id int64 | answer float64 | url string | question string | solution string | a_difficulty int64 | domains sequence |
|---|---|---|---|---|---|---|
0 | 27 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_1 | Cities $A$ and $B$ are $45$ miles apart. Alicia lives in $A$ and Beth lives in $B$. Alicia bikes towards $B$ at 18 miles per hour. Leaving at the same time, Beth bikes toward $A$ at 12 miles per hour. How many miles from City $A$ will they be when they meet? | This is a $d=st$ problem, so let $x$ be the time it takes to meet. We can write the following equation:
\[12x+18x=45\]
Solving gives us $x=1.5$. The $18x$ is Alicia so $18\times1.5=\boxed{\textbf{(E) 27}}$
| 1 | [
"Applied Mathematics -> Math Word Problems"
] |
1 | 36 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_10 | Positive real numbers $x$ and $y$ satisfy $y^3=x^2$ and $(y-x)^2=4y^2$. What is $x+y$? | Because $y^3=x^2$, set $x=a^3$, $y=a^2$ ($a\neq 0$). Put them in $(y-x)^2=4y^2$ we get $(a^2(a-1))^2=4a^4$ which implies $a^2-2a+1=4$. Solve the equation to get $a=3$ or $-1$. Since $x$ and $y$ are positive, $a=3$ and $x+y=3^3+3^2=\boxed{\textbf{(D)} 36}$.
| 1 | [
"Algebra -> Algebra -> Equations and Inequalities"
] |
2 | 45 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_11 | What is the degree measure of the acute angle formed by lines with slopes $2$ and $\frac{1}{3}$? | Remind that $\text{slope}=\dfrac{\Delta y}{\Delta x}=\tan \theta$ where $\theta$ is the angle between the slope and $x$-axis. $k_1=2=\tan \alpha$, $k_2=\dfrac{1}{3}=\tan \beta$. The angle formed by the two lines is $\alpha-\beta$. $\tan(\alpha-\beta)=\dfrac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}=\dfrac{2-1/3}{1+2... | 1 | [
"Geometry -> Plane Geometry -> Angles",
"Algebra -> Algebra -> Algebraic Expressions"
] |
3 | 3,159 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_12 | What is the value of
\[2^3 - 1^3 + 4^3 - 3^3 + 6^3 - 5^3 + \dots + 18^3 - 17^3?\] | To solve this problem, we will be using difference of cube, sum of squares and sum of arithmetic sequence formulas.
\[2^3-1^3+4^3-3^3+6^3-5^3+...+18^3-17^3\]
$=(2-1)(2^2+1 \cdot 2+1^2)+(4-3)(4^2+4 \cdot 3+3^2)+(6-5)(6^2+6 \cdot 5+5^2)+...+(18-17)(18^2+18 \cdot 17+17^2)$
$=(2^2+1 \cdot 2+1^2)+(4^2+4 \cdot 3+3^2)+(6^2+6 ... | 2 | [
"Algebra -> Algebra -> Algebraic Expressions",
"Algebra -> Algebra -> Polynomial Operations",
"Algebra -> Prealgebra -> Integers"
] |
4 | 36 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_13 | In a table tennis tournament every participant played every other participant exactly once. Although there were twice as many right-handed players as left-handed players, the number of games won by left-handed players was $40\%$ more than the number of games won by right-handed players. (There were no ties and no ambid... | We know that the total amount of games must be the sum of games won by left and right handed players. Then, we can write $g = l + r$, and since $l = 1.4r$, $g = 2.4r$. Given that $r$ and $g$ are both integers, $g/2.4$ also must be an integer. From here we can see that $g$ must be divisible by 12, leaving only answers B... | 2 | [
"Applied Mathematics -> Math Word Problems",
"Discrete Mathematics -> Combinatorics"
] |
5 | 7 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_14 | How many complex numbers satisfy the equation $z^5=\overline{z}$, where $\overline{z}$ is the conjugate of the complex number $z$? | When $z^5=\overline{z}$, there are two conditions: either $z=0$ or $z\neq 0$. When $z\neq 0$, since $|z^5|=|\overline{z}|$, $|z|=1$. $z^5\cdot z=z^6=\overline{z}\cdot z=|z|^2=1$. Consider the $r(\cos \theta +i\sin \theta)$ form, when $z^6=1$, there are 6 different solutions for $z$. Therefore, the number of complex num... | 2 | [
"Algebra -> Intermediate Algebra -> Complex Numbers"
] |
7 | 21 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_16 | Consider the set of complex numbers $z$ satisfying $|1+z+z^{2}|=4$. The maximum value of the imaginary part of $z$ can be written in the form $\tfrac{\sqrt{m}}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? | First, substitute in $z=a+bi$.
