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/2024_AIME_II_Problems/Problem_14 | null | 211 | Let \(b\ge 2\) be an integer. Call a positive integer \(n\) \(b\text-\textit{eautiful}\) if it has exactly two digits when expressed in base \(b\) and these two digits sum to \(\sqrt n\). For example, \(81\) is \(13\text-\textit{eautiful}\) because \(81 = \underline{6} \ \underline{3}_{13} \) and \(6 + 3 = \sqrt{81}... | [
"We write the base- $b$ two-digit integer as $\\left( xy \\right)_b$ .\nThus, this number satisfies \\[ \\left( x + y \\right)^2 = b x + y \\] with $x \\in \\left\\{ 1, 2, \\cdots , b-1 \\right\\}$ and $y \\in \\left\\{ 0, 1, \\cdots , b - 1 \\right\\}$\nThe above conditions imply $\\left( x + y \\right)^2 < b^2$ .... |
https://artofproblemsolving.com/wiki/index.php/2003_AIME_II_Problems/Problem_15 | null | 15 | Let \[P(x) = 24x^{24} + \sum_{j = 1}^{23}(24 - j)(x^{24 - j} + x^{24 + j}).\] Let $z_{1},z_{2},\ldots,z_{r}$ be the distinct zeros of $P(x),$ and let $z_{k}^{2} = a_{k} + b_{k}i$ for $k = 1,2,\ldots,r,$ where $a_{k}$ and $b_{k}$ are real numbers. Let
where $m, n,$ and $p$ are integers and $p$ is not divisible by the sq... | [
"This can be factored as:\n\\[P(x) = x\\left( x^{23} + x^{22} + \\cdots + x^2 + x + 1 \\right)^2\\]\nNote that $\\left( x^{23} + x^{22} + \\cdots + x^2 + x + 1 \\right) \\cdot (x-1) = x^{24} - 1$ .\nSo the roots of $x^{23} + x^{22} + \\cdots + x^2 + x + 1$ are exactly all $24$ -th complex roots of $1$ , except for ... |
https://artofproblemsolving.com/wiki/index.php/2019_AMC_12A_Problems/Problem_21 | C | 36 | Let \[z=\frac{1+i}{\sqrt{2}}.\] What is \[\left(z^{1^2}+z^{2^2}+z^{3^2}+\dots+z^{{12}^2}\right) \cdot \left(\frac{1}{z^{1^2}}+\frac{1}{z^{2^2}}+\frac{1}{z^{3^2}}+\dots+\frac{1}{z^{{12}^2}}\right)?\]
$\textbf{(A) } 18 \qquad \textbf{(B) } 72-36\sqrt2 \qquad \textbf{(C) } 36 \qquad \textbf{(D) } 72 \qquad \textbf{(E) } 7... | [
"Note that $z = \\mathrm{cis }(45^{\\circ})$\nAlso note that $z^{k} = z^{k + 8}$ for all positive integers $k$ because of De Moivre's Theorem. Therefore, we want to look at the exponents of each term modulo $8$\n$1^2, 5^2,$ and $9^2$ are all $1 \\pmod{8}$\n$2^2, 6^2,$ and $10^2$ are all $4 \\pmod{8}$\n$3^2, 7^2,$ a... |
https://artofproblemsolving.com/wiki/index.php/2019_AMC_12A_Problems/Problem_21 | null | 36 | Let \[z=\frac{1+i}{\sqrt{2}}.\] What is \[\left(z^{1^2}+z^{2^2}+z^{3^2}+\dots+z^{{12}^2}\right) \cdot \left(\frac{1}{z^{1^2}}+\frac{1}{z^{2^2}}+\frac{1}{z^{3^2}}+\dots+\frac{1}{z^{{12}^2}}\right)?\]
$\textbf{(A) } 18 \qquad \textbf{(B) } 72-36\sqrt2 \qquad \textbf{(C) } 36 \qquad \textbf{(D) } 72 \qquad \textbf{(E) } 7... | [
"It is well known that if $|z|=1$ then $\\bar{z}=\\frac{1}{z}$ . Therefore, we have that the desired expression is equal to \\[\\left(z^1+z^4+z^9+...+z^{144}\\right)\\left(\\bar{z}^1+\\bar{z}^4+\\bar{z}^9+...+\\bar{z}^{144}\\right)\\] We know that $z=e^{\\frac{i\\pi}{4}}$ so $\\bar{z}=e^{\\frac{i7\\pi}{4}}$ . Then,... |
https://artofproblemsolving.com/wiki/index.php/1969_AHSME_Problems/Problem_32 | C | 5 | Let a sequence $\{u_n\}$ be defined by $u_1=5$ and the relationship $u_{n+1}-u_n=3+4(n-1), n=1,2,3\cdots.$ If $u_n$ is expressed as a polynomial in $n$ , the algebraic sum of its coefficients is:
$\text{(A) 3} \quad \text{(B) 4} \quad \text{(C) 5} \quad \text{(D) 6} \quad \text{(E) 11}$ | [
"Note that the first differences create a linear function, so the sequence ${u_n}$ is quadratic.\nThe first three terms of the sequence are $5$ $8$ , and $15$ . From there, a system of equations can be written. \\[a+b+c=5\\] \\[4a+2b+c=8\\] \\[9a+3b+c=15\\] Solve the system to get $a=2$ $b=-3$ , and $c=6$ . The s... |
https://artofproblemsolving.com/wiki/index.php/2013_AMC_10A_Problems/Problem_18 | B | 58 | Let points $A = (0, 0)$ $B = (1, 2)$ $C=(3, 3)$ , and $D = (4, 0)$ . Quadrilateral $ABCD$ is cut into equal area pieces by a line passing through $A$ . This line intersects $\overline{CD}$ at point $\left(\frac{p}{q}, \frac{r}{s}\right)$ , where these fractions are in lowest terms. What is $p+q+r+s$
$\textbf{(A)}\ 5... | [
"First, we shall find the area of quadrilateral $ABCD$ . This can be done in any of three ways:\nPick's Theorem $[ABCD] = I + \\dfrac{B}{2} - 1 = 5 + \\dfrac{7}{2} - 1 = \\dfrac{15}{2}.$\nSplitting: Drop perpendiculars from $B$ and $C$ to the x-axis to divide the quadrilateral into triangles and trapezoids, and so ... |
https://artofproblemsolving.com/wiki/index.php/2013_AMC_12A_Problems/Problem_13 | B | 58 | Let points $A = (0,0) , \ B = (1,2), \ C = (3,3),$ and $D = (4,0)$ . Quadrilateral $ABCD$ is cut into equal area pieces by a line passing through $A$ . This line intersects $\overline{CD}$ at point $\left (\frac{p}{q}, \frac{r}{s} \right )$ , where these fractions are in lowest terms. What is $p + q + r + s$
$\textbf{(... | [
"If you have graph paper, use Pick's Theorem to quickly and efficiently find the area of the quadrilateral. If not, just find the area by other methods.\nPick's Theorem states that\n$A$ $I$ $+$ $\\frac{B}{2}$ $1$ , where $I$ is the number of lattice points in the interior of the polygon, and $B$ is the number of la... |
https://artofproblemsolving.com/wiki/index.php/2013_AMC_12A_Problems/Problem_13 | null | 58 | Let points $A = (0,0) , \ B = (1,2), \ C = (3,3),$ and $D = (4,0)$ . Quadrilateral $ABCD$ is cut into equal area pieces by a line passing through $A$ . This line intersects $\overline{CD}$ at point $\left (\frac{p}{q}, \frac{r}{s} \right )$ , where these fractions are in lowest terms. What is $p + q + r + s$
$\textbf{(... | [
"Let the point of intersection be $E$ , with coordinates $(x, y)$ . Then, $ABCD$ is cut into $ABCE$ and $AED$\nSince the areas are equal, we can use Shoelace Theorem to find the area. This gives $3 + 3x - 3y = 4y$\nThe line going through $CD$ is $y = -3x + 12$ . Since $E$ is on $CD$ , we can substitute this in, giv... |
https://artofproblemsolving.com/wiki/index.php/2006_AIME_I_Problems/Problem_2 | null | 901 | Let set $\mathcal{A}$ be a 90- element subset of $\{1,2,3,\ldots,100\},$ and let $S$ be the sum of the elements of $\mathcal{A}.$ Find the number of possible values of $S.$ | [
"The smallest $S$ is $1+2+ \\ldots +90 = 91 \\cdot 45 = 4095$ . The largest $S$ is $11+12+ \\ldots +100=111\\cdot 45=4995$ . All numbers between $4095$ and $4995$ are possible values of S, so the number of possible values of S is $4995-4095+1=901$\nAlternatively, for ease of calculation, let set $\\mathcal{B}$ be a... |
https://artofproblemsolving.com/wiki/index.php/2024_AMC_8_Problems/Problem_15 | C | 1,107 | Let the letters $F$ $L$ $Y$ $B$ $U$ $G$ represent distinct digits. Suppose $\underline{F}~\underline{L}~\underline{Y}~\underline{F}~\underline{L}~\underline{Y}$ is the greatest number that satisfies the equation
\[8\cdot\underline{F}~\underline{L}~\underline{Y}~\underline{F}~\underline{L}~\underline{Y}=\underline{B}~\u... | [
"The highest that $FLYFLY$ can be would have to be $124124$ , and it cannot be higher than that because then it would exceed the $6$ -digit limit set on $BUGBUG$\nSo, if we start at $124124\\cdot8$ , we get $992992$ , which would be wrong because both $B \\& U$ would be $9$ , and the numbers cannot be repeated betw... |
https://artofproblemsolving.com/wiki/index.php/2014_AIME_I_Problems/Problem_5 | null | 134 | Let the set $S = \{P_1, P_2, \dots, P_{12}\}$ consist of the twelve vertices of a regular $12$ -gon. A subset $Q$ of $S$ is called "communal" if there is a circle such that all points of $Q$ are inside the circle, and all points of $S$ not in $Q$ are outside of the circle. How many communal subsets are there? (Note tha... | [
"By looking at the problem and drawing a few pictures, it quickly becomes obvious that one cannot draw a circle that covers $2$ disjoint areas of the $12$ -gon without including all the vertices in between those areas. In other words, in order for a subset to be communal, all the vertices in the subset must be adja... |
https://artofproblemsolving.com/wiki/index.php/2003_AIME_I_Problems/Problem_3 | null | 484 | Let the set $\mathcal{S} = \{8, 5, 1, 13, 34, 3, 21, 2\}.$ Susan makes a list as follows: for each two-element subset of $\mathcal{S},$ she writes on her list the greater of the set's two elements. Find the sum of the numbers on the list. | [
"Order the numbers in the set from greatest to least to reduce error: $\\{34, 21, 13, 8, 5, 3, 2, 1\\}.$ Each element of the set will appear in $7$ two-element subsets , once with each other number.\nTherefore the desired sum is $34\\cdot7+21\\cdot6+13\\cdot5+8\\cdot4+5\\cdot3+3 \\cdot2+2\\cdot1+1\\cdot0=\\boxed{48... |
