ΒΆ 2019 AMC 10A Problems and Solutions
Problem Set Workbook
Access the downloadable workbook for 2019 AMC 10A problems here.
Discussion Forum
Engage in discussion about the 2019 AMC 10A math contest by visiting Random Math AMC 10A 2019 Forum
Individual Problems and Solutions
For problems and detailed solutions to each of the 2019 AMC 10A problems, please refer below:
Problem 1: What is the value of
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Problem 2: What is the hundreds digit of ?
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Problem 3: Ana and Bonita were born on the same date in different years, years apart. Last year Ana was times as old as Bonita. This year Ana's age is the square of Bonita's age. What is ?
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Problem 4: A box contains red balls, green balls, yellow balls, blue balls, white balls, and black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least balls of a single color will be drawn?
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Problem 5: What is the greatest number of consecutive integers whose sum is ?
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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?
. a square
. a rectangle that is not a square
. a rhombus that is not a square
. a parallelogram that is not a rectangle or a rhombus
. an isosceles trapezoid that is not a parallelogram
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Problem 7: Two lines with slopes and intersect at . What is the area of the triangle enclosed by these two lines and the line ?
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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?
. some rotation around a point on line
. some translation in the direction parallel to line
. the reflection across line
. some reflection across a line perpendicular to line
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Problem 9: What is the greatest three-digit positive integer for which the sum of the first positive integers is a divisor of the product of the first positive integers?
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Problem 10: A rectangular floor that is feet wide and feet long is tiled with one-foot square tiles. A bug walks from one corner to the opposite corner in a straight line. Including the first and last tile, how many tiles does the bug visit?
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Problem 11: How many positive integer divisors of are perfect squares or perfect cubes (or both)?
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Problem 12: Melanie computes the mean , the median , and the modes of the values that are the dates in the months of . Thus her data consists of , , . . . , , , , and . Let be the median of the modes. Which of the following statements is true?
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Problem 13: Let be an isosceles triangle with and Construct the circle with diameter , and let and be the other intersection points of the circle with the sides and , respectively. Let be the intersection of the diagonals of the quadrilateral . What is the degree measure of ?
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Problem 14: For a set of four distinct lines in a plane, there are exactly distinct points that lie on two or more of the lines. What is the sum of all possible values of ?
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Problem 15: A sequence of numbers is defined recursively by , and
for all . Then can be written as , where and are relatively prime positive integers. What is ?
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Problem 16: The figure below shows circles of radius within a larger circle. All the intersections occur at points of tangency. What is the area of the region, shaded in the figure, inside the larger circle but outside all of the circles of radius ?
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Problem 17: A child builds towers using identically shaped cubes of different colors. How many different towers with a height of cubes can the child build with rod cubes, blue cubes, and green cubes? (One cube will be left out.)
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Problem 18: For some positive integer , the repeating bas- representation of the (base-ten) fraction is . What is ?
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Problem 19: What is the least possible value of
where is a real number?
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Problem 20: The numbers are randomly placed into the squares of a grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and column is odd.
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Problem 21: A sphere with center has radius . A triangle with sides of length , and is situated in space so that each of its sides is tangent to the sphere. What is the distance between and the plane determined by the triangle?
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Problem 22: Real numbers between and , inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is if the second flip is heads, and if the second flip is tails. On the other hand, if the first coin flip is tails, then the number is chosen uniformly at random from the closed interval . Two random numbers are chosen independently in this manner. What is the probability that ?
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Problem 23: Travis has to babysit the terrible Thompson triplets. Knowing that they love big numbers, Travis devises a counting game for them. First Tadd will say the number , then Todd must say the next two numbers ( and ), then Tucker must say the next three numbers , then Tadd must say the next four numbers , and the process continues to rotate through the three children in order, each saying one more number than the previous child did, until the number is reached. What is the th number said by Tadd?
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Problem 24: Let , and be the distinct roots of the polynomial . There exist real numbers , and such that
for all real numbers with . What is ?
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Problem 25: For how many integers between and , inclusive, is
an integer? (Recall that .)
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The problems on this page are the property of the MAA's American Mathematics Competitions