\[|1+(a+bi)+(a+bi)^2|=4\]
\[|(1+a+a^2-b^2)+ (b+2ab)i|=4\]
\[(1+a+a^2-b^2)^2+ (b+2ab)^2=16\]
\[(1+a+a^2-b^2)^2+ b^2(1+4a+4a^2)=16\]
Let $p=b^2$ and $q=1+a+a^2$
\[(q-p)^2+ p(4q-3)=16\]
\[p^2-2pq+q^2 + 4pq -3p=16\]
We are trying to maximize $b=\sqrt p$, so we'll turn the equation into a qu... | 3 | [
"Algebra -> Intermediate Algebra -> Complex Numbers"
] |
8 | 3 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_17 | Flora the frog starts at 0 on the number line and makes a sequence of jumps to the right. In any one jump, independent of previous jumps, Flora leaps a positive integer distance $m$ with probability $\frac{1}{2^m}$.
What is the probability that Flora will eventually land at 10? Write the answer as a simplified fraction... | Initially, the probability of landing at $10$ and landing past $10$ (summing the infinte series) are exactly the same. Landing before 10 repeats this initial condition, with a different irrelevant scaling factor. Therefore, the probability must be $\boxed{\textbf{(E)}~\frac12}$.
| 3 | [
"Applied Mathematics -> Probability -> Other",
"Discrete Mathematics -> Combinatorics -> Other"
] |
10 | 1 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_19 | What is the product of all solutions to the equation
\[\log_{7x}2023\cdot \log_{289x}2023=\log_{2023x}2023\] | For $\log_{7x}2023\cdot \log_{289x}2023=\log_{2023x}2023$, transform it into $\dfrac{\ln 289+\ln 7}{\ln 7 + \ln x}\cdot \dfrac{\ln 289+\ln 7}{\ln 289 + \ln x}=\dfrac{\ln 289+\ln 7}{\ln 289+\ln 7+\ln x}$. Replace $\ln x$ with $y$. Because we want to find the product of all solutions of $x$, it is equivalent to finding t... | 2 | [
"Algebra -> Algebra -> Equations and Inequalities",
"Algebra -> Intermediate Algebra -> Exponential Functions"
] |
11 | 4 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_2 | The weight of $\frac{1}{3}$ of a large pizza together with $3 \frac{1}{2}$ cups of orange slices is the same as the weight of $\frac{3}{4}$ of a large pizza together with $\frac{1}{2}$ cup of orange slices. A cup of orange slices weighs $\frac{1}{4}$ of a pound. What is the weight, in pounds, of a large pizza? The answ... | Use a system of equations. Let $x$ be the weight of a pizza and $y$ be the weight of a cup of orange slices.
We have \[\frac{1}{3}x+\frac{7}{2}y=\frac{3}{4}x+\frac{1}{2}y.\]
Rearranging, we get \begin{align*} \frac{5}{12}x&=3y, \\ x&=\frac{36}{5}y. \end{align*}
Plugging in $\frac{1}{4}$ pounds for $y$ by the given give... | 1 | [
"Applied Mathematics -> Math Word Problems",
"Algebra -> Algebra -> Equations and Inequalities"
] |
12 | 5 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_20 | Rows 1, 2, 3, 4, and 5 of a triangular array of integers are shown below.
1
1 1
1 3 1
1 5 5 1
1 7 11 7 1
Each row after the first row is formed by placing a 1 at each end of the row, and each interior entry is 1 greater than the sum of the two numbers diagonally above it in the previous row. What is the units digits of... | First, let $R(n)$ be the sum of the $n$th row. Now, with some observation and math instinct, we can guess that $R(n) = 2^n - n$.