https://artofproblemsolving.com/wiki/index.php/1986_AIME_Problems/Problem_15 | null | 400 | Let triangle $ABC$ be a right triangle in the xy-plane with a right angle at $C_{}$ . Given that the length of the hypotenuse $AB$ is $60$ , and that the medians through $A$ and $B$ lie along the lines $y=x+3$ and $y=2x+4$ respectively, find the area of triangle $ABC$ | [
"We first seek to find the angle between the lines $y = x + 3$ and $y = 2x + 4$ Let the acute angle the red line makes with the $x-$ axis be $\\alpha$ and the acute angle the blue line makes with the $x-$ axis be $\\beta$ . Then, we know that $\\tan \\alpha = 1$ and $\\tan \\beta = 2$ . Note that the acute angle b... |
https://artofproblemsolving.com/wiki/index.php/1998_AHSME_Problems/Problem_2 | E | 17 | Letters $A,B,C,$ and $D$ represent four different digits selected from $0,1,2,\ldots ,9.$ If $(A+B)/(C+D)$ is an integer that is as large as possible, what is the value of $A+B$
$\mathrm{(A) \ }13 \qquad \mathrm{(B) \ }14 \qquad \mathrm{(C) \ } 15\qquad \mathrm{(D) \ }16 \qquad \mathrm{(E) \ } 17$ | [
"If we want $\\frac{A+B}{C+D}$ to be as large as possible, we want to try to maximize the numerator $A+B$ and minimize the denominator $C+D$ . Picking $A=9$ and $B=8$ will maximize the numerator, and picking $C=0$ and $D=1$ will minimize the denominator.\nChecking to make sure the fraction is an integer, $\\frac{A... |
https://artofproblemsolving.com/wiki/index.php/2019_AIME_II_Problems/Problem_2 | null | 107 | Lilypads $1,2,3,\ldots$ lie in a row on a pond. A frog makes a sequence of jumps starting on pad $1$ . From any pad $k$ the frog jumps to either pad $k+1$ or pad $k+2$ chosen randomly with probability $\tfrac{1}{2}$ and independently of other jumps. The probability that the frog visits pad $7$ is $\tfrac{p}{q}$ , where... | [
"Let $P_n$ be the probability the frog visits pad $7$ starting from pad $n$ . Then $P_7 = 1$ $P_6 = \\frac12$ , and $P_n = \\frac12(P_{n + 1} + P_{n + 2})$ for all integers $1 \\leq n \\leq 5$ . Working our way down, we find \\[P_5 = \\frac{3}{4}\\] \\[P_4 = \\frac{5}{8}\\] \\[P_3 = \\frac{11}{16}\\] \\[P_2 = \\fra... |
https://artofproblemsolving.com/wiki/index.php/2020_AMC_12A_Problems/Problem_12 | B | 15 | Line $l$ in the coordinate plane has equation $3x-5y+40=0$ . This line is rotated $45^{\circ}$ counterclockwise about the point $(20,20)$ to obtain line $k$ . What is the $x$ -coordinate of the $x$ -intercept of line $k?$
$\textbf{(A) } 10 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 20 \qquad \textbf{(D) } 25 \qquad \... | [
"The slope of the line is $\\frac{3}{5}$ . We must transform it by $45^{\\circ}$\n$45^{\\circ}$ creates an isosceles right triangle, since the sum of the angles of the triangle must be $180^{\\circ}$ and one angle is $90^{\\circ}$ . This means the last leg angle must also be $45^{\\circ}$\nIn the isosceles right tr... |
https://artofproblemsolving.com/wiki/index.php/2020_AMC_12A_Problems/Problem_12 | null | 15 | Line $l$ in the coordinate plane has equation $3x-5y+40=0$ . This line is rotated $45^{\circ}$ counterclockwise about the point $(20,20)$ to obtain line $k$ . What is the $x$ -coordinate of the $x$ -intercept of line $k?$
$\textbf{(A) } 10 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 20 \qquad \textbf{(D) } 25 \qquad \... | [
"Since the slope of the line is $\\frac{3}{5}$ , and the angle we are rotating around is x, then $\\tan x = \\frac{3}{5}$ $\\tan(x+45^{\\circ}) = \\frac{\\tan x + \\tan(45^{\\circ})}{1-\\tan x*\\tan(45^{\\circ})} = \\frac{0.6+1}{1-0.6} = \\frac{1.6}{0.4} = 4$\nHence, the slope of the rotated line is $4$ . Since we ... |
https://artofproblemsolving.com/wiki/index.php/2018_AMC_10B_Problems/Problem_12 | C | 50 | Line segment $\overline{AB}$ is a diameter of a circle with $AB = 24$ . Point $C$ , not equal to $A$ or $B$ , lies on the circle. As point $C$ moves around the circle, the centroid (center of mass) of $\triangle ABC$ traces out a closed curve missing two points. To the nearest positive integer, what is the area of the ... | [
"For each $\\triangle ABC,$ note that the length of one median is $OC=12.$ Let $G$ be the centroid of $\\triangle ABC.$ It follows that $OG=\\frac13 OC=4.$\nAs shown below, $\\triangle ABC_1$ and $\\triangle ABC_2$ are two shapes of $\\triangle ABC$ with centroids $G_1$ and $G_2,$ respectively: Therefore, point $G... |
https://artofproblemsolving.com/wiki/index.php/2018_AMC_12B_Problems/Problem_8 | C | 50 | Line segment $\overline{AB}$ is a diameter of a circle with $AB = 24$ . Point $C$ , not equal to $A$ or $B$ , lies on the circle. As point $C$ moves around the circle, the centroid (center of mass) of $\triangle ABC$ traces out a closed curve missing two points. To the nearest positive integer, what is the area of the ... | [
"For each $\\triangle ABC,$ note that the length of one median is $OC=12.$ Let $G$ be the centroid of $\\triangle ABC.$ It follows that $OG=\\frac13 OC=4.$\nAs shown below, $\\triangle ABC_1$ and $\\triangle ABC_2$ are two shapes of $\\triangle ABC$ with centroids $G_1$ and $G_2,$ respectively: Therefore, point $G... |
https://artofproblemsolving.com/wiki/index.php/1976_AHSME_Problems/Problem_28 | B | 4,351 | Lines $L_1,L_2,\dots,L_{100}$ are distinct. All lines $L_{4n}, n$ a positive integer, are parallel to each other.
All lines $L_{4n-3}, n$ a positive integer, pass through a given point $A.$ The maximum number of points of intersection of pairs of lines from the complete set $\{L_1,L_2,\dots,L_{100}\}$ is
$\textbf{(A) ... | [
"We partition $\\{L_1,L_2,\\dots,L_{100}\\}$ into three sets. Let \\begin{align*} X &= \\{L_n\\mid n\\equiv0\\pmod{4}\\}, \\\\ Y &= \\{L_n\\mid n\\equiv1\\pmod{4}\\}, \\\\ Z &= \\{L_n\\mid n\\equiv2,3\\pmod{4}\\}, \\\\ \\end{align*} from which $|X|=|Y|=25$ and $|Z|=50.$\nAny two distinct lines can intersect at most... |
https://artofproblemsolving.com/wiki/index.php/1992_AIME_Problems/Problem_11 | null | 945 | Lines $l_1^{}$ and $l_2^{}$ both pass through the origin and make first-quadrant angles of $\frac{\pi}{70}$ and $\frac{\pi}{54}$ radians, respectively, with the positive x-axis. For any line $l^{}_{}$ , the transformation $R(l)^{}_{}$ produces another line as follows: $l^{}_{}$ is reflected in $l_1^{}$ , and the result... | [
"Let $l$ be a line that makes an angle of $\\theta$ with the positive $x$ -axis. Let $l'$ be the reflection of $l$ in $l_1$ , and let $l''$ be the reflection of $l'$ in $l_2$\nThe angle between $l$ and $l_1$ is $\\theta - \\frac{\\pi}{70}$ , so the angle between $l_1$ and $l'$ must also be $\\theta - \\frac{\\pi}{7... |
https://artofproblemsolving.com/wiki/index.php/1967_AHSME_Problems/Problem_40 | D | 79 | Located inside equilateral triangle $ABC$ is a point $P$ such that $PA=8$ $PB=6$ , and $PC=10$ . To the nearest integer the area of triangle $ABC$ is:
$\textbf{(A)}\ 159\qquad \textbf{(B)}\ 131\qquad \textbf{(C)}\ 95\qquad \textbf{(D)}\ 79\qquad \textbf{(E)}\ 50$ | [
"\nNotice that $6^2+8^2=10^2.$ That makes us want to construct a right triangle.\nRotate $\\triangle APC$ $60^{\\circ}$ about A. Note that $\\triangle PAC \\cong \\triangle P'AB$ , so \\[\\angle P'AP = \\angle PAB + \\angle P'AB = \\angle PAB + \\angle PAC = 60^{\\circ}.\\]\nTherefore, $\\triangle APP'$ is equilate... |
https://artofproblemsolving.com/wiki/index.php/1967_AHSME_Problems/Problem_40 | null | 79 | Located inside equilateral triangle $ABC$ is a point $P$ such that $PA=8$ $PB=6$ , and $PC=10$ . To the nearest integer the area of triangle $ABC$ is:
$\textbf{(A)}\ 159\qquad \textbf{(B)}\ 131\qquad \textbf{(C)}\ 95\qquad \textbf{(D)}\ 79\qquad \textbf{(E)}\ 50$ | [
"Let $s$ be the side length of $ABC.$ Notice that $s\\le 14$ by the triangle inequality. This means that \\[[ABC]\\le\\dfrac{14^2\\sqrt{3}}{2}\\approx 84.87.\\] This automatically rules out choices $A, B,$ and $C.$ Now, we will look at if the area is $50$ . By the equilateral triangle area formula, $s$ would equal ... |
https://artofproblemsolving.com/wiki/index.php/2010_AMC_10A_Problems/Problem_12 | C | 0.4 | Logan is constructing a scaled model of his town. The city's water tower stands 40 meters high, and the top portion is a sphere that holds 100,000 liters of water. Logan's miniature water tower holds 0.1 liters. How tall, in meters, should Logan make his tower?
