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Difference between revisions of "2019 AMC 10A Problems"

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==Problem 1==
 
==Problem 1==
 +
 
What is the value of <cmath>2^{\left(0^{\left(1^9\right)}\right)}+\left(\left(2^0\right)^1\right)^9?</cmath>
 
What is the value of <cmath>2^{\left(0^{\left(1^9\right)}\right)}+\left(\left(2^0\right)^1\right)^9?</cmath>
 
<math>\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 4</math>
 
<math>\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 4</math>
 +
 +
[[2019 AMC 10A Problems/Problem 1|Solution]]
  
 
==Problem 2==
 
==Problem 2==
−
What is the hundreds digit of <math>(20!-15!)\ ?</math>
+
What is the hundreds digit of <math>(20!-15!)?</math>
  
 
<math>\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }5</math>
 
<math>\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }5</math>
 +
 +
[[2019 AMC 10A Problems/Problem 2|Solution]]
  
 
==Problem 3==
 
==Problem 3==
−
Ana and Bonita were born on the same date in different years, <math>n</math> years apart. Last year Ana was <math>5</math> times as old as Bonita. This year Ana's age is the square of Bonita's age. What is <math>n?</math>
+
Ana and Bonita are born on the same date in different years, <math>n</math> years apart. Last year Ana was <math>5</math> times as old as Bonita. This year Ana's age is the square of Bonita's age. What is <math>n?</math>
  
 
<math>\textbf{(A) } 3 \qquad\textbf{(B) } 5 \qquad\textbf{(C) } 9 \qquad\textbf{(D) } 12 \qquad\textbf{(E) } 15</math>
 
<math>\textbf{(A) } 3 \qquad\textbf{(B) } 5 \qquad\textbf{(C) } 9 \qquad\textbf{(D) } 12 \qquad\textbf{(E) } 15</math>
 +
 +
[[2019 AMC 10A Problems/Problem 3|Solution]]
  
 
==Problem 4==
 
==Problem 4==
Line 19: Line 26:
  
 
<math>\textbf{(A) } 75 \qquad\textbf{(B) } 76 \qquad\textbf{(C) } 79 \qquad\textbf{(D) } 84 \qquad\textbf{(E) } 91</math>
 
<math>\textbf{(A) } 75 \qquad\textbf{(B) } 76 \qquad\textbf{(C) } 79 \qquad\textbf{(D) } 84 \qquad\textbf{(E) } 91</math>
 +
 +
[[2019 AMC 10A Problems/Problem 4|Solution]]
  
 
==Problem 5==
 
==Problem 5==
−
What is the greatest number of consecutive integers whose sum is <math>45 ?</math>
+
What is the greatest number of consecutive integers whose sum is <math>45?</math>
  
 
<math>\textbf{(A) } 9 \qquad\textbf{(B) } 25 \qquad\textbf{(C) } 45 \qquad\textbf{(D) } 90 \qquad\textbf{(E) } 120</math>
 
<math>\textbf{(A) } 9 \qquad\textbf{(B) } 25 \qquad\textbf{(C) } 45 \qquad\textbf{(D) } 90 \qquad\textbf{(E) } 120</math>
 +
 +
[[2019 AMC 10A Problems/Problem 5|Solution]]
  
 
==Problem 6==
 
==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
 +
 +
<math>\textbf{(A) } 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 3 \qquad\textbf{(D) } 4 \qquad\textbf{(E) } 5</math>
 +
 +
[[2019 AMC 10A Problems/Problem 6|Solution]]
 +
 
==Problem 7==
 
==Problem 7==
 +
 +
Two lines with slopes <math>\dfrac{1}{2}</math> and <math>2</math> intersect at <math>(2,2)</math>. What is the area of the triangle enclosed by these two lines and the line <math>x+y=10  ?</math>
 +
 +
<math>\textbf{(A) } 4 \qquad\textbf{(B) } 4\sqrt{2} \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 8 \qquad\textbf{(E) } 6\sqrt{2}</math>
 +
 +
[[2019 AMC 10A Problems/Problem 7|Solution]]
 +
 