Now we try to prove it by induction,
$R(1) = 2^n - n = 2^1 - 1 = 1$ (works for base case)
$R(k) = 2^k - k$
$R(k+1) = 2^{k+1} - (k + 1) = 2(2^k) - k - 1$
By definition from the question, the n... | 2 | [
"Algebra -> Algebra -> Algebraic Expressions",
"Number Theory -> Congruences",
"Discrete Mathematics -> Combinatorics"
] |
13 | 29 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_21 | If $A$ and $B$ are vertices of a polyhedron, define the distance $d(A,B)$ to be the minimum number of edges of the polyhedron one must traverse in order to connect $A$ and $B$. For example, if $\overline{AB}$ is an edge of the polyhedron, then $d(A, B) = 1$, but if $\overline{AC}$ and $\overline{CB}$ are edges and $\ov... | To find the total amount of vertices we first find the amount of edges, and that is $\frac{20 \times 3}{2}$. Next, to find the amount of vertices we can use Euler's characteristic, $V - E + F = 2$, and therefore the amount of vertices is $12$
So there are $P(12,3) = 1320$ ways to choose 3 distinct points.
Now, the furt... | 3 | [
"Discrete Mathematics -> Graph Theory",
"Applied Mathematics -> Probability"
] |
14 | 96 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_22 | Let $f$ be the unique function defined on the positive integers such that \[\sum_{d\mid n}d\cdot f\left(\frac{n}{d}\right)=1\] for all positive integers $n$. What is $f(2023)$? | First, we note that $f(1) = 1$, since the only divisor of $1$ is itself.
Then, let's look at $f(p)$ for $p$ a prime. We see that
\[\sum_{d \mid p} d \cdot f\left(\frac{p}{d}\right) = 1\]
\[1 \cdot f(p) + p \cdot f(1) = 1\]
\[f(p) = 1 - p \cdot f(1)\]
\[f(p) = 1-p\]
Nice.
Now consider $f(p^k)$, for $k \in \mathbb... | 3 | [
"Number Theory -> Other"
] |
15 | 1 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_23 | How many ordered pairs of positive real numbers $(a,b)$ satisfy the equation
\[(1+2a)(2+2b)(2a+b) = 32ab?\] | Using the AM-GM inequality on the two terms in each factor on the left-hand side, we get
\[(1+2a)(2+2b)(2a+b) \ge 8\sqrt{2a \cdot 4b \cdot 2ab}= 32ab,\]
This means the equality condition must be satisfied. Therefore, we must have $1 = 2a = b$, so the only solution is $\boxed{\textbf{(B) }1}$.
| 2 | [
"Algebra -> Algebra -> Equations and Inequalities",
"Algebra -> Algebra -> Algebraic Expressions"
] |
16 | 5 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_24 | Let $K$ be the number of sequences $A_1$, $A_2$, $\dots$, $A_n$ such that $n$ is a positive integer less than or equal to $10$, each $A_i$ is a subset of $\{1, 2, 3, \dots, 10\}$, and $A_{i-1}$ is a subset of $A_i$ for each $i$ between $2$ and $n$, inclusive. For example, $\{\}$, $\{5, 7\}$, $\{2, 5, 7\}$, $\{2, 5, 7\}... | Consider any sequence with $n$ terms. Every 10 number has such choices: never appear, appear the first time in the first spot, appear the first time in the second spot… and appear the first time in the $n$th spot, which means every number has $(n+1)$ choices to show up in the sequence. Consequently, for each sequence ... | 3 | [
"Discrete Mathematics -> Combinatorics",
"Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] |
17 | -1 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_25 | There is a unique sequence of integers $a_1, a_2, \cdots a_{2023}$ such that
\[\tan2023x = \frac{a_1 \tan x + a_3 \tan^3 x + a_5 \tan^5 x + \cdots + a_{2023} \tan^{2023} x}{1 + a_2 \tan^2 x + a_4 \tan^4 x \cdots + a_{2022} \tan^{2022} x}\]whenever $\tan 2023x$ is defined. What is $a_{2023}?$ | \begin{align*} \cos 2023 x + i \sin 2023 x &= (\cos x + i \sin x)^{2023}\\ &= \cos^{2023} x + \binom{2023}{1} \cos^{2022} x (i\sin x) + \binom{2023}{2} \cos^{2021} x (i \sin x)^{2} +\binom{2020}{3} \cos^{2020} x (i \sin x)^{3}\\ &+ \dots + \binom{2023}{2022} \cos x (i \sin x)^{2022} + (i \sin x)^{2023}\\ &= \cos^{2023}... | 2 | [
"Precalculus -> Trigonometric Functions",
"Algebra -> Algebraic Expressions"
] |
18 | 8 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_3 | How many positive perfect squares less than $2023$ are divisible by $5$? | Since $\left \lfloor{\sqrt{2023}}\right \rfloor = 44$, there are $\left \lfloor{\frac{44}{5}}\right \rfloor = \boxed{\textbf{(A) 8}}$ perfect squares less than 2023 that are divisible by 5.