$\textbf{(A)}\ 0.04 \qquad \textbf{(B)}\ \frac{0.4}{\pi} \... | [
"The water tower holds $\\frac{100000}{0.1} = 1000000$ times more water than Logan's miniature. The volume of a sphere is: $V=\\dfrac{4}{3}\\pi r^3$ . Since we are comparing the heights (m), we should compare the radii (m) to find the ratio. Since, the radius is cubed, Logan should make his tower $\\sqrt[3]{1000000... |
https://artofproblemsolving.com/wiki/index.php/2010_AMC_12A_Problems/Problem_7 | C | 0.4 | Logan is constructing a scaled model of his town. The city's water tower stands 40 meters high, and the top portion is a sphere that holds 100,000 liters of water. Logan's miniature water tower holds 0.1 liters. How tall, in meters, should Logan make his tower?
$\textbf{(A)}\ 0.04 \qquad \textbf{(B)}\ \frac{0.4}{\pi} \... | [
"The water tower holds $\\frac{100000}{0.1} = 1000000$ times more water than Logan's miniature. The volume of a sphere is: $V=\\dfrac{4}{3}\\pi r^3$ . Since we are comparing the heights (m), we should compare the radii (m) to find the ratio. Since, the radius is cubed, Logan should make his tower $\\sqrt[3]{1000000... |
https://artofproblemsolving.com/wiki/index.php/2002_AMC_8_Problems/Problem_25 | B | 14 | Loki, Moe, Nick and Ott are good friends. Ott had no money, but the others did. Moe gave Ott one-fifth of his money, Loki gave Ott one-fourth of his money and Nick gave Ott one-third of his money. Each gave Ott the same amount of money. What fractional part of the group's money does Ott now have?
$\text{(A)}\ \frac{1}{... | [
"Since Ott gets equal amounts of money from each friend, we can say that he gets $x$ dollars from each friend. This means that Moe has $5x$ dollars, Loki has $4x$ dollars, and Nick has $3x$ dollars. The total amount is $12x$ dollars, and since Ott gets $3x$ dollars total, $\\frac{3x}{12x}= \\frac{3}{12} = \\boxed{1... |
https://artofproblemsolving.com/wiki/index.php/2023_AMC_8_Problems/Problem_8 | A | 101 | Lola, Lolo, Tiya, and Tiyo participated in a ping pong tournament. Each player competed against each of the other three players exactly twice. Shown below are the win-loss records for the players. The numbers $1$ and $0$ represent a win or loss, respectively. For example, Lola won five matches and lost the fourth match... | [
"We can calculate the total number of wins ( $1$ 's) by seeing how many matches were players, which is $12$ matches played. Then, we can calculate the # of wins already on the table, which is $5 + 3 + 2 = 10$ , so there are $12 - 10 = 2$ wins left in the mystery player. Now, we will make the key observation that th... |
https://artofproblemsolving.com/wiki/index.php/2020_AMC_8_Problems/Problem_1 | E | 24 | Luka is making lemonade to sell at a school fundraiser. His recipe requires $4$ times as much water as sugar and twice as much sugar as lemon juice. He uses $3$ cups of lemon juice. How many cups of water does he need?
$\textbf{(A) }6 \qquad \textbf{(B) }8 \qquad \textbf{(C) }12 \qquad \textbf{(D) }18 \qquad \textbf{(E... | [
"We have $\\text{water} : \\text{sugar} : \\text{lemon juice} = 4\\cdot 2 : 2 : 1 = 8 : 2 : 1,$ so Luka needs $3 \\cdot 8 = \\boxed{24}$ cups.",
"Since the amount of sugar is twice the amount of lemon juice, Luka uses $3\\cdot2=6$ cups of sugar.\nSince the amount of water is $4$ times the amount of sugar, he uses... |
https://artofproblemsolving.com/wiki/index.php/2023_AMC_10B_Problems/Problem_5 | A | 10 | Maddy and Lara see a list of numbers written on a blackboard. Maddy adds $3$ to each number in the list and finds that the sum of her new numbers is $45$ . Lara multiplies each number in the list by $3$ and finds that the sum of her new numbers is also $45$ . How many numbers are written on the blackboard?
$\textbf{(A)... | [
"Let there be $n$ numbers in the list of numbers, and let their sum be $S$ . Then we have the following\n\\[S+3n=45\\]\n\\[3S=45\\]\nFrom the second equation, $S=15$ . So, $15+3n=45$ $\\Rightarrow$ $n=\\boxed{10}.$",
"Let $x_1,x_2,x_3,...,x_n$ where $x_n$ represents the $n$ th number written on the board. Lara's ... |
https://artofproblemsolving.com/wiki/index.php/2010_AMC_10B_Problems/Problem_2 | C | 25 | Makarla attended two meetings during her $9$ -hour work day. The first meeting took $45$ minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings?
$\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 25 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 35$ | [
"The total number of minutes in her $9$ -hour work day is $9 \\times 60 = 540.$ The total amount of time spend in meetings in minutes is $45 + 45 \\times 2 = 135.$ The answer is then $\\frac{135}{540}$ $= \\boxed{25}$"
] |
https://artofproblemsolving.com/wiki/index.php/2010_AMC_12B_Problems/Problem_1 | C | 25 | Makarla attended two meetings during her $9$ -hour work day. The first meeting took $45$ minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings?
$\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 25 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 35$ | [
"The total number of minutes in her $9$ -hour work day is $9 \\times 60 = 540.$ The total amount of time spend in meetings in minutes is $45 + 45 \\times 2 = 135.$ The answer is then $\\frac{135}{540}$ $= \\boxed{25}$"
] |
https://artofproblemsolving.com/wiki/index.php/2023_AMC_8_Problems/Problem_9 | B | 8 | Malaika is skiing on a mountain. The graph below shows her elevation, in meters, above the base of the mountain as she skis along a trail. In total, how many seconds does she spend at an elevation between $4$ and $7$ meters? [asy] // Diagram by TheMathGuyd. Found cubic, so graph is perfect. import graph; size(8cm); int... | [
"We mark the time intervals in which Malaika's elevation is between $4$ and $7$ meters in red, as shown below: The requested time intervals are:\nIn total, Malaika spends $(4-2) + (10-6) + (14-12) = \\boxed{8}$ seconds at such elevation.",
"Notice that the entire section between the $2$ second mark and the $14$ ... |
https://artofproblemsolving.com/wiki/index.php/2017_AMC_8_Problems/Problem_8 | D | 8 | Malcolm wants to visit Isabella after school today and knows the street where she lives but doesn't know her house number. She tells him, "My house number has two digits, and exactly three of the following four statements about it are true."
(1) It is prime.
(2) It is even.
(3) It is divisible by 7.
(4) One of its digi... | [
"Notice that (1) cannot be true. Otherwise, the number would have to be prime and be either even or divisible by 7. This only happens if the number is 2 or 7, neither of which are two-digit numbers, so we run into a contradiction. Thus, we must have (2), (3), and (4) be true. By (2), the $2$ -digit number is even, ... |
https://artofproblemsolving.com/wiki/index.php/2002_AIME_I_Problems/Problem_1 | null | 59 | Many states use a sequence of three letters followed by a sequence of three digits as their standard license-plate pattern. Given that each three-letter three-digit arrangement is equally likely, the probability that such a license plate will contain at least one palindrome (a three-letter arrangement or a three-digit... | [
"Using complementary counting, we count all of the license plates that do not have the desired property. To not be a palindrome, the first and third characters of each string must be different. Therefore, there are $10\\cdot 10\\cdot 9$ three-digit non-palindromes, and there are $26\\cdot 26\\cdot 25$ three-lette... |
https://artofproblemsolving.com/wiki/index.php/2003_AMC_10B_Problems/Problem_6 | D | 21.5 | Many television screens are rectangles that are measured by the length of their diagonals. The ratio of the horizontal length to the height in a standard television screen is $4:3$ . The horizontal length of a " $27$ -inch" television screen is closest, in inches, to which of the following?
$\textbf{(A) } 20 \qquad\tex... | [
"If you divide the television screen into two right triangles, the legs are in the ratio of $4 : 3$ , and we can let one leg be $4x$ and the other be $3x$ . Then we can use the Pythagorean Theorem.\n\\begin{align*}(4x)^2+(3x)^2&=27^2\\\\ 16x^2+9x^2&=729\\\\ 25x^2&=729\\\\ x^2&=\\frac{729}{25}\\\\ x&=\\frac{27}{5}\\... |
https://artofproblemsolving.com/wiki/index.php/2003_AMC_12B_Problems/Problem_5 | D | 21.5 | Many television screens are rectangles that are measured by the length of their diagonals. The ratio of the horizontal length to the height in a standard television screen is $4:3$ . The horizontal length of a " $27$ -inch" television screen is closest, in inches, to which of the following?
