link stringlengths 75 84 | letter stringclasses 5
values | answer float64 0 2,935,363,332B | problem stringlengths 14 5.33k | solution listlengths 1 13 |
|---|---|---|---|---|
https://artofproblemsolving.com/wiki/index.php/2022_AMC_12A_Problems/Problem_19 | D | 8,178 | Suppose that $13$ cards numbered $1, 2, 3, \ldots, 13$ are arranged in a row. The task is to pick them up in numerically increasing order, working repeatedly from left to right. In the example below, cards $1, 2, 3$ are picked up on the first pass, $4$ and $5$ on the second pass, $6$ on the third pass, $7, 8, 9, 10$ on... | [
"For $1\\leq k\\leq 12,$ suppose that cards $1, 2, \\ldots, k$ are picked up on the first pass. It follows that cards $k+1,k+2,\\ldots,13$ are picked up on the second pass.\nOnce we pick the spots for the cards on the first pass, there is only one way to arrange all $\\boldsymbol{13}$ cards.\nFor each value of $k,$... |
https://artofproblemsolving.com/wiki/index.php/2009_AMC_10A_Problems/Problem_13 | E | 2 | Suppose that $P = 2^m$ and $Q = 3^n$ . Which of the following is equal to $12^{mn}$ for every pair of integers $(m,n)$
$\textbf{(A)}\ P^2Q \qquad \textbf{(B)}\ P^nQ^m \qquad \textbf{(C)}\ P^nQ^{2m} \qquad \textbf{(D)}\ P^{2m}Q^n \qquad \textbf{(E)}\ P^{2n}Q^m$ | [
"We have $12^{mn} = (2\\cdot 2\\cdot 3)^{mn} = 2^{2mn} \\cdot 3^{mn} = (2^m)^{2n} \\cdot (3^n)^m = \\boxed{2}$"
] |
https://artofproblemsolving.com/wiki/index.php/2009_AMC_12A_Problems/Problem_6 | E | 2 | Suppose that $P = 2^m$ and $Q = 3^n$ . Which of the following is equal to $12^{mn}$ for every pair of integers $(m,n)$
$\textbf{(A)}\ P^2Q \qquad \textbf{(B)}\ P^nQ^m \qquad \textbf{(C)}\ P^nQ^{2m} \qquad \textbf{(D)}\ P^{2m}Q^n \qquad \textbf{(E)}\ P^{2n}Q^m$ | [
"We have $12^{mn} = (2\\cdot 2\\cdot 3)^{mn} = 2^{2mn} \\cdot 3^{mn} = (2^m)^{2n} \\cdot (3^n)^m = \\boxed{2}$"
] |
https://artofproblemsolving.com/wiki/index.php/2021_Fall_AMC_12B_Problems/Problem_14 | B | 1 | Suppose that $P(z), Q(z)$ , and $R(z)$ are polynomials with real coefficients, having degrees $2$ $3$ , and $6$ , respectively, and constant terms $1$ $2$ , and $3$ , respectively. Let $N$ be the number of distinct complex numbers $z$ that satisfy the equation $P(z) \cdot Q(z)=R(z)$ . What is the minimum possible value... | [
"The answer cannot be $0,$ as every nonconstant polynomial has at least $1$ distinct complex root (Fundamental Theorem of Algebra). Since $P(z) \\cdot Q(z)$ has degree $2 + 3 = 5,$ we conclude that $R(z) - P(z)\\cdot Q(z)$ has degree $6$ and is thus nonconstant.\nIt now suffices to illustrate an example for which $... |
https://artofproblemsolving.com/wiki/index.php/2021_AMC_10B_Problems/Problem_19 | D | 36.8 | Suppose that $S$ is a finite set of positive integers. If the greatest integer in $S$ is removed from $S$ , then the average value (arithmetic mean) of the integers remaining is $32$ . If the least integer in $S$ is also removed, then the average value of the integers remaining is $35$ . If the greatest integer is then... | [
"Let the lowest value be $L$ and the highest $G$ , and let the sum be $Z$ and the amount of numbers $n$ . We have $\\frac{Z-G}{n-1}=32$ $\\frac{Z-L-G}{n-2}=35$ $\\frac{Z-L}{n-1}=40$ , and $G=L+72$ . Clearing denominators gives $Z-G=32n-32$ $Z-L-G=35n-70$ , and $Z-L=40n-40$ . We use $G=L+72$ to turn the first equati... |
https://artofproblemsolving.com/wiki/index.php/2021_AMC_12B_Problems/Problem_12 | D | 36.8 | Suppose that $S$ is a finite set of positive integers. If the greatest integer in $S$ is removed from $S$ , then the average value (arithmetic mean) of the integers remaining is $32$ . If the least integer in $S$ is also removed, then the average value of the integers remaining is $35$ . If the greatest integer is then... | [
"Let the lowest value be $L$ and the highest $G$ , and let the sum be $Z$ and the amount of numbers $n$ . We have $\\frac{Z-G}{n-1}=32$ $\\frac{Z-L-G}{n-2}=35$ $\\frac{Z-L}{n-1}=40$ , and $G=L+72$ . Clearing denominators gives $Z-G=32n-32$ $Z-L-G=35n-70$ , and $Z-L=40n-40$ . We use $G=L+72$ to turn the first equati... |
https://artofproblemsolving.com/wiki/index.php/2022_AMC_10B_Problems/Problem_14 | B | 13 | Suppose that $S$ is a subset of $\left\{ 1, 2, 3, \cdots , 25 \right\}$ such that the sum of any two (not necessarily distinct) elements of $S$ is never an element of $S.$ What is the maximum number of elements $S$ may contain?
$\textbf{(A)}\ 12 \qquad\textbf{(B)}\ 13 \qquad\textbf{(C)}\ 14 \qquad\textbf{(D)}\ 15 \qqua... | [
"Let $M$ be the largest number in $S$ .\nWe categorize numbers $\\left\\{ 1, 2, \\ldots , M-1 \\right\\}$ (except $\\frac{M}{2}$ if $M$ is even) into $\\left\\lfloor \\frac{M-1}{2} \\right\\rfloor$ groups, such that the $i$ th group contains two numbers $i$ and $M-i$\nRecall that $M \\in S$ and the sum of two numbe... |
https://artofproblemsolving.com/wiki/index.php/2011_AMC_12A_Problems/Problem_18 | D | 8 | Suppose that $\left|x+y\right|+\left|x-y\right|=2$ . What is the maximum possible value of $x^2-6x+y^2$
$\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$ | [
"Plugging in some values, we see that the graph of the equation $|x+y|+|x-y| = 2$ is a square bounded by $x= \\pm 1$ and $y = \\pm 1$\nNotice that $x^2 - 6x + y^2 = (x-3)^2 + y^2 - 9$ means the square of the distance from a point $(x,y)$ to point $(3,0)$ minus 9. To maximize that value, we need to choose the point... |
https://artofproblemsolving.com/wiki/index.php/1991_AIME_Problems/Problem_9 | null | 44 | Suppose that $\sec x+\tan x=\frac{22}7$ and that $\csc x+\cot x=\frac mn,$ where $\frac mn$ is in lowest terms. Find $m+n^{}_{}.$ | [
"Use the two trigonometric Pythagorean identities $1 + \\tan^2 x = \\sec^2 x$ and $1 + \\cot^2 x = \\csc^2 x$\nIf we square the given $\\sec x = \\frac{22}{7} - \\tan x$ , we find that\n\\begin{align*} \\sec^2 x &= \\left(\\frac{22}7\\right)^2 - 2\\left(\\frac{22}7\\right)\\tan x + \\tan^2 x \\\\ 1 &= \\left(\\frac... |
https://artofproblemsolving.com/wiki/index.php/2020_AMC_12A_Problems/Problem_24 | B | 7 | Suppose that $\triangle{ABC}$ is an equilateral triangle of side length $s$ , with the property that there is a unique point $P$ inside the triangle such that $AP=1$ $BP=\sqrt{3}$ , and $CP=2$ . What is $s$
$\textbf{(A) } 1+\sqrt{2} \qquad \textbf{(B) } \sqrt{7} \qquad \textbf{(C) } \frac{8}{3} \qquad \textbf{(D) } \sq... | [
"We begin by rotating $\\triangle{ APB}$ counterclockwise by $60^{\\circ}$ about $A$ , such that $P\\mapsto Q$ and $B\\mapsto C$ . We see that $\\triangle{ APQ}$ is equilateral with side length $1$ , meaning that $\\angle APQ = 60^{\\circ}$ . We also see that $\\triangle{CPQ}$ is a $30$ $60$ $90$ right triangle, me... |
https://artofproblemsolving.com/wiki/index.php/2002_AMC_10B_Problems/Problem_19 | C | 0.01 | Suppose that $\{a_n\}$ is an arithmetic sequence with \[a_1+a_2+\cdots+a_{100}=100 \text{ and } a_{101}+a_{102}+\cdots+a_{200}=200.\] What is the value of $a_2 - a_1 ?$
$\mathrm{(A) \ } 0.0001\qquad \mathrm{(B) \ } 0.001\qquad \mathrm{(C) \ } 0.01\qquad \mathrm{(D) \ } 0.1\qquad \mathrm{(E) \ } 1$ | [
"We should realize that the two equations are 100 terms apart, so by subtracting the two equations in a form like...\n\\[(a_{101} - a_1) + (a_{102} - a_2) +... + (a_{200} - a_{100}) = 200-100 = 100\\]\n...we get the value of the common difference of every hundred terms one hundred times. So we have to divide the an... |
https://artofproblemsolving.com/wiki/index.php/2016_AMC_8_Problems/Problem_10 | D | 10 | Suppose that $a * b$ means $3a-b.$ What is the value of $x$ if \[2 * (5 * x)=1\] $\textbf{(A) }\frac{1}{10} \qquad\textbf{(B) }2\qquad\textbf{(C) }\frac{10}{3} \qquad\textbf{(D) }10\qquad \textbf{(E) }14$ | [
"Let us plug in $(5 * x)=1$ into $3a-b$ . Thus it would be $3(5)-x$ . Now we have $2*(15-x)=1$ . Plugging $2*(15-x)$ into $3a-b$ , we have $6-15+x=1$ . Solving for $x$ we have \\[-9+x=1\\] \\[x=\\boxed{10}\\]",
"Let us set a variable $y$ equal to $5 * x$ . Solving for y in the equation $3(2)-y=1$ , we see that y ... |
https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_24 | C | 3 | 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\c... | [
"Denote by $\\nu_p (x)$ the number of prime factor $p$ in number $x$\nWe index Equations given in this problem from (1) to (7).\nFirst, we compute $\\nu_2 (x)$ for $x \\in \\left\\{ a, b, c, d \\right\\}$\nEquation (5) implies $\\max \\left\\{ \\nu_2 (b), \\nu_2 (c) \\right\\} = 1$ .\nEquation (2) implies $\\max \\... |
https://artofproblemsolving.com/wiki/index.php/2009_AIME_II_Problems/Problem_2 | null | 469 | Suppose that $a$ $b$ , and $c$ are positive real numbers such that $a^{\log_3 7} = 27$ $b^{\log_7 11} = 49$ , and $c^{\log_{11}25} = \sqrt{11}$ . Find \[a^{(\log_3 7)^2} + b^{(\log_7 11)^2} + c^{(\log_{11} 25)^2}.\] | [
"First, we have: \\[x^{(\\log_y z)^2} = x^{\\left( (\\log_y z)^2 \\right) } = x^{(\\log_y z) \\cdot (\\log_y z) } = \\left( x^{\\log_y z} \\right)^{\\log_y z}\\]\nNow, let $x=y^w$ , then we have: \\[x^{\\log_y z} = \\left( y^w \\right)^{\\log_y z} = y^{w\\log_y z} = y^{\\log_y (z^w)} = z^w\\]\nThis is all we ne... |
https://artofproblemsolving.com/wiki/index.php/2002_AMC_12A_Problems/Problem_20 | C | 5 | Suppose that $a$ and $b$ are digits, not both nine and not both zero, and the repeating decimal $0.\overline{ab}$ is expressed as a fraction in lowest terms. How many different denominators are possible?