| 1 | [
"Number Theory -> Factorization",
"Number Theory -> Other"
] |
19 | 18 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_4 | How many digits are in the base-ten representation of $8^5 \cdot 5^{10} \cdot 15^5$? | Prime factorizing this gives us $2^{15}\cdot3^{5}\cdot5^{15}=10^{15}\cdot3^5=243\cdot10^{15}$.
$10^{15}$ has $16$ digits and $243$ = $2.43*10^{2}$ gives us $3$ more digits. $16+2=\text{\boxed{\textbf{(E) }18}}$
$2.43*10^{17}$ has $18$ digits
| 1 | [
"Algebra -> Algebraic Expressions",
"Number Theory -> Factorization"
] |
20 | 265 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_5 | Janet rolls a standard $6$-sided die $4$ times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal $3$? The final answer can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? | There are $3$ cases where the running total will equal $3$: one roll; two rolls; or three rolls:
The chance of rolling a running total of $3$, namely $(3)$ in exactly one roll is $\frac{1}{6}$.
The chance of rolling a running total of $3$ in exactly two rolls, namely $(1, 2)$ and $(2, 1)$ is $\frac{1}{6}\cdot\frac{1}... | 2 | [
"Applied Mathematics -> Probability -> Combinations"
] |
21 | 9 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_6 | Points $A$ and $B$ lie on the graph of $y=\log_{2}x$. The midpoint of $\overline{AB}$ is $(6, 2)$. What is the positive difference between the $x$-coordinates of $A$ and $B$? The final answer can be written in the form $m \sqrt{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? | Let $A(6+m,2+n)$ and $B(6-m,2-n)$, since $(6,2)$ is their midpoint. Thus, we must find $2m$. We find two equations due to $A,B$ both lying on the function $y=\log_{2}x$. The two equations are then $\log_{2}(6+m)=2+n$ and $\log_{2}(6-m)=2-n$. Now add these two equations to obtain $\log_{2}(6+m)+\log_{2}(6-m)=4$. By loga... | 2 | [
"Algebra -> Intermediate Algebra -> Logarithmic Functions",
"Algebra -> Algebra -> Equations and Inequalities"
] |
22 | 9 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_7 | A digital display shows the current date as an $8$-digit integer consisting of a $4$-digit year, followed by a $2$-digit month, followed by a $2$-digit date within the month. For example, Arbor Day this year is displayed as 20230428. For how many dates in $2023$ will each digit appear an even number of times in the 8-d... | Do careful casework by each month. In the month and the date, we need a $0$, a $3$, and two digits repeated (which has to be $1$ and $2$ after consideration). After the casework, we get $\boxed{\textbf{(E)}~9}$.
For curious readers, the numbers (in chronological order) are:
| 2 | [
"Discrete Mathematics -> Combinatorics"
] |
23 | 7 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_8 | Maureen is keeping track of the mean of her quiz scores this semester. If Maureen scores an $11$ on the next quiz, her mean will increase by $1$. If she scores an $11$ on each of the next three quizzes, her mean will increase by $2$. What is the mean of her quiz scores currently? | Let $a$ represent the amount of tests taken previously and $x$ the mean of the scores taken previously.
We can write the following equations:
\[\frac{ax+11}{a+1}=x+1\qquad (1)\]
\[\frac{ax+33}{a+3}=x+2\qquad (2)\]
Multiplying equation $(1)$ by $(a+1)$ and solving, we get:
\[ax+11=ax+a+x+1\]
\[11=a+x+1\]
\[a+x=10\qquad... | 1 | [
"Applied Mathematics -> Math Word Problems",
"Algebra -> Algebra -> Equations and Inequalities"
] |
25 | 7 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_1 | Mrs. Jones is pouring orange juice into four identical glasses for her four sons. She fills the first three glasses completely but runs out of juice when the fourth glass is only $\frac{1}{3}$ full. What fraction of a glass must Mrs. Jones pour from each of the first three glasses into the fourth glass so that all four... | The first three glasses each have a full glass. Let's assume that each glass has "1 unit" of juice. It won't matter exactly how much juice everyone has because we're dealing with ratios, and that wouldn't affect our answer. The fourth glass has a glass that is one third. So the total amount of juice will be $1+1+1+\df... | 1 | [
"Applied Mathematics -> Math Word Problems"
] |
26 | 7 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_10 | In the $xy$-plane, a circle of radius $4$ with center on the positive $x$-axis is tangent to the $y$-axis at the origin, and a circle with radius $10$ with center on the positive $y$-axis is tangent to the $x$-axis at the origin. What is the slope of the line passing through the two points at which these circles inters... | The center of the first circle is $(4,0)$.
