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/1997_AIME_Problems/Problem_7 | null | 198 | A car travels due east at $\frac 23$ mile per minute on a long, straight road. At the same time, a circular storm, whose radius is $51$ miles, moves southeast at $\frac 12\sqrt{2}$ mile per minute. At time $t=0$ , the center of the storm is $110$ miles due north of the car. At time $t=t_1$ minutes, the car enters the s... | [
"We set up a coordinate system, with the starting point of the car at the origin . At time $t$ , the car is at $\\left(\\frac 23t,0\\right)$ and the center of the storm is at $\\left(\\frac{t}{2}, 110 - \\frac{t}{2}\\right)$ . Using the distance formula,\n\\begin{eqnarray*} \\sqrt{\\left(\\frac{2}{3}t - \\frac 12t\... |
https://artofproblemsolving.com/wiki/index.php/2021_AMC_10A_Problems/Problem_4 | D | 3,195 | A cart rolls down a hill, travelling $5$ inches the first second and accelerating so that during each successive $1$ -second time interval, it travels $7$ inches more than during the previous $1$ -second interval. The cart takes $30$ seconds to reach the bottom of the hill. How far, in inches, does it travel?
$\textbf{... | [
"Since \\[\\mathrm{Distance}=\\mathrm{Speed}\\cdot\\mathrm{Time},\\] we seek the sum \\[5\\cdot1+12\\cdot1+19\\cdot1+26\\cdot1+\\cdots=5+12+19+26+\\cdots,\\] in which there are $30$ terms.\nThe last term is $5+7\\cdot(30-1)=208.$ Therefore, the requested sum is \\[5+12+19+26+\\cdots+208=\\frac{5+208}{2}\\cdot30=\\b... |
https://artofproblemsolving.com/wiki/index.php/2011_AMC_10A_Problems/Problem_1 | D | 28 | A cell phone plan costs $20$ dollars each month, plus $5$ cents per text message sent, plus $10$ cents for each minute used over $30$ hours. In January Michelle sent $100$ text messages and talked for $30.5$ hours. How much did she have to pay?
$\textbf{(A)}\ 24.00 \qquad \textbf{(B)}\ 24.50 \qquad \textbf{(C)}\ 25.50 ... | [
"The base price of Michelle's cell phone plan is $20$ dollars. \nIf she sent $100$ text messages and it costs $5$ cents per text, then she must have spent $500$ cents for texting, or $5$ dollars. She talked for $30.5$ hours, but $30.5-30$ will give us the amount of time exceeded past 30 hours. $30.5-30=.5$ hours $=... |
https://artofproblemsolving.com/wiki/index.php/2011_AMC_12A_Problems/Problem_1 | D | 28 | A cell phone plan costs $20$ dollars each month, plus $5$ cents per text message sent, plus $10$ cents for each minute used over $30$ hours. In January Michelle sent $100$ text messages and talked for $30.5$ hours. How much did she have to pay?