$\text{(A) }3 \qquad \text{(B) }4 \qquad \text{(C) }5 \qquad \text{(D) }8 \qquad \text{(E) }9$ | [
"The repeating decimal $0.\\overline{ab}$ is equal to \\[\\frac{10a+b}{100} + \\frac{10a+b}{10000} + \\cdots = (10a+b)\\cdot\\left(\\frac 1{10^2} + \\frac 1{10^4} + \\cdots \\right) = (10a+b) \\cdot \\frac 1{99} = \\frac{10a+b}{99}\\]\nWhen expressed in the lowest terms, the denominator of this fraction will alway... |
https://artofproblemsolving.com/wiki/index.php/2002_AMC_12A_Problems/Problem_20 | null | 5 | Suppose that $a$ and $b$ are digits, not both nine and not both zero, and the repeating decimal $0.\overline{ab}$ is expressed as a fraction in lowest terms. How many different denominators are possible?
$\text{(A) }3 \qquad \text{(B) }4 \qquad \text{(C) }5 \qquad \text{(D) }8 \qquad \text{(E) }9$ | [
"Since $\\frac{1}{99}=0.\\overline{01}$ , we know that $0.\\overline{ab} = \\frac{ab}{99}$ . From here, we wish to find the number of factors of $99$ , which is $6$ . However, notice that $1$ is not a possible denominator, so our answer is $6-1=\\boxed{5}$ \\[\\] ~AopsUser101",
"Since $0.\\overline{ab} = \\frac{a... |
https://artofproblemsolving.com/wiki/index.php/2009_AMC_12A_Problems/Problem_9 | null | 2 | Suppose that $f(x+3)=3x^2 + 7x + 4$ and $f(x)=ax^2 + bx + c$ . What is $a+b+c$
$\textbf{(A)}\ -1 \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 3$ | [
"As $f(x)=ax^2 + bx + c$ , we have $f(1)=a\\cdot 1^2 + b\\cdot 1 + c = a+b+c$\nTo compute $f(1)$ , set $x=-2$ in the first formula. We get $f(1) = f(-2+3) = 3(-2)^2 + 7(-2) + 4 = 12 - 14 + 4 = \\boxed{2}$",
"Combining the two formulas, we know that $f(x+3) = a(x+3)^2 + b(x+3) + c$\nWe can rearrange the right hand... |
https://artofproblemsolving.com/wiki/index.php/2007_AMC_10A_Problems/Problem_17 | D | 60 | Suppose that $m$ and $n$ are positive integers such that $75m = n^{3}$ . What is the minimum possible value of $m + n$
$\text{(A)}\ 15 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 50 \qquad \text{(D)}\ 60 \qquad \text{(E)}\ 5700$ | [
"$3 \\cdot 5^2m$ must be a perfect cube, so each power of a prime in the factorization for $3 \\cdot 5^2m$ must be divisible by $3$ . Thus the minimum value of $m$ is $3^2 \\cdot 5 = 45$ , which makes $n = \\sqrt[3]{3^3 \\cdot 5^3} = 15$ . The minimum possible value for the sum of $m$ and $n$ is $\\boxed{60}.$",
... |
https://artofproblemsolving.com/wiki/index.php/2001_AMC_10_Problems/Problem_12 | D | 28 | Suppose that $n$ is the product of three consecutive integers and that $n$ is divisible by $7$ . Which of the following is not necessarily a divisor of $n$
$\textbf{(A)}\ 6 \qquad \textbf{(B)}\ 14 \qquad \textbf{(C)}\ 21 \qquad \textbf{(D)}\ 28 \qquad \textbf{(E)}\ 42$ | [
"Whenever $n$ is the product of three consecutive integers, $n$ is divisible by $3!$ , meaning it is divisible by $6$\nIt also mentions that it is divisible by $7$ , so the number is definitely divisible by all the factors of $42$\nIn our answer choices, the one that is not a factor of $42$ is $\\boxed{28}$",
"We... |
https://artofproblemsolving.com/wiki/index.php/2018_AIME_II_Problems/Problem_5 | null | 74 | Suppose that $x$ $y$ , and $z$ are complex numbers such that $xy = -80 - 320i$ $yz = 60$ , and $zx = -96 + 24i$ , where $i$ $=$ $\sqrt{-1}$ . Then there are real numbers $a$ and $b$ such that $x + y + z = a + bi$ . Find $a^2 + b^2$ | [
"The First (pun intended) thing to notice is that $xy$ and $zx$ have a similar structure, but not exactly conjugates, but instead once you take out the magnitudes of both, simply multiples of a root of unity. It turns out that root of unity is $e^{\\frac{3\\pi i}{2}}$ . Anyway this results in getting that $\\left(\... |
https://artofproblemsolving.com/wiki/index.php/2000_AIME_I_Problems/Problem_7 | null | 5 | Suppose that $x,$ $y,$ and $z$ are three positive numbers that satisfy the equations $xyz = 1,$ $x + \frac {1}{z} = 5,$ and $y + \frac {1}{x} = 29.$ Then $z + \frac {1}{y} = \frac {m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$
note: this is the type of problem that makes you think symme... | [
"We can rewrite $xyz=1$ as $\\frac{1}{z}=xy$\nSubstituting into one of the given equations, we have \\[x+xy=5\\] \\[x(1+y)=5\\] \\[\\frac{1}{x}=\\frac{1+y}{5}.\\]\nWe can substitute back into $y+\\frac{1}{x}=29$ to obtain \\[y+\\frac{1+y}{5}=29\\] \\[5y+1+y=145\\] \\[y=24.\\]\nWe can then substitute once again to g... |
https://artofproblemsolving.com/wiki/index.php/2010_AIME_I_Problems/Problem_3 | null | 529 | Suppose that $y = \frac34x$ and $x^y = y^x$ . The quantity $x + y$ can be expressed as a rational number $\frac {r}{s}$ , where $r$ and $s$ are relatively prime positive integers. Find $r + s$ | [
"Substitute $y = \\frac34x$ into $x^y = y^x$ and solve. \\[x^{\\frac34x} = \\left(\\frac34x\\right)^x\\] \\[x^{\\frac34x} = \\left(\\frac34\\right)^x \\cdot x^x\\] \\[x^{-\\frac14x} = \\left(\\frac34\\right)^x\\] \\[x^{-\\frac14} = \\frac34\\] \\[x = \\frac{256}{81}\\] \\[y = \\frac34x = \\frac{192}{81}\\] \\[x + y... |
https://artofproblemsolving.com/wiki/index.php/1988_AIME_Problems/Problem_4 | null | 20 | Suppose that $|x_i| < 1$ for $i = 1, 2, \dots, n$ . Suppose further that $|x_1| + |x_2| + \dots + |x_n| = 19 + |x_1 + x_2 + \dots + x_n|.$ What is the smallest possible value of $n$ | [
"Since $|x_i| < 1$ then\n\\[|x_1| + |x_2| + \\dots + |x_n| = 19 + |x_1 + x_2 + \\dots + x_n| < n.\\]\nSo $n \\ge 20$ . We now just need to find an example where $n = 20$ : suppose $x_{2k-1} = \\frac{19}{20}$ and $x_{2k} = -\\frac{19}{20}$ ; then on the left hand side we have $\\left|\\frac{19}{20}\\right| + \\left|... |
https://artofproblemsolving.com/wiki/index.php/1989_AHSME_Problems/Problem_30 | A | 9 | Suppose that 7 boys and 13 girls line up in a row. Let $S$ be the number of places in the row where a boy and a girl are standing next to each other. For example, for the row $\text{GBBGGGBGBGGGBGBGGBGG}$ we have that $S=12$ . The average value of $S$ (if all possible orders of these 20 people are considered) is closes... | [
"We approach this problem using Linearity of Expectation. Consider a pair of two people standing next to each other. Ignoring all other people, the probability that a boy is standing on the left position and a girl is standing on the right position is $\\frac7{20}\\cdot\\frac{13}{19}$ . Similarly, if a girl is stan... |
https://artofproblemsolving.com/wiki/index.php/1992_AJHSME_Problems/Problem_6 | D | 1 | Suppose that [asy] unitsize(18); draw((0,0)--(2,0)--(1,sqrt(3))--cycle); label("$a$",(1,sqrt(3)-0.2),S); label("$b$",(sqrt(3)/10,0.1),ENE); label("$c$",(2-sqrt(3)/10,0.1),WNW); [/asy] means $a+b-c$ .
