2019 AMC 12A Problems
2019 AMC 12A (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
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Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 See also
Problem 1
The area of a pizza with radius inches is percent larger than the area of a pizza with radius inches. What is the integer closest to ?
Problem 2
Suppose is of . What percent of is ?
Problem 3
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?
Problem 4
What is the greatest number of consecutive integers whose sum is ?
Problem 5
Two lines with slopes and intersect at . What is the area of the triangle enclosed by these two lines and the line ?
Problem 6
The figure below shows line with a regular, infinite, recurring pattern of squares and line segments.
size(300); defaultpen(linewidth(0.8)); real r = 0.35; path P = (0,0)--(0,1)--(1,1)--(1,0), Q = (1,1)--(1+r,1+r); path Pp = (0,0)--(0,-1)--(1,-1)--(1,0), Qp = (-1,-1)--(-1-r,-1-r); for(int i=0;i <= 4;i=i+1) { draw(shift((4*i,0)) * P); draw(shift((4*i,0)) * Q); } for(int i=1;i <= 4;i=i+1) { draw(shift((4*i-2,0)) * Pp); draw(shift((4*i-1,0)) * Qp); } draw((-1,0)--(18.5,0),Arrows(TeXHead)); (Error making remote request. Unexpected URL sent back)
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
Problem 7
Melanie computes the mean , the median , and the modes of the values that are the dates in the months of . Thus her data consist of , , . . . , , , , and . Let be the median of the modes. Which of the following statements is true?
Problem 8
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 ?
Problem 9
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
Problem 10
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 ?
Problem 11
For some positive integer , the repeating base- representation of the (base-ten) fraction is . What is ?
Problem 12
Positive real numbers and satisfy and . What is ?
Problem 13
How many ways are there to paint each of the integers either red, green, or blue so that each number has a different color from each of its proper divisors?
Problem 14
For a certain complex number , the polynomial has exactly 4 distinct roots. What is ?
Problem 15
Positive real numbers and have the property that
and all four terms on the left are positive integers, where denotes the base- logarithm. What is ?
Problem 16
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?
Problem 17
Let denote the sum of the th powers of the roots of the polynomial . In particular, , , and . Let , , and be real numbers such that for , , What is ?
Problem 18
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?
Problem 19
In with integer side lengths, What is the least possible perimeter for ?
Problem 20
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 and are chosen independently in this manner. What is the probability that ?
Problem 21
Let What is
Problem 22
Circles and , both centered at , have radii and , respectively. Equilateral triangle , whose interior lies in the interior of but in the exterior of , has vertex on , and the line containing side is tangent to . Segments and intersect at , and . Then can be written in the form for positive integers , , , with . What is ?
Problem 23
Define binary operations and by for all real numbers and for which these expressions are defined. The sequence is defined recursively by and for all integers . To the nearest integer, what is ?
Problem 24
For how many integers between and , inclusive, is an integer? (Recall that .)
Problem 25
Let be a triangle whose angle measures are exactly , , and . For each positive integer , define to be the foot of the altitude from to line . Likewise, define to be the foot of the altitude from to line , and to be the foot of the altitude from to line . What is the least positive integer for which is obtuse?
See also
2019 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by 2018 AMC 12B Problems |
Followed by 2019 AMC 12B Problems |
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All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.