$\textbf{(A)}\ 24.00 \qquad \textbf{(B)}\ 24.50 \qquad \textbf{(C)}\ 25.50 ... | [
"The base price of Michelle's cell phone plan is $20$ dollars. \nIf she sent $100$ text messages and it costs $5$ cents per text, then she must have spent $500$ cents for texting, or $5$ dollars. She talked for $30.5$ hours, but $30.5-30$ will give us the amount of time exceeded past 30 hours. $30.5-30=.5$ hours $=... |
https://artofproblemsolving.com/wiki/index.php/2005_AMC_8_Problems/Problem_24 | B | 9 | A certain calculator has only two keys [+1] and [x2]. When you press one of the keys, the calculator automatically displays the result. For instance, if the calculator originally displayed "9" and you pressed [+1], it would display "10." If you then pressed [x2], it would display "20." Starting with the display "1," w... | [
"We can start at $200$ and work our way down to $1$ . We want to press the button that multiplies by $2$ the most, but since we are going down instead of up, we divide by $2$ instead. If we come across an odd number, then we will subtract that number by $1$ . Notice\nSince we've reached $1$ , it's clear that the an... |
https://artofproblemsolving.com/wiki/index.php/2001_AIME_II_Problems/Problem_8 | null | 429 | A certain function $f$ has the properties that $f(3x) = 3f(x)$ for all positive real values of $x$ , and that $f(x) = 1-|x-2|$ for $1\le x \le 3$ . Find the smallest $x$ for which $f(x) = f(2001)$ | [
"Iterating the condition $f(3x) = 3f(x)$ , we find that $f(x) = 3^kf\\left(\\frac{x}{3^k}\\right)$ for positive integers $k$ . We know the definition of $f(x)$ from $1 \\le x \\le 3$ , so we would like to express $f(2001) = 3^kf\\left(\\frac{2001}{3^k}\\right),\\ 1 \\le \\frac{2001}{3^k} \\le 3 \\Longrightarrow k =... |
https://artofproblemsolving.com/wiki/index.php/1993_AJHSME_Problems/Problem_25 | E | 12 | A checkerboard consists of one-inch squares. A square card, $1.5$ inches on a side, is placed on the board so that it covers part or all of the area of each of $n$ squares. The maximum possible value of $n$ is
$\text{(A)}\ 4\text{ or }5 \qquad \text{(B)}\ 6\text{ or }7\qquad \text{(C)}\ 8\text{ or }9 \qquad \text{(D)... | [
"Using the Pythagorean Theorem , the diagonal of the square $\\sqrt{(1.5)^2+(1.5)^2}=\\sqrt{4.5}>2$ . Because this is longer than $2$ ( length of the sides of two adjacent squares), the card can be placed like so, covering $12$ squares. $\\rightarrow \\boxed{12}$"
] |
https://artofproblemsolving.com/wiki/index.php/2000_AMC_12_Problems/Problem_16 | D | 555 | A checkerboard of $13$ rows and $17$ columns has a number written in each square, beginning in the upper left corner, so that the first row is numbered $1,2,\ldots,17$ , the second row $18,19,\ldots,34$ , and so on down the board. If the board is renumbered so that the left column, top to bottom, is $1,2,\ldots,13,$ , ... | [
"Index the rows with $i = 1, 2, 3, ..., 13$ Index the columns with $j = 1, 2, 3, ..., 17$\nFor the first row number the cells $1, 2, 3, ..., 17$ For the second, $18, 19, ..., 34$ and so on\nSo the number in row = $i$ and column = $j$ is $f(i, j) = 17(i-1) + j = 17i + j - 17$\nSimilarly, numbering the same cells col... |
https://artofproblemsolving.com/wiki/index.php/2024_AMC_8_Problems/Problem_17 | E | 32 | A chess king is said to attack all the squares one step away from it, horizontally, vertically, or diagonally. For instance, a king on the center square of a $3$ $3$ grid attacks all $8$ other squares, as shown below. Suppose a white king and a black king are placed on different squares of a $3$ $3$ grid so that they d... | [
"Corners have $5$ spots to go and there are $4$ corners, so $5 \\times 4=20$ .\nEdges have $3$ spots to go and there are $4$ sides so, $3 \\times 4=12$ . \nThat gives us $20+12=32$ spots to go into totally.\nSo $\\boxed{32}$ is the answer.\n~andliu766"
] |
https://artofproblemsolving.com/wiki/index.php/2019_AMC_10A_Problems/Problem_17 | D | 1,260 | A child builds towers using identically shaped cubes of different colors. How many different towers with a height $8$ cubes can the child build with $2$ red cubes, $3$ blue cubes, and $4$ green cubes? (One cube will be left out.)