==Problem 8==
 
==Problem 8==
 +
 +
The figure below shows line <math>\ell</math> with a regular, infinite, recurring pattern of squares and line segments.
 +
<asy>
 +
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));
 +
</asy>
 +
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 <math>\ell</math>
 +
*some translation in the direction parallel to line <math>\ell</math>
 +
*the reflection across line <math>\ell</math>
 +
*some reflection across a line perpendicular to line <math>\ell</math>
 +
<math>\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 4</math>
 +
 +
[[2019 AMC 10A Problems/Problem 8|Solution]]
 +
 
==Problem 9==
 
==Problem 9==
 +
 +
What is the greatest three-digit positive integer <math>n</math> for which the sum of the first <math>n</math> positive integers is <math>\underline{\text{not}}</math> a divisor of the product of the first <math>n</math> positive integers?
 +
 +
<math>\textbf{(A) } 995 \qquad\textbf{(B) } 996 \qquad\textbf{(C) } 997 \qquad\textbf{(D) } 998 \qquad\textbf{(E) } 999</math>
 +
 +
[[2019 AMC 10A Problems/Problem 9|Solution]]
 +
 
==Problem 10==
 
==Problem 10==
 +
 +
A rectangular floor that is <math>10</math> feet wide and <math>17</math> feet long is tiled with <math>170</math> 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?
 +
 +
<math>\textbf{(A) } 17 \qquad\textbf{(B) } 25 \qquad\textbf{(C) } 26 \qquad\textbf{(D) } 27 \qquad\textbf{(E) } 28</math>
 +
 +
[[2019 AMC 10A Problems/Problem 10|Solution]]
 +
 
==Problem 11==
 
==Problem 11==
 +
 +
How many positive integer divisors of <math>201^9</math> are perfect squares or perfect cubes (or both)?
 +
 +
<math>\textbf{(A) } 32 \qquad\textbf{(B) } 36 \qquad\textbf{(C) } 37 \qquad\textbf{(D) } 39 \qquad\textbf{(E) } 41</math>
 +
 +
[[2019 AMC 10A Problems/Problem 11|Solution]]
 +
 
==Problem 12==
 
==Problem 12==
 +
 +
Melanie computes the mean <math>\mu</math>, the median <math>M</math>, and the modes of the <math>365</math> values that are the dates in the months of <math>2019</math>. Thus her data consists of <math>12</math> <math>1\text{s}</math>, <math>12</math> <math>2\text{s}</math>, . . . , <math>12</math> <math>28\text{s}</math>, <math>11</math> <math>29\text{s}</math>, <math>11</math> <math>30\text{s}</math>, and <math>7</math> <math>31\text{s}</math>. Let <math>d</math> be the median of the modes. Which of the following statements is true?
 +
 +
<math>\textbf{(A) } \mu < d < M \qquad\textbf{(B) } M < d < \mu \qquad\textbf{(C) } d = M =\mu \qquad\textbf{(D) } d < M < \mu \qquad\textbf{(E) } d < \mu < M</math>
 +
 +
[[2019 AMC 10A Problems/Problem 12|Solution]]
 +
 
==Problem 13==
 
==Problem 13==
−
Let <math>\Delta ABC</math> be an isosceles triangle with <math>BC = AC</math> and <math>\angle ACB = 40^{\circ}</math>. Contruct the circle with diameter <math>\overline{BC}</math>, and let <math>D</math> and  <math>E</math> be the other intersection points of the circle with the sides <math>\overline{AC}</math> and <math>\overline{AB}</math>, respectively. Let <math>F</math> be the intersection of the diagonals of the quadrilateral <math>BCDE</math>. What is the degree measure of <math>\angle BFC ?</math>
+
 
 +
Let <math>\triangle ABC</math> be an isosceles triangle with <math>BC = AC</math> and <math>\angle ACB = 40^{\circ}</math>. Construct the circle with diameter <math>\overline{BC}</math>, and let <math>D</math> and  <math>E</math> be the other intersection points of the circle with the sides <math>\overline{AC}</math> and <math>\overline{AB}</math>, respectively. Let <math>F</math> be the intersection of the diagonals of the quadrilateral <math>BCDE</math>. What is the degree measure of <math>\angle BFC ?</math>
  