For example, [asy] unitsize(18); draw((0,0)--(2,0)--(1,sqrt(3))--cycle); label("$5$",(1,sqrt(3)-0.2),S); label("$4$",(s... | [
"The first triangle represents $1+3-4$ The 2nd triangle represents $2+5-6$\nSolving the first triangle, we get $0$ Solving the 2nd triangle, we get $1$\nSince we have to add the 2 triangles the final answer is $1$ , which is $\\boxed{1}$"
] |
https://artofproblemsolving.com/wiki/index.php/2011_AIME_I_Problems/Problem_6 | null | 11 | Suppose that a parabola has vertex $\left(\frac{1}{4},-\frac{9}{8}\right)$ and equation $y = ax^2 + bx + c$ , where $a > 0$ and $a + b + c$ is an integer. The minimum possible value of $a$ can be written in the form $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p + q$ | [
"If the vertex is at $\\left(\\frac{1}{4}, -\\frac{9}{8}\\right)$ , the equation of the parabola can be expressed in the form \\[y=a\\left(x-\\frac{1}{4}\\right)^2-\\frac{9}{8}.\\] Expanding, we find that \\[y=a\\left(x^2-\\frac{x}{2}+\\frac{1}{16}\\right)-\\frac{9}{8},\\] and \\[y=ax^2-\\frac{ax}{2}+\\frac{a}{16}-... |
https://artofproblemsolving.com/wiki/index.php/2014_AIME_II_Problems/Problem_12 | null | 399 | Suppose that the angles of $\triangle ABC$ satisfy $\cos(3A)+\cos(3B)+\cos(3C)=1.$ Two sides of the triangle have lengths 10 and 13. There is a positive integer $m$ so that the maximum possible length for the remaining side of $\triangle ABC$ is $\sqrt{m}.$ Find $m.$ | [
"WLOG, let C be the largest angle in the triangle.\nAs above, we can see that $\\cos3A+\\cos3B-\\cos(3A+3B)=1$\nExpanding, we get\n$\\cos3A+\\cos3B-\\cos3A\\cos3B+\\sin3A\\sin3B=1$\n$\\cos3A\\cos3B-\\cos3A-\\cos3B+1=\\sin3A\\sin3B$\n$(\\cos3A-1)(\\cos3B-1)=\\sin3A\\sin3B$\nCASE 1: If $\\sin 3A = 0$ or $\\sin 3B = 0... |
https://artofproblemsolving.com/wiki/index.php/2012_AMC_12B_Problems/Problem_4 | B | 4 | Suppose that the euro is worth 1.3 dollars. If Diana has 500 dollars and Etienne has 400 euros, by what percent is the value of Etienne's money greater that the value of Diana's money?
$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 6.5\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 13$ | [
"The ratio $\\frac{400 \\text{ euros}}{500 \\text{ dollars}}$ can be simplified using conversion factors: \\[\\frac{400 \\text{ euros}}{500 \\text{ dollars}} \\cdot \\frac{1.3 \\text{ dollars}}{1 \\text{ euro}} = \\frac{520}{500} = 1.04\\] which means the money is greater by $\\boxed{4}$ percent.",
"If we divide ... |
https://artofproblemsolving.com/wiki/index.php/2013_AIME_II_Problems/Problem_1 | null | 275 | Suppose that the measurement of time during the day is converted to the metric system so that each day has $10$ metric hours, and each metric hour has $100$ metric minutes. Digital clocks would then be produced that would read $\text{9:99}$ just before midnight, $\text{0:00}$ at midnight, $\text{1:25}$ at the former $\... | [
"There are $24 \\cdot 60=1440$ normal minutes in a day , and $10 \\cdot 100=1000$ metric minutes in a day. The ratio of normal to metric minutes in a day is $\\frac{1440}{1000}$ , which simplifies to $\\frac{36}{25}$ . This means that every time 36 normal minutes pass, 25 metric minutes pass. From midnight to $\\te... |
https://artofproblemsolving.com/wiki/index.php/2007_AMC_10A_Problems/Problem_20 | D | 194 | Suppose that the number $a$ satisfies the equation $4 = a + a^{ - 1}$ . What is the value of $a^{4} + a^{ - 4}$
$\textbf{(A)}\ 164 \qquad \textbf{(B)}\ 172 \qquad \textbf{(C)}\ 192 \qquad \textbf{(D)}\ 194 \qquad \textbf{(E)}\ 212$ | [
"Note that for all real numbers $k,$ we have $a^{2k} + a^{-2k} + 2 = (a^{k} + a^{-k})^2,$ from which \\[a^{2k} + a^{-2k} = (a^{k} + a^{-k})^2-2.\\] We apply this result twice to get the answer: \\begin{align*} a^4 + a^{-4} &= (a^2 + a^{-2})^2 - 2 \\\\ &= [(a + a^{-1})^2 - 2]^2 - 2 \\\\ &= \\boxed{194} ~Azjps (Funda... |
https://artofproblemsolving.com/wiki/index.php/1983_AIME_Problems/Problem_5 | null | 4 | Suppose that the sum of the squares of two complex numbers $x$ and $y$ is $7$ and the sum of the cubes is $10$ . What is the largest real value that $x + y$ can have? | [
"One way to solve this problem is by substitution . We have\n$x^2+y^2=(x+y)^2-2xy=7$ and $x^3+y^3=(x+y)(x^2-xy+y^2)=(7-xy)(x+y)=10$\nHence observe that we can write $w=x+y$ and $z=xy$\nThis reduces the equations to $w^2-2z=7$ and $w(7-z)=10$\nBecause we want the largest possible $w$ , let's find an expression for $... |
https://artofproblemsolving.com/wiki/index.php/1988_AJHSME_Problems/Problem_12 | C | 80 | Suppose the estimated $20$ billion dollar cost to send a person to the planet Mars is shared equally by the $250$ million people in the U.S. Then each person's share is
$\text{(A)}\ 40\text{ dollars} \qquad \text{(B)}\ 50\text{ dollars} \qquad \text{(C)}\ 80\text{ dollars} \qquad \text{(D)}\ 100\text{ dollars} \qquad \... | [
"We want the cost per person, which is \\begin{align*} \\frac{20\\text{ billion}}{250\\text{ million}} &= \\frac{20000\\text{ million}}{250\\text{ million}} \\\\ &= 80 \\rightarrow \\boxed{80}"
] |
https://artofproblemsolving.com/wiki/index.php/1996_AJHSME_Problems/Problem_20 | A | 0.25 | Suppose there is a special key on a calculator that replaces the number $x$ currently displayed with the number given by the formula $1/(1-x)$ . For example, if the calculator is displaying 2 and the special key is pressed, then the calculator will display -1 since $1/(1-2)=-1$ . Now suppose that the calculator is di... | [
"We look for a pattern, hoping this sequence either settles down to one number, or that it forms a cycle that repeats.\nAfter $1$ press, the calculator displays $\\frac{1}{1 - 5} = -\\frac{1}{4}$\nAfter $2$ presses, the calculator displays $\\frac{1}{1 - (-\\frac{1}{4})} = \\frac{1}{\\frac{5}{4}} = \\frac{4}{5}$\nA... |
https://artofproblemsolving.com/wiki/index.php/2017_AMC_10B_Problems/Problem_4 | D | 2 | Supposed that $x$ and $y$ are nonzero real numbers such that $\frac{3x+y}{x-3y}=-2$ . What is the value of $\frac{x+3y}{3x-y}$
$\textbf{(A)}\ -3\qquad\textbf{(B)}\ -1\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ 3$ | [
"Rearranging, we find $3x+y=-2x+6y$ , or $5x=5y\\implies x=y$ .\nSubstituting, we can convert the second equation into $\\frac{x+3x}{3x-x}=\\frac{4x}{2x}=\\boxed{2}$",
"Substituting each $x$ and $y$ with $1$ , we see that the given equation holds true, as $\\frac{3(1)+1}{1-3(1)} = -2$ . Thus, $\\frac{x+3y}{3x-y}=... |
https://artofproblemsolving.com/wiki/index.php/2017_AMC_12B_Problems/Problem_3 | D | 2 | Supposed that $x$ and $y$ are nonzero real numbers such that $\frac{3x+y}{x-3y}=-2$ . What is the value of $\frac{x+3y}{3x-y}$
$\textbf{(A)}\ -3\qquad\textbf{(B)}\ -1\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ 3$ | [
"Rearranging, we find $3x+y=-2x+6y$ , or $5x=5y\\implies x=y$ .\nSubstituting, we can convert the second equation into $\\frac{x+3x}{3x-x}=\\frac{4x}{2x}=\\boxed{2}$",
"Substituting each $x$ and $y$ with $1$ , we see that the given equation holds true, as $\\frac{3(1)+1}{1-3(1)} = -2$ . Thus, $\\frac{x+3y}{3x-y}=... |
https://artofproblemsolving.com/wiki/index.php/2008_AMC_8_Problems/Problem_1 | B | 14 | Susan had 50 dollars to spend at the carnival. She spent 12 dollars on food and twice as much on rides. How many dollars did she have left to spend?
$\textbf{(A)}\ 12 \qquad \textbf{(B)}\ 14 \qquad \textbf{(C)}\ 26 \qquad \textbf{(D)}\ 38 \qquad \textbf{(E)}\ 50$ | [
"If Susan spent 12 dollars, then twice that much on rides, then she spent $12+12 \\times 2=36$ dollars in total. We subtract $36$ from $50$ to get $\\boxed{14}$"
] |
https://artofproblemsolving.com/wiki/index.php/2023_AMC_10B_Problems/Problem_11 | B | 21 | Suzanne went to the bank and withdrew $$800$ . The teller gave her this amount using $$20$ bills, $$50$ bills, and $$100$ bills, with at least one of each denomination. How many different collections of bills could Suzanne have received?