$\textbf{(A) } 24 \qquad\textbf{(B) } 288 \qquad\textbf{(C) } 312 \qquad\textbf{(D) } 1,26... | [
"Arranging eight cubes is the same as arranging the nine cubes first, and then removing the last cube. In other words, there is a one-to-one correspondence between every arrangement of nine cubes, and every actual valid arrangement. Thus, we initially get $9!$ . However, we have overcounted, because the red cubes c... |
https://artofproblemsolving.com/wiki/index.php/1989_AHSME_Problems/Problem_22 | null | 29 | A child has a set of $96$ distinct blocks. Each block is one of $2$ materials (plastic, wood), $3$ sizes (small, medium, large), $4$ colors (blue, green, red, yellow), and $4$ shapes (circle, hexagon, square, triangle). How many blocks in the set differ from the 'plastic medium red circle' in exactly $2$ ways? (The 'wo... | [
"The process of choosing a block can be represented by a generating function. Each choice we make can match the 'plastic medium red circle' in one of its qualities $(1)$ or differ from it in $k$ different ways $(kx)$ . Choosing the material is represented by the factor $(1+1x)$ , choosing the size by the factor $(1... |
https://artofproblemsolving.com/wiki/index.php/1998_AJHSME_Problems/Problem_8 | C | 185 | A child's wading pool contains 200 gallons of water. If water evaporates at the rate of 0.5 gallons per day and no other water is added or removed, how many gallons of water will be in the pool after 30 days?
$\text{(A)}\ 140 \qquad \text{(B)}\ 170 \qquad \text{(C)}\ 185 \qquad \text{(D)}\ 198.5 \qquad \text{(E)}\ 199... | [
"$30$ days multiplied by $0.5$ gallons a day results in $15$ gallons of water loss.\nThe remaining water is $200-15=185=\\boxed{185}$"
] |
https://artofproblemsolving.com/wiki/index.php/2001_AMC_12_Problems/Problem_18 | null | 49 | A circle centered at $A$ with a radius of 1 and a circle centered at $B$ with a radius of 4 are externally tangent. A third circle is tangent to the first two and to one of their common external tangents as shown. What is the radius of the third circle?
[asy] unitsize(0.75cm); pair A=(0,1), B=(4,4); dot(A); dot(B); dra... | [
"\nIn the triangle $ABC$ we have $AB = 1+4 = 5$ and $BC=4-1 = 3$ , thus by the Pythagorean theorem we have $AC=4$\nLet $r$ be the radius of the small circle, and let $s$ be the perpendicular distance from $S$ to $\\overline{AC}$ . Moreover, the small circle is tangent to both other circles, hence we have $SA=1+r$ a... |
https://artofproblemsolving.com/wiki/index.php/2018_AMC_12B_Problems/Problem_4 | B | 50 | A circle has a chord of length $10$ , and the distance from the center of the circle to the chord is $5$ . What is the area of the circle?
$\textbf{(A) }25\pi \qquad \textbf{(B) }50\pi \qquad \textbf{(C) }75\pi \qquad \textbf{(D) }100\pi \qquad \textbf{(E) }125\pi \qquad$ | [
"Let $O$ be the center of the circle, $\\overline{AB}$ be the chord, and $M$ be the midpoint of $\\overline{AB},$ as shown below. Note that $\\overline{OM}\\perp\\overline{AB}.$ Since $OM=AM=BM=5,$ we conclude that $\\triangle OMA$ and $\\triangle OMB$ are congruent isosceles right triangles. It follows that $r=5\... |
https://artofproblemsolving.com/wiki/index.php/2017_AMC_12B_Problems/Problem_9 | A | 3 | A circle has center $(-10, -4)$ and has radius $13$ . Another circle has center $(3, 9)$ and radius $\sqrt{65}$ . The line passing through the two points of intersection of the two circles has equation $x+y=c$ . What is $c$
$\textbf{(A)}\ 3\qquad\textbf{(B)}\ 3\sqrt{3}\qquad\textbf{(C)}\ 4\sqrt{2}\qquad\textbf{(D)}\ 6\... | [
"The equations of the two circles are $(x+10)^2+(y+4)^2=169$ and $(x-3)^2+(y-9)^2=65$ . Rearrange them to $(x+10)^2+(y+4)^2-169=0$ and $(x-3)^2+(y-9)^2-65=0$ , respectively. Their intersection points are where these two equations gain equality. The two points lie on the line with the equation $(x+10)^2+(y+4)^2-169=... |
https://artofproblemsolving.com/wiki/index.php/2010_AMC_10B_Problems/Problem_6 | B | 25 | A circle is centered at $O$ $\overline{AB}$ is a diameter and $C$ is a point on the circle with $\angle COB = 50^\circ$ . What is the degree measure of $\angle CAB$
