2016 AMC12 A
Problem 1
What is the value of
Solution
Problem 2
For what value of does
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Problem 3
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The remainder can be defined for all real numbers 吉林省
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and with bywhere denotes the greatest integer less than or equal to . What is the value of Solution
Problem 4
The mean, median, and mode of the data values are all equal to . What is the value of
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Problem 5
Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two prime numbers (for example, ). So far, no one has been able to prove that the conjecture is true, and no one has found a counterexample to show that the conjecture is false. What would a counterexample consist of?
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A triangular array of coins has coin in the first row, coins in the second row, coins in the third row, and so on up to coins in the th row. What is the sum of the digits of ?
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Problem 7
Which of these describes the graph of ?
Solution
Problem 8
What is the area of the shaded region of the given 烟台大学研究生院 rectangle? Solution
Problem 9
The five small shaded squares inside this unit square are congruent and have disjoint interiors. The midpoint of each side of the middle square coincides with one of the vertices of the other four small squares as shown. The common side length is , where and are positive integers. What is ?
Solution
Problem 10
Five friends sat in a movie theater in a row containing seats, numbered to from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?
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Problem 11
Each of the students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are students who cannot sing, students who cannot dance, and students who cannot act. How many students have two of these talents?
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Problem 12
In , , , and . Point lies on , and bisects . Point lies on , and 广东自驾旅游景点大全bisects . The bisectors intersect at . What is the ratio :
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Problem 13
碧峰峡动物园Let be a positive multiple of . One red ball and green balls are arranged in a line in random order. Let be the probability that at least of the green balls are on the same side of the red ball. Observe that and that approaches as grows large. What is the sum of the digits of the least value of such that
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