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_10B_Problems/Problem_12 | C | 4 | A pair of fair $6$ -sided dice is rolled $n$ times. What is the least value of $n$ such that the probability that the sum of the numbers face up on a roll equals $7$ at least once is greater than $\frac{1}{2}$
$\textbf{(A) } 2 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 4 \qquad \textbf{(D) } 5 \qquad \textbf{(E) } 6$ | [
"Rolling a pair of fair $6$ -sided dice, the probability of getting a sum of $7$ is $\\frac16:$ Regardless what the first die shows, the second die has exactly one outcome to make the sum $7.$ We consider the complement: The probability of not getting a sum of $7$ is $1-\\frac16=\\frac56.$ Rolling the pair of dice ... |
https://artofproblemsolving.com/wiki/index.php/2012_AMC_10A_Problems/Problem_9 | D | 13 | A pair of six-sided dice are labeled so that one die has only even numbers (two each of 2, 4, and 6), and the other die has only odd numbers (two of each 1, 3, and 5). The pair of dice is rolled. What is the probability that the sum of the numbers on the tops of the two dice is 7?
$\textbf{(A)}\ \frac{1}{6}\qquad\textbf{(B)}\ \frac{1}{5}\qquad\textbf{(C)}\ \frac{1}{4}\qquad\textbf{(D)}\ \frac{1}{3}\qquad\textbf{(E)}\ \frac{1}{2}$ | [
"The total number of combinations when rolling two dice is $6*6 = 36$\nThere are three ways that a sum of 7 can be rolled. $2+5$ $4+3$ , and $6+1$ . There are two 2's on one die and two 5's on the other, so there are a total of 4 ways to roll the combination of 2 and 5. There are two 4's on one die and two 3's on t... |
https://artofproblemsolving.com/wiki/index.php/2013_AIME_I_Problems/Problem_9 | null | 113 | A paper equilateral triangle $ABC$ has side length $12$ . The paper triangle is folded so that vertex $A$ touches a point on side $\overline{BC}$ a distance $9$ from point $B$ . The length of the line segment along which the triangle is folded can be written as $\frac{m\sqrt{p}}{n}$ , where $m$ $n$ , and $p$ are positive integers, $m$ and $n$ are relatively prime, and $p$ is not divisible by the square of any prime. Find $m+n+p$
[asy] import cse5; size(12cm); pen tpen = defaultpen + 1.337; real a = 39/5.0; real b = 39/7.0; pair B = MP("B", (0,0), dir(200)); pair A = MP("A", (9,0), dir(-80)); pair C = MP("C", (12,0), dir(-20)); pair K = (6,10.392); pair M = (a*B+(12-a)*K) / 12; pair N = (b*C+(12-b)*K) / 12; draw(B--M--N--C--cycle, tpen); draw(M--A--N--cycle); fill(M--A--N--cycle, mediumgrey); pair shift = (-20.13, 0); pair B1 = MP("B", B+shift, dir(200)); pair A1 = MP("A", K+shift, dir(90)); pair C1 = MP("C", C+shift, dir(-20)); draw(A1--B1--C1--cycle, tpen);[/asy] | [
"Let $M$ and $N$ be the points on $\\overline{AB}$ and $\\overline{AC}$ , respectively, where the paper is folded. Let $D$ be the point on $\\overline{BC}$ where the folded $A$ touches it. We have $AF=6\\sqrt{3}$ and $FD=3$ , so $AD=3\\sqrt{13}$ . Denote $\\angle DAF = \\theta$ ; we get $\\cos\\theta = 2\\sqrt{3}/... |
https://artofproblemsolving.com/wiki/index.php/1986_AHSME_Problems/Problem_13 | E | 12 | A parabola $y = ax^{2} + bx + c$ has vertex $(4,2)$ . If $(2,0)$ is on the parabola, then $abc$ equals
$\textbf{(A)}\ -12\qquad \textbf{(B)}\ -6\qquad \textbf{(C)}\ 0\qquad \textbf{(D)}\ 6\qquad \textbf{(E)}\ 12$ | [
"Consider the quadratic in completed square form: it must be $y=a(x-4)^{2}+2$ . Now substitute $x=2$ and $y=0$ to give $a=-\\frac{1}{2}$ . Now expanding gives $y=-\\frac{1}{2}x^{2}+4x-6$ , so the product is $-\\frac{1}{2} \\cdot 4 \\cdot -6 = 3 \\cdot 4 = 12$ , which is $\\boxed{12}$"
] |
https://artofproblemsolving.com/wiki/index.php/1986_AHSME_Problems/Problem_19 | A | 13 | A park is in the shape of a regular hexagon $2$ km on a side. Starting at a corner,
Alice walks along the perimeter of the park for a distance of $5$ km.
How many kilometers is she from her starting point?
$\textbf{(A)}\ \sqrt{13}\qquad \textbf{(B)}\ \sqrt{14}\qquad \textbf{(C)}\ \sqrt{15}\qquad \textbf{(D)}\ \sqrt{16}\qquad \textbf{(E)}\ \sqrt{17}$ | [
"We imagine this problem on a coordinate plane and let Alice's starting position be the origin. We see that she will travel along two edges and then go halfway along a third. Therefore, her new $x$ -coordinate will be $1 + 2 + \\frac{1}{2} = \\frac{7}{2}$ because she travels along a distance of $2 \\cdot \\frac{1}{... |
https://artofproblemsolving.com/wiki/index.php/2008_AIME_II_Problems/Problem_9 | null | 19 | A particle is located on the coordinate plane at $(5,0)$ . Define a move for the particle as a counterclockwise rotation of $\pi/4$ radians about the origin followed by a translation of $10$ units in the positive $x$ -direction. Given that the particle's position after $150$ moves is $(p,q)$ , find the greatest integer less than or equal to $|p| + |q|$ | [
"Let the particle's position be represented by a complex number. Recall that multiplying a number by cis $\\left( \\theta \\right)$ rotates the object in the complex plane by $\\theta$ counterclockwise. In this case, we use $a = cis(\\frac{\\pi}{4})$ . Therefore, applying the rotation and shifting the coordinates b... |
https://artofproblemsolving.com/wiki/index.php/2005_AIME_I_Problems/Problem_13 | null | 83 | A particle moves in the Cartesian plane according to the following rules:
How many different paths can the particle take from $(0,0)$ to $(5,5)$ | [
"The length of the path (the number of times the particle moves) can range from $l = 5$ to $9$ ; notice that $d = 10-l$ gives the number of diagonals. Let $R$ represent a move to the right, $U$ represent a move upwards, and $D$ to be a move that is diagonal. Casework upon the number of diagonal moves:\nTogether, th... |
https://artofproblemsolving.com/wiki/index.php/1989_AHSME_Problems/Problem_23 | D | 4,435 | A particle moves through the first quadrant as follows. During the first minute it moves from the origin to $(1,0)$ . Thereafter, it continues to follow the directions indicated in the figure, going back and forth between the positive x and y axes, moving one unit of distance parallel to an axis in each minute. At which point will the particle be after exactly 1989 minutes?
[asy] import graph; Label f; f.p=fontsize(6); xaxis(0,3.5,Ticks(f, 1.0)); yaxis(0,4.5,Ticks(f, 1.0)); draw((0,0)--(1,0)--(1,1)--(0,1)--(0,2)--(2,2)--(2,0)--(3,0)--(3,3)--(0,3)--(0,4)--(1.5,4),blue+linewidth(2)); arrow((2,4),dir(180),blue); [/asy]
$\text{(A)}\ (35,44)\qquad\text{(B)}\ (36,45)\qquad\text{(C)}\ (37,45)\qquad\text{(D)}\ (44,35)\qquad\text{(E)}\ (45,36)$ | [
"Squares of size $1\\times1,\\ 2\\times2,\\ 3\\times3,\\ ...$ are successively enclosed between the path and the axes.\n\nIt takes $1+1+1$ minutes to enclose the first square, $1+2+2$ minutes to enclose the second, $1+3+3$ minutes to enclose the third, and so on. After odd squares, the particle is on the Y axis; af... |
https://artofproblemsolving.com/wiki/index.php/1963_AHSME_Problems/Problem_29 | C | 400 | A particle projected vertically upward reaches, at the end of $t$ seconds, an elevation of $s$ feet where $s = 160 t - 16t^2$ . The highest elevation is:
$\textbf{(A)}\ 800 \qquad \textbf{(B)}\ 640\qquad \textbf{(C)}\ 400 \qquad \textbf{(D)}\ 320 \qquad \textbf{(E)}\ 160$ | [
"The highest elevation a particle can reach is the vertex of the quadratic. The x-value that can get the maximum is $\\frac{-160}{-2 \\cdot 16} = 5$ , so the highest elevation is $160(5) - 16(5^2) = 400$ feet, which is answer choice $\\boxed{400}$"
] |
https://artofproblemsolving.com/wiki/index.php/2009_AMC_10B_Problems/Problem_19 | A | 12 | A particular $12$ -hour digital clock displays the hour and minute of a day. Unfortunately, whenever it is supposed to display a $1$ , it mistakenly displays a $9$ . For example, when it is 1:16 PM the clock incorrectly shows 9:96 PM. What fraction of the day will the clock show the correct time?
$\mathrm{(A)}\ \frac 12\qquad \mathrm{(B)}\ \frac 58\qquad \mathrm{(C)}\ \frac 34\qquad \mathrm{(D)}\ \frac 56\qquad \mathrm{(E)}\ \frac {9}{10}$ | [
"The clock will display the incorrect time for the entire hours of $1, 10, 11$ and $12$ . So the correct hour is displayed $\\frac 23$ of the time. The minutes will not display correctly whenever either the tens digit or the ones digit is a $1$ , so the minutes that will not display correctly are $10, 11, 12, \\d... |
https://artofproblemsolving.com/wiki/index.php/2009_AMC_12B_Problems/Problem_10 | A | 12 | A particular $12$ -hour digital clock displays the hour and minute of a day. Unfortunately, whenever it is supposed to display a $1$ , it mistakenly displays a $9$ . For example, when it is 1:16 PM the clock incorrectly shows 9:96 PM. What fraction of the day will the clock show the correct time?
$\mathrm{(A)}\ \frac 12\qquad \mathrm{(B)}\ \frac 58\qquad \mathrm{(C)}\ \frac 34\qquad \mathrm{(D)}\ \frac 56\qquad \mathrm{(E)}\ \frac {9}{10}$ | [
"The clock will display the incorrect time for the entire hours of $1, 10, 11$ and $12$ . So the correct hour is displayed $\\frac 23$ of the time. The minutes will not display correctly whenever either the tens digit or the ones digit is a $1$ , so the minutes that will not display correctly are $10, 11, 12, \\d... |
https://artofproblemsolving.com/wiki/index.php/2008_AMC_12A_Problems/Problem_21 | D | 48 | A permutation $(a_1,a_2,a_3,a_4,a_5)$ of $(1,2,3,4,5)$ is heavy-tailed if $a_1 + a_2 < a_4 + a_5$ . What is the number of heavy-tailed permutations?