$\textbf{(A) } 45 \qquad \textbf{(B) } 21 \qquad \text{(C) } 36 \qquad \text{(D) }... | [
"Denote by $x$ $y$ $z$ the amount of $20 bills, $50 bills and $100 bills, respectively.\nThus, we need to find the number of tuples $\\left( x , y, z \\right)$ with $x, y, z \\in \\Bbb N$ that satisfy \\[ 20 x + 50 y + 100 z = 800. \\]\nFirst, this equation can be simplified as \\[ 2 x + 5 y + 10 z = 80. \\]\nSeco... |
https://artofproblemsolving.com/wiki/index.php/2017_AMC_10A_Problems/Problem_3 | B | 78 | Tamara has three rows of two $6$ -feet by $2$ -feet flower beds in her garden. The beds are separated and also surrounded by $1$ -foot-wide walkways, as shown on the diagram. What is the total area of the walkways, in square feet?
[asy] draw((0,0)--(0,10)--(15,10)--(15,0)--cycle); fill((0,0)--(0,10)--(15,10)--(15,0)--c... | [
"Finding the area of the shaded walkway can be achieved by computing the total area of Tamara's garden and then subtracting the combined area of her six flower beds.\nSince the width of Tamara's garden contains three margins, the total width is $2\\cdot 6+3\\cdot 1 = 15$ feet.\nSimilarly, the height of Tamara's gar... |
https://artofproblemsolving.com/wiki/index.php/2020_AMC_12B_Problems/Problem_5 | C | 42 | Teams $A$ and $B$ are playing in a basketball league where each game results in a win for one team and a loss for the other team. Team $A$ has won $\tfrac{2}{3}$ of its games and team $B$ has won $\tfrac{5}{8}$ of its games. Also, team $B$ has won $7$ more games and lost $7$ more games than team $A.$ How many games has... | [
"Suppose team $A$ has played $g$ games in total so that it has won $\\frac23g$ games.\nIt follows that team $B$ has played $g+14$ games in total so that it has won $\\frac23g+7$ games.\nWe set up and solve an equation for team $B$ 's win ratio: \\begin{align*} \\frac{\\frac23g+7}{g+14}&=\\frac58 \\\\ \\frac{16}{3}g... |
https://artofproblemsolving.com/wiki/index.php/2022_AMC_10A_Problems/Problem_11 | C | 7 | Ted mistakenly wrote $2^m\cdot\sqrt{\frac{1}{4096}}$ as $2\cdot\sqrt[m]{\frac{1}{4096}}.$ What is the sum of all real numbers $m$ for which these two expressions have the same value?
$\textbf{(A) } 5 \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 7 \qquad \textbf{(D) } 8 \qquad \textbf{(E) } 9$ | [
"We are given that \\[2^m\\cdot\\sqrt{\\frac{1}{4096}} = 2\\cdot\\sqrt[m]{\\frac{1}{4096}}.\\] Converting everything into powers of $2,$ we have \\begin{align*} 2^m\\cdot(2^{-12})^{\\frac12} &= 2\\cdot (2^{-12})^{\\frac1m} \\\\ 2^{m-6} &= 2^{1-\\frac{12}{m}} \\\\ m-6 &= 1-\\frac{12}{m}. \\end{align*} We multiply bo... |
https://artofproblemsolving.com/wiki/index.php/2013_AMC_8_Problems/Problem_11 | D | 4 | Ted's grandfather used his treadmill on 3 days this week. He went 2 miles each day. On Monday he jogged at a speed of 5 miles per hour. He walked at the rate of 3 miles per hour on Wednesday and at 4 miles per hour on Friday. If Grandfather had always walked at 4 miles per hour, he would have spent less time on the tre... | [
"We use that fact that $d=rt$ . Let d= distance, r= rate or speed, and t=time. In this case, let $x$ represent the time.\nOn Monday, he was at a rate of $5 \\text{ m.p.h}$ . So, $5x = 2 \\text{ miles}\\implies x = \\frac{2}{5} \\text { hours}$\nFor Wednesday, he walked at a rate of $3 \\text{ m.p.h}$ . Therefore, $... |
https://artofproblemsolving.com/wiki/index.php/2014_AIME_II_Problems/Problem_13 | null | 28 | Ten adults enter a room, remove their shoes, and toss their shoes into a pile. Later, a child randomly pairs each left shoe with a right shoe without regard to which shoes belong together. The probability that for every positive integer $k<5$ , no collection of $k$ pairs made by the child contains the shoes from exactl... | [
"Label the left shoes be $L_1,\\dots, L_{10}$ and the right shoes $R_1,\\dots, R_{10}$ . Notice that there are $10!$ possible pairings.\nLet a pairing be \"bad\" if it violates the stated condition. We would like a better condition to determine if a given pairing is bad.\nNote that, in order to have a bad pairing, ... |
https://artofproblemsolving.com/wiki/index.php/2014_AIME_II_Problems/Problem_9 | null | 581 | Ten chairs are arranged in a circle. Find the number of subsets of this set of chairs that contain at least three adjacent chairs. | [
"We know that a subset with less than $3$ chairs cannot contain $3$ adjacent chairs. There are only $10$ sets of $3$ chairs so that they are all $3$ adjacent. There are $10$ subsets of $4$ chairs where all $4$ are adjacent, and $10 \\cdot 5$ or $50$ where there are only $3.$ If there are $5$ chairs, $10$ have all... |
https://artofproblemsolving.com/wiki/index.php/2008_AMC_10B_Problems/Problem_21 | C | 480 | Ten chairs are evenly spaced around a round table and numbered clockwise from $1$ through $10$ . Five married couples are to sit in the chairs with men and women alternating, and no one is to sit either next to or across from his/her spouse. How many seating arrangements are possible?
$\mathrm{(A)}\ 240\qquad\mathrm{(... | [
"For the first man, there are $10$ possible seats. For each subsequent man, there are $4$ $3$ $2$ , or $1$ possible seats. After the men are seated, there are only two possible arrangements for the five women. The answer is $10\\cdot 4\\cdot 3\\cdot 2\\cdot 1\\cdot 2 = \\boxed{480}$",
"Label the seats ABCDEFGHIJ,... |
https://artofproblemsolving.com/wiki/index.php/2008_AIME_I_Problems/Problem_9 | null | 190 | Ten identical crates each of dimensions $3\mathrm{ft}\times 4\mathrm{ft}\times 6\mathrm{ft}$ . The first crate is placed flat on the floor. Each of the remaining nine crates is placed, in turn, flat on top of the previous crate, and the orientation of each crate is chosen at random. Let $\frac {m}{n}$ be the probabi... | [
"Only the heights matter, and each crate is either 3, 4, or 6 feet tall with equal probability. We have the following:\n\\begin{align*}3a + 4b + 6c &= 41\\\\ a + b + c &= 10\\end{align*}\nSubtracting 3 times the second from the first gives $b + 3c = 11$ , or $(b,c) = (2,3),(5,2),(8,1),(11,0)$ . The last doesn't wor... |
https://artofproblemsolving.com/wiki/index.php/1990_AHSME_Problems/Problem_26 | A | 1 | Ten people form a circle. Each picks a number and tells it to the two neighbors adjacent to them in the circle. Then each person computes and announces the average of the numbers of their two neighbors. The figure shows the average announced by each person ( not the original number the person picked.) [asy] unitsize(... | [
"For $i\\in\\{1,2,3,\\ldots,10\\},$ suppose Person $i$ picks the number $a_i$ and announces the number $i.$ We wish to find $a_6.$\nTaking the indices modulo $10,$ we are given that $\\frac{a_{i-1}+a_{i+1}}{2}=i,$ from which $a_{i-1}+a_{i+1}=2i.$\nWe have ten equations: five with odd-numbered indices and five with ... |
https://artofproblemsolving.com/wiki/index.php/1989_AIME_Problems/Problem_2 | null | 968 | Ten points are marked on a circle . How many distinct convex polygons of three or more sides can be drawn using some (or all) of the ten points as vertices | [
"Any subset of the ten points with three or more members can be made into exactly one such polygon. Thus, we need to count the number of such subsets. There are $2^{10} = 1024$ total subsets of a ten-member set , but of these ${10 \\choose 0} = 1$ have 0 members, ${10 \\choose 1} = 10$ have 1 member and ${10 \\ch... |
https://artofproblemsolving.com/wiki/index.php/1999_AIME_Problems/Problem_10 | null | 489 | Ten points in the plane are given, with no three collinear . Four distinct segments joining pairs of these points are chosen at random, all such segments being equally likely. The probability that some three of the segments form a triangle whose vertices are among the ten given points is $m/n,$ where $m_{}$ and $n_{}... | [
"Note that 4 points can NEVER form 2 triangles. Therefore, we just need to multiply the probability that the first three segments picked form a triangle by 4. We can pick any segment for the first choice, then only segments that share an endpoint with the first one, then the one segment that completes the triangle.... |
https://artofproblemsolving.com/wiki/index.php/2009_AMC_12B_Problems/Problem_21 | null | 89 | Ten women sit in $10$ seats in a line. All of the $10$ get up and then reseat themselves using all $10$ seats, each sitting in the seat she was in before or a seat next to the one she occupied before. In how many ways can the women be reseated?
$\textbf{(A)}\ 89\qquad \textbf{(B)}\ 90\qquad \textbf{(C)}\ 120\qquad \tex... | [
"Notice that either a woman stays in her own seat after the rearrangement, or two adjacent women swap places. Thus, our answer is counting the number of ways to arrange 1x1 and 2x1 blocks to form a 1x10 rectangle. This can be done via casework depending on the number of 2x1 blocks. The cases of 0, 1, 2, 3, 4, 5 2x1... |
https://artofproblemsolving.com/wiki/index.php/1998_AJHSME_Problems/Problem_22 | D | 27 | Terri produces a sequence of positive integers by following three rules. She starts with a positive integer, then applies the appropriate rule to the result, and continues in this fashion.