$\textbf{(A) } 20 \qquad\tex... | [
"If you divide the television screen into two right triangles, the legs are in the ratio of $4 : 3$ , and we can let one leg be $4x$ and the other be $3x$ . Then we can use the Pythagorean Theorem.\n\\begin{align*}(4x)^2+(3x)^2&=27^2\\\\ 16x^2+9x^2&=729\\\\ 25x^2&=729\\\\ x^2&=\\frac{729}{25}\\\\ x&=\\frac{27}{5}\\... |
https://artofproblemsolving.com/wiki/index.php/2011_AMC_8_Problems/Problem_1 | E | 3.5 | Margie bought $3$ apples at a cost of $50$ cents per apple. She paid with a 5-dollar bill. How much change did Margie receive?
$\textbf{(A)}\ \textdollar 1.50 \qquad \textbf{(B)}\ \textdollar 2.00 \qquad \textbf{(C)}\ \textdollar 2.50 \qquad \textbf{(D)}\ \textdollar 3.00 \qquad \textbf{(E)}\ \textdollar 3.50$ | [
"$50$ cents is equivalent to $\\textdollar 0.50.$ Then the three apples cost $3 \\times \\textdollar 0.50 = \\textdollar 1.50.$ The change Margie receives is $\\textdollar 5.00 - \\textdollar 1.50 = \\boxed{3.50}$"
] |
https://artofproblemsolving.com/wiki/index.php/2014_AMC_8_Problems/Problem_5 | C | 160 | Margie's car can go $32$ miles on a gallon of gas, and gas currently costs $ $4$ per gallon. How many miles can Margie drive on $\textdollar 20$ worth of gas?
$\textbf{(A) }64\qquad\textbf{(B) }128\qquad\textbf{(C) }160\qquad\textbf{(D) }320\qquad \textbf{(E) }640$ | [
"Margie can afford $20/4=5$ gallons of gas. She can go $32\\cdot5=\\boxed{160}$ miles on this amount of gas."
] |
https://artofproblemsolving.com/wiki/index.php/2008_AMC_8_Problems/Problem_25 | A | 42 | Margie's winning art design is shown. The smallest circle has radius 2 inches, with each successive circle's radius increasing by 2 inches. Which of the following is closest to the percent of the design that is black?
[asy] real d=320; pair O=origin; pair P=O+8*dir(d); pair A0 = origin; pair A1 = O+1*dir(d); pair A2 = ... | [
"Let the smallest circle be 1, the second smallest circle be 2, the third smallest circle be 3, etc. \\[\\begin{array}{c|cc} \\text{circle \\#} & \\text{radius} & \\text{area} \\\\ \\hline 1 & 2 & 4\\pi \\\\ 2 & 4 & 16\\pi \\\\ 3 & 6 & 36\\pi \\\\ 4 & 8 & 64\\pi \\\\ 5 & 10 & 100\\pi \\\\ 6 & 12 & 144\\pi \\end{arr... |
https://artofproblemsolving.com/wiki/index.php/1988_AJHSME_Problems/Problem_23 | D | 240 | Maria buys computer disks at a price of $4$ for $$5$ and sells them at a price of $3$ for $$5$ . How many computer disks must she sell in order to make a profit of $$100$
$\text{(A)}\ 100 \qquad \text{(B)}\ 120 \qquad \text{(C)}\ 200 \qquad \text{(D)}\ 240 \qquad \text{(E)}\ 1200$ | [
"This is the equivalent of saying she buys $12$ for $$15$ and sells $12$ for $$20$ , so for every dozen disks she sells, she profits $$5$\nShe needs to profit $$100$ , so she needs to sell $\\frac{100}{5}=20$ dozen disks, which is $240\\rightarrow \\boxed{240}$"
] |
https://artofproblemsolving.com/wiki/index.php/2012_AMC_8_Problems/Problem_21 | D | 50 | Marla has a large white cube that has an edge of 10 feet. She also has enough green paint to cover 300 square feet. Marla uses all the paint to create a white square centered on each face, surrounded by a green border. What is the area of one of the white squares, in square feet? $\textbf{(A)}\hspace{.05in}5\sqrt2\qqua... | [
"If Marla evenly distributes her $300$ square feet of paint between the 6 faces, each face will get $300\\div6 = 50$ square feet of paint. The surface area of one of the faces of the cube is $10^2 = 100$ square feet. Therefore, there will be $100-50 = \\boxed{50}$ square feet of white on each side."
] |
https://artofproblemsolving.com/wiki/index.php/1987_AJHSME_Problems/Problem_4 | C | 125 | Martians measure angles in clerts. There are $500$ clerts in a full circle. How many clerts are there in a right angle?
$\text{(A)}\ 90 \qquad \text{(B)}\ 100 \qquad \text{(C)}\ 125 \qquad \text{(D)}\ 180 \qquad \text{(E)}\ 250$ | [
"The right angle is $1/4$ of the circle, hence it contains $500/4=125\\rightarrow \\boxed{125}$ clerts."
] |
https://artofproblemsolving.com/wiki/index.php/2018_AMC_10B_Problems/Problem_21 | C | 340 | Mary chose an even $4$ -digit number $n$ . She wrote down all the divisors of $n$ in increasing order from left to right: $1,2,\ldots,\dfrac{n}{2},n$ . At some moment Mary wrote $323$ as a divisor of $n$ . What is the smallest possible value of the next divisor written to the right of $323$
$\textbf{(A) } 324 \qquad \t... | [
"Let $d$ be the next divisor written to the right of $323.$\nIf $\\gcd(323,d)=1,$ then \\[n\\geq323d>323^2>100^2=10000,\\] which contradicts the precondition that $n$ is a $4$ -digit number.\nIt follows that $\\gcd(323,d)>1.$ Since $323=17\\cdot19,$ the smallest possible value of $d$ is $17\\cdot20=\\boxed{340}(323... |
https://artofproblemsolving.com/wiki/index.php/2018_AMC_12B_Problems/Problem_19 | C | 340 | Mary chose an even $4$ -digit number $n$ . She wrote down all the divisors of $n$ in increasing order from left to right: $1,2,\ldots,\dfrac{n}{2},n$ . At some moment Mary wrote $323$ as a divisor of $n$ . What is the smallest possible value of the next divisor written to the right of $323$
$\textbf{(A) } 324 \qquad \t... | [
"Let $d$ be the next divisor written to the right of $323.$\nIf $\\gcd(323,d)=1,$ then \\[n\\geq323d>323^2>100^2=10000,\\] which contradicts the precondition that $n$ is a $4$ -digit number.\nIt follows that $\\gcd(323,d)>1.$ Since $323=17\\cdot19,$ the smallest possible value of $d$ is $17\\cdot20=\\boxed{340}(323... |
https://artofproblemsolving.com/wiki/index.php/2012_AMC_10A_Problems/Problem_10 | C | 8 | Mary divides a circle into 12 sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle?
$\textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 12... | [
"Let $a_1$ be the first term of the arithmetic progression and $a_{12}$ be the last term of the arithmetic progression. From the formula of the sum of an arithmetic progression (or arithmetic series), we have $12*\\frac{a_1+a_{12}}{2}=360$ , which leads us to $a_1 + a_{12} = 60$ $a_{12}$ , the largest term of the p... |
https://artofproblemsolving.com/wiki/index.php/2012_AMC_12A_Problems/Problem_7 | C | 8 | Mary divides a circle into 12 sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle?
$\textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 12... | [
"Let $a_1$ be the first term of the arithmetic progression and $a_{12}$ be the last term of the arithmetic progression. From the formula of the sum of an arithmetic progression (or arithmetic series), we have $12*\\frac{a_1+a_{12}}{2}=360$ , which leads us to $a_1 + a_{12} = 60$ $a_{12}$ , the largest term of the p... |
https://artofproblemsolving.com/wiki/index.php/2017_AMC_10B_Problems/Problem_1 | B | 12 | Mary thought of a positive two-digit number. She multiplied it by $3$ and added $11$ . Then she switched the digits of the result, obtaining a number between $71$ and $75$ , inclusive. What was Mary's number?
$\textbf{(A)}\ 11\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 13\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 15$ | [
"Let her $2$ -digit number be $x$ . Multiplying by $3$ makes it a multiple of $3$ , meaning that the sum of its digits is divisible by $3$ . Adding on $11$ increases the sum of the digits by $1+1 = 2,$ (we can ignore numbers such as $39+11=50$ ) and reversing the digits keeps the sum of the digits the same; this me... |
https://artofproblemsolving.com/wiki/index.php/1984_AIME_Problems/Problem_10 | null | 119 | Mary told John her score on the American High School Mathematics Examination (AHSME), which was over $80$ . From this, John was able to determine the number of problems Mary solved correctly. If Mary's score had been any lower, but still over $80$ , John could not have determined this. What was Mary's score? (Recall th... | [
"Let Mary's score, number correct, and number wrong be $s,c,w$ respectively. Then \\begin{align*} s&=30+4c-w \\\\ &=30+4(c-1)-(w-4) \\\\ &=30+4(c+1)-(w+4). \\end{align*} Therefore, Mary could not have left at least five blank; otherwise, one more correct and four more wrong would produce the same score. Similarly, ... |
https://artofproblemsolving.com/wiki/index.php/2010_AMC_10A_Problems/Problem_1 | D | 4 | Mary's top book shelf holds five books with the following widths, in centimeters: $6$ $\dfrac{1}{2}$ $1$ $2.5$ , and $10$ . What is the average book width, in centimeters?