The center of the second circle is $(0,10)$.
Thus, the slope of the line that passes through these two centers is $- \frac{10}{4} = - \frac{5}{2}$.
Because this line is the perpendicular bisector of the line that passes through two intersecting points of two circles, the slope... | 2 | [
"Geometry -> Plane Geometry -> Circles"
] |
27 | 13 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_11 | Calculate the maximum area of an isosceles trapezoid that has legs of length $1$ and one base twice as long as the other. The final answer can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m^2+n^2$? | Let the trapezoid be $ABCD$ with $AD = BC = 1, \; AB = x, CD = 2x$. Extend $AD$ and $BC$ to meet at point $E$. Then, notice $\triangle ABE \sim \triangle DCE$ with side length ratio $1:2$ and $AE = BE = 1$. Thus, $[DCE] = 4 \cdot [ABE]$ and $[ABCD] = [DCE] - [ABE] = \frac{3}{4} \cdot [DCE]$.
The problem reduces to maxi... | 3 | [
"Geometry -> Plane Geometry -> Area",
"Geometry -> Plane Geometry -> Polygons"
] |
28 | 50 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_12 | For complex number $u = a+bi$ and $v = c+di$ (where $i=\sqrt{-1}$), define the binary operation
$u \otimes v = ac + bdi$
Suppose $z$ is a complex number such that $z\otimes z = z^{2}+40$. What is $|z|^2$? | let $z$ = $a+bi$.
$z \otimes z = a^{2}+b^{2}i$.
This is equal to $z^{2} + 40 = a^{2}-b^{2}+40+2abi$
Since the real values have to be equal to each other, $a^{2}-b^{2}+40 = a^{2}$.
Simple algebra shows $b^{2} = 40$, so $b$ is $2\sqrt{10}$.
The imaginary components must also equal each other, meaning $b^{2} = 2ab$, or $b... | 2 | [
"Algebra -> Intermediate Algebra -> Complex Numbers"
] |
29 | 13 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_13 | A rectangular box $P$ has distinct edge lengths $a$, $b$, and $c$. The sum of the lengths of all $12$ edges of $P$ is $13$, the areas of all $6$ faces of $P$ is $\frac{11}{2}$, and the volume of $P$ is $\frac{1}{2}$. Find the length of the longest interior diagonal connecting two vertices of $P$. The final answer can b... | [asy] import geometry; pair A = (-3, 4); pair B = (-3, 5); pair C = (-1, 4); pair D = (-1, 5); pair AA = (0, 0); pair BB = (0, 1); pair CC = (2, 0); pair DD = (2, 1); draw(D--AA,dashed); draw(A--B); draw(A--C); draw(B--D); draw(C--D); draw(A--AA); draw(B--BB); draw(C--CC); draw(D--DD); // Dotted vertices dot(A... | 2 | [
"Geometry -> Solid Geometry -> 3D Shapes",
"Algebra -> Algebra -> Equations and Inequalities"
] |
30 | 5 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_14 | For how many ordered pairs $(a,b)$ of integers does the polynomial $x^3+ax^2+bx+6$ have $3$ distinct integer roots? | Denote three roots as $r_1 < r_2 < r_3$.
Following from Vieta's formula, $r_1r_2r_3 = -6$.
Case 1: All roots are negative.
We have the following solution: $\left( -3, -2, -1 \right)$.
Case 2: One root is negative and two roots are positive.