Rule 1: If the integer is less than 10, multiply it by 9.
Rule 2: If the integer is even and greater than 9, divide it by 2.
Rule ... | [
"We could start by looking for a pattern.\n$98, 49, 44, 22, 11, 6, 54, 27, 22, 11, 6, \\ldots .$\nFrom here, we see that we have a pattern of $22, 11, 6, 54, 27, \\ldots .$ after $98, 49, 44$\nOur problem is now really\nFind the $95^\\text{th}$ term of the sequence that goes $22, 11, 6, 54, 27, 22, 11, 6, 54, 27, 2... |
https://artofproblemsolving.com/wiki/index.php/2017_AIME_II_Problems/Problem_15 | null | 682 | Tetrahedron $ABCD$ has $AD=BC=28$ $AC=BD=44$ , and $AB=CD=52$ . For any point $X$ in space, suppose $f(X)=AX+BX+CX+DX$ . The least possible value of $f(X)$ can be expressed as $m\sqrt{n}$ , where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$ | [
"Let $M$ and $N$ be midpoints of $\\overline{AB}$ and $\\overline{CD}$ . The given conditions imply that $\\triangle ABD\\cong\\triangle BAC$ and $\\triangle CDA\\cong\\triangle DCB$ , and therefore $MC=MD$ and $NA=NB$ . It follows that $M$ and $N$ both lie on the common perpendicular bisector of $\\overline{AB}$ a... |
https://artofproblemsolving.com/wiki/index.php/2014_AMC_8_Problems/Problem_16 | B | 88 | The "Middle School Eight" basketball conference has $8$ teams. Every season, each team plays every other conference team twice (home and away), and each team also plays $4$ games against non-conference opponents. What is the total number of games in a season involving the "Middle School Eight" teams?
$\textbf{(A) }60\q... | [
"Within the conference, there are 8 teams, so there are $\\dbinom{8}{2}=28$ pairings of teams, and each pair must play two games, for a total of $28\\cdot 2=56$ games within the conference.\nEach team also plays 4 games outside the conference, and there are 8 teams, so there are a total of $4\\cdot 8 =32$ games out... |
https://artofproblemsolving.com/wiki/index.php/1991_AJHSME_Problems/Problem_16 | B | 9 | The $16$ squares on a piece of paper are numbered as shown in the diagram. While lying on a table, the paper is folded in half four times in the following sequence:
(1) fold the top half over the bottom half
(2) fold the bottom half over the top half
(3) fold the right half over the left half
(4) fold the left half ov... | [
"Suppose we undo each of the four folds, considering just the top square until we completely unfold the paper. $x$ will be marked in the square if the face that shows after all the folds is face up, $y$ if that face is facing down.\nStep 0: Step 1: Step 2: Step 3: Step 4: The marked square is in the same spot ... |
https://artofproblemsolving.com/wiki/index.php/2007_AMC_10B_Problems/Problem_6 | D | 16 | The $2007 \text{ AMC }10$ will be scored by awarding $6$ points for each correct response, $0$ points for each incorrect response, and $1.5$ points for each problem left unanswered. After looking over the $25$ problems, Sarah has decided to attempt the first $22$ and leave only the last $3$ unanswered. How many of the ... | [
"Sarah is leaving $3$ questions unanswered, guaranteeing her $3 \\times 1.5 = 4.5$ points. She will either get $6$ points or $0$ points for the rest of the questions. Let $x$ be the number of questions Sarah answers correctly. \\begin{align*} 6x+4.5 &\\ge 100\\\\ 6x &\\ge 95.5\\\\ x &\\ge 15.92 \\end{align*} The nu... |
https://artofproblemsolving.com/wiki/index.php/2020_AMC_10A_Problems/Problem_7 | C | 10 | The $25$ integers from $-10$ to $14,$ inclusive, can be arranged to form a $5$ -by- $5$ square in which the sum of the numbers in each row, the sum of the numbers in each column, and the sum of the numbers along each of the main diagonals are all the same. What is the value of this common sum?
$\textbf{(A) }2 \qquad\te... | [
"Without loss of generality, consider the five rows in the square. Each row must have the same sum of numbers, meaning that the sum of all the numbers in the square divided by $5$ is the total value per row. The sum of the $25$ integers is $-10+-9+...+14=11+12+13+14=50$ , and the common sum is $\\frac{50}{5}=\\boxe... |
https://artofproblemsolving.com/wiki/index.php/2020_AMC_10A_Problems/Problem_7 | null | 10 | The $25$ integers from $-10$ to $14,$ inclusive, can be arranged to form a $5$ -by- $5$ square in which the sum of the numbers in each row, the sum of the numbers in each column, and the sum of the numbers along each of the main diagonals are all the same. What is the value of this common sum?
$\textbf{(A) }2 \qquad\te... | [
"If the sum of each row, column, and diagonal is x, then we have a total of 12x for the sum. The sum of the rows and columns is the sum of all the numbers doubled, which is $50\\cdot2=100$ . Therefore $100+2x=12x$ $100=10x$ , and $x=\\boxed{10}$ .\n~MC413551"
] |
https://artofproblemsolving.com/wiki/index.php/2020_AMC_12A_Problems/Problem_5 | C | 10 | The $25$ integers from $-10$ to $14,$ inclusive, can be arranged to form a $5$ -by- $5$ square in which the sum of the numbers in each row, the sum of the numbers in each column, and the sum of the numbers along each of the main diagonals are all the same. What is the value of this common sum?
$\textbf{(A) }2 \qquad\te... | [
"Without loss of generality, consider the five rows in the square. Each row must have the same sum of numbers, meaning that the sum of all the numbers in the square divided by $5$ is the total value per row. The sum of the $25$ integers is $-10+-9+...+14=11+12+13+14=50$ , and the common sum is $\\frac{50}{5}=\\boxe... |
https://artofproblemsolving.com/wiki/index.php/2020_AMC_12A_Problems/Problem_5 | null | 10 | The $25$ integers from $-10$ to $14,$ inclusive, can be arranged to form a $5$ -by- $5$ square in which the sum of the numbers in each row, the sum of the numbers in each column, and the sum of the numbers along each of the main diagonals are all the same. What is the value of this common sum?
$\textbf{(A) }2 \qquad\te... | [
"If the sum of each row, column, and diagonal is x, then we have a total of 12x for the sum. The sum of the rows and columns is the sum of all the numbers doubled, which is $50\\cdot2=100$ . Therefore $100+2x=12x$ $100=10x$ , and $x=\\boxed{10}$ .\n~MC413551"
] |
https://artofproblemsolving.com/wiki/index.php/2018_AMC_8_Problems/Problem_7 | B | 3 | The $5$ -digit number $\underline{2}$ $\underline{0}$ $\underline{1}$ $\underline{8}$ $\underline{U}$ is divisible by $9$ . What is the remainder when this number is divided by $8$
$\textbf{(A) }1\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }7$ | [
"We use the property that the digits of a number must sum to a multiple of $9$ if it are divisible by $9$ . This means $2+0+1+8+U$ must be divisible by $9$ . The only possible value for $U$ then must be $7$ . Since we are looking for the remainder when divided by $8$ , we can ignore the thousands. The remainder whe... |
https://artofproblemsolving.com/wiki/index.php/2010_AIME_II_Problems/Problem_13 | null | 263 | The $52$ cards in a deck are numbered $1, 2, \cdots, 52$ . Alex, Blair, Corey, and Dylan each picks a card from the deck without replacement and with each card being equally likely to be picked, The two persons with lower numbered cards form a team, and the two persons with higher numbered cards form another team. Let ... | [
"Once the two cards are drawn, there are $\\dbinom{50}{2} = 1225$ ways for the other two people to draw. Alex and Dylan are the team with higher numbers if Blair and Corey both draw below $a$ , which occurs in $\\dbinom{a-1}{2}$ ways. Alex and Dylan are the team with lower numbers if Blair and Corey both draw above... |
https://artofproblemsolving.com/wiki/index.php/2004_AMC_10A_Problems/Problem_16 | D | 19 | The $5\times 5$ grid shown contains a collection of squares with sizes from $1\times 1$ to $5\times 5$ . How many of these squares contain the black center square?