$\textbf{(A)}\ 20 \qquad \textbf{(B)}\ 25 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 50 \qquad \textbf{(E)}\ 65$ | [
"Assuming we do not already know an inscribed angle is always half of its central angle, we will try a different approach. Since $O$ is the center, $OC$ and $OA$ are radii and they are congruent. Thus, $\\triangle COA$ is an isosceles triangle. Also, note that $\\angle COB$ and $\\angle COA$ are supplementary, then... |
https://artofproblemsolving.com/wiki/index.php/2010_AMC_10B_Problems/Problem_6 | null | 25 | A circle is centered at $O$ $\overline{AB}$ is a diameter and $C$ is a point on the circle with $\angle COB = 50^\circ$ . What is the degree measure of $\angle CAB$
$\textbf{(A)}\ 20 \qquad \textbf{(B)}\ 25 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 50 \qquad \textbf{(E)}\ 65$ | [
"Note that $\\angle AOC = 180^\\circ - 50^\\circ = 130^\\circ$ . Because triangle $AOC$ is isosceles, $\\angle CAB = (180^\\circ - 130^\\circ)/2 = \\boxed{25}$"
] |
https://artofproblemsolving.com/wiki/index.php/2017_AIME_I_Problems/Problem_6 | null | 48 | A circle is circumscribed around an isosceles triangle whose two congruent angles have degree measure $x$ . Two points are chosen independently and uniformly at random on the circle, and a chord is drawn between them. The probability that the chord intersects the triangle is $\frac{14}{25}$ . Find the difference betwee... | [
"The probability that the chord doesn't intersect the triangle is $\\frac{11}{25}$ . The only way this can happen is if the two points are chosen on the same arc between two of the triangle vertices. The probability that a point is chosen on one of the arcs opposite one of the base angles is $\\frac{2x}{360}=\\frac... |
https://artofproblemsolving.com/wiki/index.php/2005_AMC_10B_Problems/Problem_7 | B | 8 | A circle is inscribed in a square, then a square is inscribed in this circle, and finally, a circle is inscribed in this square. What is the ratio of the area of the smallest circle to the area of the largest square?
$\textbf{(A) } \frac{\pi}{16} \qquad \textbf{(B) } \frac{\pi}{8} \qquad \textbf{(C) } \frac{3\pi}{16} \... | [
"Let the side of the largest square be $x$ . It follows that the diameter of the inscribed circle is also $x$ . Therefore, the diagonal of the square inscribed inscribed in the circle is $x$ . The side length of the smaller square is $\\dfrac{x}{\\sqrt{2}}=\\dfrac{x\\sqrt{2}}{2}$ . Similarly, the diameter of the sm... |
https://artofproblemsolving.com/wiki/index.php/1955_AHSME_Problems/Problem_9 | D | 3 | A circle is inscribed in a triangle with sides $8, 15$ , and $17$ . The radius of the circle is:
$\textbf{(A)}\ 6 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 7$ | [
"We know that $A = sr$ , where $A$ is the triangle's area, $s$ its semiperimeter, and $r$ its inradius. Since this particular triangle is a right triangle (which we can verify by the Pythagorean theorem), the area is half of $8*15 = 120$ , and the semiperimeter is half of $8 + 15 + 17 = 40$ . Therefore, the inradiu... |
https://artofproblemsolving.com/wiki/index.php/1992_AJHSME_Problems/Problem_5 | E | 5 | A circle of diameter $1$ is removed from a $2\times 3$ rectangle, as shown. Which whole number is closest to the area of the shaded region?
[asy] fill((0,0)--(0,2)--(3,2)--(3,0)--cycle,gray); draw((0,0)--(0,2)--(3,2)--(3,0)--cycle,linewidth(1)); fill(circle((1,5/4),1/2),white); draw(circle((1,5/4),1/2),linewidth(1)); ... | [
"The area of the shaded region is the area of the circle subtracted from the area of the rectangle.\nThe diameter of the circle is $1$ , so the radius is $1/2$ and the area is \\[(1/2)^2\\pi = \\pi /4.\\]\nThe rectangle obviously has area $2\\times 3= 6$ , so the area of the shaded region is $6-\\pi / 4$ .\nThis is... |
https://artofproblemsolving.com/wiki/index.php/2012_AMC_8_Problems/Problem_24 | A | 4 | A circle of radius $2$ is cut into four congruent arcs. The four arcs are joined to form the star figure shown. What is the ratio of the area of the star figure to the area of the original circle?