$\mathrm{(A)}\ 36\qquad\mathrm{(B)}\ 40\qquad\textbf{(C)}\ 44\qquad\mathrm{(D)}\ 48\qquad\mathrm{(E)}\ 52$ | [
"We use case work on the value of $a_3$\nCase 1: $a_3 = 1$ . Since $a_1 + a_2 < a_4 + a_5$ $(a_1, a_2)$ can only be a permutation of $(2, 3)$ or $(2, 4)$ . The values of $a_1$ and $a_2$ , as well as the values of $a_4$ and $a_5$ , are interchangeable, so this case produces a total of $2(2 \\cdot 2) = 8$ solutions.\... |
https://artofproblemsolving.com/wiki/index.php/2010_USAJMO_Problems/Problem_1 | null | 4,489 | A permutation of the set of positive integers $[n] = \{1, 2, \ldots, n\}$ is a sequence $(a_1, a_2, \ldots, a_n)$ such that each element of $[n]$ appears precisely one time as a term of the sequence. For example, $(3, 5, 1, 2, 4)$ is a permutation of $[5]$ . Let $P(n)$ be the number of permutations of $[n]$ for which $ka_k$ is a perfect square for all $1\leq k\leq n$ . Find with proof the smallest $n$ such that $P(n)$ is a multiple of $2010$ | [
"We claim that the smallest $n$ is $67^2 = \\boxed{4489}$",
"This proof can also be rephrased as follows, in a longer way, but with fewer highly technical words such as \"equivalence relation\":\nIt is possible to write all positive integers $n$ in the form $p\\cdot m^2$ , where $m^2$ is the largest perfect squar... |
https://artofproblemsolving.com/wiki/index.php/2007_AMC_12A_Problems/Problem_13 | null | 10 | A piece of cheese is located at $(12,10)$ in a coordinate plane . A mouse is at $(4,-2)$ and is running up the line $y=-5x+18$ . At the point $(a,b)$ the mouse starts getting farther from the cheese rather than closer to it. What is $a+b$
$\mathrm{(A)}\ 6\qquad \mathrm{(B)}\ 10\qquad \mathrm{(C)}\ 14\qquad \mathrm{(D)}\ 18\qquad \mathrm{(E)}\ 22$ | [
"The point $(a,b)$ is the foot of the perpendicular from $(12,10)$ to the line $y=-5x+18$ . The perpendicular has slope $\\frac{1}{5}$ , so its equation is $y=10+\\frac{1}{5}(x-12)=\\frac{1}{5}x+\\frac{38}{5}$ . The $x$ -coordinate at the foot of the perpendicular satisfies the equation $\\frac{1}{5}x+\\frac{38}{5}... |
https://artofproblemsolving.com/wiki/index.php/1998_AHSME_Problems/Problem_25 | B | 6.8 | A piece of graph paper is folded once so that $(0,2)$ is matched with $(4,0)$ , and $(7,3)$ is matched with $(m,n)$ . Find $m+n$
$\mathrm{(A) \ }6.7 \qquad \mathrm{(B) \ }6.8 \qquad \mathrm{(C) \ }6.9 \qquad \mathrm{(D) \ }7.0 \qquad \mathrm{(E) \ }8.0$ | [
"Note that the fold is the perpendicular bisector of $(0,2)$ and $(4,0)$ . Thus, the fold goes through the midpoint $(2,1)$\nThe fold also has a slope of $-\\frac{1}{m}$ , where the $m$ is the slope of the line connecting these two points. We find $m = \\frac{0 - 2}{4 - 0} = -\\frac{1}{2}$ . Thus, the slope of th... |
https://artofproblemsolving.com/wiki/index.php/1998_AHSME_Problems/Problem_25 | null | 6.8 | A piece of graph paper is folded once so that $(0,2)$ is matched with $(4,0)$ , and $(7,3)$ is matched with $(m,n)$ . Find $m+n$
$\mathrm{(A) \ }6.7 \qquad \mathrm{(B) \ }6.8 \qquad \mathrm{(C) \ }6.9 \qquad \mathrm{(D) \ }7.0 \qquad \mathrm{(E) \ }8.0$ | [
"The line of the fold is the perpendicular bisector of the segment that connects $(0,2)$ and $(4,0)$ . \nThe point $(m,n)$ is the image of the point $(7,3)$ according to this axis.\nThe situation looks as follows.\nNow, we will compute the coordinates of the point $D$ , using the following facts:\nAs the triangles ... |
https://artofproblemsolving.com/wiki/index.php/2023_AIME_I_Problems/Problem_3 | null | 607 | A plane contains $40$ lines, no $2$ of which are parallel. Suppose that there are $3$ points where exactly $3$ lines intersect, $4$ points where exactly $4$ lines intersect, $5$ points where exactly $5$ lines intersect, $6$ points where exactly $6$ lines intersect, and no points where more than $6$ lines intersect. Find the number of points where exactly $2$ lines intersect. | [
"In this solution, let $\\boldsymbol{n}$ -line points be the points where exactly $n$ lines intersect. We wish to find the number of $2$ -line points.\nThere are $\\binom{40}{2}=780$ pairs of lines. Among them:\nIt follows that the $2$ -line points account for $780-9-24-50-90=\\boxed{607}$ pairs of lines, where eac... |
https://artofproblemsolving.com/wiki/index.php/2006_AMC_10A_Problems/Problem_13 | D | 60 | A player pays $\textdollar 5$ to play a game. A die is rolled. If the number on the die is odd, the game is lost. If the number on the die is even, the die is rolled again. In this case the player wins if the second number matches the first and loses otherwise. How much should the player win if the game is fair? (In a fair game the probability of winning times the amount won is what the player should pay.)
$\textbf{(A) } \textdollar12\qquad\textbf{(B) } \textdollar30\qquad\textbf{(C) } \textdollar50\qquad\textbf{(D) } \textdollar60\qquad\textbf{(E) } \textdollar 100\qquad$ | [
"The probability of rolling an even number on the first turn is $\\frac{1}{2}$ and the probability of rolling the same number on the next turn is $\\frac{1}{6}$ . The probability of winning is $\\frac{1}{2}\\cdot \\frac{1}{6} =\\frac{1}{12}$ . If the game is to be fair, the amount paid, $5$ dollars, must be $\\frac... |
https://artofproblemsolving.com/wiki/index.php/1998_AHSME_Problems/Problem_29 | D | 5 | A point $(x,y)$ in the plane is called a lattice point if both $x$ and $y$ are integers. The area of the largest square that contains exactly three lattice points in its interior is closest to
$\mathrm{(A) \ } 4.0 \qquad \mathrm{(B) \ } 4.2 \qquad \mathrm{(C) \ } 4.5 \qquad \mathrm{(D) \ } 5.0 \qquad \mathrm{(E) \ } 5.6$ | [
" The best square's side length is slightly less than $\\sqrt 5$ , yielding an answer of $\\boxed{5.0}$"
] |
https://artofproblemsolving.com/wiki/index.php/2010_AIME_II_Problems/Problem_2 | null | 281 | A point $P$ is chosen at random in the interior of a unit square $S$ . Let $d(P)$ denote the distance from $P$ to the closest side of $S$ . The probability that $\frac{1}{5}\le d(P)\le\frac{1}{3}$ is equal to $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ | [
"Any point outside the square with side length $\\frac{1}{3}$ that has the same center and orientation as the unit square and inside the square with side length $\\frac{3}{5}$ that has the same center and orientation as the unit square has $\\frac{1}{5}\\le d(P)\\le\\frac{1}{3}$\nSince the area of the unit square i... |
https://artofproblemsolving.com/wiki/index.php/1954_AHSME_Problems/Problem_9 | C | 5 | A point $P$ is outside a circle and is $13$ inches from the center. A secant from $P$ cuts the circle at $Q$ and $R$ so that the external segment of the secant $PQ$ is $9$ inches and $QR$ is $7$ inches. The radius of the circle is:
$\textbf{(A)}\ 3" \qquad \textbf{(B)}\ 4" \qquad \textbf{(C)}\ 5" \qquad \textbf{(D)}\ 6"\qquad\textbf{(E)}\ 7"$ | [
"Using the Secant-Secant Power Theorem, you can get $9(16)=(13-r)(13+r)$ , where $r$ is the radius of the given circle. Solving the equation, you get a quadratic: $r^2-25$ . A radius cannot be negative so the answer is $\\boxed{5}$"
] |
https://artofproblemsolving.com/wiki/index.php/2020_AMC_10A_Problems/Problem_16 | B | 0.4 | A point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0), (2020, 0), (2020, 2020),$ and $(0, 2020)$ . The probability that the point is within $d$ units of a lattice point is $\tfrac{1}{2}$ . (A point $(x, y)$ is a lattice point if $x$ and $y$ are both integers.) What is $d$ to the nearest tenth $?$
$\textbf{(A) } 0.3 \qquad \textbf{(B) } 0.4 \qquad \textbf{(C) } 0.5 \qquad \textbf{(D) } 0.6 \qquad \textbf{(E) } 0.7$ | [
"\nThe diagram represents each unit square of the given $2020 \\times 2020$ square.\nWe consider an individual one-by-one block.\nIf we draw a quarter of a circle from each corner (where the lattice points are located), each with radius $d$ , the area covered by the circles should be $0.5$ . Because of this, and th... |
https://artofproblemsolving.com/wiki/index.php/2020_AMC_12A_Problems/Problem_16 | B | 0.4 | A point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0), (2020, 0), (2020, 2020),$ and $(0, 2020)$ . The probability that the point is within $d$ units of a lattice point is $\tfrac{1}{2}$ . (A point $(x, y)$ is a lattice point if $x$ and $y$ are both integers.) What is $d$ to the nearest tenth $?$
$\textbf{(A) } 0.3 \qquad \textbf{(B) } 0.4 \qquad \textbf{(C) } 0.5 \qquad \textbf{(D) } 0.6 \qquad \textbf{(E) } 0.7$ | [
"\nThe diagram represents each unit square of the given $2020 \\times 2020$ square.\nWe consider an individual one-by-one block.\nIf we draw a quarter of a circle from each corner (where the lattice points are located), each with radius $d$ , the area covered by the circles should be $0.5$ . Because of this, and th... |
https://artofproblemsolving.com/wiki/index.php/2000_AIME_II_Problems/Problem_2 | null | 98 | A point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola $x^2 - y^2 = 2000^2$ | [
"\\[(x-y)(x+y)=2000^2=2^8 \\cdot 5^6\\]\nNote that $(x-y)$ and $(x+y)$ have the same parities , so both must be even. We first give a factor of $2$ to both $(x-y)$ and $(x+y)$ . We have $2^6 \\cdot 5^6$ left. Since there are $7 \\cdot 7=49$ factors of $2^6 \\cdot 5^6$ , and since both $x$ and $y$ can be negative, t... |
https://artofproblemsolving.com/wiki/index.php/2008_AMC_10B_Problems/Problem_17 | B | 0.189 | A poll shows that $70\%$ of all voters approve of the mayor's work. On three separate occasions a pollster selects a voter at random. What is the probability that on exactly one of these three occasions the voter approves of the mayor's work?
$\mathrm{(A)}\ {{{0.063}}} \qquad \mathrm{(B)}\ {{{0.189}}} \qquad \mathrm{(C)}\ {{{0.233}}} \qquad \mathrm{(D)}\ {{{0.333}}} \qquad \mathrm{(E)}\ {{{0.441}}}$ | [
"Letting Y stand for a voter who approved of the work, and N stand for a person who didn't approve of the work, the pollster could select responses in $3$ different ways: $\\text{YNN, NYN, and NNY}$ . The probability of each of these is $(0.7)(0.3)^2=0.063$ . Thus, the answer is $3\\cdot0.063=\\boxed{0.189}$",
"I... |
https://artofproblemsolving.com/wiki/index.php/2020_AIME_I_Problems/Problem_3 | null | 621 | A positive integer $N$ has base-eleven representation $\underline{a}\kern 0.1em\underline{b}\kern 0.1em\underline{c}$ and base-eight representation $\underline1\kern 0.1em\underline{b}\kern 0.1em\underline{c}\kern 0.1em\underline{a},$ where $a,b,$ and $c$ represent (not necessarily distinct) digits. Find the least such $N$ expressed in base ten. | [
"From the given information, $121a+11b+c=512+64b+8c+a \\implies 120a=512+53b+7c$ . Since $a$ $b$ , and $c$ have to be positive, $a \\geq 5$ . Since we need to minimize the value of $n$ , we want to minimize $a$ , so we have $a = 5$ . Then we know $88=53b+7c$ , and we can see the only solution is $b=1$ $c=5$ . Final... |
https://artofproblemsolving.com/wiki/index.php/1991_AHSME_Problems/Problem_17 | D | 4 | A positive integer $N$ is a palindrome if the integer obtained by reversing the sequence of digits of $N$ is equal to $N$ . The year 1991 is the only year in the current century with the following 2 properties:
(a) It is a palindrome
(b) It factors as a product of a 2-digit prime palindrome and a 3-digit prime palindrome.
How many years in the millenium between 1000 and 2000 have properties (a) and (b)?
$\text{(A) } 1\quad \text{(B) } 2\quad \text{(C) } 3\quad \text{(D) } 4\quad \text{(E) } 5$ | [
"Solution by e_power_pi_times_i\nNotice that all four-digit palindromes are divisible by $11$ , so that is our two-digit prime. Because the other factor is a three-digit number, we are looking at palindromes between $1100$ and $2000$ , which also means that the last digit of the three-digit number is $1$ . Checking... |
https://artofproblemsolving.com/wiki/index.php/2005_AMC_12B_Problems/Problem_21 | C | 2 | A positive integer $n$ has $60$ divisors and $7n$ has $80$ divisors. What is the greatest integer $k$ such that $7^k$ divides $n$
$\mathrm{(A)}\ {{{0}}} \qquad \mathrm{(B)}\ {{{1}}} \qquad \mathrm{(C)}\ {{{2}}} \qquad \mathrm{(D)}\ {{{3}}} \qquad \mathrm{(E)}\ {{{4}}}$ | [
"We may let $n = 7^k \\cdot m$ , where $m$ is not divisible by 7. Using the fact that the number of divisors function $d(n)$ is multiplicative, we have $d(n) = d(7^k)d(m) = (k+1)d(m) = 60$ . Also, $d(7n) = d(7^{k+1})d(m) = (k+2)d(m) = 80$ . These numbers are in the ratio 3:4, so $\\frac{k+1}{k+2} = \\frac{3}{4} \\i... |
https://artofproblemsolving.com/wiki/index.php/2013_AMC_10B_Problems/Problem_24 | A | 1 | A positive integer $n$ is nice if there is a positive integer $m$ with exactly four positive divisors (including $1$ and $m$ ) such that the sum of the four divisors is equal to $n$ . How many numbers in the set $\{ 2010,2011,2012,\dotsc,2019 \}$ are nice?