 
<math>\textbf{(A) } 90 \qquad\textbf{(B) } 100 \qquad\textbf{(C) } 105 \qquad\textbf{(D) } 110 \qquad\textbf{(E) } 120</math>
 
<math>\textbf{(A) } 90 \qquad\textbf{(B) } 100 \qquad\textbf{(C) } 105 \qquad\textbf{(D) } 110 \qquad\textbf{(E) } 120</math>
 +
 +
[[2019 AMC 10A Problems/Problem 13|Solution]]
  
 
==Problem 14==
 
==Problem 14==
Line 42: Line 132:
  
 
<math>\textbf{(A) } 14 \qquad \textbf{(B) } 16 \qquad \textbf{(C) } 18 \qquad \textbf{(D) } 19 \qquad \textbf{(E) } 21</math>
 
<math>\textbf{(A) } 14 \qquad \textbf{(B) } 16 \qquad \textbf{(C) } 18 \qquad \textbf{(D) } 19 \qquad \textbf{(E) } 21</math>
 +
 +
[[2019 AMC 10A Problems/Problem 14|Solution]]
  
 
==Problem 15==
 
==Problem 15==
  
 
A sequence of numbers is defined recursively by <math>a_1 = 1</math>, <math>a_2 = \frac{3}{7}</math>, and
 
A sequence of numbers is defined recursively by <math>a_1 = 1</math>, <math>a_2 = \frac{3}{7}</math>, and
−
<cmath>a_n=\frac{a_{n-2} \cdot a_{n-1}}{2a_{n-2} - a_{n-1}}</cmath>for all <math>n \geq 3</math> Then <math>a_{2019}</math> can be written as <math>\frac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive inegers. What is <math>p+q ?</math>
+
<cmath>a_n=\frac{a_{n-2} \cdot a_{n-1}}{2a_{n-2} - a_{n-1}}</cmath>for all <math>n \geq 3</math>. Then <math>a_{2019}</math> can be written as <math>\frac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. What is <math>p+q ?</math>
 +
 
 +
<math>\textbf{(A) } 2020 \qquad\textbf{(B) } 4039 \qquad\textbf{(C) } 6057 \qquad\textbf{(D) } 6061 \qquad\textbf{(E) } 8078</math>
  
−
<math>\textbf{(A) } 2020 \qquad\textbf{(B) } 39 \qquad\textbf{(C) } 6057 \qquad\textbf{(D) } 6061 \qquad\textbf{(E) } 8078</math>
+
[[2019 AMC 10A Problems/Problem 15|Solution]]
  
 
==Problem 16==
 
==Problem 16==
Line 56: Line 150:
 
<asy>unitsize(20);filldraw(circle((0,0),2*sqrt(3)+1),rgb(0.5,0.5,0.5));filldraw(circle((-2,0),1),white);filldraw(circle((0,0),1),white);filldraw(circle((2,0),1),white);filldraw(circle((1,sqrt(3)),1),white);filldraw(circle((3,sqrt(3)),1),white);filldraw(circle((-1,sqrt(3)),1),white);filldraw(circle((-3,sqrt(3)),1),white);filldraw(circle((1,-1*sqrt(3)),1),white);filldraw(circle((3,-1*sqrt(3)),1),white);filldraw(circle((-1,-1*sqrt(3)),1),white);filldraw(circle((-3,-1*sqrt(3)),1),white);filldraw(circle((0,2*sqrt(3)),1),white);filldraw(circle((0,-2*sqrt(3)),1),white);</asy>
 
<asy>unitsize(20);filldraw(circle((0,0),2*sqrt(3)+1),rgb(0.5,0.5,0.5));filldraw(circle((-2,0),1),white);filldraw(circle((0,0),1),white);filldraw(circle((2,0),1),white);filldraw(circle((1,sqrt(3)),1),white);filldraw(circle((3,sqrt(3)),1),white);filldraw(circle((-1,sqrt(3)),1),white);filldraw(circle((-3,sqrt(3)),1),white);filldraw(circle((1,-1*sqrt(3)),1),white);filldraw(circle((3,-1*sqrt(3)),1),white);filldraw(circle((-1,-1*sqrt(3)),1),white);filldraw(circle((-3,-1*sqrt(3)),1),white);filldraw(circle((0,2*sqrt(3)),1),white);filldraw(circle((0,-2*sqrt(3)),1),white);</asy>
  