We have the following solutions: $\left( -3, 1, 2 \right)$, $\left( -2, 1, 3 \r... | 2 | [
"Algebra -> Algebra -> Equations and Inequalities",
"Number Theory -> Factorization"
] |
32 | 11 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_16 | In the state of Coinland, coins have values $6,10,$ and $15$ cents. Suppose $x$ is the value in cents of the most expensive item in Coinland that cannot be purchased using these coins with exact change. What is the sum of the digits of $x?$ | This problem asks to find largest $x$ that cannot be written as
\[6 a + 10 b + 15 c = x, \hspace{1cm} (1)\]
where $a, b, c \in \Bbb Z_+$.
Denote by $r \in \left\{ 0, 1 \right\}$ the remainder of $x$ divided by 2.
Modulo 2 on Equation (1), we get
By using modulus $m \in \left\{ 2, 3, 5 \right\}$ on the equation above, w... | 2 | [
"Number Theory -> Factorization",
"Algebra -> Algebra -> Equations and Inequalities"
] |
33 | 18 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_17 | Triangle $ABC$ has side lengths in arithmetic progression, and the smallest side has length $6.$ If the triangle has an angle of $120^\circ,$ Find the area of $ABC$. The final answer can be simplified in the form $m \sqrt{n}$, where $m$ and $n$ are positive integers and $n$ without square factore. What is $m+n$? | The length of the side opposite to the angle with $120^\circ$ is longest.
We denote its value as $x$.
Because three side lengths form an arithmetic sequence, the middle-valued side length is $\frac{x + 6}{2}$.
Following from the law of cosines, we have
\begin{align*} 6^2 + \left( \frac{x + 6}{2} \right)^2 - 2 \cdot 6... | 2 | [
"Geometry -> Plane Geometry -> Triangulations",
"Algebra -> Algebra -> Equations and Inequalities"
] |
36 | 50 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_2 | Carlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by $20\%$ on every pair of shoes. Carlos also knew that he had to pay a $7.5\%$ sales tax on the discounted price. He had $$43$ dollars. What is the original (before discount) price of the most expensive shoes he could a... | We can create the equation:
\[0.8x \cdot 1.075 = 43\]
using the information given. This is because x, the original price, got reduced by 20%, or multiplied by 0.8, and it also got multiplied by 1.075 on the discounted price. Solving that equation, we get
\[\frac{4}{5} \cdot x \cdot \frac{43}{40} = 43\]
\[\frac{4}{5} \c... | 1 | [
"Applied Mathematics -> Math Word Problems",
"Algebra -> Prealgebra -> Simple Equations"
] |
40 | 11 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_23 | When $n$ standard six-sided dice are rolled, the product of the numbers rolled can be any of $936$ possible values. What is $n$? | We start by trying to prove a function of $n$, and then we can apply the function and equate it to $936$ to find the value of $n$.
It is helpful to think of this problem in the format $(1+2+3+4+5+6) \cdot (1+2+3+4+5+6) \dots$. Note that if we represent the scenario in this manner, we can think of picking a $1$ for one ... | 5 | [
"Discrete Mathematics -> Combinatorics",
"Number Theory -> Factorization"
] |
41 | 3 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_24 | Suppose that $a$, $b$, $c$ and $d$ are positive integers satisfying all of the following relations.
\[abcd=2^6\cdot 3^9\cdot 5^7\]
\[\text{lcm}(a,b)=2^3\cdot 3^2\cdot 5^3\]
\[\text{lcm}(a,c)=2^3\cdot 3^3\cdot 5^3\]
\[\text{lcm}(a,d)=2^3\cdot 3^3\cdot 5^3\]
\[\text{lcm}(b,c)=2^1\cdot 3^3\cdot 5^2\]
\[\text{lcm}(b,d)=2^2... | Denote by $\nu_p (x)$ the number of prime factor $p$ in number $x$.
We index Equations given in this problem from (1) to (7).
First, we compute $\nu_2 (x)$ for $x \in \left\{ a, b, c, d \right\}$.
Equation (5) implies $\max \left\{ \nu_2 (b), \nu_2 (c) \right\} = 1$.