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5$ | [
"A positive integer with only four positive divisors has its prime factorization in the form of $a \\cdot b$ , where $a$ and $b$ are both prime positive integers or $c^3$ where $c$ is a prime. One can easily deduce that none of the numbers are even near a cube so the second case is not possible. We now look at the ... |
https://artofproblemsolving.com/wiki/index.php/1978_AHSME_Problems/Problem_19 | C | 0.08 | A positive integer $n$ not exceeding $100$ is chosen in such a way that if $n\le 50$ , then the probability of choosing $n$ is $p$ , and if $n > 50$ , then the probability of choosing $n$ is $3p$ . The probability that a perfect square is chosen is
$\textbf{(A) }.05\qquad \textbf{(B) }.065\qquad \textbf{(C) }.08\qquad \textbf{(D) }.09\qquad \textbf{(E) }.1$ | [
"Let's say that we will have $3$ slips for every number not exceeding $100$ but bigger than $50.$ This is to account for the $3p$ probability part. Let's now say that we will only have one slip for each number below or equal to $50.$ The probability(or $p$ ) will then be $\\frac{1}{200}.$ Now let's have all the squ... |
https://artofproblemsolving.com/wiki/index.php/2020_AMC_10A_Problems/Problem_15 | E | 23 | A positive integer divisor of $12!$ is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. What is $m+n$
$\textbf{(A)}\ 3\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 23$ | [
"The prime factorization of $12!$ is $2^{10} \\cdot 3^5 \\cdot 5^2 \\cdot 7 \\cdot 11$ . \nThis yields a total of $11 \\cdot 6 \\cdot 3 \\cdot 2 \\cdot 2$ divisors of $12!.$ In order to produce a perfect square divisor, there must be an even exponent for each number in the prime factorization. Note that the divisor... |
https://artofproblemsolving.com/wiki/index.php/2005_AMC_10B_Problems/Problem_2 | D | 20 | A positive number $x$ has the property that $x\%$ of $x$ is $4$ . What is $x$
$\textbf{(A) }\ 2 \qquad \textbf{(B) }\ 4 \qquad \textbf{(C) }\ 10 \qquad \textbf{(D) }\ 20 \qquad \textbf{(E) }\ 40$ | [
"Since $x\\%$ means $0.01x$ , the statement \" $x\\% \\text{ of } x \\text{ is 4}$ \" can be rewritten as \" $0.01x \\cdot x = 4$ \":\n$0.01x \\cdot x=4 \\Rightarrow x^2 = 400 \\Rightarrow x = \\boxed{20}.$",
"Try the answer choices one by one. Upon examination, it is quite obvious that the answer is $\\boxed{20}... |
https://artofproblemsolving.com/wiki/index.php/2005_AMC_12B_Problems/Problem_2 | D | 20 | A positive number $x$ has the property that $x\%$ of $x$ is $4$ . What is $x$
$\textbf{(A) }\ 2 \qquad \textbf{(B) }\ 4 \qquad \textbf{(C) }\ 10 \qquad \textbf{(D) }\ 20 \qquad \textbf{(E) }\ 40$ | [
"Since $x\\%$ means $0.01x$ , the statement \" $x\\% \\text{ of } x \\text{ is 4}$ \" can be rewritten as \" $0.01x \\cdot x = 4$ \":\n$0.01x \\cdot x=4 \\Rightarrow x^2 = 400 \\Rightarrow x = \\boxed{20}.$",
"Try the answer choices one by one. Upon examination, it is quite obvious that the answer is $\\boxed{20}... |
https://artofproblemsolving.com/wiki/index.php/1952_AHSME_Problems/Problem_25 | D | 245 | A powderman set a fuse for a blast to take place in $30$ seconds. He ran away at a rate of $8$ yards per second. Sound travels at the rate of $1080$ feet per second. When the powderman heard the blast, he had run approximately: $\textbf{(A)}\ \text{200 yd.}\qquad\textbf{(B)}\ \text{352 yd.}\qquad\textbf{(C)}\ \text{300 yd.}\qquad\textbf{(D)}\ \text{245 yd.}\qquad\textbf{(E)}\ \text{512 yd.}$ | [
"Let $p(t)=24t$ be the number of feet the powderman is from the blast at $t$ seconds after the fuse is lit, and let $q(t)=1080t-32400$ be the number of feet the sound has traveled. We want to solve for $p(t)=q(t)$ \\[24t=1080t-32400\\] \\[1056t=32400\\] \\[t=\\frac{32400}{1056}\\] \\[t=\\frac{675}{22}=30.6\\overlin... |
https://artofproblemsolving.com/wiki/index.php/2011_AMC_12A_Problems/Problem_12 | D | 4.5 | A power boat and a raft both left dock $A$ on a river and headed downstream. The raft drifted at the speed of the river current. The power boat maintained a constant speed with respect to the river. The power boat reached dock $B$ downriver, then immediately turned and traveled back upriver. It eventually met the raft on the river 9 hours after leaving dock $A.$ How many hours did it take the power boat to go from $A$ to $B$
$\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 3.5 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 4.5 \qquad \textbf{(E)}\ 5$ | [
"WLOG let the speed of the river be 0. This is allowed because the problem never states that the speed of the current has to have a magnitude greater than 0. In this case, when the powerboat travels from $A$ to $B$ , the raft remains at $A$ . Thus the trip from $A$ to $B$ takes the same time as the trip from $B$ to... |
https://artofproblemsolving.com/wiki/index.php/2008_AMC_12B_Problems/Problem_18 | E | 784 | A pyramid has a square base $ABCD$ and vertex $E$ . The area of square $ABCD$ is $196$ , and the areas of $\triangle ABE$ and $\triangle CDE$ are $105$ and $91$ , respectively. What is the volume of the pyramid?
$\textbf{(A)}\ 392 \qquad \textbf{(B)}\ 196\sqrt {6} \qquad \textbf{(C)}\ 392\sqrt {2} \qquad \textbf{(D)}\ 392\sqrt {3} \qquad \textbf{(E)}\ 784$ | [
"Let $h$ be the height of the pyramid and $a$ be the distance from $h$ to $CD$ . The side length of the base is $14$ . The heights of $\\triangle ABE$ and $\\triangle CDE$ are $2\\cdot105\\div14=15$ and $2\\cdot91\\div14=13$ , respectively. Consider a side view of the pyramid from $\\triangle BCE$ . We have a syst... |
https://artofproblemsolving.com/wiki/index.php/2017_AIME_I_Problems/Problem_4 | null | 803 | A pyramid has a triangular base with side lengths $20$ $20$ , and $24$ . The three edges of the pyramid from the three corners of the base to the fourth vertex of the pyramid all have length $25$ . The volume of the pyramid is $m\sqrt{n}$ , where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$ | [
"Let the triangular base be $\\triangle ABC$ , with $\\overline {AB} = 24$ . We find that the altitude to side $\\overline {AB}$ is $16$ , so the area of $\\triangle ABC$ is $(24*16)/2 = 192$\nLet the fourth vertex of the tetrahedron be $P$ , and let the midpoint of $\\overline {AB}$ be $M$ . Since $P$ is equidista... |
https://artofproblemsolving.com/wiki/index.php/2023_AMC_10A_Problems/Problem_4 | D | 12 | A quadrilateral has all integer sides lengths, a perimeter of $26$ , and one side of length $4$ . What is the greatest possible length of one side of this quadrilateral?
$\textbf{(A) }9\qquad\textbf{(B) }10\qquad\textbf{(C) }11\qquad\textbf{(D) }12\qquad\textbf{(E) }13$ | [
"Let's use the triangle inequality. We know that for a triangle, the sum of the 2 shorter sides must always be longer than the longest side. This is because if the longest side were to be as long as the sum of the other sides, or longer, we would only have a line.\nSimilarly, for a convex quadrilateral, the sum of ... |
https://artofproblemsolving.com/wiki/index.php/2016_AMC_12B_Problems/Problem_10 | A | 4 | A quadrilateral has vertices $P(a,b)$ $Q(b,a)$ $R(-a, -b)$ , and $S(-b, -a)$ , where $a$ and $b$ are integers with $a>b>0$ . The area of $PQRS$ is $16$ . What is $a+b$
$\textbf{(A)}\ 4 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 13$ | [
"Note that the slope of $PQ$ is $\\frac{a-b}{b-a}=-1$ and the slope of $PS$ is $\\frac{b+a}{a+b}=1$ . Hence, $PQ\\perp PS$ and we can similarly prove that the other angles are right angles. This means that $PQRS$ is a rectangle. By distance formula we have $(a-b)^2+(b-a)^2*2*(a+b)^2 = 256$ . Simplifying we get $(... |
https://artofproblemsolving.com/wiki/index.php/2016_AMC_10A_Problems/Problem_24 | E | 500 | A quadrilateral is inscribed in a circle of radius $200\sqrt{2}$ . Three of the sides of this quadrilateral have length $200$ . What is the length of the fourth side?
$\textbf{(A) }200\qquad \textbf{(B) }200\sqrt{2}\qquad\textbf{(C) }200\sqrt{3}\qquad\textbf{(D) }300\sqrt{2}\qquad\textbf{(E) } 500$ | [
"\nLet $AD$ intersect $OB$ at $E$ and $OC$ at $F.$\n$\\overarc{AB}= \\overarc{BC}= \\overarc{CD}=\\theta$\n$\\angle{BAD}=\\frac{1}{2} \\cdot \\overarc{BCD}=\\theta=\\angle{AOB}$\nFrom there, $\\triangle{OAB} \\sim \\triangle{ABE}$ , thus:\n$\\frac{OA}{AB} = \\frac{AB}{BE} = \\frac{OB}{AE}$\n$OA = OB$ because they a... |
https://artofproblemsolving.com/wiki/index.php/2016_AMC_10A_Problems/Problem_24 | null | 500 | A quadrilateral is inscribed in a circle of radius $200\sqrt{2}$ . Three of the sides of this quadrilateral have length $200$ . What is the length of the fourth side?
$\textbf{(A) }200\qquad \textbf{(B) }200\sqrt{2}\qquad\textbf{(C) }200\sqrt{3}\qquad\textbf{(D) }300\sqrt{2}\qquad\textbf{(E) } 500$ | [
"\nLet quadrilateral $ABCD$ be inscribed in circle $O$ , where $AD$ is the side of unknown length. Draw the radii from center $O$ to all four vertices of the quadrilateral, and draw the altitude of $\\triangle BOC$ such that it passes through side $AD$ at the point $G$ and meets side $BC$ at point $H$\nBy the Pytha... |
https://artofproblemsolving.com/wiki/index.php/2016_AMC_12A_Problems/Problem_21 | E | 500 | A quadrilateral is inscribed in a circle of radius $200\sqrt{2}$ . Three of the sides of this quadrilateral have length $200$ . What is the length of the fourth side?
$\textbf{(A) }200\qquad \textbf{(B) }200\sqrt{2}\qquad\textbf{(C) }200\sqrt{3}\qquad\textbf{(D) }300\sqrt{2}\qquad\textbf{(E) } 500$ | [
"\nLet $AD$ intersect $OB$ at $E$ and $OC$ at $F.$\n$\\overarc{AB}= \\overarc{BC}= \\overarc{CD}=\\theta$\n$\\angle{BAD}=\\frac{1}{2} \\cdot \\overarc{BCD}=\\theta=\\angle{AOB}$\nFrom there, $\\triangle{OAB} \\sim \\triangle{ABE}$ , thus:\n$\\frac{OA}{AB} = \\frac{AB}{BE} = \\frac{OB}{AE}$\n$OA = OB$ because they a... |
https://artofproblemsolving.com/wiki/index.php/2016_AMC_12A_Problems/Problem_21 | null | 500 | A quadrilateral is inscribed in a circle of radius $200\sqrt{2}$ . Three of the sides of this quadrilateral have length $200$ . What is the length of the fourth side?