−
<math>\textbf{(A) } 4 \pi \sqrt{3} \qquad\textbf{(B) } 7 \pi \qquad\textbf{(C) } \pi(3\sqrt{3} +2) \qquad\textbf{(D) } 10 \pi (\sqrt{3} - 1) \qquad\textbf{(E) } \pi(\sqrt{3} + 6)</math>
+
<math>\textbf{(A) } 4 \pi \sqrt{3} \qquad\textbf{(B) } 7 \pi \qquad\textbf{(C) } \pi\left(3\sqrt{3} +2\right) \qquad\textbf{(D) } 10 \pi \left(\sqrt{3} - 1\right) \qquad\textbf{(E) } \pi\left(\sqrt{3} + 6\right)</math>
 +
 
 +
[[2019 AMC 10A Problems/Problem 16|Solution]]
  
 
==Problem 17==
 
==Problem 17==
  
−
A child builds towers using identically shaped cubes of different color. How many different towers with a height <math>8</math> cubes can the child build with <math>2</math> red cubes, <math>3</math> blue cubes, and <math>4</math> green cubes? (One cube will be left out.)
+
A child builds towers using identically shaped cubes of different colors. How many different towers with a height <math>8</math> cubes can the child build with <math>2</math> red cubes, <math>3</math> blue cubes, and <math>4</math> green cubes? (One cube will be left out.)
  
 
<math>\textbf{(A) } 24 \qquad\textbf{(B) } 288 \qquad\textbf{(C) } 312 \qquad\textbf{(D) } 1,260 \qquad\textbf{(E) } 40,320</math>
 
<math>\textbf{(A) } 24 \qquad\textbf{(B) } 288 \qquad\textbf{(C) } 312 \qquad\textbf{(D) } 1,260 \qquad\textbf{(E) } 40,320</math>
 +
 +
[[2019 AMC 10A Problems/Problem 17|Solution]]
  
 
==Problem 18==
 
==Problem 18==
Line 68: Line 166:
 
For some positive integer <math>k</math>, the repeating base-<math>k</math> representation of the (base-ten) fraction <math>\frac{7}{51}</math> is <math>0.\overline{23}_k = 0.232323..._k</math>. What is <math>k</math>?
 
For some positive integer <math>k</math>, the repeating base-<math>k</math> representation of the (base-ten) fraction <math>\frac{7}{51}</math> is <math>0.\overline{23}_k = 0.232323..._k</math>. What is <math>k</math>?
  
 +
<math>\textbf{(A) } 13 \qquad\textbf{(B) } 14 \qquad\textbf{(C) } 15 \qquad\textbf{(D) } 16 \qquad\textbf{(E) } 17</math>
  
−
<math>\textbf{(A) } 13 \qquad\textbf{(B) } 14 \qquad\textbf{(C) } 15 \qquad\textbf{(D) } 16 \qquad\textbf{(E) } 17</math>
+
[[2019 AMC 10A Problems/Problem 18|Solution]]
  
 
==Problem 19==
 
==Problem 19==
Line 77: Line 176:
  
 
<math>\textbf{(A) } 2017 \qquad\textbf{(B) } 2018 \qquad\textbf{(C) } 2019 \qquad\textbf{(D) } 2020 \qquad\textbf{(E) } 2021</math>
 
<math>\textbf{(A) } 2017 \qquad\textbf{(B) } 2018 \qquad\textbf{(C) } 2019 \qquad\textbf{(D) } 2020 \qquad\textbf{(E) } 2021</math>
 +
 +
[[2019 AMC 10A Problems/Problem 19|Solution]]
  