$\mathrm{(A)}\ 1 \qquad \mathrm{(B)}\ 2 \qquad \mathrm{(C)}\ 3 \qquad \mathrm{(D)}\ 4 \qquad \mathrm{(E)}\ 5$ | [
"To find the average, we add up the widths $6$ $\\dfrac{1}{2}$ $1$ $2.5$ , and $10$ , to get a total sum of $20$ . Since there are $5$ books, the average book width is $\\frac{20}{5}=4$ The answer is $\\boxed{4}$"
] |
https://artofproblemsolving.com/wiki/index.php/2023_AMC_10A_Problems/Problem_10 | D | 7 | 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? $\textbf{(A) }4\qquad\textbf{(B) }5\q... | [
"Let $a$ represent the amount of tests taken previously and $x$ the mean of the scores taken previously.\nWe can write the following equations:\n\\[\\frac{ax+11}{a+1}=x+1\\qquad (1)\\] \\[\\frac{ax+33}{a+3}=x+2\\qquad (2)\\]\nMultiplying $(x+1)$ by $(a+1)$ and solving, we get: \\[ax+11=ax+a+x+1\\] \\[11=a+x+1\\] \\... |
https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_8 | D | 7 | 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? $\textbf{(A) }4\qquad\textbf{(B) }5\q... | [
"Let $a$ represent the amount of tests taken previously and $x$ the mean of the scores taken previously.\nWe can write the following equations:\n\\[\\frac{ax+11}{a+1}=x+1\\qquad (1)\\] \\[\\frac{ax+33}{a+3}=x+2\\qquad (2)\\]\nMultiplying $(x+1)$ by $(a+1)$ and solving, we get: \\[ax+11=ax+a+x+1\\] \\[11=a+x+1\\] \\... |
https://artofproblemsolving.com/wiki/index.php/2010_AIME_I_Problems/Problem_1 | null | 107 | Maya lists all the positive divisors of $2010^2$ . She then randomly selects two distinct divisors from this list. Let $p$ be the probability that exactly one of the selected divisors is a perfect square . The probability $p$ can be expressed in the form $\frac {m}{n}$ , where $m$ and $n$ are relatively prime positive ... | [
"$2010^2 = 2^2\\cdot3^2\\cdot5^2\\cdot67^2$ . Thus there are $(2+1)^4$ divisors, $(1+1)^4$ of which are squares (the exponent of each prime factor must either be $0$ or $2$ ). Therefore the probability is \\[\\frac {2\\cdot2^4\\cdot(3^4 - 2^4)}{3^4(3^4 - 1)} = \\frac {26}{81} \\Longrightarrow 26+ 81 = \\boxed{107}.... |
https://artofproblemsolving.com/wiki/index.php/1997_AHSME_Problems/Problem_15 | D | 64 | Medians $BD$ and $CE$ of triangle $ABC$ are perpendicular, $BD=8$ , and $CE=12$ . The area of triangle $ABC$ is
[asy] defaultpen(linewidth(.8pt)); dotfactor=4; pair A = origin; pair B = (1.25,1); pair C = (2,0); pair D = midpoint(A--C); pair E = midpoint(A--B); pair G = intersectionpoint(E--C,B--D); dot(A);dot(B);dot(... | [
"\nOne median divides a triangle into $2$ equal areas, so all three medians will divide a triangle into $6$ equal areas.\nThe median $CE$ is divided into a $2:1$ ratio at centroid $G$ , so $GE = \\frac{1}{3}\\cdot CE = \\frac{1}{3}\\cdot 12 = 4$\nSimilarly, $BG = \\frac{2}{3}\\cdot 8 = \\frac{16}{3}$\nThe area of t... |
https://artofproblemsolving.com/wiki/index.php/2013_AIME_I_Problems/Problem_6 | null | 47 | Melinda has three empty boxes and $12$ textbooks, three of which are mathematics textbooks. One box will hold any three of her textbooks, one will hold any four of her textbooks, and one will hold any five of her textbooks. If Melinda packs her textbooks into these boxes in random order, the probability that all three ... | [
"The total ways the textbooks can be arranged in the 3 boxes is $12\\textbf{C}3\\cdot 9\\textbf{C}4$ , which is equivalent to $\\frac{12\\cdot 11\\cdot 10\\cdot 9\\cdot 8\\cdot 7\\cdot 6}{144}=12\\cdot11\\cdot10\\cdot7\\cdot3$ . If all of the math textbooks are put into the box that can hold $3$ textbooks, there ... |
https://artofproblemsolving.com/wiki/index.php/2003_AMC_10A_Problems/Problem_2 | B | 91 | Members of the Rockham Soccer League buy socks and T-shirts. Socks cost $4 per pair and each T-shirt costs $5 more than a pair of socks. Each member needs one pair of socks and a shirt for home games and another pair of socks and a shirt for away games. If the total cost is $2366, how many members are in the League?
$\... | [
"Since T-shirts cost $5$ dollars more than a pair of socks, T-shirts cost $5+4=9$ dollars.\nSince each member needs $2$ pairs of socks and $2$ T-shirts, the total cost for $1$ member is $2(4+9)=26$ dollars.\nSince $2366$ dollars was the cost for the club, and $26$ was the cost per member, the number of members in t... |
https://artofproblemsolving.com/wiki/index.php/2003_AMC_12A_Problems/Problem_2 | B | 91 | Members of the Rockham Soccer League buy socks and T-shirts. Socks cost $4 per pair and each T-shirt costs $5 more than a pair of socks. Each member needs one pair of socks and a shirt for home games and another pair of socks and a shirt for away games. If the total cost is $2366, how many members are in the League?
$\... | [
"Since T-shirts cost $5$ dollars more than a pair of socks, T-shirts cost $5+4=9$ dollars.\nSince each member needs $2$ pairs of socks and $2$ T-shirts, the total cost for $1$ member is $2(4+9)=26$ dollars.\nSince $2366$ dollars was the cost for the club, and $26$ was the cost per member, the number of members in t... |
https://artofproblemsolving.com/wiki/index.php/2021_Fall_AMC_10A_Problems/Problem_2 | E | 20 | Menkara has a $4 \times 6$ index card. If she shortens the length of one side of this card by $1$ inch, the card would have area $18$ square inches. What would the area of the card be in square inches if instead she shortens the length of the other side by $1$ inch?
$\textbf{(A) } 16 \qquad\textbf{(B) } 17 \qquad\textb... | [
"We construct the following table: \\[\\begin{array}{c||c|c||c} & & & \\\\ [-2.5ex] \\textbf{Scenario} & \\textbf{Length} & \\textbf{Width} & \\textbf{Area} \\\\ [0.5ex] \\hline & & & \\\\ [-2ex] \\text{Initial} & 4 & 6 & 24 \\\\ \\text{Menkara shortens one side.} & 3 & 6 & 18 \\\\ \\text{Menkara shortens other sid... |
https://artofproblemsolving.com/wiki/index.php/2021_Fall_AMC_12A_Problems/Problem_2 | E | 20 | Menkara has a $4 \times 6$ index card. If she shortens the length of one side of this card by $1$ inch, the card would have area $18$ square inches. What would the area of the card be in square inches if instead she shortens the length of the other side by $1$ inch?
$\textbf{(A) } 16 \qquad\textbf{(B) } 17 \qquad\textb... | [
"We construct the following table: \\[\\begin{array}{c||c|c||c} & & & \\\\ [-2.5ex] \\textbf{Scenario} & \\textbf{Length} & \\textbf{Width} & \\textbf{Area} \\\\ [0.5ex] \\hline & & & \\\\ [-2ex] \\text{Initial} & 4 & 6 & 24 \\\\ \\text{Menkara shortens one side.} & 3 & 6 & 18 \\\\ \\text{Menkara shortens other sid... |
https://artofproblemsolving.com/wiki/index.php/2017_AMC_10A_Problems/Problem_4 | B | 14 | Mia is "helping" her mom pick up $30$ toys that are strewn on the floor. Mia’s mom manages to put $3$ toys into the toy box every $30$ seconds, but each time immediately after those $30$ seconds have elapsed, Mia takes $2$ toys out of the box. How much time, in minutes, will it take Mia and her mom to put all $30$ toys... | [
"Every $30$ seconds, $3$ toys are put in the box and $2$ toys are taken out, so the number of toys in the box increases by $3-2=1$ every $30$ seconds. Then after $27 \\times 30 = 810$ seconds (or $13 \\frac{1}{2}$ minutes), there are $27$ toys in the box. Mia's mom will then put the remaining $3$ toys into the box ... |
https://artofproblemsolving.com/wiki/index.php/2008_AMC_10B_Problems/Problem_25 | B | 5 | Michael walks at the rate of $5$ feet per second on a long straight path. Trash pails are located every $200$ feet along the path. A garbage truck traveling at $10$ feet per second in the same direction as Michael stops for $30$ seconds at each pail. As Michael passes a pail, he notices the truck ahead of him just leav... | [
"Pick a coordinate system where Michael's starting pail is $0$ and the one where the truck starts is $200$ .\nLet $M(t)$ and $T(t)$ be the coordinates of Michael and the truck after $t$ seconds.\nLet $D(t)=T(t)-M(t)$ be their (signed) distance after $t$ seconds.\nMeetings occur whenever $D(t)=0$ .\nWe have $D(0)=20... |
https://artofproblemsolving.com/wiki/index.php/2008_AMC_12B_Problems/Problem_20 | B | 5 | Michael walks at the rate of $5$ feet per second on a long straight path. Trash pails are located every $200$ feet along the path. A garbage truck traveling at $10$ feet per second in the same direction as Michael stops for $30$ seconds at each pail. As Michael passes a pail, he notices the truck ahead of him just leav... | [
"Pick a coordinate system where Michael's starting pail is $0$ and the one where the truck starts is $200$ .\nLet $M(t)$ and $T(t)$ be the coordinates of Michael and the truck after $t$ seconds.\nLet $D(t)=T(t)-M(t)$ be their (signed) distance after $t$ seconds.\nMeetings occur whenever $D(t)=0$ .\nWe have $D(0)=20... |
https://artofproblemsolving.com/wiki/index.php/1997_AHSME_Problems/Problem_8 | D | 6 | Mientka Publishing Company prices its bestseller Where's Walter? as follows:
$C(n) =\left\{\begin{matrix}12n, &\text{if }1\le n\le 24\\ 11n, &\text{if }25\le n\le 48\\ 10n, &\text{if }49\le n\end{matrix}\right.$
where $n$ is the number of books ordered, and $C(n)$ is the cost in dollars of $n$ books. Notice that $25$ b... | [
"Clearly, the areas of concern are where the piecewise function shifts value.\nSince $C(25) = 11\\cdot 25 = 275$ , we want to find the least value of $n$ for which $C(n) > 275$\nIf $n \\le 24$ , then $C(n) = 12n$ , so for $C(n) > 275$ $12n > 275$ , which is equivalent to $n > 22.91$ . Thus, both $n=23$ and $n=24$ ... |
https://artofproblemsolving.com/wiki/index.php/2022_AMC_10A_Problems/Problem_2 | B | 7 | Mike cycled $15$ laps in $57$ minutes. Assume he cycled at a constant speed throughout. Approximately how many laps did he complete in the first $27$ minutes?