$\textbf{(A) }200\qquad \textbf{(B) }200\sqrt{2}\qquad\textbf{(C) }200\sqrt{3}\qquad\textbf{(D) }300\sqrt{2}\qquad\textbf{(E) } 500$ | [
"\nLet quadrilateral $ABCD$ be inscribed in circle $O$ , where $AD$ is the side of unknown length. Draw the radii from center $O$ to all four vertices of the quadrilateral, and draw the altitude of $\\triangle BOC$ such that it passes through side $AD$ at the point $G$ and meets side $BC$ at point $H$\nBy the Pytha... |
https://artofproblemsolving.com/wiki/index.php/2017_AIME_I_Problems/Problem_5 | null | 321 | A rational number written in base eight is $\underline{ab} . \underline{cd}$ , where all digits are nonzero. The same number in base twelve is $\underline{bb} . \underline{ba}$ . Find the base-ten number $\underline{abc}$ | [
"First, note that the first two digits will always be a positive number. We will start with base twelve because of its repetition. List all the positive numbers in base eight that have equal tens and ones digits in base 12.\n$11_{12}=15_8$\n$22_{12}=32_8$\n$33_{12}=47_8$\n$44_{12}=64_8$\n$55_{12}=101_8$\nWe stop be... |
https://artofproblemsolving.com/wiki/index.php/1981_AHSME_Problems/Problem_20 | B | 10 | A ray of light originates from point $A$ and travels in a plane, being reflected $n$ times between lines $AD$ and $CD$ before striking a point $B$ (which may be on $AD$ or $CD$ ) perpendicularly and retracing its path back to $A$ (At each point of reflection the light makes two equal angles as indicated in the adjoining figure. The figure shows the light path for $n=3$ ). If $\measuredangle CDA=8^\circ$ , what is the largest value $n$ can have?
[asy] unitsize(1.5cm); pair D=origin, A=(-6,0), C=6*dir(160), E=3.2*dir(160), F=(-2.1,0), G=1.5*dir(160), B=(-1.4095,0); draw((-6.5,0)--D--C,black); draw(A--E--F--G--B,black); dotfactor=4; dot("$A$",A,S); dot("$C$",C,N); dot("$R_1$",E,N); dot("$R_2$",F,S); dot("$R_3$",G,N); dot("$B$",B,S); markscalefactor=0.015; draw(rightanglemark(G,B,D)); draw(anglemark(C,E,A,12)); draw(anglemark(F,E,G,12)); draw(anglemark(E,F,A)); draw(anglemark(E,F,A,12)); draw(anglemark(B,F,G)); draw(anglemark(B,F,G,12)); draw(anglemark(E,G,F)); draw(anglemark(E,G,F,12)); draw(anglemark(E,G,F,16)); draw(anglemark(B,G,D)); draw(anglemark(B,G,D,12)); draw(anglemark(B,G,D,16)); [/asy]
$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 38\qquad\textbf{(D)}\ 98\qquad\textbf{(E)}\ \text{There is no largest value.}$ | [
"Notice that when we start, we want the smallest angle possible of reflection. The ideal reflection would be $0$ , but that would be impossible. Therefore we start by working backwards. Since angle $CDA$ is $8$ , the reflection would give us a triangle with angles $16, 90$ , and $74$ . Then, when we reflect again, ... |
https://artofproblemsolving.com/wiki/index.php/2018_AIME_II_Problems/Problem_6 | null | 37 | A real number $a$ is chosen randomly and uniformly from the interval $[-20, 18]$ . The probability that the roots of the polynomial
$x^4 + 2ax^3 + (2a - 2)x^2 + (-4a + 3)x - 2$
are all real can be written in the form $\dfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ | [
"The polynomial we are given is rather complicated, so we could use Rational Root Theorem to turn the given polynomial into a degree-2 polynomial. With Rational Root Theorem, $x = 1, -1, 2, -2$ are all possible rational roots. Upon plugging these roots into the polynomial, $x = -2$ and $x = 1$ make the polynomial e... |
https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_22 | E | 2 | A real-valued function $f$ has the property that for all real numbers $a$ and $b,$ \[f(a + b) + f(a - b) = 2f(a) f(b).\] Which one of the following cannot be the value of $f(1)?$
$\textbf{(A) } 0 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } -1 \qquad \textbf{(D) } 2 \qquad \textbf{(E) } -2$ | [
"Substituting $a = b$ we get \\[f(2a) + f(0) = 2f(a)^2\\] Substituting $a= 0$ we find \\[2f(0) = 2f(0)^2 \\implies f(0) \\in \\{0, 1\\}.\\] This gives \\[f(2a) = 2f(a)^2 - f(0) \\geq 0-1\\] Plugging in $a = \\frac{1}{2}$ implies $f(1) \\geq -1$ , so answer choice $\\boxed{2}$ is impossible.",
"First, we set $a \\... |
https://artofproblemsolving.com/wiki/index.php/1985_AJHSME_Problems/Problem_6 | D | 750 | A ream of paper containing $500$ sheets is $5$ cm thick. Approximately how many sheets of this type of paper would there be in a stack $7.5$ cm high?
$\text{(A)}\ 250 \qquad \text{(B)}\ 550 \qquad \text{(C)}\ 667 \qquad \text{(D)}\ 750 \qquad \text{(E)}\ 1250$ | [
"We could solve the first equation for the thickness of one sheet of paper, and divide into the 2nd equation (which is one way to do the problem), but there are other ways, too.\nLet's say that $500\\text{ sheets}=5\\text{ cm}\\Rightarrow \\frac{500 \\text{ sheets}}{5 \\text{ cm}} = 1$ . So by multiplying $7.5 \\te... |
https://artofproblemsolving.com/wiki/index.php/2014_AIME_II_Problems/Problem_3 | null | 720 | A rectangle has sides of length $a$ and 36. A hinge is installed at each vertex of the rectangle, and at the midpoint of each side of length 36. The sides of length $a$ can be pressed toward each other keeping those two sides parallel so the rectangle becomes a convex hexagon as shown. When the figure is a hexagon with the sides of length $a$ parallel and separated by a distance of 24, the hexagon has the same area as the original rectangle. Find $a^2$
[asy] pair A,B,C,D,E,F,R,S,T,X,Y,Z; dotfactor = 2; unitsize(.1cm); A = (0,0); B = (0,18); C = (0,36); // don't look here D = (12*2.236, 36); E = (12*2.236, 18); F = (12*2.236, 0); draw(A--B--C--D--E--F--cycle); dot(" ",A,NW); dot(" ",B,NW); dot(" ",C,NW); dot(" ",D,NW); dot(" ",E,NW); dot(" ",F,NW); //don't look here R = (12*2.236 +22,0); S = (12*2.236 + 22 - 13.4164,12); T = (12*2.236 + 22,24); X = (12*4.472+ 22,24); Y = (12*4.472+ 22 + 13.4164,12); Z = (12*4.472+ 22,0); draw(R--S--T--X--Y--Z--cycle); dot(" ",R,NW); dot(" ",S,NW); dot(" ",T,NW); dot(" ",X,NW); dot(" ",Y,NW); dot(" ",Z,NW); // sqrt180 = 13.4164 // sqrt5 = 2.236[/asy] | [
"When we squish the rectangle, the hexagon is composed of a rectangle and two isosceles triangles with side lengths 18, 18, and 24 as shown below.\n\nBy Heron's Formula, the area of each isosceles triangle is $\\sqrt{(30)(12)(12)(6)}=\\sqrt{180\\times 12^2}=72\\sqrt{5}$ . So the area of both is $144\\sqrt{5}$ . Fro... |
https://artofproblemsolving.com/wiki/index.php/1984_AHSME_Problems/Problem_4 | null | 7 | A rectangle intersects a circle as shown: $AB=4$ $BC=5$ , and $DE=3$ . Then $EF$ equals:
[asy]defaultpen(linewidth(0.7)+fontsize(10)); pair D=origin, E=(3,0), F=(10,0), G=(12,0), H=(12,1), A=(0,1), B=(4,1), C=(9,1), O=circumcenter(B,C,F); draw(D--G--H--A--cycle); draw(Circle(O, abs(O-C))); label("$A$", A, NW); label("$B$", B, NW); label("$C$", C, NE); label("$D$", D, SW); label("$E$", E, SE); label("$F$", F, SW); label("4", (2,0.85), N); label("3", D--E, S); label("5", (6.5,0.85), N); [/asy] $\mathbf{(A)}\; 6\qquad \mathbf{(B)}\; 7\qquad \mathbf{(C)}\; \frac{20}3\qquad \mathbf{(D)}\; 8\qquad \mathbf{(E)}\; 9$ | [
"\nDraw $BE$ and $CF$ , forming a trapezoid . Since it's cyclic, this trapezoid must be isosceles . Also, drop altitudes from $E$ to $AC$ $B$ to $DF$ , and $C$ to $DF$ , and let the feet of these altitudes be $G$ $H$ , and $I$ respectively. $AGED$ is a rectangle since it has $4$ right angles . Therefore, $AG=DE=3$ ... |
https://artofproblemsolving.com/wiki/index.php/2022_AMC_10A_Problems/Problem_9 | D | 540 | A rectangle is partitioned into $5$ regions as shown. Each region is to be painted a solid color - red, orange, yellow, blue, or green - so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible?
[asy] size(5.5cm); draw((0,0)--(0,2)--(2,2)--(2,0)--cycle); draw((2,0)--(8,0)--(8,2)--(2,2)--cycle); draw((8,0)--(12,0)--(12,2)--(8,2)--cycle); draw((0,2)--(6,2)--(6,4)--(0,4)--cycle); draw((6,2)--(12,2)--(12,4)--(6,4)--cycle); [/asy]
$\textbf{(A) }120\qquad\textbf{(B) }270\qquad\textbf{(C) }360\qquad\textbf{(D) }540\qquad\textbf{(E) }720$ | [
"The top left rectangle can be $5$ possible colors. Then the bottom left region can only be $4$ possible colors, and the bottom middle can only be $3$ colors since it is next to the top left and bottom left. Similarly, we have $3$ choices for the top right and $3$ choices for the bottom right, which gives us a tota... |
https://artofproblemsolving.com/wiki/index.php/2022_AMC_12A_Problems/Problem_7 | D | 540 | A rectangle is partitioned into $5$ regions as shown. Each region is to be painted a solid color - red, orange, yellow, blue, or green - so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible?
