### 2020 AMC 10B Solutions

Problems used with permission of the Mathematical Association of America. Scroll down to view all solutions, or:

1.

What is the value of \[1 - (-2) - 3 - (-4) - 5 - (-6)?\]

\(-20\)

\(-3\)

\(3\)

\(5\)

\(21\)

###### Solution(s):

Let's first notice that when we subtract a negative number, it is the same as adding it's positive counterpart. In other words: \[a-(-b) = a+b\] With that in mind, we can rewrite the above expression as: \[1+2-3+4-5+6\] Which can be solved to yield \(5\) as our final answer.

Thus, **D** is the correct answer.

2.

Carl has \(5\) cubes each having side length \(1\), and Kate has \(5\) cubes each having side length \(2\). What is the total volume of the \(10\) cubes?

\(24\)

\(25\)

\(28\)

\(40\)

\(45\)

###### Solution(s):

Recall that a cube with a side length of \(a\) has a volume of \(a^3.\) With this in mind, Carl's cubes (each with side length \(1\)) have a volume of \(1^3=1\) each. Therefore, all of Carl's \(5\) cubes will have a total volume of \(5\cdot 1=5\).

Similarly, Kate's cubes (each with side length \(2\)) have a volume of \(2^3=8\) each. Therefore, all of Kates's \(5\) cubes will have a total volume of \(5\cdot 8=40\).

Therefore, the total volume of these 10 cubes is \(5+40=45\).

Thus, the correct answer is **E**.

3.

The ratio of \(w\) to \(x\) is \(4:3\), the ratio of \(y\) to \(z\) is \(3:2\), and the ratio of \(z\) to \(x\) is \(1:6\). What is the ratio of \(w\) to \(y?\)

\(4:3\)

\(3:2\)

\(8:3\)

\(4:1\)

\(16:3\)

###### Solution(s):

Let's begin by restating the following: \[w:x = \dfrac{w}{x}=\dfrac43=4:3\] \[y:z = \dfrac{y}{z}=\dfrac32=3:2\] \[z:x = \dfrac{z}{x}=\dfrac16=1:6\] Armed with these three equations, we want to find \(w:y= \dfrac wy.\) To do this, we must try and represent \(\dfrac wy\) using \(\dfrac wx, \dfrac yz,\) and \(\dfrac zy.\)

Notice that we can represent \(\dfrac wy\) as \(\dfrac wz \cdot \dfrac zy.\)

Further notice that \(\dfrac wz = \dfrac wx \cdot \dfrac xz.\) Therefore, combining these two facts shows us that: \[\begin{align*}\dfrac wy &= \dfrac wx \cdot \dfrac xz \cdot \dfrac zy\\ &= \dfrac wx \cdot \dfrac {1}{(\dfrac zx)} \cdot \dfrac {1}{(\dfrac yz)} \\ &= \dfrac 43 \cdot \dfrac{1}{(\dfrac 16)} \cdot \dfrac{1}{(\dfrac 32)} \\ &= \dfrac 43 \cdot \dfrac 61 \cdot \dfrac 23\\ &= \dfrac{48}{9}\\ &= \dfrac{16}{3} \end{align*}\] As such: \[w:y = \dfrac wy = \dfrac {16}{3} = 16:3\] Thus, **E** is the correct answer.

4.

The acute angles of a right triangle are \(a^{\circ}\) and \(b^{\circ}\), where \(a>b\) and both \(a\) and \(b\) are prime numbers. What is the least possible value of \(b\)?

\(2\)

\(3\)

\(5\)

\(7\)

\(11\)

###### Solution(s):

We know that the interior angles of a triangle add up to \(180^{\circ}\), and since the triangle in question is a right triangle, by definition one of the interior angles must measure \(90^{\circ}.\) The remaining two acute angles, \(a^{\circ}\) and \(b^{\circ}\), must therefore have a sum of \(180^{\circ}-90^{\circ}=90^{\circ}.\)

Let's begin by exploring the largest values of \(a^{\circ}\) and going from there, as those will naturally yield the smallest values of \(b^{\circ}.\)

The greatest possible value of \(a^{\circ}\) is \(89^{\circ}.\) This makes \(b^{\circ}=90^{\circ}-89^{\circ}=1^{\circ},\) which is not prime, so this is not a valid possible case.

Moving on, the next largest possible value of \(a^{\circ}\) is \(83^{\circ}.\) This makes \(b^{\circ}=90^{\circ}-83^{\circ}=7^{\circ},\) which is prime! Therefore, this is the smallest possible value of \(b.\)

Thus, **D** is the correct answer.

5.

How many distinguishable arrangements are there of \(1\) brown tile, \(1\) purple tile, \(2\) green tiles, and \(3\) yellow tiles in a row from left to right? (Tiles of the same color are indistinguishable.)