Equation (2) implies $\max \left\{ \nu_2 (a), \nu_2... | 4 | [
"Number Theory -> Greatest Common Divisors (GCD)",
"Number Theory -> Factorization"
] |
43 | 194 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_3 | A $3-4-5$ right triangle is inscribed in circle $A$, and a $5-12-13$ right triangle is inscribed in circle $B$. Find the ratio of the area of circle $A$ to the area of circle $B$. The final answer can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? | Because the triangles are right triangles, we know the hypotenuses are diameters of circles $A$ and $B$. Thus, their radii are 2.5 and 6.5 (respectively). Square the two numbers and multiply $\pi$ to get $6.25\pi$ and $42.25\pi$ as the areas of the circles. Multiply 4 on both numbers to get $25\pi$ and $169\pi$. Cancel... | 1 | [
"Geometry -> Plane Geometry -> Circles"
] |
44 | 1,625 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_4 | Jackson's paintbrush makes a narrow strip with a width of $6.5$ millimeters. Jackson has enough paint to make a strip $25$ meters long. How many square centimeters of paper could Jackson cover with paint? | $6.5$ millimeters is equal to $0.65$ centimeters. $25$ meters is $2500$ centimeters. The answer is $0.65 \times 2500$, so the answer is $\boxed{\textbf{(C) 1,625}}$.
| 1 | [
"Applied Mathematics -> Math Word Problems"
] |
45 | 4 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_5 | You are playing a game. A $2 \times 1$ rectangle covers two adjacent squares (oriented either horizontally or vertically) of a $3 \times 3$ grid of squares, but you are not told which two squares are covered. Your goal is to find at least one square that is covered by the rectangle. A "turn" consists of you guessing a ... | Notice that the $3\times3$ square grid has a total of $12$ possible $2\times1$ rectangles.
Suppose you choose the middle square for one of your turns. The middle square is covered by $4$ rectangles, and each of the remaining $8$ squares is covered by a maximum of $2$ uncounted rectangles. This means that the number of ... | 3 | [
"Discrete Mathematics -> Combinatorics",
"Applied Mathematics -> Math Word Problems"
] |
46 | 6 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_6 | When the roots of the polynomial
\[P(x) = (x-1)^1 (x-2)^2 (x-3)^3 \cdot \cdot \cdot (x-10)^{10}\]
are removed from the number line, what remains is the union of $11$ disjoint open intervals. On how many of these intervals is $P(x)$ positive? | The expressions to the power of even powers are always positive, so we don't need to care about those. We only need to care about $(x-1)^1(x-3)^3(x-5)^5(x-7)^7(x-9)^9$. We need 0, 2, or 4 of the expressions to be negative. The 9 through 10 interval and 10 plus interval make all of the expressions positive. The 5 throug... | 2 | [
"Algebra -> Algebra -> Polynomial Operations"
] |
47 | 901 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_7 | For how many integers $n$ does the expression\[\sqrt{\frac{\log (n^2) - (\log n)^2}{\log n - 3}}\]represent a real number, where log denotes the base $10$ logarithm? | We have
\begin{align*} \sqrt{\frac{\log \left( n^2 \right) - \left( \log n \right)^2}{\log n - 3}} & = \sqrt{\frac{2 \log n - \left( \log n \right)^2}{\log n - 3}} \\ & = \sqrt{\frac{\left( \log n \right) \left( 2 - \log n\right)}{\log n - 3}} . \end{align*}
Because $n$ is an integer and $\log n$ is well defined, $n$ m... | 2 | [
"Algebra -> Algebraic Expressions -> Other",
"Precalculus -> Functions -> Other"
] |
48 | 144 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_8 | How many nonempty subsets $B$ of ${0, 1, 2, 3, \cdots, 12}$ have the property that the number of elements in $B$ is equal to the least element of $B$? For example, $B = {4, 6, 8, 11}$ satisfies the condition. | There is no way to have a set with 0. If a set is to have its lowest element as 1, it must have only 1 element: 1. If a set is to have its lowest element as 2, it must have 2, and the other element will be chosen from the natural numbers between 3 and 12, inclusive. To calculate this, we do $\binom{10}{1}$. If the set ... | 2 | [
"Discrete Mathematics -> Combinatorics"
] |
49 | 8 | https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_9 | What is the area of the region in the coordinate plane defined by
$| | x | - 1 | + | | y | - 1 | \le 1$? | First consider, $|x-1|+|y-1| \le 1.$
We can see that it is a square with a side length of $\sqrt{2}$ (diagonal $2$). The area of the square is $\sqrt{2}^2 = 2.$
Next, we insert an absolute value sign into the equation and get $|x-1|+||y|-1| \le 1.$ This will double the square reflecting over x-axis.
So now we have $2$ ... | 2 | [
"Geometry -> Plane Geometry -> Area",
"Algebra -> Algebra -> Equations and Inequalities"
] |
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