[asy] size(5.5cm); draw((0,0)--(0,2)--(2,2)--(2,0)--cycle); draw((2,0)--(8,0)--(8,2)--(2,2)--cycle); draw((8,0)--(12,0)--(12,2)--(8,2)--cycle); draw((0,2)--(6,2)--(6,4)--(0,4)--cycle); draw((6,2)--(12,2)--(12,4)--(6,4)--cycle); [/asy]
$\textbf{(A) }120\qquad\textbf{(B) }270\qquad\textbf{(C) }360\qquad\textbf{(D) }540\qquad\textbf{(E) }720$ | [
"The top left rectangle can be $5$ possible colors. Then the bottom left region can only be $4$ possible colors, and the bottom middle can only be $3$ colors since it is next to the top left and bottom left. Similarly, we have $3$ choices for the top right and $3$ choices for the bottom right, which gives us a tota... |
https://artofproblemsolving.com/wiki/index.php/1993_AIME_Problems/Problem_14 | null | 448 | A rectangle that is inscribed in a larger rectangle (with one vertex on each side) is called unstuck if it is possible to rotate (however slightly) the smaller rectangle about its center within the confines of the larger. Of all the rectangles that can be inscribed unstuck in a 6 by 8 rectangle, the smallest perimeter has the form $\sqrt{N}\,$ , for a positive integer $N\,$ . Find $N\,$ | [
"Put the rectangle on the coordinate plane so its vertices are at $(\\pm4,\\pm3)$ , for all four combinations of positive and negative. Then by symmetry, the other rectangle is also centered at the origin, $O$\nNote that such a rectangle is unstuck if its four vertices are in or on the edge of all four quadrants, a... |
https://artofproblemsolving.com/wiki/index.php/1997_AHSME_Problems/Problem_5 | C | 80 | A rectangle with perimeter $176$ is divided into five congruent rectangles as shown in the diagram. What is the perimeter of one of the five congruent rectangles? [asy] defaultpen(linewidth(.8pt)); draw(origin--(0,3)--(4,3)--(4,0)--cycle); draw((0,1)--(4,1)); draw((2,0)--midpoint((0,1)--(4,1))); real r = 4/3; draw((r,3)--foot((r,3),(0,1),(4,1))); draw((2r,3)--foot((2r,3),(0,1),(4,1)));[/asy]
$\mathrm{(A)\ } 35.2 \qquad \mathrm{(B) \ }76 \qquad \mathrm{(C) \ } 80 \qquad \mathrm{(D) \ } 84 \qquad \mathrm{(E) \ }86$ | [
"Let $l$ represent the length of one of the smaller rectangles, and let $w$ represent the width of one of the smaller rectangles, with $w < l$\nFrom the large rectangle, we see that the top has length $3w$ , the right has length $l + w$ , the bottom has length $2l$ , and the left has length $l + 2$\nSince the perim... |
https://artofproblemsolving.com/wiki/index.php/2015_AMC_10A_Problems/Problem_20 | B | 102 | A rectangle with positive integer side lengths in $\text{cm}$ has area $A$ $\text{cm}^2$ and perimeter $P$ $\text{cm}$ . Which of the following numbers cannot equal $A+P$
$\textbf{(A) }100\qquad\textbf{(B) }102\qquad\textbf{(C) }104\qquad\textbf{(D) }106\qquad\textbf{(E) }108$ | [
"Let the rectangle's length be $a$ and its width be $b$ . Its area is $ab$ and the perimeter is $2a+2b$\nThen $A + P = ab + 2a + 2b$ . Factoring, we have $(a + 2)(b + 2) - 4$\nThe only one of the answer choices that cannot be expressed in this form is $102$ , as $102 + 4$ is twice a prime. There would then be no wa... |
https://artofproblemsolving.com/wiki/index.php/2021_Fall_AMC_10B_Problems/Problem_25 | E | 67 | A rectangle with side lengths $1{ }$ and $3,$ a square with side length $1,$ and a rectangle $R$ are inscribed inside a larger square as shown. The sum of all possible values for the area of $R$ can be written in the form $\tfrac mn$ , where $m$ and $n$ are relatively prime positive integers. What is $m+n?$ [asy] size(8cm); draw((0,0)--(10,0)); draw((0,0)--(0,10)); draw((10,0)--(10,10)); draw((0,10)--(10,10)); draw((1,6)--(0,9)); draw((0,9)--(3,10)); draw((3,10)--(4,7)); draw((4,7)--(1,6)); draw((0,3)--(1,6)); draw((1,6)--(10,3)); draw((10,3)--(9,0)); draw((9,0)--(0,3)); draw((6,13/3)--(10,22/3)); draw((10,22/3)--(8,10)); draw((8,10)--(4,7)); draw((4,7)--(6,13/3)); label("$3$",(9/2,3/2),N); label("$3$",(11/2,9/2),S); label("$1$",(1/2,9/2),E); label("$1$",(19/2,3/2),W); label("$1$",(1/2,15/2),E); label("$1$",(3/2,19/2),S); label("$1$",(5/2,13/2),N); label("$1$",(7/2,17/2),W); label("$R$",(7,43/6),W); [/asy] $(\textbf{A})\: 14\qquad(\textbf{B}) \: 23\qquad(\textbf{C}) \: 46\qquad(\textbf{D}) \: 59\qquad(\textbf{E}) \: 67$ | [
"We use Image:2021_AMC_10B_(Nov)_Problem_25,_sol.png to facilitate our analysis.\nDenote $\\angle AFE = \\theta$ . Thus, $\\angle FIB = \\angle CEF = \\angle EKG = \\angle KLC = \\theta$\nHence, \\begin{align*} AB & = AF + FB \\\\ & = EF \\cos \\angle EFA + IF \\sin \\angle FIB \\\\ & = 3 \\cos \\theta + \\sin \\th... |
https://artofproblemsolving.com/wiki/index.php/2023_AMC_8_Problems/Problem_7 | B | 1 | A rectangle, with sides parallel to the $x$ -axis and $y$ -axis, has opposite vertices located at $(15, 3)$ and $(16, 5)$ . A line is drawn through points $A(0, 0)$ and $B(3, 1)$ . Another line is drawn through points $C(0, 10)$ and $D(2, 9)$ . How many points on the rectangle lie on at least one of the two lines? [asy] usepackage("mathptmx"); size(9cm); draw((0,-.5)--(0,11),EndArrow(size=.15cm)); draw((1,0)--(1,11),mediumgray); draw((2,0)--(2,11),mediumgray); draw((3,0)--(3,11),mediumgray); draw((4,0)--(4,11),mediumgray); draw((5,0)--(5,11),mediumgray); draw((6,0)--(6,11),mediumgray); draw((7,0)--(7,11),mediumgray); draw((8,0)--(8,11),mediumgray); draw((9,0)--(9,11),mediumgray); draw((10,0)--(10,11),mediumgray); draw((11,0)--(11,11),mediumgray); draw((12,0)--(12,11),mediumgray); draw((13,0)--(13,11),mediumgray); draw((14,0)--(14,11),mediumgray); draw((15,0)--(15,11),mediumgray); draw((16,0)--(16,11),mediumgray); draw((-.5,0)--(17,0),EndArrow(size=.15cm)); draw((0,1)--(17,1),mediumgray); draw((0,2)--(17,2),mediumgray); draw((0,3)--(17,3),mediumgray); draw((0,4)--(17,4),mediumgray); draw((0,5)--(17,5),mediumgray); draw((0,6)--(17,6),mediumgray); draw((0,7)--(17,7),mediumgray); draw((0,8)--(17,8),mediumgray); draw((0,9)--(17,9),mediumgray); draw((0,10)--(17,10),mediumgray); draw((-.13,1)--(.13,1)); draw((-.13,2)--(.13,2)); draw((-.13,3)--(.13,3)); draw((-.13,4)--(.13,4)); draw((-.13,5)--(.13,5)); draw((-.13,6)--(.13,6)); draw((-.13,7)--(.13,7)); draw((-.13,8)--(.13,8)); draw((-.13,9)--(.13,9)); draw((-.13,10)--(.13,10)); draw((1,-.13)--(1,.13)); draw((2,-.13)--(2,.13)); draw((3,-.13)--(3,.13)); draw((4,-.13)--(4,.13)); draw((5,-.13)--(5,.13)); draw((6,-.13)--(6,.13)); draw((7,-.13)--(7,.13)); draw((8,-.13)--(8,.13)); draw((9,-.13)--(9,.13)); draw((10,-.13)--(10,.13)); draw((11,-.13)--(11,.13)); draw((12,-.13)--(12,.13)); draw((13,-.13)--(13,.13)); draw((14,-.13)--(14,.13)); draw((15,-.13)--(15,.13)); draw((16,-.13)--(16,.13)); label(scale(.7)*"$1$", (1,-.13), S); label(scale(.7)*"$2$", (2,-.13), S); label(scale(.7)*"$3$", (3,-.13), S); label(scale(.7)*"$4$", (4,-.13), S); label(scale(.7)*"$5$", (5,-.13), S); label(scale(.7)*"$6$", (6,-.13), S); label(scale(.7)*"$7$", (7,-.13), S); label(scale(.7)*"$8$", (8,-.13), S); label(scale(.7)*"$9$", (9,-.13), S); label(scale(.7)*"$10$", (10,-.13), S); label(scale(.7)*"$11$", (11,-.13), S); label(scale(.7)*"$12$", (12,-.13), S); label(scale(.7)*"$13$", (13,-.13), S); label(scale(.7)*"$14$", (14,-.13), S); label(scale(.7)*"$15$", (15,-.13), S); label(scale(.7)*"$16$", (16,-.13), S); label(scale(.7)*"$1$", (-.13,1), W); label(scale(.7)*"$2$", (-.13,2), W); label(scale(.7)*"$3$", (-.13,3), W); label(scale(.7)*"$4$", (-.13,4), W); label(scale(.7)*"$5$", (-.13,5), W); label(scale(.7)*"$6$", (-.13,6), W); label(scale(.7)*"$7$", (-.13,7), W); label(scale(.7)*"$8$", (-.13,8), W); label(scale(.7)*"$9$", (-.13,9), W); label(scale(.7)*"$10$", (-.13,10), W); dot((0,0),linewidth(4)); label(scale(.75)*"$A$", (0,0), NE); dot((3,1),linewidth(4)); label(scale(.75)*"$B$", (3,1), NE); dot((0,10),linewidth(4)); label(scale(.75)*"$C$", (0,10), NE); dot((2,9),linewidth(4)); label(scale(.75)*"$D$", (2,9), NE); draw((15,3)--(16,3)--(16,5)--(15,5)--cycle,linewidth(1.125)); dot((15,3),linewidth(4)); dot((16,3),linewidth(4)); dot((16,5),linewidth(4)); dot((15,5),linewidth(4)); [/asy] $\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$ | [
"If we extend the lines, we have the following diagram: Therefore, we see that the answer is $\\boxed{1}.$",
"Note that the $y$ -intercepts of line $AB$ and line $CD$ are $0$ and $10$ . If the analytic expression for line $AB$ is $y=k_{1}x$ , and the analytic expression for line $CD$ is $y=k_{2}x+10$ , we have e... |
https://artofproblemsolving.com/wiki/index.php/1998_AJHSME_Problems/Problem_24 | E | 120 | A rectangular board of 8 columns has squares numbered beginning in the upper left corner and moving left to right so row one is numbered 1 through 8, row two is 9 through 16, and so on. A student shades square 1, then skips one square and shades square 3, skips two squares and shades square 6, skips 3 squares and shades square 10, and continues in this way until there is at least one shaded square in each column. What is the number of the shaded square that first achieves this result?
[asy] unitsize(20); for(int a = 0; a < 10; ++a) { draw((0,a)--(8,a)); } for (int b = 0; b < 9; ++b) { draw((b,0)--(b,9)); } draw((0,0)--(0,-.5)); draw((1,0)--(1,-1.5)); draw((.5,-1)--(1.5,-1)); draw((2,0)--(2,-.5)); draw((4,0)--(4,-.5)); draw((5,0)--(5,-1.5)); draw((4.5,-1)--(5.5,-1)); draw((6,0)--(6,-.5)); draw((8,0)--(8,-.5)); fill((0,8)--(1,8)--(1,9)--(0,9)--cycle,black); fill((2,8)--(3,8)--(3,9)--(2,9)--cycle,black); fill((5,8)--(6,8)--(6,9)--(5,9)--cycle,black); fill((1,7)--(2,7)--(2,8)--(1,8)--cycle,black); fill((6,7)--(7,7)--(7,8)--(6,8)--cycle,black); label("$2$",(1.5,8.2),N); label("$4$",(3.5,8.2),N); label("$5$",(4.5,8.2),N); label("$7$",(6.5,8.2),N); label("$8$",(7.5,8.2),N); label("$9$",(0.5,7.2),N); label("$11$",(2.5,7.2),N); label("$12$",(3.5,7.2),N); label("$13$",(4.5,7.2),N); label("$14$",(5.5,7.2),N); label("$16$",(7.5,7.2),N); [/asy]
$\text{(A)}\ 36\qquad\text{(B)}\ 64\qquad\text{(C)}\ 78\qquad\text{(D)}\ 91\qquad\text{(E)}\ 120$ | [
"The numbers that are shaded are the triangular numbers, which are numbers in the form $\\frac{(n)(n+1)}{2}$ for positive integers. They can also be generated by starting with $1$ , and adding $1, 2, 3, 4...$ as in the description of the problem.\nSquares that have the same remainder after being divided by $8$ wil... |
https://artofproblemsolving.com/wiki/index.php/2005_AMC_12A_Problems/Problem_22 | B | 10 | A rectangular box $P$ is inscribed in a sphere of radius $r$ . The surface area of $P$ is 384, and the sum of the lengths of its 12 edges is 112. What is $r$
$\mathrm{(A)}\ 8\qquad \mathrm{(B)}\ 10\qquad \mathrm{(C)}\ 12\qquad \mathrm{(D)}\ 14\qquad \mathrm{(E)}\ 16$ | [
"Box P has dimensions $l$ $w$ , and $h$ . \nIts surface area is \\[2lw+2lh+2wh=384,\\] and the sum of all its edges is \\[l + w + h = \\dfrac{4l+4w+4h}{4} = \\dfrac{112}{4} = 28.\\]\nThe diameter of the sphere is the space diagonal of the prism, which is \\[\\sqrt{l^2 + w^2 +h^2}.\\] Notice that \\[(l + w + h)^2 - ... |
https://artofproblemsolving.com/wiki/index.php/2016_AMC_10A_Problems/Problem_5 | D | 96 | A rectangular box has integer side lengths in the ratio $1: 3: 4$ . Which of the following could be the volume of the box?