2004 AMC 10A problem 16.png
$\mathrm{(A) \ } 12 \qquad \mathrm{(B) \ } 15 \qquad \mathrm{(C) \ } 17 \qquad \mathrm{(D) \ } 19\qquad \mathrm{(E) \ } 20$ | [
"Since there are five types of squares: $1 \\times 1, 2 \\times 2, 3 \\times 3, 4 \\times 4,$ and $5 \\times 5.$ We must find how many of each square contain the black shaded square in the center.\nIf we list them, we get that\nThus, the answer is $1+4+9+4+1=19\\Rightarrow\\boxed{19}$",
"We use complementary coun... |
https://artofproblemsolving.com/wiki/index.php/2014_AMC_8_Problems/Problem_21 | A | 1 | The $7$ -digit numbers $\underline{7} \underline{4} \underline{A} \underline{5} \underline{2} \underline{B} \underline{1}$ and $\underline{3} \underline{2} \underline{6} \underline{A} \underline{B} \underline{4} \underline{C}$ are each multiples of $3$ . Which of the following could be the value of $C$
$\textbf{(A) }1\... | [
"The number $\\mod{3}$ is congruent to sum of a number's digits $\\mod{3}$ is congruent to the number $\\mod{3}$ $74A52B1 \\pmod{3}$ must be congruent to 0, since it is divisible by 3. Therefore, $7+4+A+5+2+B+1 \\pmod{3}$ is also congruent to 0. $7+4+5+2+1 \\equiv 1 \\pmod{3}$ , so $A+B\\equiv 2 \\pmod{3}$ . As we ... |
https://artofproblemsolving.com/wiki/index.php/2006_AMC_10A_Problems/Problem_7 | A | 6 | The $8\times18$ rectangle $ABCD$ is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is $y$
[asy] unitsize(3mm); defaultpen(fontsize(10pt)+linewidth(.8pt)); dotfactor=4; draw((0,4)--(18,4)--(18,-4)--(0,-4)--cycle); draw((6,4)--(6,... | [
"Since the two hexagons are going to be repositioned to form a square without overlap, the area will remain the same. The rectangle's area is $18\\cdot8=144$ . This means the square will have four sides of length 12. The only way to do this is shown below.\n\nAs you can see from the diagram, the line segment denote... |
https://artofproblemsolving.com/wiki/index.php/2006_AMC_12A_Problems/Problem_6 | A | 6 | The $8\times18$ rectangle $ABCD$ is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is $y$
[asy] unitsize(3mm); defaultpen(fontsize(10pt)+linewidth(.8pt)); dotfactor=4; draw((0,4)--(18,4)--(18,-4)--(0,-4)--cycle); draw((6,4)--(6,... | [
"Since the two hexagons are going to be repositioned to form a square without overlap, the area will remain the same. The rectangle's area is $18\\cdot8=144$ . This means the square will have four sides of length 12. The only way to do this is shown below.\n\nAs you can see from the diagram, the line segment denote... |
https://artofproblemsolving.com/wiki/index.php/2022_AMC_12A_Problems/Problem_5 | C | 841 | The $\textit{taxicab distance}$ between points $(x_1, y_1)$ and $(x_2, y_2)$ in the coordinate plane is given by \[|x_1 - x_2| + |y_1 - y_2|.\] For how many points $P$ with integer coordinates is the taxicab distance between $P$ and the origin less than or equal to $20$
$\textbf{(A)} \, 441 \qquad\textbf{(B)} \, 761 \q... | [
"Let us consider the number of points for a certain $x$ -coordinate. For any $x$ , the viable points are in the range $[-20 + |x|, 20 - |x|]$ . This means that our total sum is equal to \\begin{align*} 1 + 3 + 5 + \\cdots + 41 + 39 + 37 + \\cdots + 1 &= (1 + 3 + 5 + \\cdots + 39) + (1 + 3 + 5 + \\cdots + 41) \\\\ &... |
https://artofproblemsolving.com/wiki/index.php/2014_AMC_10A_Problems/Problem_14 | D | 60 | The $y$ -intercepts, $P$ and $Q$ , of two perpendicular lines intersecting at the point $A(6,8)$ have a sum of zero. What is the area of $\triangle APQ$
$\textbf{(A)}\ 45\qquad\textbf{(B)}\ 48\qquad\textbf{(C)}\ 54\qquad\textbf{(D)}\ 60\qquad\textbf{(E)}\ 72$ | [
" Note that if the $y$ -intercepts have a sum of $0$ , the distance from the origin to each of the intercepts must be the same. Call this distance $a$ . Since the $\\angle PAQ = 90^\\circ$ , the length of the median to the midpoint of the hypotenuse is equal to half the length of the hypotenuse. Since the median's ... |
https://artofproblemsolving.com/wiki/index.php/1996_AJHSME_Problems/Problem_3 | A | 130 | The 64 whole numbers from 1 through 64 are written, one per square, on a checkerboard (an 8 by 8 array of 64 squares). The first 8 numbers are written in order across the first row, the next 8 across the second row, and so on. After all 64 numbers are written, the sum of the numbers in the four corners will be
$\text... | [
"Obviously $1$ is in the top left corner, $8$ is in the top right corner, and $64$ is in the bottom right corner. To find the bottom left corner, subtract $7$ from $64$ which is $57$ . Adding the results gives $1+8+57+64=130$ which is answer $\\boxed{130}$"
] |
https://artofproblemsolving.com/wiki/index.php/2014_AIME_I_Problems/Problem_1 | null | 790 | The 8 eyelets for the lace of a sneaker all lie on a rectangle, four equally spaced on each of the longer sides. The rectangle has a width of 50 mm and a length of 80 mm. There is one eyelet at each vertex of the rectangle. The lace itself must pass between the vertex eyelets along a width side of the rectangle and the... | [
"The rectangle is divided into three smaller rectangles with a width of 50 mm and a length of $\\dfrac{80}{3}$ mm. According to the Pythagorean Theorem (or by noticing the 8-15-17 Pythagorean triple), the diagonal of the rectangle is $\\sqrt{50^2+\\left(\\frac{80}{3}\\right)^2}=\\frac{170}{3}$ mm. Since that on t... |
https://artofproblemsolving.com/wiki/index.php/2013_AIME_I_Problems/Problem_1 | null | 150 | The AIME Triathlon consists of a half-mile swim, a 30-mile bicycle ride, and an eight-mile run. Tom swims, bicycles, and runs at constant rates. He runs fives times as fast as he swims, and he bicycles twice as fast as he runs. Tom completes the AIME Triathlon in four and a quarter hours. How many minutes does he spend... | [
"Let $r$ represent the rate Tom swims in miles per minute. Then we have\n$\\frac{1/2}{r} + \\frac{8}{5r} + \\frac{30}{10r} = 255$\nSolving for $r$ , we find $r = 1/50$ , so the time Tom spends biking is $\\frac{30}{(10)(1/50)} = \\boxed{150}$ minutes."
] |
https://artofproblemsolving.com/wiki/index.php/2009_AMC_8_Problems/Problem_11 | D | 4 | The Amaco Middle School bookstore sells pencils costing a whole number of cents. Some seventh graders each bought a pencil, paying a total of $1.43$ dollars. Some of the $30$ sixth graders each bought a pencil, and they paid a total of $1.95$ dollars. How many more sixth graders than seventh graders bought a pencil?
$\... | [
"Because the pencil costs a whole number of cents, the cost must be a factor of both $143$ and $195$ . They can be factored into $11\\cdot13$ and $3\\cdot5\\cdot13$ . The common factor cannot be $1$ or there would have to be more than $30$ sixth graders, so the pencil costs $13$ cents. The difference in costs that ... |
https://artofproblemsolving.com/wiki/index.php/2009_AIME_I_Problems/Problem_10 | null | 346 | The Annual Interplanetary Mathematics Examination (AIME) is written by a committee of five Martians, five Venusians, and five Earthlings. At meetings, committee members sit at a round table with chairs numbered from $1$ to $15$ in clockwise order. Committee rules state that a Martian must occupy chair $1$ and an Eart... | [
"Since the 5 members of each planet committee are distinct we get that the number of arrangement of sittings is in the form $N*(5!)^3$ because for each $M, V, E$ sequence we have $5!$ arrangements within the Ms, Vs, and Es.\nPretend the table only seats $3$ \"people\", with $1$ \"person\" from each planet. Counting... |
https://artofproblemsolving.com/wiki/index.php/2002_AIME_I_Problems/Problem_7 | null | 428 | The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers $x,y$ and $r$ with $|x|>|y|$
\[(x+y)^r=x^r+rx^{r-1}y+\dfrac{r(r-1)}{2}x^{r-2}y^2+\dfrac{r(r-1)(r-2)}{3!}x^{r-3}y^3 \cdots\]
What are the first three digits to the right of the decimal point in the decimal representation o... | [
"$1^n$ will always be 1, so we can ignore those terms, and using the definition ( $2002 / 7 = 286$ ):\n\\[(10^{2002} + 1)^{\\frac {10}7} = 10^{2860}+\\dfrac{10}{7}10^{858}+\\dfrac{15}{49}10^{-1144}+\\cdots\\]\nSince the exponent of the $10$ goes down extremely fast, it suffices to consider the first few terms. Also... |
https://artofproblemsolving.com/wiki/index.php/2015_AMC_8_Problems/Problem_4 | E | 12 | The Blue Bird High School chess team consists of two boys and three girls. A photographer wants to take a picture of the team to appear in the local newspaper. She decides to have them sit in a row with a boy at each end and the three girls in the middle. How many such arrangements are possible?
$\textbf{(A) }2\qquad\t... | [
"There are $2! = 2$ ways to order the boys on the ends, and there are $3!=6$ ways to order the girls in the middle. We get the answer to be $2 \\cdot 6 = \\boxed{12}$"
] |
https://artofproblemsolving.com/wiki/index.php/2007_AMC_10A_Problems/Problem_10 | E | 6 | The Dunbar family consists of a mother, a father, and some children. The average age of the members of the family is $20$ , the father is $48$ years old, and the average age of the mother and children is $16$ . How many children are in the family?
$\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D... | [
"Let $x$ be the number of the children and the mom. The father, who is $48$ , plus the sum of the ages of the kids and mom divided by the number of kids and mom plus $1$ (for the dad) = $20$ . This is because the average age of the entire family is $20.$ This statement, written as an equation, is: \\[\\frac{48+16x}... |
https://artofproblemsolving.com/wiki/index.php/2000_AMC_10_Problems/Problem_6 | C | 6 | The Fibonacci sequence $1,1,2,3,5,8,13,21,\ldots$ starts with two 1s, and each term afterwards is the sum of its two predecessors. Which one of the ten digits is the last to appear in the units position of a number in the Fibonacci sequence?
$\textbf{(A)} \ 0 \qquad \textbf{(B)} \ 4 \qquad \textbf{(C)} \ 6 \qquad \t... | [
"Note that any digits other than the units digit will not affect the answer. So to make computation quicker, we can just look at the Fibonacci sequence in $\\bmod{10}$\n$1,1,2,3,5,8,3,1,4,5,9,4,3,7,0,7,7,4,1,5,6,....$\nThe last digit to appear in the units position of a number in the Fibonacci sequence is $6 \\Lon... |
https://artofproblemsolving.com/wiki/index.php/2000_AMC_12_Problems/Problem_4 | C | 6 | The Fibonacci sequence $1,1,2,3,5,8,13,21,\ldots$ starts with two 1s, and each term afterwards is the sum of its two predecessors. Which one of the ten digits is the last to appear in the units position of a number in the Fibonacci sequence?