 
==Problem 20==
 
==Problem 20==
 +
 +
The numbers <math>1,2,\dots,9</math> are randomly placed into the <math>9</math> squares of a <math>3 \times 3</math> 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?
 +
 +
<math>\textbf{(A) }\dfrac{1}{21}\qquad\textbf{(B) }\dfrac{1}{14}\qquad\textbf{(C) }\dfrac{5}{63}\qquad\textbf{(D) }\dfrac{2}{21}\qquad\textbf{(E) }\dfrac{1}{7}</math>
 +
 +
[[2019 AMC 10A Problems/Problem 20|Solution]]
 +
 
==Problem 21==
 
==Problem 21==
−
A sphere with center <math>O</math> has radius 6. A triangle with sides of length <math>15</math>, <math>15</math>, and <math>24</math> is situated in space so that each of its sides is tangent to the sphere. What is the distance between <math>O</math> and the plane determined by the triangle?
+
A sphere with center <math>O</math> has radius 6. A triangle with sides of length <math>15</math>, <math>15</math>, and <math>24</math> is situated in space so that each of its sides are tangent to the sphere. What is the distance between <math>O</math> and the plane determined by the triangle?
  
 
<math>\textbf{(A) } 2\sqrt{3} \qquad \textbf{(B) }4 \qquad \textbf{(C) } 3\sqrt{2} \qquad \textbf{(D) } 2\sqrt{5} \qquad \textbf{(E) } 5</math>
 
<math>\textbf{(A) } 2\sqrt{3} \qquad \textbf{(B) }4 \qquad \textbf{(C) } 3\sqrt{2} \qquad \textbf{(D) } 2\sqrt{5} \qquad \textbf{(E) } 5</math>
 +
 +
[[2019 AMC 10A Problems/Problem 21|Solution]]
  
 
==Problem 22==
 
==Problem 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 <math>[0,1]</math>. Two random numbers <math>x</math> and <math>y</math> are chosen independently in this manner. What is the probability that <math>|x-y| > \tfrac{1}{2}</math>?
 +
 +
<math>\textbf{(A) } \frac{1}{3} \qquad \textbf{(B) } \frac{7}{16} \qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) } \frac{9}{16} \qquad \textbf{(E) } \frac{2}{3}</math>
 +
 +
[[2019 AMC 10A Problems/Problem 22|Solution]]
 +
 
==Problem 23==
 
==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 <math>1</math>, then Todd must say the next two numbers (<math>2</math> and <math>3</math>), then Tucker must say the next three numbers  (<math>4</math>, <math>5</math>, <math>6</math>), then Tadd must say the next four numbers (<math>7</math>, <math>8</math>, <math>9</math>, <math>10</math>), 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 <math>10,000</math> is reached. What is the <math>2019</math>th number said by Tadd?
 +
 +
<math> \textbf{(A)}\ 5743
 +
\qquad\textbf{(B)}\ 5885
 +
\qquad\textbf{(C)}\ 5979
 +
\qquad\textbf{(D)}\ 6001
 +
\qquad\textbf{(E)}\ 6011
 +
</math>
 +
 +
[[2019 AMC 10A Problems/Problem 23|Solution]]
 +
 
==Problem 24==
 
==Problem 24==
 +
 +
Let <math>p</math>, <math>q</math>, and <math>r</math> be the distinct roots of the polynomial <math>x^3 - 22x^2 + 80x - 67</math>. It is given that there exist real numbers <math>A</math>, <math>B</math>, and <math>C</math> such that <cmath>\dfrac{1}{s^3 - 22s^2 + 80s - 67} = \dfrac{A}{s-p} + \dfrac{B}{s-q} + \frac{C}{s-r}</cmath>for all <math>s\not\in\{p,q,r\}</math>. What is <math>\tfrac1A+\tfrac1B+\tfrac1C</math>?
 +
 +
<math>\textbf{(A) }243\qquad\textbf{(B) }244\qquad\textbf{(C) }245\qquad\textbf{(D) }246\qquad\textbf{(E) } 247</math>
 +
 +
[[2019 AMC 10A Problems/Problem 24|Solution]]
 +
 
==Problem 25==
 
==Problem 25==
 
For how many integers <math>n</math> between <math>1</math> and <math>50</math>, inclusive, is <cmath>\frac{(n^2-1)!}{(n!)^{n}}</cmath> an integer? (Recall that <math>0!=1</math>.)
 