$\textbf{(A)}\ 48\qquad\textbf{(B)}\ 56\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 96\qquad\textbf{(E)}\ 144$ | [
"Let the smallest side length be $x$ . Then the volume is $x \\cdot 3x \\cdot 4x =12x^3$ . If $x=2$ , then $12x^3 = 96 \\implies \\boxed{96.}$"
] |
https://artofproblemsolving.com/wiki/index.php/2013_AIME_I_Problems/Problem_7 | null | 41 | A rectangular box has width $12$ inches, length $16$ inches, and height $\frac{m}{n}$ inches, where $m$ and $n$ are relatively prime positive integers. Three faces of the box meet at a corner of the box. The center points of those three faces are the vertices of a triangle with an area of $30$ square inches. Find $m+n$ | [
"Let the height of the box be $x$\nAfter using the Pythagorean Theorem three times, we can quickly see that the sides of the triangle are 10, $\\sqrt{\\left(\\frac{x}{2}\\right)^2 + 64}$ , and $\\sqrt{\\left(\\frac{x}{2}\\right)^2 + 36}$ . Since the area of the triangle is $30$ , the altitude of the triangle from t... |
https://artofproblemsolving.com/wiki/index.php/2015_AMC_10B_Problems/Problem_25 | B | 10 | A rectangular box measures $a \times b \times c$ , where $a$ $b$ , and $c$ are integers and $1\leq a \leq b \leq c$ . The volume and the surface area of the box are numerically equal. How many ordered triples $(a,b,c)$ are possible?
$\textbf{(A)}\; 4 \qquad\textbf{(B)}\; 10 \qquad\textbf{(C)}\; 12 \qquad\textbf{(D)}\; 21 \qquad\textbf{(E)}\; 26$ | [
"We need \\[abc = 2(ab+bc+ac) \\quad \\text{ or } \\quad (a-2)bc = 2a(b+c).\\] Since $a\\le b, ac \\le bc$ , from the first equation we get $abc \\le 6bc$ . Thus $a\\le 6$ . From the second equation we see that $a > 2$ . Thus $a\\in \\{3, 4, 5, 6\\}$\nThus, there are $5+3+1+1 = \\boxed{10}$ solutions.",
"The surf... |
https://artofproblemsolving.com/wiki/index.php/2015_AMC_12B_Problems/Problem_23 | B | 10 | A rectangular box measures $a \times b \times c$ , where $a$ $b$ , and $c$ are integers and $1\leq a \leq b \leq c$ . The volume and the surface area of the box are numerically equal. How many ordered triples $(a,b,c)$ are possible?
$\textbf{(A)}\; 4 \qquad\textbf{(B)}\; 10 \qquad\textbf{(C)}\; 12 \qquad\textbf{(D)}\; 21 \qquad\textbf{(E)}\; 26$ | [
"We need \\[abc = 2(ab+bc+ac) \\quad \\text{ or } \\quad (a-2)bc = 2a(b+c).\\] Since $a\\le b, ac \\le bc$ , from the first equation we get $abc \\le 6bc$ . Thus $a\\le 6$ . From the second equation we see that $a > 2$ . Thus $a\\in \\{3, 4, 5, 6\\}$\nThus, there are $5+3+1+1 = \\boxed{10}$ solutions.",
"The surf... |
https://artofproblemsolving.com/wiki/index.php/2008_AMC_10B_Problems/Problem_23 | B | 2 | A rectangular floor measures $a$ by $b$ feet, where $a$ and $b$ are positive integers and $b > a$ . An artist paints a rectangle on the floor with the sides of the rectangle parallel to the floor. The unpainted part of the floor forms a border of width $1$ foot around the painted rectangle and occupies half the area of the whole floor. How many possibilities are there for the ordered pair $(a,b)$
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$ | [
"Because the unpainted part of the floor covers half the area, then the painted rectangle covers half the area as well. Since the border width is 1 foot, the dimensions of the rectangle are $a-2$ by $b-2$ . With this information we can make the equation:\n\\begin{eqnarray*} ab &=& 2\\left((a-2)(b-2)\\right) \\\\ ab... |
https://artofproblemsolving.com/wiki/index.php/2019_AMC_10A_Problems/Problem_10 | C | 26 | A rectangular floor that is $10$ feet wide and $17$ feet long is tiled with $170$ one-foot square tiles. A bug walks from one corner to the opposite corner in a straight line. Including the first and the last tile, how many tiles does the bug visit?
$\textbf{(A) } 17 \qquad\textbf{(B) } 25 \qquad\textbf{(C) } 26 \qquad\textbf{(D) } 27 \qquad\textbf{(E) } 28$ | [
"The number of tiles the bug visits is equal to $1$ plus the number of times it crosses a horizontal or vertical line. As it must cross $16$ horizontal lines and $9$ vertical lines, it must be that the bug visits a total of $16+9+1 = \\boxed{26}$ squares.",
"We can also draw a diagram or scale model of the entir... |
https://artofproblemsolving.com/wiki/index.php/1999_AMC_8_Problems/Problem_5 | D | 400 | A rectangular garden 60 feet long and 20 feet wide is enclosed by a fence. To make the garden larger, while using the same fence, its shape is changed to a square. By how many square feet does this enlarge the garden?
$\text{(A)}\ 100 \qquad \text{(B)}\ 200 \qquad \text{(C)}\ 300 \qquad \text{(D)}\ 400 \qquad \text{(E)}\ 500$ | [
"We need the same perimeter as a $60$ by $20$ rectangle, but the greatest area we can get. right now the perimeter is $160$ . To get the greatest area while keeping a perimeter of $160$ , the sides should all be $40$ . that means an area of $1600$ . Right now, the area is $20 \\times 60$ which is $1200$ $1600-1200=... |
https://artofproblemsolving.com/wiki/index.php/1986_AJHSME_Problems/Problem_18 | B | 12 | A rectangular grazing area is to be fenced off on three sides using part of a $100$ meter rock wall as the fourth side. Fence posts are to be placed every $12$ meters along the fence including the two posts where the fence meets the rock wall. What is the fewest number of posts required to fence an area $36$ m by $60$ m?
[asy] unitsize(12); draw((0,0)--(16,12)); draw((10.66666,8)--(6.66666,13.33333)--(1.33333,9.33333)--(5.33333,4)); label("WALL",(7,4),SE); [/asy]
$\text{(A)}\ 11 \qquad \text{(B)}\ 12 \qquad \text{(C)}\ 13 \qquad \text{(D)}\ 14 \qquad \text{(E)}\ 16$ | [
"Since we want to minimize the amount of fence that we use, we should have the longer side of the rectangle have one side as the wall. The grazing area is a $36$ m by $60$ m rectangle, so the $60$ m side should be parallel to the wall. That means the two fences perpendicular to the wall are $36$ m. We can start by ... |
https://artofproblemsolving.com/wiki/index.php/2011_AMC_10B_Problems/Problem_14 | C | 62 | A rectangular parking lot has a diagonal of $25$ meters and an area of $168$ square meters. In meters, what is the perimeter of the parking lot?
$\textbf{(A)}\ 52 \qquad\textbf{(B)}\ 58 \qquad\textbf{(C)}\ 62 \qquad\textbf{(D)}\ 68 \qquad\textbf{(E)}\ 70$ | [
"Let the sides of the rectangular parking lot be $a$ and $b$ . Then $a^2 + b^2 = 625$ and $ab = 168$ . Add the two equations together, then factor. \\begin{align*} a^2 + 2ab + b^2 &= 625 + 168 \\times 2\\\\ (a + b)^2 &= 961\\\\ a + b &= 31 \\end{align*} The perimeter of a rectangle is $2 (a + b) = 2 (31) = \\boxed{... |
https://artofproblemsolving.com/wiki/index.php/2012_AMC_8_Problems/Problem_6 | E | 88 | A rectangular photograph is placed in a frame that forms a border two inches wide on all sides of the photograph. The photograph measures $8$ inches high and $10$ inches wide. What is the area of the border, in square inches?
$\textbf{(A)}\hspace{.05in}36\qquad\textbf{(B)}\hspace{.05in}40\qquad\textbf{(C)}\hspace{.05in}64\qquad\textbf{(D)}\hspace{.05in}72\qquad\textbf{(E)}\hspace{.05in}88$ | [
"In order to find the area of the frame, we need to subtract the area of the photograph from the area of the photograph and the frame together. The area of the photograph is $8 \\times 10 = 80$ square inches. The height of the whole frame (including the photograph) would be $8+2+2 = 12$ , and the width of the whole... |
https://artofproblemsolving.com/wiki/index.php/2009_AMC_10B_Problems/Problem_4 | C | 15 | A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths $15$ and $25$ meters. What fraction of the yard is occupied by the flower beds?
$\mathrm{(A)}\frac {1}{8}\qquad \mathrm{(B)}\frac {1}{6}\qquad \mathrm{(C)}\frac {1}{5}\qquad \mathrm{(D)}\frac {1}{4}\qquad \mathrm{(E)}\frac {1}{3}$ | [
"Each triangle has leg length $\\frac 12 \\cdot (25 - 15) = 5$ meters and area $\\frac 12 \\cdot 5^2 = \\frac {25}{2}$ square meters. Thus the flower beds have a total area of $25$ square meters. The entire yard has length $25$ m and width $5$ m, so its area is $125$ square meters. The fraction of the yard occup... |
https://artofproblemsolving.com/wiki/index.php/2009_AMC_12B_Problems/Problem_4 | C | 15 | A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths $15$ and $25$ meters. What fraction of the yard is occupied by the flower beds?
$\mathrm{(A)}\frac {1}{8}\qquad \mathrm{(B)}\frac {1}{6}\qquad \mathrm{(C)}\frac {1}{5}\qquad \mathrm{(D)}\frac {1}{4}\qquad \mathrm{(E)}\frac {1}{3}$ | [
"Each triangle has leg length $\\frac 12 \\cdot (25 - 15) = 5$ meters and area $\\frac 12 \\cdot 5^2 = \\frac {25}{2}$ square meters. Thus the flower beds have a total area of $25$ square meters. The entire yard has length $25$ m and width $5$ m, so its area is $125$ square meters. The fraction of the yard occup... |
https://artofproblemsolving.com/wiki/index.php/2009_AMC_12B_Problems/Problem_23 | D | 79 | A region $S$ in the complex plane is defined by \[S = \{x + iy: - 1\le x\le1, - 1\le y\le1\}.\] A complex number $z = x + iy$ is chosen uniformly at random from $S$ . What is the probability that $\left(\frac34 + \frac34i\right)z$ is also in $S$
$\textbf{(A)}\ \frac12\qquad \textbf{(B)}\ \frac23\qquad \textbf{(C)}\ \frac34\qquad \textbf{(D)}\ \frac79\qquad \textbf{(E)}\ \frac78$ | [
"We multiply $z$ and $(\\frac{3}{4}+\\frac{3}{4}i)$ to get \\[(\\frac{3}{4}x-\\frac{3}{4}y)+(\\frac{3}{4}xi+\\frac{3}{4}yi).\\] Since we want to find the probability that this number is in $S$ , we need the real and complex coefficients of this number to be less than or equal to $1$ or greater than or equal to $-1.... |
https://artofproblemsolving.com/wiki/index.php/2006_AMC_10B_Problems/Problem_6 | D | 4 | A region is bounded by semicircular arcs constructed on the side of a square whose sides measure $\frac{2}{\pi}$ , as shown. What is the perimeter of this region?
[asy] size(90); defaultpen(linewidth(0.7)); filldraw((0,0)--(2,0)--(2,2)--(0,2)--cycle,gray(0.5)); filldraw(arc((1,0),1,180,0, CCW)--cycle,gray(0.7)); filldraw(arc((0,1),1,90,270)--cycle,gray(0.7)); filldraw(arc((1,2),1,0,180)--cycle,gray(0.7)); filldraw(arc((2,1),1,270,90, CCW)--cycle,gray(0.7)); [/asy]
$\textbf{(A) } \frac{4}{\pi}\qquad \textbf{(B) } 2\qquad \textbf{(C) } \frac{8}{\pi}\qquad \textbf{(D) } 4\qquad \textbf{(E) } \frac{16}{\pi}$ | [
"Since the side of the square is the diameter of the semicircle, the radius of the semicircle is $\\frac{1}{2}\\cdot\\frac{2}{\\pi}=\\frac{1}{\\pi}$\nSince the length of one of the semicircular arcs is half the circumference of the corresponding circle, the length of one arc is $\\frac{1}{2}\\cdot2\\cdot\\pi\\cdot\... |
https://artofproblemsolving.com/wiki/index.php/2015_AMC_12B_Problems/Problem_7 | D | 39 | A regular 15-gon has $L$ lines of symmetry, and the smallest positive angle for which it has rotational symmetry is $R$ degrees. What is $L+R$
$\textbf{(A)}\; 24 \qquad\textbf{(B)}\; 27 \qquad\textbf{(C)}\; 32 \qquad\textbf{(D)}\; 39 \qquad\textbf{(E)}\; 54$ | [
"From consideration of a smaller regular polygon with an odd number of sides (e.g. a pentagon), we see that the lines of symmetry go through a vertex of the polygon and bisect the opposite side. Hence $L=15$ , the number of sides / vertices. The smallest angle for a rotational symmetry transforms one side into an a... |
https://artofproblemsolving.com/wiki/index.php/1998_AHSME_Problems/Problem_15 | C | 6 | A regular hexagon and an equilateral triangle have equal areas. What is the ratio of the length of a side of the triangle to the length of a side of the hexagon?