\(210\)

\(420\)

\(630\)

\(840\)

\(1050\)

###### Solution(s):

Just do it.

6.

Driving along a highway, Megan noticed that her odometer showed \(15951\) (miles). This number is a palindrome — it reads the same forward and backward. Then \(2\) hours later, the odometer displayed the next higher palindrome. What was her average speed, in miles per hour, during this \(2\)-hour period?

50

55

60

65

70

###### Solution(s):

Just do it.

7.

How many positive even multiples of \(3\) less than \(2020\) are perfect squares?

\(7\)

\(8\)

\(9\)

\(10\)

\(12\)

###### Solution(s):

Just do it.

8.

Points \(P\) and \(Q\) lie in a plane with \(PQ=8\). How many locations for point \(R\) in this plane are there such that the triangle with vertices \(P\), \(Q\), and \(R\) is a right triangle with area \(12\) square units?

\(2\)

\(4\)

\(6\)

\(8\)

\(12\)

###### Solution(s):

Just do it.

9.

How many ordered pairs of integers \((x, y)\) satisfy the equation \[x^{2020}+y^2=2y?\]

\(1\)

\(2\)

\(3\)

\(4\)

infinitely many

###### Solution(s):

Just do it.

10.

A three-quarter sector of a circle of radius \(4\) inches together with its interior can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches?

\(3\pi\sqrt{5}\)

\(4\pi\sqrt{3}\)

\(3\pi\sqrt{7}\)

\(6\pi\sqrt{3}\)

\(6\pi\sqrt{7}\)

###### Solution(s):

Just do it.

11.

Ms. Carr asks her students to read any \(5\) of the \(10\) books on a reading list. Harold randomly selects \(5\) books from this list, and Betty does the same. What is the probability that there are exactly \(2\) books that they both select?

\(\dfrac{1}{8}\)

\(\dfrac{5}{36}\)

\(\dfrac{14}{45}\)

\(\dfrac{25}{63}\)

\(\dfrac{1}{2}\)

###### Solution(s):

Just do it.

12.

The decimal representation of \[\frac{1}{20^{20}}\] consists of a string of zeros after the decimal point, followed by a \(9\) and then several more digits. How many zeros are in that initial string of zeros after the decimal point?

- \(23\)
- \(24\)
- \(25\)
- \(26\)
- \(27\)

Just do it.

\(23\)

\(24\)

\(25\)

\(26\)

\(27\)

###### Solution(s):

Just do it.

13.

Andy the Ant lives on a coordinate plane and is currently at \((-20, 20)\) facing east (that is, in the positive \(x\)-direction). Andy moves \(1\) unit and then turns \(90^{\circ}\) left. From there, Andy moves \(2\) units (north) and then turns \(90^{\circ}\) left. He then moves \(3\) units (west) and again turns \(90^{\circ}\) left. Andy continues his progress, increasing his distance each time by \(1\) unit and always turning left. What is the location of the point at which Andy makes the \(2020\text{th}\) left turn?

\((-1030,-994)\)

\((-1030,-990)\)

\((-1026,-994)\)

\((-1026,-990)\)

\((-1022,-994)\)

###### Solution(s):

Just do it.

14.

As shown in the figure below, six semicircles lie in the interior of a regular hexagon with side length 2 so that the diameters of the semicircles coincide with the sides of the hexagon. What is the area of the shaded region — inside the hexagon but outside all of the semicircles?

\(6\sqrt3-3\pi\)

\(\dfrac{9\sqrt3}{2}-2\pi\)

\(\dfrac{3\sqrt3}{2}-\dfrac{\pi}{3}\)

\(3\sqrt3-\pi\)

\(\dfrac{9\sqrt3}{2}-\pi\)

###### Solution(s):

Just do it.

15.

Steve wrote the digits \(1\), \(2\), \(3\), \(4\), and \(5\) in order repeatedly from left to right, forming a list of \(10,000\) digits, beginning \(123451234512\ldots\)

He then erased every third digit from his list (that is, the \(3\)rd, \(6\)th, \(9\)th, \(\ldots\) digits from the left), then erased every fourth digit from the resulting list (that is, the \(4\)th, \(8\)th, \(12\)th, \(\ldots\) digits from the left in what remained), and then erased every fifth digit from what remained at that point.

What is the sum of the three digits that were then in the positions \(2019, 2020, 2021\)?

\(7\)

\(9\)

\(10\)

\(11\)

\(12\)

###### Solution(s):

Just do it.

16.

Bela and Jenn play the following game on the closed interval \([0, n]\) of the real number line, where \(n\) is a fixed integer greater than \(4\). They take turns playing, with Bela going first. At his first turn, Bela chooses any real number in the interval \([0, n]\). Thereafter, the player whose turn it is chooses a real number that is more than one unit away from all numbers previously chosen by either player. A player unable to choose such a number loses. Using optimal strategy, which player will win the game?