$\textbf{(A) } 5 \qquad\textbf{(B) } 7 \qquad\textbf{(C) } 9 \qquad\textbf{(D) } 11 \qquad\textbf{(E) } 13$ | [
"Mike's speed is $\\frac{15}{57}=\\frac{5}{19}$ laps per minute.\nIn the first $27$ minutes, he completed approximately $\\frac{5}{19}\\cdot27\\approx\\frac{1}{4}\\cdot28=\\boxed{7}$ laps.",
"Mike runs $1$ lap in $\\frac{57}{15}=\\frac{19}{5}$ minutes. So, in $27$ minutes, Mike ran about $\\frac{27}{\\frac{19}{5}... |
https://artofproblemsolving.com/wiki/index.php/2002_AMC_8_Problems/Problem_24 | B | 40 | Miki has a dozen oranges of the same size and a dozen pears of the same size. Miki uses her juicer to extract 8 ounces of pear juice from 3 pears and 8 ounces of orange juice from 2 oranges. She makes a pear-orange juice blend from an equal number of pears and oranges. What percent of the blend is pear juice?
$\text{(A... | [
"A pear gives $8/3$ ounces of juice per pear. An orange gives $8/2=4$ ounces of juice per orange. If the pear-orange juice blend used one pear and one orange each, the percentage of pear juice would be\n\\[\\frac{8/3}{8/3+4} \\times 100 = \\frac{8}{8+12} \\times 100 = \\boxed{40}\\]",
"Since it doesn't matter ho... |
https://artofproblemsolving.com/wiki/index.php/2006_AMC_8_Problems/Problem_1 | D | 17 | Mindy made three purchases for $\textdollar 1.98$ dollars, $\textdollar 5.04$ dollars, and $\textdollar 9.89$ dollars. What was her total, to the nearest dollar?
$\textbf{(A)}\ 10\qquad\textbf{(B)}\ 15\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 17\qquad\textbf{(E)}\ 18$ | [
"The three prices round to $\\textdollar 2$ $\\textdollar 5$ , and $\\textdollar 10$ , which have a sum of $\\boxed{17}$"
] |
https://artofproblemsolving.com/wiki/index.php/2024_AMC_8_Problems/Problem_16 | D | 11 | Minh enters the numbers $1$ through $81$ into the cells of a $9 \times 9$ grid in some order. She calculates the product of the numbers in each row and column. What is the least number of rows and columns that could have a product divisible by $3$
$\textbf{(A) } 8\qquad\textbf{(B) } 9\qquad\textbf{(C) } 10\qquad\textbf... | [
"\nWe know that if a row/column of numbers has a single multiple of $3$ , that entire row/column will be divisible by $3$ . Since there are $27$ multiples of $3$ from $1$ to $81$ , We need to find a way to place the $54$ non-multiples of $3$ such that they take up as many entire rows and columns as possible.\nIf we... |
https://artofproblemsolving.com/wiki/index.php/2004_AMC_10B_Problems/Problem_8 | A | 13 | Minneapolis-St. Paul International Airport is $8$ miles southwest of downtown St. Paul and $10$ miles southeast of downtown Minneapolis. Which of the following is closest to the number of miles between downtown St. Paul and downtown Minneapolis?
$\mathrm{(A)\ }13\qquad\mathrm{(B)\ }14\qquad\mathrm{(C)\ }15\qquad\mathrm... | [
"The directions \"southwest\" and \"southeast\" are orthogonal. Thus the described situation is a right triangle with legs $8$ miles and $10$ miles long. The hypotenuse length is $\\sqrt{8^2 + 10^2}\\approx12.8$ , and thus the answer is $\\boxed{13}$"
] |
https://artofproblemsolving.com/wiki/index.php/2004_AMC_12B_Problems/Problem_6 | A | 13 | Minneapolis-St. Paul International Airport is $8$ miles southwest of downtown St. Paul and $10$ miles southeast of downtown Minneapolis. Which of the following is closest to the number of miles between downtown St. Paul and downtown Minneapolis?
$\mathrm{(A)\ }13\qquad\mathrm{(B)\ }14\qquad\mathrm{(C)\ }15\qquad\mathrm... | [
"The directions \"southwest\" and \"southeast\" are orthogonal. Thus the described situation is a right triangle with legs $8$ miles and $10$ miles long. The hypotenuse length is $\\sqrt{8^2 + 10^2}\\approx12.8$ , and thus the answer is $\\boxed{13}$"
] |
https://artofproblemsolving.com/wiki/index.php/2017_AMC_10A_Problems/Problem_9 | C | 65 | Minnie rides on a flat road at $20$ kilometers per hour (kph), downhill at $30$ kph, and uphill at $5$ kph. Penny rides on a flat road at $30$ kph, downhill at $40$ kph, and uphill at $10$ kph. Minnie goes from town $A$ to town $B$ , a distance of $10$ km all uphill, then from town $B$ to town $C$ , a distance of $15$ ... | [
"The distance from town $A$ to town $B$ is $10$ km uphill, and since Minnie rides uphill at a speed of $5$ kph, it will take her $2$ hours. Next, she will ride from town $B$ to town $C$ , a distance of $15$ km all downhill. Since Minnie rides downhill at a speed of $30$ kph, it will take her half an hour. Finally, ... |
https://artofproblemsolving.com/wiki/index.php/2018_AIME_II_Problems/Problem_13 | null | 647 | Misha rolls a standard, fair six-sided die until she rolls $1-2-3$ in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is $\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$ | [
"Let $P_n$ be the probability of getting consecutive $1,2,3$ rolls in $n$ rolls and not rolling $1,2,3$ prior to the nth roll.\nLet $x = P_3+P_5+...=1-(P_4+P_6+..)$ . Following Solution 2, one can see that \\[P_{n+1}=P_{n}-\\frac{P_{n-2}}{6^3}\\] for all positive integers $n \\ge 5$ . Summing for $n=5,7,...$ gives ... |
https://artofproblemsolving.com/wiki/index.php/2003_AMC_10B_Problems/Problem_5 | C | 1.35 | Moe uses a mower to cut his rectangular $90$ -foot by $150$ -foot lawn. The swath he cuts is $28$ inches wide, but he overlaps each cut by $4$ inches to make sure that no grass is missed. He walks at the rate of $5000$ feet per hour while pushing the mower. Which of the following is closest to the number of hours it wi... | [
"Since the swath Moe actually mows is $24$ inches, or $2$ feet wide, he mows $10000$ square feet in one hour. His lawn has an area of $13500$ , so it will take Moe $1.35$ hours to finish mowing the lawn. Thus the answer is $\\boxed{1.35}$",
"Let's assume that the swath moves back and forth; parallel to the $90$ f... |
https://artofproblemsolving.com/wiki/index.php/2003_AMC_12B_Problems/Problem_4 | C | 1.35 | Moe uses a mower to cut his rectangular $90$ -foot by $150$ -foot lawn. The swath he cuts is $28$ inches wide, but he overlaps each cut by $4$ inches to make sure that no grass is missed. He walks at the rate of $5000$ feet per hour while pushing the mower. Which of the following is closest to the number of hours it wi... | [
"Since the swath Moe actually mows is $24$ inches, or $2$ feet wide, he mows $10000$ square feet in one hour. His lawn has an area of $13500$ , so it will take Moe $1.35$ hours to finish mowing the lawn. Thus the answer is $\\boxed{1.35}$",
"Let's assume that the swath moves back and forth; parallel to the $90$ f... |
https://artofproblemsolving.com/wiki/index.php/2010_AMC_12B_Problems/Problem_23 | A | 100 | Monic quadratic polynomial $P(x)$ and $Q(x)$ have the property that $P(Q(x))$ has zeros at $x=-23, -21, -17,$ and $-15$ , and $Q(P(x))$ has zeros at $x=-59,-57,-51$ and $-49$ . What is the sum of the minimum values of $P(x)$ and $Q(x)$
$\textbf{(A)}\ -100 \qquad \textbf{(B)}\ -82 \qquad \textbf{(C)}\ -73 \qquad \textbf... | [
"$P(x) = (x - a)^2 - b, Q(x) = (x - c)^2 - d$ . Notice that $P(x)$ has roots $a\\pm \\sqrt {b}$ , so that the roots of $P(Q(x))$ are the roots of $Q(x) = a + \\sqrt {b}, a - \\sqrt {b}$ . For each individual equation, the sum of the roots will be $2c$ (symmetry or Vieta's). Thus, we have $4c = - 23 - 21 - 17 - 15$ ... |
https://artofproblemsolving.com/wiki/index.php/2010_AMC_12B_Problems/Problem_23 | null | 100 | Monic quadratic polynomial $P(x)$ and $Q(x)$ have the property that $P(Q(x))$ has zeros at $x=-23, -21, -17,$ and $-15$ , and $Q(P(x))$ has zeros at $x=-59,-57,-51$ and $-49$ . What is the sum of the minimum values of $P(x)$ and $Q(x)$
$\textbf{(A)}\ -100 \qquad \textbf{(B)}\ -82 \qquad \textbf{(C)}\ -73 \qquad \textbf... | [
"Let $P(x) = x^2 + Bx + C$ and $Q(x) = x^2 + Ex + F$\nThen $P(Q(x))$ is $(x^2 + Ex + F)^2 + B(x^2 + Ex + F) + C$ , which simplifies to:\n$P(Q(x)) = x^4 + 2Ex^3 + (E^2 + 2F + B)x^2 + (2EF + BE)x + (F^2 + BF + C)$\nWe can find $Q(P(x))$ by simply doing $B\\Leftrightarrow E$ and $C \\Leftrightarrow F$ to get:\n$Q(P(x)... |
https://artofproblemsolving.com/wiki/index.php/2018_AMC_8_Problems/Problem_9 | B | 87 | Monica is tiling the floor of her 12-foot by 16-foot living room. She plans to place one-foot by one-foot square tiles to form a border along the edges of the room and to fill in the rest of the floor with two-foot by two-foot square tiles. How many tiles will she use?