For how many integers <math>n</math> between <math>1</math> and <math>50</math>, inclusive, is <cmath>\frac{(n^2-1)!}{(n!)^{n}}</cmath> an integer? (Recall that <math>0!=1</math>.)
  
 
<math>\textbf{(A) } 31 \qquad \textbf{(B) } 32 \qquad \textbf{(C) } 33 \qquad \textbf{(D) } 34 \qquad \textbf{(E) } 35</math>
 
<math>\textbf{(A) } 31 \qquad \textbf{(B) } 32 \qquad \textbf{(C) } 33 \qquad \textbf{(D) } 34 \qquad \textbf{(E) } 35</math>
 +
 +
[[2019 AMC 10A Problems/Problem 25|Solution]]
  
 
==See also==
 
==See also==
 
{{AMC10 box|year=2019|ab=A|before=[[2018 AMC 10B Problems]]|after=[[2019 AMC 10B Problems]]}}
 
{{AMC10 box|year=2019|ab=A|before=[[2018 AMC 10B Problems]]|after=[[2019 AMC 10B Problems]]}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 12:17, 28 June 2024

2019 AMC 10A (Answer Key)
Printable versions: Wiki • AoPS Resources • PDF

Instructions

  1. This is a 25-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct.
  2. You will receive 6 points for each correct answer, 2.5 points for each problem left unanswered if the year is before 2006, 1.5 points for each problem left unanswered if the year is after 2006, and 0 points for each incorrect answer.
  3. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers (and calculators that are accepted for use on the SAT if before 2006. No problems on the test will require the use of a calculator).
  4. Figures are not necessarily drawn to scale.
  5. You will have 75 minutes working time to complete the test.
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25

Problem 1

What is the value of \[2^{\left(0^{\left(1^9\right)}\right)}+\left(\left(2^0\right)^1\right)^9?\] $\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 4$

Solution

Problem 2

What is the hundreds digit of $(20!-15!)?$

$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }5$

Solution

Problem 3

Ana and Bonita are 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 $n?$

$\textbf{(A) } 3 \qquad\textbf{(B) } 5 \qquad\textbf{(C) } 9 \qquad\textbf{(D) } 12 \qquad\textbf{(E) } 15$

Solution

Problem 4

A box contains $28$ red balls, $20$ green balls, $19$ 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$?$

$\textbf{(A) } 75 \qquad\textbf{(B) } 76 \qquad\textbf{(C) } 79 \qquad\textbf{(D) } 84 \qquad\textbf{(E) } 91$

Solution

Problem 5

What is the greatest number of consecutive integers whose sum is $45?$

$\textbf{(A) } 9 \qquad\textbf{(B) } 25 \qquad\textbf{(C) } 45 \qquad\textbf{(D) } 90 \qquad\textbf{(E) } 120$

Solution

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

$\textbf{(A) } 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 3 \qquad\textbf{(D) } 4 \qquad\textbf{(E) } 5$

Solution

Problem 7

Two lines with slopes $\dfrac{1}{2}$ and $2$ intersect at $(2,2)$. What is the area of the triangle enclosed by these two lines and the line $x+y=10  ?$

$\textbf{(A) } 4 \qquad\textbf{(B) } 4\sqrt{2} \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 8 \qquad\textbf{(E) } 6\sqrt{2}$

Solution

Problem 8

The figure below shows line $\ell$ with a regular, infinite, recurring pattern of squares and line segments. [asy] 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)); [/asy] 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 $\ell$
  • some translation in the direction parallel to line $\ell$
  • the reflection across line $\ell$
  • some reflection across a line perpendicular to line $\ell$

$\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 4$

Solution

Problem 9

What is the greatest three-digit positive integer $n$ for which the sum of the first $n$ positive integers is $\underline{\text{not}}$ a divisor of the product of the first $n$ positive integers?

$\textbf{(A) } 995 \qquad\textbf{(B) } 996 \qquad\textbf{(C) } 997 \qquad\textbf{(D) } 998 \qquad\textbf{(E) } 999$

Solution

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?