Bela will always win.

Jenn will always win.

Bela will win if and only if \(n\) is odd.

Jenn will win if and only if \(n\) is odd.

Bela will win if and only if \(n >8\).

###### Solution(s):

Just do it.

17.

There are \(10\) people standing equally spaced around a circle. Each person knows exactly \(3\) of the other \(9\) people: the \(2\) people standing next to her or him, as well as the person directly across the circle. How many ways are there for the \(10\) people to split up into \(5\) pairs so that the members of each pair know each other?

\(11\)

\(12\)

\(13\)

\(14\)

\(15\)

###### Solution(s):

Just do it.

18.

An urn contains one red ball and one blue ball. A box of extra red and blue balls lie nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations the urn contains six balls. What is the probability that the urn contains three balls of each color?

\(\dfrac16\)

\(\dfrac15\)

\(\dfrac14\)

\(\dfrac13\)

\(\dfrac12\)

###### Solution(s):

Just do it.

19.

In a certain card game, a player is dealt a hand of \(10\) cards from a deck of \(52\) distinct cards. The number of distinct (unordered) hands that can be dealt to the player can be written as \(158A00A4AA0\). What is the digit \(A\)?

\(2\)

\(3\)

\(4\)

\(6\)

\(7\)

###### Solution(s):

Just do it.

20.

Let \(B\) be a right rectangular prism (box) with edges lengths \(1,\) \(3,\) and \(4\), together with its interior. For real \(r\geq0\), let \(S(r)\) be the set of points in \(3\)-dimensional space that lie within a distance \(r\) of some point in \(B\). The volume of \(S(r)\) can be expressed as \[ar^{3} + br^{2} + cr +d,\] where \(a,\) \(b,\) \(c,\) and \(d\) are positive real numbers. What is \(\dfrac{bc}{ad}?\)

\(6\)

\(19\)

\(24\)

\(26\)

\(38\)

###### Solution(s):

Just do it.

21.

In square \(ABCD\), points \(E\) and \(H\) lie on \(\overline{AB}\) and \(\overline{DA}\), respectively, so that \(AE=AH.\) Points \(F\) and \(G\) lie on \(\overline{BC}\) and \(\overline{CD}\), respectively, and points \(I\) and \(J\) lie on \(\overline{EH}\) so that \(\overline{FI} \perp \overline{EH}\) and \(\overline{GJ} \perp \overline{EH}\). See the figure below. Triangle \(AEH\), quadrilateral \(BFIE\), quadrilateral \(DHJG\), and pentagon \(FCGJI\) each has area \(1.\) What is \(FI^2\)?

\(\dfrac73\)

\(8-4\sqrt2\)

\(1+\sqrt2\)

\(\dfrac74\sqrt2\)

\(2\sqrt2\)

###### Solution(s):

Just do it.

22.

What is the remainder when \(2^{202} +202\) is divided by \(2^{101}+2^{51}+1\)?

\(100\)

\(101\)

\(200\)

\(201\)

\(202\)

###### Solution(s):

Just do it.

23.

Square \(ABCD\) in the coordinate plane has vertices at the points \(A(1,1), B(-1,1), C(-1,-1),\) and \(D(1,-1).\) Consider the following four transformations:

Each of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying \(R\) and then \(V\) would send the vertex \(A\) at \((1,1)\) to \((-1,-1)\) and would send the vertex \(B\) at \((-1,1)\) to itself. How many sequences of \(20\) transformations chosen from \(\{L, R, H, V\}\) will send all of the labeled vertices back to their original positions? (For example, \(R, R, V, H\) is one sequence of \(4\) transformations that will send the vertices back to their original positions.)

\(2^{37}\)

\(3\cdot 2^{36}\)

\(2^{38}\)

\(3\cdot2^{37}\)

\(2^{39}\)

###### Solution(s):

Just do it.

24.

How many positive integers \(n\) satisfy \[\dfrac{n+1000}{70} = \lfloor \sqrt{n} \rfloor?\]

(Recall that \(\lfloor x\rfloor\) is the greatest integer not exceeding \(x\).)

\(2\)

\(4\)

\(6\)

\(30\)

\(32\)

###### Solution(s):

Just do it.

25.

Let \(D(n)\) denote the number of ways of writing the positive integer \(n\) as a product \[n = f_1\cdot f_2\cdots f_k,\] where \(k\ge1\), the \(f_i\) are integers strictly greater than \(1\), and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number \(6\) can be written as \(6\), \(2\cdot 3\), and \(3\cdot2\), so \(D(6) = 3\). What is \(D(96)\)?

\(112\)

\(128\)

\(144\)

\(172\)

\(184\)

###### Solution(s):

Just do it.