$\textbf{(A) }48\qquad\textbf{(B) }87\qquad\textbf... | [
"She will place $(12\\cdot2)+(14\\cdot2)=52$ tiles around the border. For the inner part of the room, we have $10\\cdot14=140$ square feet. Each tile takes up $4$ square feet, so he will use $\\frac{140}{4}=35$ tiles for the inner part of the room. Thus, the answer is $52+35= \\boxed{87}$"
] |
https://artofproblemsolving.com/wiki/index.php/2018_AMC_8_Problems/Problem_9 | null | 87 | Monica is tiling the floor of her 12-foot by 16-foot living room. She plans to place one-foot by one-foot square tiles to form a border along the edges of the room and to fill in the rest of the floor with two-foot by two-foot square tiles. How many tiles will she use?
$\textbf{(A) }48\qquad\textbf{(B) }87\qquad\textbf... | [
"The area around the border: $(12 \\cdot 2) + (14 \\cdot 2) = 52$ . The area of tiles around the border: $1 \\cdot 1 = 1$ . Therefore, $\\frac{52}{1} = 52$ is the number of tiles around the border.\nThe inner part will have $(12 - 2)(16 - 2) = 140$ . The area of those tiles are $2 \\cdot 2 = 4$ $\\frac{140}{4} = 35... |
https://artofproblemsolving.com/wiki/index.php/1951_AHSME_Problems/Problem_5 | B | 1,100 | Mr. $A$ owns a home worth $ $10,000$ . He sells it to Mr. $B$ at a $10\%$ profit based on the worth of the house. Mr. $B$ sells the house back to Mr. $A$ at a $10\%$ loss. Then:
$\mathrm{(A) \ A\ comes\ out\ even } \qquad$ $\mathrm{(B) \ A\ makes\ 1100\ on\ the\ deal}$ $\qquad \mathrm{(C) \ A\ makes\ 1000\ on\ the\ ... | [
"Mr. $A$ sells his home for $(1 + 10$ $)$ $\\cdot$ $10,000$ dollars $=$ $1.1$ $\\cdot$ $10,000$ dollars $=$ $11,000$ dollars to Mr. $B$ . Then, Mr. $B$ sells it at a price of $(1-10$ $)$ $\\cdot$ $11,000$ dollars $=$ $0.9$ $\\cdot$ $11,000$ dollars $=$ $9,900$ dollars, thus $11,000 - 9,900$ $=$ $\\boxed{1100}$"
] |
https://artofproblemsolving.com/wiki/index.php/2002_AMC_10A_Problems/Problem_12 | B | 48 | Mr. Earl E. Bird gets up every day at 8:00 AM to go to work. If he drives at an average speed of 40 miles per hour, he will be late by 3 minutes. If he drives at an average speed of 60 miles per hour, he will be early by 3 minutes. How many miles per hour does Mr. Bird need to drive to get to work exactly on time?
$\te... | [
"Let the time he needs to get there in be $t$ and the distance he travels be $d$ . From the given equations, we know that $d=\\left(t+\\frac{1}{20}\\right)40$ and $d=\\left(t-\\frac{1}{20}\\right)60$ . Setting the two equal, we have $40t+2=60t-3$ and we find $t=\\frac{1}{4}$ of an hour. Substituting t back in, we f... |
https://artofproblemsolving.com/wiki/index.php/2002_AMC_12A_Problems/Problem_11 | B | 48 | Mr. Earl E. Bird gets up every day at 8:00 AM to go to work. If he drives at an average speed of 40 miles per hour, he will be late by 3 minutes. If he drives at an average speed of 60 miles per hour, he will be early by 3 minutes. How many miles per hour does Mr. Bird need to drive to get to work exactly on time?
$\te... | [
"Let the time he needs to get there in be $t$ and the distance he travels be $d$ . From the given equations, we know that $d=\\left(t+\\frac{1}{20}\\right)40$ and $d=\\left(t-\\frac{1}{20}\\right)60$ . Setting the two equal, we have $40t+2=60t-3$ and we find $t=\\frac{1}{4}$ of an hour. Substituting t back in, we f... |
https://artofproblemsolving.com/wiki/index.php/2018_AMC_8_Problems/Problem_8 | C | 4.36 | Mr. Garcia asked the members of his health class how many days last week they exercised for at least 30 minutes. The results are summarized in the following bar graph, where the heights of the bars represent the number of students.
[asy] size(8cm); void drawbar(real x, real h) { fill((x-0.15,0.5)--(x+0.15,0.5)--(x+0... | [
"The mean, or average number of days is the total number of days divided by the total number of students. The total number of days is $1\\cdot 1+2\\cdot 3+3\\cdot 2+4\\cdot 6+5\\cdot 8+6\\cdot 3+7\\cdot 2=109$ . The total number of students is $1+3+2+6+8+3+2=25$ . Hence, $\\frac{109}{25}=\\boxed{4.36}$"
] |
https://artofproblemsolving.com/wiki/index.php/2013_AMC_10B_Problems/Problem_2 | A | 600 | Mr. Green measures his rectangular garden by walking two of the sides and finds that it is $15$ steps by $20$ steps. Each of Mr. Green's steps is $2$ feet long. Mr. Green expects a half a pound of potatoes per square foot from his garden. How many pounds of potatoes does Mr. Green expect from his garden?
$\textbf{(A)}\... | [
"Since each step is $2$ feet, his garden is $30$ by $40$ feet. Thus, the area of $30(40) = 1200$ square feet. Since he is expecting $\\frac{1}{2}$ of a pound per square foot, the total amount of potatoes expected is $1200 \\times \\frac{1}{2} = \\boxed{600}$"
] |
https://artofproblemsolving.com/wiki/index.php/2013_AMC_12B_Problems/Problem_2 | A | 600 | Mr. Green measures his rectangular garden by walking two of the sides and finds that it is $15$ steps by $20$ steps. Each of Mr. Green's steps is $2$ feet long. Mr. Green expects a half a pound of potatoes per square foot from his garden. How many pounds of potatoes does Mr. Green expect from his garden?
$\textbf{(A)}\... | [
"Since each step is $2$ feet, his garden is $30$ by $40$ feet. Thus, the area of $30(40) = 1200$ square feet. Since he is expecting $\\frac{1}{2}$ of a pound per square foot, the total amount of potatoes expected is $1200 \\times \\frac{1}{2} = \\boxed{600}$"
] |
https://artofproblemsolving.com/wiki/index.php/1985_AJHSME_Problems/Problem_21 | E | 45 | Mr. Green receives a $10\%$ raise every year. His salary after four such raises has gone up by what percent?
$\text{(A)}\ \text{less than }40\% \qquad \text{(B)}\ 40\% \qquad \text{(C)}\ 44\% \qquad \text{(D)}\ 45\% \qquad \text{(E)}\ \text{more than }45\%$ | [
"Assume his salary is originally $100$ dollars. Then, in the next year, he would have $110$ dollars, and in the next, he would have $121$ dollars. The next year he would have $133.1$ dollars and in the final year, he would have $146.41$ . As the total increase is greater than $45\\%$ , the answer is $\\boxed{45}$"
... |
https://artofproblemsolving.com/wiki/index.php/2008_AMC_8_Problems/Problem_13 | C | 187 | Mr. Harman needs to know the combined weight in pounds of three boxes he wants to mail. However, the only available scale is not accurate for weights less than $100$ pounds or more than $150$ pounds. So the boxes are weighed in pairs in every possible way. The results are $122$ $125$ and $127$ pounds. What is the combi... | [
"Each box is weighed twice during this, so the combined weight of the three boxes is half the weight of these separate measures:\n\\[\\frac{122+125+127}{2} = \\frac{374}{2} = \\boxed{187}.\\]"
] |
https://artofproblemsolving.com/wiki/index.php/1956_AHSME_Problems/Problem_28 | D | 7,000 | Mr. J left his entire estate to his wife, his daughter, his son, and the cook.
His daughter and son got half the estate, sharing in the ratio of $4$ to $3$ .