$\textbf{(A) } 17 \qquad\textbf{(B) } 25 \qquad\textbf{(C) } 26 \qquad\textbf{(D) } 27 \qquad\textbf{(E) } 28$

Solution

Problem 11

How many positive integer divisors of $201^9$ are perfect squares or perfect cubes (or both)?

$\textbf{(A) } 32 \qquad\textbf{(B) } 36 \qquad\textbf{(C) } 37 \qquad\textbf{(D) } 39 \qquad\textbf{(E) } 41$

Solution

Problem 12

Melanie computes the mean $\mu$, the median $M$, and the modes of the $365$ values that are the dates in the months of $2019$. Thus her data consists of $12$ $1\text{s}$, $12$ $2\text{s}$, . . . , $12$ $28\text{s}$, $11$ $29\text{s}$, $11$ $30\text{s}$, and $7$ $31\text{s}$. Let $d$ be the median of the modes. Which of the following statements is true?

$\textbf{(A) } \mu < d < M \qquad\textbf{(B) } M < d < \mu \qquad\textbf{(C) } d = M =\mu \qquad\textbf{(D) } d < M < \mu \qquad\textbf{(E) } d < \mu < M$

Solution

Problem 13

Let $\triangle ABC$ be an isosceles triangle with $BC = AC$ and $\angle ACB = 40^{\circ}$. Construct the circle with diameter $\overline{BC}$, and let $D$ and $E$ be the other intersection points of the circle with the sides $\overline{AC}$ and $\overline{AB}$, respectively. Let $F$ be the intersection of the diagonals of the quadrilateral $BCDE$. What is the degree measure of $\angle BFC ?$

$\textbf{(A) } 90 \qquad\textbf{(B) } 100 \qquad\textbf{(C) } 105 \qquad\textbf{(D) } 110 \qquad\textbf{(E) } 120$

Solution

Problem 14

For a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $N$?

$\textbf{(A) } 14 \qquad \textbf{(B) } 16 \qquad \textbf{(C) } 18 \qquad \textbf{(D) } 19 \qquad \textbf{(E) } 21$

Solution

Problem 15

A sequence of numbers is defined recursively by $a_1 = 1$, $a_2 = \frac{3}{7}$, and \[a_n=\frac{a_{n-2} \cdot a_{n-1}}{2a_{n-2} - a_{n-1}}\]for all $n \geq 3$. Then $a_{2019}$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p+q ?$

$\textbf{(A) } 2020 \qquad\textbf{(B) } 4039 \qquad\textbf{(C) } 6057 \qquad\textbf{(D) } 6061 \qquad\textbf{(E) } 8078$

Solution

Problem 16

The figure below shows $13$ circles of radius $1$ 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 $1 ?$

[asy]unitsize(20);filldraw(circle((0,0),2*sqrt(3)+1),rgb(0.5,0.5,0.5));filldraw(circle((-2,0),1),white);filldraw(circle((0,0),1),white);filldraw(circle((2,0),1),white);filldraw(circle((1,sqrt(3)),1),white);filldraw(circle((3,sqrt(3)),1),white);filldraw(circle((-1,sqrt(3)),1),white);filldraw(circle((-3,sqrt(3)),1),white);filldraw(circle((1,-1*sqrt(3)),1),white);filldraw(circle((3,-1*sqrt(3)),1),white);filldraw(circle((-1,-1*sqrt(3)),1),white);filldraw(circle((-3,-1*sqrt(3)),1),white);filldraw(circle((0,2*sqrt(3)),1),white);filldraw(circle((0,-2*sqrt(3)),1),white);[/asy]

$\textbf{(A) } 4 \pi \sqrt{3} \qquad\textbf{(B) } 7 \pi \qquad\textbf{(C) } \pi\left(3\sqrt{3} +2\right) \qquad\textbf{(D) } 10 \pi \left(\sqrt{3} - 1\right) \qquad\textbf{(E) } \pi\left(\sqrt{3} + 6\right)$

Solution

Problem 17

A child builds towers using identically shaped cubes of different colors. How many different towers with a height $8$ cubes can the child build with $2$ red cubes, $3$ blue cubes, and $4$ green cubes? (One cube will be left out.)