$\mathrm{(A) \ }\sqrt{3} \qquad \mathrm{(B) \ }2 \qquad \mathrm{(C) \ }\sqrt{6} \qquad \mathrm{(D) \ }3 \qquad \mathrm{(E) \ }6$ | [
"$A_{\\triangle} = \\frac{s_t^2\\sqrt{3}}{4}$\n$A_{hex} = \\frac{6s_h^2\\sqrt{3}}{4}$ since a regular hexagon is just six equilateral triangles.\nSetting the areas equal, we get:\n$s_t^2 = 6s_h^2$\n$\\left(\\frac{s_t}{s_h}\\right)^2 = 6$\n$\\frac{s_t}{s_h} = \\sqrt{6}$ , and the answer is $\\boxed{6}$"
] |
https://artofproblemsolving.com/wiki/index.php/1994_AHSME_Problems/Problem_26 | A | 5 | A regular polygon of $m$ sides is exactly enclosed (no overlaps, no gaps) by $m$ regular polygons of $n$ sides each. (Shown here for $m=4, n=8$ .) If $m=10$ , what is the value of $n$ [asy] size(200); defaultpen(linewidth(0.8)); draw(unitsquare); path p=(0,1)--(1,1)--(1+sqrt(2)/2,1+sqrt(2)/2)--(1+sqrt(2)/2,2+sqrt(2)/2)--(1,2+sqrt(2))--(0,2+sqrt(2))--(-sqrt(2)/2,2+sqrt(2)/2)--(-sqrt(2)/2,1+sqrt(2)/2)--cycle; draw(p); draw(shift((1+sqrt(2)/2,-sqrt(2)/2-1))*p); draw(shift((0,-2-sqrt(2)))*p); draw(shift((-1-sqrt(2)/2,-sqrt(2)/2-1))*p);[/asy] $\textbf{(A)}\ 5 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 14 \qquad\textbf{(D)}\ 20 \qquad\textbf{(E)}\ 26$ | [
"To find the number of sides on the regular polygons that surround the decagon, we can find the interior angles and work from there. Knowing that the measure of the interior angle of any regular polygon is $\\frac{(n-2)*180}{n}$ , the measure of the decagon's interior angle is $\\frac{8*180}{10} = 144$ degrees.\nTh... |
https://artofproblemsolving.com/wiki/index.php/1961_AHSME_Problems/Problem_32 | C | 12 | A regular polygon of $n$ sides is inscribed in a circle of radius $R$ . The area of the polygon is $3R^2$ . Then $n$ equals:
$\textbf{(A)}\ 8\qquad \textbf{(B)}\ 10\qquad \textbf{(C)}\ 12\qquad \textbf{(D)}\ 15\qquad \textbf{(E)}\ 18$ | [
"Note that the distance from the center of the circle to each of the vertices of the inscribed regular polygon equals the radius $R$ . Since each side of a regular polygon is the same length, all the angles between the two lines from the center to the two vertices of a side is the same.\nThat means each of these a... |
https://artofproblemsolving.com/wiki/index.php/2003_AMC_10B_Problems/Problem_16 | E | 8 | A restaurant offers three desserts, and exactly twice as many appetizers as main courses. A dinner consists of an appetizer, a main course, and a dessert. What is the least number of main courses that a restaurant should offer so that a customer could have a different dinner each night in the year $2003$
$\textbf{(A) } 4 \qquad\textbf{(B) } 5 \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 7 \qquad\textbf{(E) } 8$ | [
"Let $m$ be the number of main courses the restaurant serves, so $2m$ is the number of appetizers. Then the number of dinner combinations is $2m\\times m\\times3=6m^2$ . Since the customer wants to eat a different dinner in all $365$ days of $2003$ , we must have\n\\begin{align*} 6m^2 &\\geq 365\\\\ m^2 &\\geq 60.8... |
https://artofproblemsolving.com/wiki/index.php/2023_AMC_10A_Problems/Problem_18 | D | 8 | A rhombic dodecahedron is a solid with $12$ congruent rhombus faces. At every vertex, $3$ or $4$ edges meet, depending on the vertex. How many vertices have exactly $3$ edges meet?
$\textbf{(A) }5\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$ | [
"Note Euler's formula where $\\text{Vertices}+\\text{Faces}-\\text{Edges}=2$ . There are $12$ faces and the number of edges is $24$ because there are 12 faces each with four edges and each edge is shared by two faces. Now we know that there are $14$ vertices on the figure. Now note that the sum of the degrees of al... |
https://artofproblemsolving.com/wiki/index.php/2023_AMC_10A_Problems/Problem_18 | null | 8 | A rhombic dodecahedron is a solid with $12$ congruent rhombus faces. At every vertex, $3$ or $4$ edges meet, depending on the vertex. How many vertices have exactly $3$ edges meet?
$\textbf{(A) }5\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$ | [
"Let $m$ be the number of $4$ -edge vertices, and $n$ be the number of $3$ -edge vertices. The total number of vertices is $m+n$ . Now, we know that there are $4 \\cdot 12 = 48$ vertices, but we have overcounted. We have overcounted $m$ vertices $3$ times and overcounted $n$ vertices $2$ times. Therefore, we subtra... |
https://artofproblemsolving.com/wiki/index.php/2008_AIME_I_Problems/Problem_5 | null | 14 | A right circular cone has base radius $r$ and height $h$ . The cone lies on its side on a flat table. As the cone rolls on the surface of the table without slipping, the point where the cone's base meets the table traces a circular arc centered at the point where the vertex touches the table. The cone first returns to its original position on the table after making $17$ complete rotations. The value of $h/r$ can be written in the form $m\sqrt {n}$ , where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m + n$ | [
"The path is a circle with radius equal to the slant height of the cone, which is $\\sqrt {r^{2} + h^{2}}$ . Thus, the length of the path is $2\\pi\\sqrt {r^{2} + h^{2}}$\nAlso, the length of the path is 17 times the circumference of the base, which is $34r\\pi$ . Setting these equal gives $\\sqrt {r^{2} + h^{2}} =... |
https://artofproblemsolving.com/wiki/index.php/2018_AIME_I_Problems/Problem_7 | null | 52 | A right hexagonal prism has height $2$ . The bases are regular hexagons with side length $1$ . Any $3$ of the $12$ vertices determine a triangle. Find the number of these triangles that are isosceles (including equilateral triangles). | [
"We can consider two cases: when the three vertices are on one base, and when the vertices are on two bases.\nCase 1: vertices are on one base. Then we can call one of the vertices $A$ for distinction. Either the triangle can have sides $1, 1, \\sqrt{3}$ with 6 cases or $\\sqrt{3}, \\sqrt{3}, \\sqrt{3}$ with 2 case... |
https://artofproblemsolving.com/wiki/index.php/2016_AIME_I_Problems/Problem_4 | null | 108 | A right prism with height $h$ has bases that are regular hexagons with sides of length $12$ . A vertex $A$ of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and the face of the pyramid that does not contain $A$ measures $60$ degrees. Find $h^2$ | [
"Let $B$ and $C$ be the vertices adjacent to $A$ on the same base as $A$ , and let $D$ be the last vertex of the triangular pyramid. Then $\\angle CAB = 120^\\circ$ . Let $X$ be the foot of the altitude from $A$ to $\\overline{BC}$ . Then since $\\triangle ABX$ is a $30-60-90$ triangle, $AX = 6$ . Since the dihed... |
https://artofproblemsolving.com/wiki/index.php/1995_AIME_Problems/Problem_11 | null | 40 | A right rectangular prism $P_{}$ (i.e., a rectangular parallelpiped) has sides of integral length $a, b, c,$ with $a\le b\le c.$ A plane parallel to one of the faces of $P_{}$ cuts $P_{}$ into two prisms, one of which is similar to $P_{},$ and both of which have nonzero volume. Given that $b=1995,$ for how many ordered triples $(a, b, c)$ does such a plane exist? | [
"Let $P'$ be the prism similar to $P$ , and let the sides of $P'$ be of length $x,y,z$ , such that $x \\le y \\le z$ . Then\n\\[\\frac{x}{a} = \\frac{y}{b} = \\frac zc < 1.\\]\nNote that if the ratio of similarity was equal to $1$ , we would have a prism with zero volume. As one face of $P'$ is a face of $P$ , it f... |
https://artofproblemsolving.com/wiki/index.php/2021_Fall_AMC_12A_Problems/Problem_9 | E | 576 | A right rectangular prism whose surface area and volume are numerically equal has edge lengths $\log_{2}x, \log_{3}x,$ and $\log_{4}x.$ What is $x?$
$\textbf{(A)}\ 2\sqrt{6} \qquad\textbf{(B)}\ 6\sqrt{6} \qquad\textbf{(C)}\ 24 \qquad\textbf{(D)}\ 48 \qquad\textbf{(E)}\ 576$ | [
"The surface area of this right rectangular prism is $2(\\log_{2}x\\log_{3}x+\\log_{2}x\\log_{4}x+\\log_{3}x\\log_{4}x).$\nThe volume of this right rectangular prism is $\\log_{2}x\\log_{3}x\\log_{4}x.$\nEquating the numerical values of the surface area and the volume, we have \\[2(\\log_{2}x\\log_{3}x+\\log_{2}x\\... |
https://artofproblemsolving.com/wiki/index.php/2022_AIME_II_Problems/Problem_3 | null | 21 | A right square pyramid with volume $54$ has a base with side length $6.$ The five vertices of the pyramid all lie on a sphere with radius $\frac mn$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ | [
"Although I can't draw the exact picture of this problem, but it is quite easy to imagine that four vertices of the base of this pyramid is on a circle (Radius $\\frac{6}{\\sqrt{2}} = 3\\sqrt{2}$ ). Since all five vertices are on the sphere, the distances of the spherical center and the vertices are the same: $l$ .... |
https://artofproblemsolving.com/wiki/index.php/1997_AHSME_Problems/Problem_24 | B | 5 | A rising number, such as $34689$ , is a positive integer each digit of which is larger than each of the digits to its left. There are $\binom{9}{5} = 126$ five-digit rising numbers. When these numbers are arranged from smallest to largest, the $97^{\text{th}}$ number in the list does not contain the digit
$\textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 8$ | [
"The list starts with $12345$ . There are $\\binom{8}{4} = 70$ four-digit rising numbers that do not begin with $1$ , and thus also $70$ five digit rising numbers that do begin with $1$ that are formed by simply putting a $1$ before the four digit number.\nThus, the $71^{\\text{st}}$ number is $23456$ . There are... |
https://artofproblemsolving.com/wiki/index.php/2024_AMC_8_Problems/Problem_22 | null | 600 | A roll of tape is $4$ inches in diameter and is wrapped around a ring that is $2$ inches in diameter. A cross section of the tape is shown in the figure below. The tape is $0.015$ inches thick. If the tape is completely unrolled, approximately how long would it be? Round your answer to the nearest $100$ inches.
$\textbf{(A) } 300\qquad\textbf{(B) } 600\qquad\textbf{(C) } 1200\qquad\textbf{(D) } 1500\qquad\textbf{(E) } 1800$ | [
"The roll of tape is $1/0.015=$ 66 layers thick. In order to find the total length, we have to find the average of each concentric circle and multiply it by $66$ . Since the diameter of the small circle is $2$ inches and the diameter of the large one is $4$ inches, the \"middle value\" is $3$ . Therefore, the avera... |
https://artofproblemsolving.com/wiki/index.php/2016_AMC_10A_Problems/Problem_10 | B | 2 | A rug is made with three different colors as shown. The areas of the three differently colored regions form an arithmetic progression. The inner rectangle is one foot wide, and each of the two shaded regions is $1$ foot wide on all four sides. What is the length in feet of the inner rectangle?
