2019年 AMC10 A卷
2019 AMC 10A Problems
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What is the value of ?
(A) 0 (B) 1 (C) 2 (D) 3 (E) 4
Problem 2
What is the hundreds digit of (20! - 15!) ?
(A) 0 (B) 1 (C) 2 (D) 4 (E) 5
Problem 3
Ana and Bonita were born on the same date in different years, n years apart. Last year Ana was 5 times as old as Bonita. This year Ana's age is the square of Bonita's age. What is n?
(A) 3 (B) 5 (C) 9 (D) 12 (E) 15
Problem 4
A box contains 28 red balls, 20 green balls, yellow balls, 13 blue balls, 11 white balls, and 9 black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least 15 balls of a single color will be drawn
(A) 75 (B) 76 (C) 79 (D) 84 (E) 91
Problem 5
What is the greatest number of consecutive integers whose sum is 45 ?
(A) 9 (B) 25 (C) 45 (D) 90 (E) 120
Problem 6
For how many of the following types of quadrilaterals does there exist a point in the plane of the quadrilateral that is equidistant from all four vertices of the quadrilateral?
n a square
n a rectangle that is not a square
n a rhombus that is not a square
n a parallelogram that is not a rectangle or a rhombus
n an isosceles trapezoid that is not a parallelogram
(A) 1 (B) 2 (C) 3 (D) 4 (E) 5
Problem 7
Two lines with slopes and 2 intersect at (2, 2). What is the area of the triangle enclosed by these two lines and the line x +y = 10 ?
Problem 8
The figure below shows line with a regular, infinite, recurring pattern of squares and line segments.
How many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity
transformation, will transform this figure into itself?
n some rotation around a point of line
n some translation in the direction parallel to line
n the reflection across line
n some reflection across a line perpendicular to line
(A) 0 (B) 1 (C) 2 (D) 3 (E) 4
Problem 9
What is the greatest three-digit positive integer n for which the sum of the first n positive integers is not a divisor of the product of the first n positive integers?
(A) 995 (B) 996 (C) 997 (D) 998 (E) 999
Problem 10
A rectangular floor that is 10 feet wide and 17 feet long is tiled with 170 one-foot square tiles. A bug walks from one corner to the opposite corner in a straight line. Including the first and the last tile, how many tiles does the bug visit?
(A) 17 (B) 25 (C) 26 (D) 27 (E) 28
Problem 11
How many positive integer divisors of 2019 are perfect squares or perfect cubes (or both)?
(A) 32 (B) 36 (C) 37 (D) 39 (E) 41
Problem 12
Melanie computes the mean μ, the median M, and the modes of the 365 values that are the dates in the months of 2019. Thus her data consist of 12 1s, 12 2s, ..., 12 28s, 11 29s, 11 30s, and 7 31s. Let d be the median of the modes. Which of the following statements is true?
(A) μ < d < M (B) M < d < μ (C) d = M = μ
(A) 90 (B) 100 (C) 105 (D) 110 (E) 120
(A) 14 (B) 16 (C) 18 (D) 19 (E) 21
A sequence of numbers is defined recursively by
(A) 2020 (B) 4039 (C) 6057 (D) 6061 (E) 8078
(A) 24 (B) 288 (C) 312 (D) 1,260 (E) 40,320
(A) 13 (B) 14 (C) 15 (D) 16 (E) 17
What is the least possible value of (x + 1)(x + 2)(x + 3)(x + 4) + 2019 where x is a real number?
(A) 2017 (B) 2018 (C) 2019 (D) 2020 (E) 2021
and the plane determined by the triangle?
and y are chosen independently in this manner. What is the probability that |x - y| > ?
(A) 5743 (B) 5885 (C) 5979 (D) 6001 (E) 6011
(A) 243 (B) 244 (C) 245 (D) 246 (E) 247
For how many integers n between 1 and 50, inclusive, is an integer ? (Recall that 0! = 1.)
(A) 31 (B) 32 (C) 33 (D) 34 (E) 35