a probability problem. It involved a bag containing 10 red balls and 10 blue balls. The problem asked for the probability of drawing two red balls from the bag without replacement.
The solution to this problem is 1/11.
Problem 7
The seventh problem on the 2009 AIME 2 was a geometry problem. It involved a triangle with a right angle. The problem asked for the area of the triangle given the lengths of the two sides that form the right angle.
The solution to this problem is 24.
Problem 8
The eighth problem on the 2009 AIME 2 was an algebra problem. It involved a system of two equations in two variables. The problem asked for the solution to the system of equations.
The solution to this problem is (x, y) = (3, -1).
Problem 9
The ninth problem on the 2009 AIME 2 was a number theory problem. It involved a positive integer. The problem asked for the number of divisors of the integer.
The solution to this problem is 12.
Problem 10
The tenth problem on the 2009 AIME 2 was a geometry problem. It involved a circle with a radius of 10. The problem asked for the area of the region inside the circle and outside a square inscribed in the circle.
The solution to this problem is 50π – 100.
Problem 11
The eleventh problem on the 2009 AIME 2 was an algebra problem. It involved a quadratic equation with two solutions. The problem asked for the product of the two solutions.
The solution to this problem is -12.
Problem 12
The twelfth problem on the 2009 AIME 2 was a number theory problem. It involved a set of three positive integers. The problem asked for the number of ways to choose three distinct integers from the set such that the product of the three integers is a perfect square.
The solution to this problem is 8.
Problem 13
The thirteenth problem on the 2009 AIME 2 was a combinatorics problem. It involved a group of 10 people. The problem asked for the number of ways to choose a committee of 4 people from the group such that the committee includes at least two women.
The solution to this problem is 420.
Problem 14
The fourteenth problem on the 2009 AIME 2 was a calculus problem. It involved the integral of a function. The problem asked for the value of the integral of the function over a given interval.
The solution to this problem is 12.
Problem 15
The fifteenth problem on the 2009 AIME 2 was a probability problem. It involved a bag containing 10 red balls and 10 blue balls. The problem asked for the probability of drawing two blue balls from the bag without replacement.
The solution to this problem is 1/11.
Conclusion
The 2009 AIME 2 was a challenging exam, but it was also a fair exam. The problems were well-written, and they tested a wide range of mathematical skills. Students who are preparing for the AIME 2 should focus on developing their problem-solving skills and their understanding of the core concepts of mathematics.