[asy] size(6cm); defaultpen(fontsize(9pt)); path rectangle(pair X, pair Y){ return X--(X.x,Y.y)--Y--(Y.x,X.y)--cycle; } filldraw(rectangle((0,0),(7,5)),gray(0.5)); filldraw(rectangle((1,1),(6,4)),gray(0.75)); filldraw(rectangle((2,2),(5,3)),white); label("$1$",(0.5,2.5)); draw((0.3,2.5)--(0,2.5),EndArrow(TeXHead)); draw((0.7,2.5)--(1,2.5),EndArrow(TeXHead)); label("$1$",(1.5,2.5)); draw((1.3,2.5)--(1,2.5),EndArrow(TeXHead)); draw((1.7,2.5)--(2,2.5),EndArrow(TeXHead)); label("$1$",(4.5,2.5)); draw((4.5,2.7)--(4.5,3),EndArrow(TeXHead)); draw((4.5,2.3)--(4.5,2),EndArrow(TeXHead)); label("$1$",(4.1,1.5)); draw((4.1,1.7)--(4.1,2),EndArrow(TeXHead)); draw((4.1,1.3)--(4.1,1),EndArrow(TeXHead)); label("$1$",(3.7,0.5)); draw((3.7,0.7)--(3.7,1),EndArrow(TeXHead)); draw((3.7,0.3)--(3.7,0),EndArrow(TeXHead)); [/asy]
$\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 4 \qquad \textbf{(D) } 6 \qquad \textbf{(E) }8$ | [
"Let the length of the inner rectangle be $x$\nThen the area of that rectangle is $x\\cdot1 = x$\nThe second largest rectangle has dimensions of $x+2$ and $3$ , making its area $3x+6$ . The area of the second shaded area, therefore, is $3x+6-x = 2x+6$\nThe largest rectangle has dimensions of $x+4$ and $5$ , making ... |
https://artofproblemsolving.com/wiki/index.php/1994_AHSME_Problems/Problem_24 | C | 5 | A sample consisting of five observations has an arithmetic mean of $10$ and a median of $12$ . The smallest value that the range (largest observation minus smallest) can assume for such a sample is
$\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ 10$ | [
"The minimum range occurs in the set $\\{7,7,12,12,12\\}$ , so the answer is $\\boxed{5}$"
] |
https://artofproblemsolving.com/wiki/index.php/1989_AIME_Problems/Problem_11 | null | 947 | A sample of 121 integers is given, each between 1 and 1000 inclusive, with repetitions allowed. The sample has a unique mode (most frequent value). Let $D$ be the difference between the mode and the arithmetic mean of the sample. What is the largest possible value of $\lfloor D\rfloor$ ? (For real $x$ $\lfloor x\rfloor$ is the greatest integer less than or equal to $x$ .) | [
"Let the mode be $x$ , which we let appear $n > 1$ times. We let the arithmetic mean be $M$ , and the sum of the numbers $\\neq x$ be $S$ . Then \\begin{align*} D &= \\left|M-x\\right| = \\left|\\frac{S+xn}{121}-x\\right| = \\left|\\frac{S}{121}-\\left(\\frac{121-n}{121}\\right)x\\right| \\end{align*} As $S$ is ess... |
https://artofproblemsolving.com/wiki/index.php/2018_AMC_10A_Problems/Problem_20 | B | 1,022 | A scanning code consists of a $7 \times 7$ grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of $49$ squares. A scanning code is called $\textit{symmetric}$ if its look does not change when the entire square is rotated by a multiple of $90 ^{\circ}$ counterclockwise around its center, nor when it is reflected across a line joining opposite corners or a line joining midpoints of opposite sides. What is the total number of possible symmetric scanning codes?
$\textbf{(A)} \text{ 510} \qquad \textbf{(B)} \text{ 1022} \qquad \textbf{(C)} \text{ 8190} \qquad \textbf{(D)} \text{ 8192} \qquad \textbf{(E)} \text{ 65,534}$ | [
"\nImagine folding the scanning code along its lines of symmetry. There will be $10$ regions which you have control over coloring. Since we must subtract off $2$ cases for the all-black and all-white cases, the answer is $2^{10}-2=\\boxed{1022.}$",
"\\[\\begin{tabular}{|c|c|c|c|c|c|c|} \\hline T & T & T & X & T &... |
https://artofproblemsolving.com/wiki/index.php/2018_AMC_12A_Problems/Problem_15 | B | 1,022 | A scanning code consists of a $7 \times 7$ grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of $49$ squares. A scanning code is called $\textit{symmetric}$ if its look does not change when the entire square is rotated by a multiple of $90 ^{\circ}$ counterclockwise around its center, nor when it is reflected across a line joining opposite corners or a line joining midpoints of opposite sides. What is the total number of possible symmetric scanning codes?
$\textbf{(A)} \text{ 510} \qquad \textbf{(B)} \text{ 1022} \qquad \textbf{(C)} \text{ 8190} \qquad \textbf{(D)} \text{ 8192} \qquad \textbf{(E)} \text{ 65,534}$ | [
"\nImagine folding the scanning code along its lines of symmetry. There will be $10$ regions which you have control over coloring. Since we must subtract off $2$ cases for the all-black and all-white cases, the answer is $2^{10}-2=\\boxed{1022.}$",
"\\[\\begin{tabular}{|c|c|c|c|c|c|c|} \\hline T & T & T & X & T &... |
https://artofproblemsolving.com/wiki/index.php/2021_Fall_AMC_10A_Problems/Problem_10 | B | 13.5 | A school has $100$ students and $5$ teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are $50, 20, 20, 5,$ and $5$ . Let $t$ be the average value obtained if a teacher is picked at random and the number of students in their class is noted. Let $s$ be the average value obtained if a student was picked at random and the number of students in their class, including the student, is noted. What is $t-s$
$\textbf{(A)}\ {-}18.5 \qquad\textbf{(B)}\ {-}13.5 \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 13.5 \qquad\textbf{(E)}\ 18.5$ | [
"The formula for expected values is \\[\\text{Expected Value}=\\sum(\\text{Outcome}\\cdot\\text{Probability}).\\] We have \\begin{align*} t &= 50\\cdot\\frac15 + 20\\cdot\\frac15 + 20\\cdot\\frac15 + 5\\cdot\\frac15 + 5\\cdot\\frac15 \\\\ &= (50+20+20+5+5)\\cdot\\frac15 \\\\ &= 100\\cdot\\frac15 \\\\ &= 20, \\\\ s ... |
https://artofproblemsolving.com/wiki/index.php/2021_Fall_AMC_12A_Problems/Problem_7 | B | 13.5 | A school has $100$ students and $5$ teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are $50, 20, 20, 5,$ and $5$ . Let $t$ be the average value obtained if a teacher is picked at random and the number of students in their class is noted. Let $s$ be the average value obtained if a student was picked at random and the number of students in their class, including the student, is noted. What is $t-s$
$\textbf{(A)}\ {-}18.5 \qquad\textbf{(B)}\ {-}13.5 \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 13.5 \qquad\textbf{(E)}\ 18.5$ | [
"The formula for expected values is \\[\\text{Expected Value}=\\sum(\\text{Outcome}\\cdot\\text{Probability}).\\] We have \\begin{align*} t &= 50\\cdot\\frac15 + 20\\cdot\\frac15 + 20\\cdot\\frac15 + 5\\cdot\\frac15 + 5\\cdot\\frac15 \\\\ &= (50+20+20+5+5)\\cdot\\frac15 \\\\ &= 100\\cdot\\frac15 \\\\ &= 20, \\\\ s ... |
https://artofproblemsolving.com/wiki/index.php/2020_AMC_8_Problems/Problem_20 | B | 24.2 | A scientist walking through a forest recorded as integers the heights of $5$ trees standing in a row. She observed that each tree was either twice as tall or half as tall as the one to its right. Unfortunately some of her data was lost when rain fell on her notebook. Her notes are shown below, with blanks indicating the missing numbers. Based on her observations, the scientist was able to reconstruct the lost data. What was the average height of the trees, in meters?
\[\begingroup \setlength{\tabcolsep}{10pt} \renewcommand{\arraystretch}{1.5} \begin{tabular}{|c|c|} \hline Tree 1 & \rule{0.4cm}{0.15mm} meters \\ Tree 2 & 11 meters \\ Tree 3 & \rule{0.5cm}{0.15mm} meters \\ Tree 4 & \rule{0.5cm}{0.15mm} meters \\ Tree 5 & \rule{0.5cm}{0.15mm} meters \\ \hline Average height & \rule{0.5cm}{0.15mm}\text{ .}2 meters \\ \hline \end{tabular} \endgroup\] $\textbf{(A) }22.2 \qquad \textbf{(B) }24.2 \qquad \textbf{(C) }33.2 \qquad \textbf{(D) }35.2 \qquad \textbf{(E) }37.2$ | [
"We will show that $22$ $11$ $22$ $44$ , and $22$ meters are the heights of the trees from left to right. We are given that all tree heights are integers, so since Tree 2 has height $11$ meters, we can deduce that Trees 1 and 3 both have a height of $22$ meters. There are now three possible cases for the heights of... |
https://artofproblemsolving.com/wiki/index.php/2005_AMC_10B_Problems/Problem_1 | A | 100 | A scout troop buys $1000$ candy bars at a price of five for $2$ dollars. They sell all the candy bars at the price of two for $1$ dollar. What was their profit, in dollars?
$\textbf{(A) }\ 100 \qquad \textbf{(B) }\ 200 \qquad \textbf{(C) }\ 300 \qquad \textbf{(D) }\ 400 \qquad \textbf{(E) }\ 500$ | [
"\\begin{align*} \\mbox{Expenses} &= 1000 \\cdot \\frac25 = 400 \\\\ \\mbox{Revenue} &= 1000 \\cdot \\frac12 = 500 \\\\ \\mbox{Profit} &= \\mbox{Revenue} - \\mbox{Expenses} = 500-400 = \\boxed{100} Note: Revenue is a gain.",
"Note that the troop buys $10$ candy bars at a price of $4$ dollars and sells $10$ bar... |
https://artofproblemsolving.com/wiki/index.php/2005_AMC_12B_Problems/Problem_1 | A | 100 | A scout troop buys $1000$ candy bars at a price of five for $2$ dollars. They sell all the candy bars at the price of two for $1$ dollar. What was their profit, in dollars?
$\textbf{(A) }\ 100 \qquad \textbf{(B) }\ 200 \qquad \textbf{(C) }\ 300 \qquad \textbf{(D) }\ 400 \qquad \textbf{(E) }\ 500$ | [
"\\begin{align*} \\mbox{Expenses} &= 1000 \\cdot \\frac25 = 400 \\\\ \\mbox{Revenue} &= 1000 \\cdot \\frac12 = 500 \\\\ \\mbox{Profit} &= \\mbox{Revenue} - \\mbox{Expenses} = 500-400 = \\boxed{100} Note: Revenue is a gain.",
"Note that the troop buys $10$ candy bars at a price of $4$ dollars and sells $10$ bar... |
https://artofproblemsolving.com/wiki/index.php/2008_AMC_10B_Problems/Problem_4 | C | 400,000 | A semipro baseball league has teams with 21 players each. League rules state that a player must be paid at least $15,000 and that the total of all players' salaries for each team cannot exceed $700,000. What is the maximum possible salary, in dollars, for a single player?
$\mathrm{(A)}\ 270,000\qquad\mathrm{(B)}\ 385,000\qquad\mathrm{(C)}\ 400,000\qquad\mathrm{(D)}\ 430,000\qquad\mathrm{(E)}\ 700,000$ | [
"The maximum salary for a single player occurs when the other 20 players receive the minimum salary. The total of all players' salaries is 700000. The answer is $700000-15000*20=400000\\Rightarrow \\boxed{400,000}$"
] |
https://artofproblemsolving.com/wiki/index.php/2006_AMC_12B_Problems/Problem_25 | B | 324 | A sequence $a_1,a_2,\dots$ of non-negative integers is defined by the rule $a_{n+2}=|a_{n+1}-a_n|$ for $n\geq 1$ . If $a_1=999$ $a_2<999$ and $a_{2006}=1$ , how many different values of $a_2$ are possible?
$\mathrm{(A)}\ 165 \qquad \mathrm{(B)}\ 324 \qquad \mathrm{(C)}\ 495 \qquad \mathrm{(D)}\ 499 \qquad \mathrm{(E)}\ 660$ | [
"We say the sequence $(a_n)$ completes at $i$ if $i$ is the minimal positive integer such that $a_i = a_{i + 1} = 1$ . Otherwise, we say $(a_n)$ does not complete.\nNote that if $d = \\gcd(999, a_2) \\neq 1$ , then $d|a_n$ for all $n \\geq 1$ , and $d$ does not divide $1$ , so if $\\gcd(999, a_2) \\neq 1$ , then $(... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.