His wife got twice as much as the son. If the cook received a bequest of $\textdollar{500}$ , then the entire estate was:
$\textbf{(A)}\ \textdollar{3500}\qquad... | [
"The wife, daughter, son, and cook received estates in the ratio $6:4:3:1.$ The estate is worth $6+4+3+1 = 14$ units of $$500.,$ which is $\\boxed{7000}$"
] |
https://artofproblemsolving.com/wiki/index.php/2006_AMC_10B_Problems/Problem_25 | B | 5 | Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and th... | [
"Let $S$ be the set of the ages of Mr. Jones' children (in other words $i \\in S$ if Mr. Jones has a child who is $i$ years old). Then $|S| = 8$ and $9 \\in S$ . Let $m$ be the positive integer seen on the license plate. Since at least one of $4$ or $8$ is contained in $S$ , we have $4 | m$\nWe would like to prove ... |
https://artofproblemsolving.com/wiki/index.php/1956_AHSME_Problems/Problem_2 | D | 10 | Mr. Jones sold two pipes at $\textdollar{ 1.20}$ each. Based on the cost, his profit on one was
$20$ % and his loss on the other was $20$ %.
On the sale of the pipes, he:
$\textbf{(A)}\ \text{broke even}\qquad \textbf{(B)}\ \text{lost }4\text{ cents} \qquad\textbf{(C)}\ \text{gained }4\text{ cents}\qquad \\ \textbf{(D... | [
"For the first pipe, we are told that his profit was 20%. In other words, he sold the pipe for 120% or $\\frac{6}{5}$ of its original value. This tells us that the original price was $\\frac{5}{6}\\cdot1.20 = $1.00$\nFor the second pipe, we are told that his loss was 20%. In other words, he sold the pipe for 80% or... |
https://artofproblemsolving.com/wiki/index.php/2015_AMC_10A_Problems/Problem_5 | E | 95 | Mr. Patrick teaches math to $15$ students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was $80$ . After he graded Payton's test, the test average became $81$ . What was Payton's score on the test?
$\textbf{(A)}\ 81\qquad\textbf{(B)}\ 85\qquad\textb... | [
"If the average of the first $14$ peoples' scores was $80$ , then the sum of all of their tests is $14 \\cdot 80 = 1120$ . When Payton's score was added, the sum of all of the scores became $15 \\cdot 81 = 1215$ . So, Payton's score must be $1215-1120 = \\boxed{95}$",
"The average of a set of numbers is the val... |
https://artofproblemsolving.com/wiki/index.php/2015_AMC_12A_Problems/Problem_3 | E | 95 | Mr. Patrick teaches math to $15$ students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was $80$ . After he graded Payton's test, the test average became $81$ . What was Payton's score on the test?
$\textbf{(A)}\ 81\qquad\textbf{(B)}\ 85\qquad\textb... | [
"If the average of the first $14$ peoples' scores was $80$ , then the sum of all of their tests is $14 \\cdot 80 = 1120$ . When Payton's score was added, the sum of all of the scores became $15 \\cdot 81 = 1215$ . So, Payton's score must be $1215-1120 = \\boxed{95}$",
"The average of a set of numbers is the val... |
https://artofproblemsolving.com/wiki/index.php/2022_AMC_8_Problems/Problem_19 | C | 4 | Mr. Ramos gave a test to his class of $20$ students. The dot plot below shows the distribution of test scores. [asy] //diagram by pog . give me 1,000,000,000 dollars for this diagram size(5cm); defaultpen(0.7); dot((0.5,1)); dot((0.5,1.5)); dot((1.5,1)); dot((1.5,1.5)); dot((2.5,1)); dot((2.5,1.5)); dot((2.5,2)); dot((... | [
"We set up our cases as solution 1 showed, realizing that only the second case is possible.\nWe notice that $13$ students have scores under $85$ currently and only $5$ have scores over $85$ . We find the median of these two numbers, getting:\n\\[13-5=8\\] \\[\\frac{8}{2}=4\\] \\[13-4=9\\]\nThus, we realize that $4$... |
https://artofproblemsolving.com/wiki/index.php/2021_AMC_10B_Problems/Problem_8 | A | 367 | Mr. Zhou places all the integers from $1$ to $225$ into a $15$ by $15$ grid. He places $1$ in the middle square (eighth row and eighth column) and places other numbers one by one clockwise, as shown in part in the diagram below. What is the sum of the greatest number and the least number that appear in the second row f... | [
"In the diagram below, the red arrows indicate the progression of numbers. In the second row from the top, the greatest number and the least number are $D$ and $E,$ respectively. Note that the numbers in the yellow cells are consecutive odd perfect squares, as we can prove by induction. By observations, we proceed... |
https://artofproblemsolving.com/wiki/index.php/2006_AMC_12B_Problems/Problem_7 | null | 12 | Mr. and Mrs. Lopez have two children. When they get into their family car, two people sit in the front, and the other two sit in the back. Either Mr. Lopez or Mrs. Lopez must sit in the driver's seat. How many seating arrangements are possible?
$\text {(A) } 4 \qquad \text {(B) } 12 \qquad \text {(C) } 16 \qquad \te... | [
"There are only two possible occupants for the driver's seat. After the driver is chosen, any of the remaining three people can sit in the front, and there are two arrangements for the other two people in the back. Thus, there are $2\\cdot 3\\cdot 2 = \\boxed{12}$ possible seating arrangements. ~ aopsav (Credit to ... |
https://artofproblemsolving.com/wiki/index.php/1989_AHSME_Problems/Problem_9 | B | 300 | Mr. and Mrs. Zeta want to name their baby Zeta so that its monogram (first, middle, and last initials) will be in alphabetical order with no letter repeated. How many such monograms are possible?
$\textrm{(A)}\ 276\qquad\textrm{(B)}\ 300\qquad\textrm{(C)}\ 552\qquad\textrm{(D)}\ 600\qquad\textrm{(E)}\ 15600$ | [
"We see that for any combination of two distinct letters other than Z (as the last name will automatically be Z), there is only one possible way to arrange them in alphabetical order, thus the answer is just $\\dbinom{25}{2}=\\boxed{300}$"
] |
https://artofproblemsolving.com/wiki/index.php/2023_AMC_10B_Problems/Problem_1 | C | 16 | 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... | [
"Given that the first three glasses are full and the fourth is only $\\frac{1}{3}$ full, let's represent their contents with a common denominator, which we'll set as 6. This makes the first three glasses $\\dfrac{6}{6}$ full, and the fourth glass $\\frac{2}{6}$ full.\nTo equalize the amounts, Mrs. Jones needs to po... |
https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_1 | C | 16 | 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... | [
"Given that the first three glasses are full and the fourth is only $\\frac{1}{3}$ full, let's represent their contents with a common denominator, which we'll set as 6. This makes the first three glasses $\\dfrac{6}{6}$ full, and the fourth glass $\\frac{2}{6}$ full.\nTo equalize the amounts, Mrs. Jones needs to po... |
https://artofproblemsolving.com/wiki/index.php/2017_AMC_8_Problems/Problem_24 | D | 146 | Mrs. Sanders has three grandchildren, who call her regularly. One calls her every three days, one calls her every four days, and one calls her every five days. All three called her on December 31, 2016. On how many days during the next year did she not receive a phone call from any of her grandchildren?
$\textbf{(A) }7... | [
"We use Principle of Inclusion-Exclusion. There are $365$ days in the year, and we subtract the days that she gets at least $1$ phone call, which is \\[\\left \\lfloor \\frac{365}{3} \\right \\rfloor + \\left \\lfloor \\frac{365}{4} \\right \\rfloor + \\left \\lfloor \\frac{365}{5} \\right \\rfloor.\\]\nTo this r... |
https://artofproblemsolving.com/wiki/index.php/2021_AMC_10B_Problems/Problem_6 | C | 76 | Ms. Blackwell gives an exam to two classes. The mean of the scores of the students in the morning class is $84$ , and the afternoon class's mean score is $70$ . The ratio of the number of students in the morning class to the number of students in the afternoon class is $\frac{3}{4}$ . What is the mean of the scores of ... | [
"Let there be $3x$ students in the morning class and $4x$ students in the afternoon class. The total number of students is $3x + 4x = 7x$ . The average is $\\frac{3x\\cdot84 + 4x\\cdot70}{7x}=76$ . Therefore, the answer is $\\boxed{76}$",
"Suppose the morning class has $m$ students and the afternoon class has $a$... |
https://artofproblemsolving.com/wiki/index.php/2021_AMC_12B_Problems/Problem_4 | C | 76 | Ms. Blackwell gives an exam to two classes. The mean of the scores of the students in the morning class is $84$ , and the afternoon class's mean score is $70$ . The ratio of the number of students in the morning class to the number of students in the afternoon class is $\frac{3}{4}$ . What is the mean of the scores of ... | [
"Let there be $3x$ students in the morning class and $4x$ students in the afternoon class. The total number of students is $3x + 4x = 7x$ . The average is $\\frac{3x\\cdot84 + 4x\\cdot70}{7x}=76$ . Therefore, the answer is $\\boxed{76}$",
"Suppose the morning class has $m$ students and the afternoon class has $a$... |
https://artofproblemsolving.com/wiki/index.php/2004_AMC_8_Problems/Problem_5 | D | 15 | Ms. Hamilton's eighth-grade class wants to participate in the annual three-person-team basketball tournament. The losing team of each game is eliminated from the tournament. If sixteen teams compete, how many games will be played to determine the winner?
$\textbf{(A)}\ 4\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\... | [
"The remaining team will be the only undefeated one. The other $\\boxed{15}$ teams must have lost a game before getting out, thus fifteen games yielding fifteen losers.",
"There will be $8$ games the first round, $4$ games the second round, $2$ games the third round, and $1$ game in the final round, giving us a t... |
https://artofproblemsolving.com/wiki/index.php/2004_AMC_8_Problems/Problem_4 | B | 4 | Ms. Hamilton’s eighth-grade class wants to participate in the annual three-person-team basketball tournament.
Lance, Sally, Joy, and Fred are chosen for the team. In how many ways can the three starters be chosen?
$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 10$ | [
"There are $\\binom{4}{3}$ ways to choose three starters. Thus the answer is $\\boxed{4}$",
"We can choose $3$ people by eliminating one from a set of $4$ one at a time and the other three get selected. There are $4$ ways to remove a person from a group of four (without considering order), so there are $\\boxed{4... |
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