$\textbf{(A)} \ 0 \qquad \textbf{(B)} \ 4 \qquad \textbf{(C)} \ 6 \qquad \t... | [
"Note that any digits other than the units digit will not affect the answer. So to make computation quicker, we can just look at the Fibonacci sequence in $\\bmod{10}$\n$1,1,2,3,5,8,3,1,4,5,9,4,3,7,0,7,7,4,1,5,6,....$\nThe last digit to appear in the units position of a number in the Fibonacci sequence is $6 \\Lon... |
https://artofproblemsolving.com/wiki/index.php/2012_AMC_8_Problems/Problem_9 | C | 139 | The Fort Worth Zoo has a number of two-legged birds and a number of four-legged mammals. On one visit to the zoo, Margie counted 200 heads and 522 legs. How many of the animals that Margie counted were two-legged birds?
$\textbf{(A)}\hspace{.05in}61\qquad\textbf{(B)}\hspace{.05in}122\qquad\textbf{(C)}\hspace{.05in}139\... | [
"Let the number of two-legged birds be $x$ and the number of four-legged mammals be $y$ . We can now use systems of equations to solve this problem.\nWrite two equations:\n$2x + 4y = 522$\n$x + y = 200$\nNow multiply the latter equation by $2$\n$2x + 4y = 522$\n$2x + 2y = 400$\nBy subtracting the second equation fr... |
https://artofproblemsolving.com/wiki/index.php/2013_AMC_8_Problems/Problem_9 | C | 11 | The Incredible Hulk can double the distance he jumps with each succeeding jump. If his first jump is 1 meter, the second jump is 2 meters, the third jump is 4 meters, and so on, then on which jump will he first be able to jump more than 1 kilometer (1,000 meters)?
$\textbf{(A)}\ 9^\text{th} \qquad \textbf{(B)}\ 10^\tex... | [
"This is a geometric sequence in which the common ratio is 2. To find the jump that would be over a 1000 meters, we note that $2^{10}=1024$\nHowever, because the first term is $2^0=1$ and not $2^1=2$ , the solution to the problem is $10-0+1=\\boxed{11}$"
] |
https://artofproblemsolving.com/wiki/index.php/2005_AMC_8_Problems/Problem_14 | B | 96 | The Little Twelve Basketball Conference has two divisions, with six teams in each division. Each team plays each of the other teams in its own division twice and every team in the other division once. How many conference games are scheduled?
$\textbf{(A)}\ 80\qquad\textbf{(B)}\ 96\qquad\textbf{(C)}\ 100\qquad\textbf{(D... | [
"Within each division, there are $\\binom {6}{2} = 15$ pairings, and each of these games happens twice. The same goes for the other division so that there are $4(15)=60$ games within their own divisions. The number of games between the two divisions is $(6)(6)=36$ . Together there are $60+36=\\boxed{96}$ conference... |
https://artofproblemsolving.com/wiki/index.php/2005_AMC_8_Problems/Problem_14 | null | 96 | The Little Twelve Basketball Conference has two divisions, with six teams in each division. Each team plays each of the other teams in its own division twice and every team in the other division once. How many conference games are scheduled?
$\textbf{(A)}\ 80\qquad\textbf{(B)}\ 96\qquad\textbf{(C)}\ 100\qquad\textbf{(D... | [
"Each team plays 10 games in its own division and 6 games against teams in the other division. So each of the 12 teams plays 16 conference games. Because each game involves two teams, there are $\\frac{12\\times 16}{2}=\\boxed{96}$ games scheduled. ~aopsav (Credit to AoPS Alcumus)"
] |
https://artofproblemsolving.com/wiki/index.php/2022_AMC_8_Problems/Problem_1 | A | 10 | The Math Team designed a logo shaped like a multiplication symbol, shown below on a grid of 1-inch squares. What is the area of the logo in square inches?
[asy] defaultpen(linewidth(0.5)); size(5cm); defaultpen(fontsize(14pt)); label("$\textbf{Math}$", (2.1,3.7)--(3.9,3.7)); label("$\textbf{Team}$", (2.1,3)--(3.9,3)); ... | [
"Draw the following four lines as shown: \nWe see these lines split the figure into five squares with side length $\\sqrt2$ . Thus, the area is $5\\cdot\\left(\\sqrt2\\right)^2=5\\cdot 2 = \\boxed{10}$",
"There are $5$ lattice points in the interior of the logo and $12$ lattice points on the boundary of the logo.... |
https://artofproblemsolving.com/wiki/index.php/1991_AJHSME_Problems/Problem_23 | A | 10 | The Pythagoras High School band has $100$ female and $80$ male members. The Pythagoras High School orchestra has $80$ female and $100$ male members. There are $60$ females who are members in both band and orchestra. Altogether, there are $230$ students who are in either band or orchestra or both. The number of male... | [
"There are $100+80-60=120$ females in either band or orchestra, so there are $230-120=110$ males in either band or orchestra. Suppose $x$ males are in both band and orchestra. \\[80+100-x=110\\Rightarrow x=70.\\] Thus, the number of males in band but not orchestra is $80-70=10\\rightarrow \\boxed{10}$"
] |
https://artofproblemsolving.com/wiki/index.php/2020_AMC_10B_Problems/Problem_4 | D | 7 | 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 $b$
$\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ 11$ | [
"Since the three angles of a triangle add up to $180^{\\circ}$ and one of the angles is $90^{\\circ}$ because it's a right triangle, $a^{\\circ} + b^{\\circ} = 90^{\\circ}$\nThe greatest prime number less than $90$ is $89$ . If $a=89^{\\circ}$ , then $b=90^{\\circ}-89^{\\circ}=1^{\\circ}$ , which is not prime.\nThe... |
https://artofproblemsolving.com/wiki/index.php/2020_AMC_12B_Problems/Problem_4 | D | 7 | 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 $b$
$\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ 11$ | [
"Since the three angles of a triangle add up to $180^{\\circ}$ and one of the angles is $90^{\\circ}$ because it's a right triangle, $a^{\\circ} + b^{\\circ} = 90^{\\circ}$\nThe greatest prime number less than $90$ is $89$ . If $a=89^{\\circ}$ , then $b=90^{\\circ}-89^{\\circ}=1^{\\circ}$ , which is not prime.\nThe... |
https://artofproblemsolving.com/wiki/index.php/1995_AHSME_Problems/Problem_13 | C | 8 | The addition below is incorrect. The display can be made correct by changing one digit $d$ , wherever it occurs, to another digit $e$ . Find the sum of $d$ and $e$
$\begin{tabular}{ccccccc} & 7 & 4 & 2 & 5 & 8 & 6 \\ + & 8 & 2 & 9 & 4 & 3 & 0 \\ \hline 1 & 2 & 1 & 2 & 0 & 1 & 6 \end{tabular}$
$\mathrm{(A) \ 4 } \qquad ... | [
"If we change $0$ , the units column would be incorrect.\nIf we change $1$ , then the leading $1$ in the sum would be incorrect.\nHowever, looking at the $2$ in the hundred-thousands column, it would be possible to change the $2$ to either a $5$ (no carry) or a $6$ (carry) to create a correct statement.\nChanging ... |
https://artofproblemsolving.com/wiki/index.php/1996_AHSME_Problems/Problem_1 | D | 7 | The addition below is incorrect. What is the largest digit that can be changed to make the addition correct?
$\begin{tabular}{rr}&\ \texttt{6 4 1}\\ &\texttt{8 5 2}\\ &+\texttt{9 7 3}\\ \hline &\texttt{2 4 5 6}\end{tabular}$
$\text{(A)}\ 4\qquad\text{(B)}\ 5\qquad\text{(C)}\ 6\qquad\text{(D)}\ 7\qquad\text{(E)}\ 8$ | [
"Doing the addition as is, we get $641 + 852 + 973 = 2466$ . This number is $10$ larger than the desired sum of $2456$ . Therefore, we must make one of the three numbers $10$ smaller.\nWe may either change $641 \\rightarrow 631$ $852 \\rightarrow 842$ , or $973 \\rightarrow 963$ . Either change results in a vali... |
https://artofproblemsolving.com/wiki/index.php/1997_AHSME_Problems/Problem_2 | D | 44 | The adjacent sides of the decagon shown meet at right angles. What is its perimeter?
[asy] defaultpen(linewidth(.8pt)); dotfactor=4; dot(origin);dot((12,0));dot((12,1));dot((9,1));dot((9,7));dot((7,7));dot((7,10));dot((3,10));dot((3,8));dot((0,8)); draw(origin--(12,0)--(12,1)--(9,1)--(9,7)--(7,7)--(7,10)--(3,10)--(3,8... | [
"The three unlabelled vertical sides have the same sum as the two labelled vertical sides, which is $10$\nThe four unlabelled horizontal sides have the same sum as the one large horizontal side, which is $12$\nThus, the perimeter is $2(12+10) = 44$ , which is option $\\boxed{44}$"
] |
https://artofproblemsolving.com/wiki/index.php/1983_AIME_Problems/Problem_15 | null | 175 | The adjoining figure shows two intersecting chords in a circle, with $B$ on minor arc $AD$ . Suppose that the radius of the circle is $5$ , that $BC=6$ , and that $AD$ is bisected by $BC$ . Suppose further that $AD$ is the only chord starting at $A$ which is bisected by $BC$ . It follows that the sine of the central an... | [
"Let $M$ be the midpoint of the chord $BC$ . From right triangle $OMB$ , we have $OM = \\sqrt{OB^2 - BM^2} =4$ . This gives $\\tan \\angle BOM = \\frac{BM}{OM} = \\frac 3 4$\nNotice that the distance $OM$ equals $PN + PO \\cos \\angle AOM = r(1 + \\cos \\angle AOM)$ , where $r$ is the radius of circle $P$\nHence \\... |
https://artofproblemsolving.com/wiki/index.php/2021_AMC_10B_Problems/Problem_5 | B | 22 | The ages of Jonie's four cousins are distinct single-digit positive integers. Two of the cousins' ages multiplied together give $24$ , while the other two multiply to $30$ . What is the sum of the ages of Jonie's four cousins?