$\textbf{(A) } 24 \qquad\textbf{(B) } 288 \qquad\textbf{(C) } 312 \qquad\textbf{(D) } 1,260 \qquad\textbf{(E) } 40,320$

Solution

Problem 18

For some positive integer $k$, the repeating base-$k$ representation of the (base-ten) fraction $\frac{7}{51}$ is $0.\overline{23}_k = 0.232323..._k$. What is $k$?

$\textbf{(A) } 13 \qquad\textbf{(B) } 14 \qquad\textbf{(C) } 15 \qquad\textbf{(D) } 16 \qquad\textbf{(E) } 17$

Solution

Problem 19

What is the least possible value of \[(x+1)(x+2)(x+3)(x+4)+2019\]where $x$ is a real number?

$\textbf{(A) } 2017 \qquad\textbf{(B) } 2018 \qquad\textbf{(C) } 2019 \qquad\textbf{(D) } 2020 \qquad\textbf{(E) } 2021$

Solution

Problem 20

The numbers $1,2,\dots,9$ are randomly placed into the $9$ squares of a $3 \times 3$ 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?

$\textbf{(A) }\dfrac{1}{21}\qquad\textbf{(B) }\dfrac{1}{14}\qquad\textbf{(C) }\dfrac{5}{63}\qquad\textbf{(D) }\dfrac{2}{21}\qquad\textbf{(E) }\dfrac{1}{7}$

Solution

Problem 21

A sphere with center $O$ has radius 6. A triangle with sides of length $15$, $15$, and $24$ is situated in space so that each of its sides are tangent to the sphere. What is the distance between $O$ and the plane determined by the triangle?

$\textbf{(A) } 2\sqrt{3} \qquad \textbf{(B) }4 \qquad \textbf{(C) } 3\sqrt{2} \qquad \textbf{(D) } 2\sqrt{5} \qquad \textbf{(E) } 5$

Solution

Problem 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 $[0,1]$. Two random numbers $x$ and $y$ are chosen independently in this manner. What is the probability that $|x-y| > \tfrac{1}{2}$?

$\textbf{(A) } \frac{1}{3} \qquad \textbf{(B) } \frac{7}{16} \qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) } \frac{9}{16} \qquad \textbf{(E) } \frac{2}{3}$

Solution

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 $1$, then Todd must say the next two numbers ($2$ and $3$), then Tucker must say the next three numbers ($4$, $5$, $6$), then Tadd must say the next four numbers ($7$, $8$, $9$, $10$), 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 $10,000$ is reached. What is the $2019$th number said by Tadd?

$\textbf{(A)}\ 5743 \qquad\textbf{(B)}\ 5885 \qquad\textbf{(C)}\ 5979 \qquad\textbf{(D)}\ 6001 \qquad\textbf{(E)}\ 6011$

Solution

Problem 24

Let $p$, $q$, and $r$ be the distinct roots of the polynomial $x^3 - 22x^2 + 80x - 67$. It is given that there exist real numbers $A$, $B$, and $C$ such that \[\dfrac{1}{s^3 - 22s^2 + 80s - 67} = \dfrac{A}{s-p} + \dfrac{B}{s-q} + \frac{C}{s-r}\]for all $s\not\in\{p,q,r\}$. What is $\tfrac1A+\tfrac1B+\tfrac1C$?

$\textbf{(A) }243\qquad\textbf{(B) }244\qquad\textbf{(C) }245\qquad\textbf{(D) }246\qquad\textbf{(E) } 247$

Solution

Problem 25

For how many integers $n$ between $1$ and $50$, inclusive, is \[\frac{(n^2-1)!}{(n!)^{n}}\] an integer? (Recall that $0!=1$.)

$\textbf{(A) } 31 \qquad \textbf{(B) } 32 \qquad \textbf{(C) } 33 \qquad \textbf{(D) } 34 \qquad \textbf{(E) } 35$

Solution

See also

2019 AMC 10A (Problems • Answer Key • Resources)
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All AMC 10 Problems and Solutions

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