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AMC 10 2019 A

Question 1

What is the value of

Solution

  
  2020-07-09 06:36:10

Question 2

What is the hundreds digit of

Solution

  
  2020-07-09 06:36:10

Question 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

Solution

  
  2020-07-09 06:36:10

Question 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

Solution

  
  2020-07-09 06:36:10

Question 5

What is the greatest number of consecutive integers whose sum is

Solution

  
  2020-07-09 06:36:10

Question 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

Solution

  
  2020-07-09 06:36:10

Question 7

Two lines with slopes and intersect at . What is the area of the triangle enclosed by these two lines and the line

Solution

  
  2020-07-09 06:36:10

Question 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 of line
  • some translation in the direction parallel to line
  • the reflection across line
  • some reflection across a line perpendicular to line

Solution

  
  2020-07-09 06:36:10

Question 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?

Solution

  
  2020-07-09 06:36:10

Question 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 the last tile, how many tiles does the bug visit?

Solution

  
  2020-07-09 06:36:10

Question 11

How many positive integer divisors of are perfect squares or perfect cubes (or both)?

Solution

  
  2020-07-09 06:36:10

Question 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?

Solution

  
  2020-07-09 06:36:10

Question 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

Solution

  
  2020-07-09 06:36:10

Question 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 ?

Solution

  
  2020-07-09 06:36:10

Question 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

Solution

  
  2020-07-09 06:36:10

Question 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 the circles of radius

Solution

  
  2020-07-09 06:36:10

Question 17

A child builds towers using identically shaped cubes of different color. How many different towers with a height cubes can the child build with red cubes, blue cubes, and green cubes? (One cube will be left out.)

Solution

  
  2020-07-09 06:36:10

Question 18

For some positive integer , the repeating base- representation of the (base-ten) fraction is . What is ?

Solution

  
  2020-07-09 06:36:10

Question 19

What is the least possible value of where is a real number?

Solution

  
  2020-07-09 06:36:10

Question 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 each column is odd?

Solution

  
  2020-07-09 06:36:10

Question 21

A sphere with center has radius 6. 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?

Solution

  
  2020-07-09 06:36:10

Question 22

Real numbers between 0 and 1, 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 0 if the second flip is heads and 1 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 and are chosen independently in this manner. What is the probability that ?

Solution

  
  2020-07-09 06:36:10

Question 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?

Solution

  
  2020-07-09 06:36:10

Question 24

Let , , and be the distinct roots of the polynomial . It is given that there exist real numbers , , and such that for all . What is ?

Solution

  
  2020-07-09 06:36:10

Question 25

For how many integers between and , inclusive, is an integer? (Recall that .)

Solution

  
  2020-07-09 06:36:10

Answer Keys


Question 1: C
Question 2: A
Question 3: D
Question 4: B
Question 5: D
Question 6: C
Question 7: C
Question 8: C
Question 9: B
Question 10: C
Question 11: C
Question 12: E
Question 13: D
Question 14: D
Question 15: E
Question 16: A
Question 17: D
Question 18: D
Question 19: B
Question 20: B
Question 21: D
Question 22: B
Question 23: C
Question 24: B
Question 25: D