$\textbf{(A)} ~21 \qquad\textbf{(B)} ~22 \qquad\textbf{(C)} ~23 \qquad\textbf{(D)} ~24 \qquad... | [
"First look at the two cousins' ages that multiply to $24$ . Since the ages must be single-digit, the ages must either be $3 \\text{ and } 8$ or $4 \\text{ and } 6.$\nNext, look at the two cousins' ages that multiply to $30$ . Since the ages must be single-digit, the only ages that work are $5 \\text{ and } 6.$ Rem... |
https://artofproblemsolving.com/wiki/index.php/2002_AMC_12P_Problems/Problem_16 | C | 90 | The altitudes of a triangle are $12, 15,$ and $20.$ The largest angle in this triangle is
$\text{(A) }72^\circ \qquad \text{(B) }75^\circ \qquad \text{(C) }90^\circ \qquad \text{(D) }108^\circ \qquad \text{(E) }120^\circ$ | [
"Let $a, b,$ and $c$ denote the bases of altitudes $12, 15,$ and $20,$ respectively. Since they are all altitudes and bases of the same triangle, they have the same area, so $\\frac{12a}{2}=\\frac{15b}{2}=\\frac{20c}{2}.$ Multiplying by $2$ , we get $12a=15b=20c.$ Notice that a simple solution to the equation is if... |
https://artofproblemsolving.com/wiki/index.php/1962_AHSME_Problems/Problem_20 | A | 108 | The angles of a pentagon are in arithmetic progression. One of the angles in degrees, must be:
$\textbf{(A)}\ 108\qquad\textbf{(B)}\ 90\qquad\textbf{(C)}\ 72\qquad\textbf{(D)}\ 54\qquad\textbf{(E)}\ 36$ | [
"If the angles are in an arithmetic progression, they can be expressed as $a$ $a+n$ $a+2n$ $a+3n$ , and $a+4n$ for some real numbers $a$ and $n$ .\nNow we know that the sum of the degree measures of the angles of a pentagon is $180(5-2)=540$ .\nAdding our expressions for the five angles together, we get $5a+10n=540... |
https://artofproblemsolving.com/wiki/index.php/2007_AMC_10B_Problems/Problem_15 | D | 173 | The angles of quadrilateral $ABCD$ satisfy $\angle A=2 \angle B=3 \angle C=4 \angle D.$ What is the degree measure of $\angle A,$ rounded to the nearest whole number?
$\textbf{(A) } 125 \qquad\textbf{(B) } 144 \qquad\textbf{(C) } 153 \qquad\textbf{(D) } 173 \qquad\textbf{(E) } 180$ | [
"The sum of the interior angles of any quadrilateral is $360^\\circ.$ \\begin{align*} 360 &= \\angle A + \\angle B + \\angle C + \\angle D\\\\ &= \\angle A + \\frac{1}{2}A + \\frac{1}{3}A + \\frac{1}{4}A\\\\ &= \\frac{12}{12}A + \\frac{6}{12}A + \\frac{4}{12}A + \\frac{3}{12}A\\\\ &= \\frac{25}{12}A \\end{align*} \... |
https://artofproblemsolving.com/wiki/index.php/2007_AMC_12B_Problems/Problem_11 | D | 173 | The angles of quadrilateral $ABCD$ satisfy $\angle A=2 \angle B=3 \angle C=4 \angle D.$ What is the degree measure of $\angle A,$ rounded to the nearest whole number?
$\textbf{(A) } 125 \qquad\textbf{(B) } 144 \qquad\textbf{(C) } 153 \qquad\textbf{(D) } 173 \qquad\textbf{(E) } 180$ | [
"The sum of the interior angles of any quadrilateral is $360^\\circ.$ \\begin{align*} 360 &= \\angle A + \\angle B + \\angle C + \\angle D\\\\ &= \\angle A + \\frac{1}{2}A + \\frac{1}{3}A + \\frac{1}{4}A\\\\ &= \\frac{12}{12}A + \\frac{6}{12}A + \\frac{4}{12}A + \\frac{3}{12}A\\\\ &= \\frac{25}{12}A \\end{align*} \... |
https://artofproblemsolving.com/wiki/index.php/1990_AJHSME_Problems/Problem_20 | A | 882 | The annual incomes of $1,000$ families range from $8200$ dollars to $98,000$ dollars. In error, the largest income was entered on the computer as $980,000$ dollars. The difference between the mean of the incorrect data and the mean of the actual data is
$\text{(A)}\ \text{882 dollars} \qquad \text{(B)}\ \text{980 dol... | [
"Let $S$ be the sum of all the incomes but the largest one. For the actual data, the mean is $\\frac{S+98000}{1000}$ , and for the incorrect data the mean is $\\frac{S+980000}{1000}$ . The difference is $882, or \\rightarrow \\boxed{882}$"
] |
https://artofproblemsolving.com/wiki/index.php/1991_AJHSME_Problems/Problem_10 | B | 8 | The area in square units of the region enclosed by parallelogram $ABCD$ is
[asy] unitsize(24); pair A,B,C,D; A=(-1,0); B=(0,2); C=(4,2); D=(3,0); draw(A--B--C--D); draw((0,-1)--(0,3)); draw((-2,0)--(6,0)); draw((-.25,2.75)--(0,3)--(.25,2.75)); draw((5.75,.25)--(6,0)--(5.75,-.25)); dot(origin); dot(A); dot(B); dot(C); ... | [
"The base is $\\overline{BC}=4$ . The height has a length of the difference of the y-coordinates of A and B, which is 2. Therefore the area is $4\\cdot 2=8\\Rightarrow \\boxed{8}$"
] |
https://artofproblemsolving.com/wiki/index.php/1957_AHSME_Problems/Problem_7 | E | 72 | The area of a circle inscribed in an equilateral triangle is $48\pi$ . The perimeter of this triangle is:
$\textbf{(A)}\ 72\sqrt{3} \qquad \textbf{(B)}\ 48\sqrt{3}\qquad \textbf{(C)}\ 36\qquad \textbf{(D)}\ 24\qquad \textbf{(E)}\ 72$ | [
" We can see that the radius of the circle is $4\\sqrt{3}$ . We know that the radius is $\\frac{1}{3}$ of each median line of the triangle; each median line is therefore $12\\sqrt{3}$ . Since the median line completes a $30$ $60$ $90$ triangle, we can conclude that one of the sides of the triangle is $24$ . Triple ... |
https://artofproblemsolving.com/wiki/index.php/2010_AMC_10A_Problems/Problem_5 | E | 144 | The area of a circle whose circumference is $24\pi$ is $k\pi$ . What is the value of $k$
$\mathrm{(A)}\ 6 \qquad \mathrm{(B)}\ 12 \qquad \mathrm{(C)}\ 24 \qquad \mathrm{(D)}\ 36 \qquad \mathrm{(E)}\ 144$ | [
"If the circumference of a circle is $24\\pi$ , the radius would be $12$ . Since the area of a circle is $\\pi r^2$ , the area is $144\\pi$ . The answer is $\\boxed{144}$",
"By definition, $\\pi$ is the ratio of the circumference to the diameter. Since the circumference is $24\\pi$ , the diameter must be $24$ and... |
https://artofproblemsolving.com/wiki/index.php/2019_AMC_12A_Problems/Problem_1 | E | 78 | The area of a pizza with radius $4$ is $N$ percent larger than the area of a pizza with radius $3$ inches. What is the integer closest to $N$
$\textbf{(A) } 25 \qquad\textbf{(B) } 33 \qquad\textbf{(C) } 44\qquad\textbf{(D) } 66 \qquad\textbf{(E) } 78$ | [
"The area of the larger pizza is $16\\pi$ , while the area of the smaller pizza is $9\\pi$ . Therefore, the larger pizza is $\\frac{7\\pi}{9\\pi} \\cdot 100\\%$ bigger than the smaller pizza. $\\frac{7\\pi}{9\\pi} \\cdot 100\\% = 77.777....$ , which is closest to $\\boxed{78}$"
] |
https://artofproblemsolving.com/wiki/index.php/1985_AJHSME_Problems/Problem_4 | C | 46 | The area of polygon $ABCDEF$ , in square units, is
$\text{(A)}\ 24 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 46 \qquad \text{(D)}\ 66 \qquad \text{(E)}\ 74$
[asy] draw((0,9)--(6,9)--(6,0)--(2,0)--(2,4)--(0,4)--cycle); label("A",(0,9),NW); label("B",(6,9),NE); label("C",(6,0),SE); label("D",(2,0),SW); label("E",(2,4),NE)... | [
"\nObviously, there are no formulas to find the area of such a messed up shape, but we do recognize some shapes we do know how to find the area of.\nIf we continue segment $\\overline{FE}$ until it reaches the right side at $G$ , we create two rectangles - one on the top and one on the bottom.\nWe know how to find ... |
https://artofproblemsolving.com/wiki/index.php/2005_AMC_8_Problems/Problem_13 | C | 9 | The area of polygon $ABCDEF$ is 52 with $AB=8$ $BC=9$ and $FA=5$ . What is $DE+EF$
[asy] pair a=(0,9), b=(8,9), c=(8,0), d=(4,0), e=(4,4), f=(0,4); draw(a--b--c--d--e--f--cycle); draw(shift(0,-.25)*a--shift(.25,-.25)*a--shift(.25,0)*a); draw(shift(-.25,0)*b--shift(-.25,-.25)*b--shift(0,-.25)*b); draw(shift(-.25,0)*c--s... | [
"Notice that $AF + DE = BC$ , so $DE=4$ . Let $O$ be the intersection of the extensions of $AF$ and $DC$ , which makes rectangle $ABCO$ . The area of the polygon is the area of $FEDO$ subtracted from the area of $ABCO$\n\\[\\text{Area} = 52 = 8 \\cdot 9- EF \\cdot 4\\]\nSolving for the unknown, $EF=5$ , therefore $... |
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