[asy] size(0,50); draw((-1,1)..(-2,2)..(-3,1)..(-2,0)..cycle); dot((-1,1)); dot((-2,2)); dot((-3,1)); dot((-2,0)); draw((1,... | [
"\nDraw a square around the star figure. The side length of this square is $4$ , because the side length is the diameter of the circle. The square forms $4$ -quarter circles around the star figure. This is the equivalent of one large circle with radius $2$ , meaning that the total area of the quarter circles is $4\... |
https://artofproblemsolving.com/wiki/index.php/2012_AMC_10B_Problems/Problem_2 | E | 200 | A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is 2:1. What is the area of the rectangle?
[asy] draw((0,0)--(0,10)--(20,10)--(20,0)--cycle); draw(circle((10,5),5));[/asy]
$\textbf{(A)}\ 50\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 125\qquad\textbf{(D)... | [
"Note that the diameter of the circle is equal to the shorter side of the rectangle. Since the radius is $5$ , the diameter is $2\\cdot 5 = 10$ .\nSince the sides of the rectangle are in a $2:1$ ratio, the longer side has length $2\\cdot 10 = 20$ .\nTherefore the area is $20\\cdot 10 = 200$ or $\\boxed{200}$"
] |
https://artofproblemsolving.com/wiki/index.php/2012_AMC_12B_Problems/Problem_2 | E | 200 | A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is 2:1. What is the area of the rectangle?
[asy] draw((0,0)--(0,10)--(20,10)--(20,0)--cycle); draw(circle((10,5),5)); [/asy]
$\textbf{(A)}\ 50\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 125\qquad\textbf{(D)}... | [
"If the radius is $5$ , then the width is $10$ , hence the length is $20$ $10\\times20= \\boxed{200}.$"
] |
https://artofproblemsolving.com/wiki/index.php/1979_AHSME_Problems/Problem_16 | E | 3 | A circle with area $A_1$ is contained in the interior of a larger circle with area $A_1+A_2$ . If the radius of the larger circle is $3$ ,
and if $A_1 , A_2, A_1 + A_2$ is an arithmetic progression, then the radius of the smaller circle is
$\textbf{(A) }\frac{\sqrt{3}}{2}\qquad \textbf{(B) }1\qquad \textbf{(C) }\frac{... | [
"Solution by e_power_pi_times_i\nThe area of the larger circle is $A_1 + A_2 = 9\\pi$ . Then $A_1 , 9\\pi-A_1 , 9\\pi$ are in an arithmetic progression. Thus $9\\pi-(9\\pi-A_1) = 9\\pi-A_1-A_1$ . This simplifies to $3A_1 = 9\\pi$ , or $A_1 = 3\\pi$ . The radius of the smaller circle is $\\boxed{3}$"
] |
https://artofproblemsolving.com/wiki/index.php/2009_AMC_12A_Problems/Problem_16 | null | 8 | A circle with center $C$ is tangent to the positive $x$ and $y$ -axes and externally tangent to the circle centered at $(3,0)$ with radius $1$ . What is the sum of all possible radii of the circle with center $C$
$\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ ... | [
"Let $r$ be the radius of our circle. For it to be tangent to the positive $x$ and $y$ axes, we must have $C=(r,r)$ . For the circle to be externally tangent to the circle centered at $(3,0)$ with radius $1$ , the distance between $C$ and $(3,0)$ must be exactly $r+1$\nBy the Pythagorean theorem the distance betwee... |
https://artofproblemsolving.com/wiki/index.php/2010_AMC_10B_Problems/Problem_19 | B | 6 | A circle with center $O$ has area $156\pi$ . Triangle $ABC$ is equilateral, $\overline{BC}$ is a chord on the circle, $OA = 4\sqrt{3}$ , and point $O$ is outside $\triangle ABC$ . What is the side length of $\triangle ABC$
$\textbf{(A)}\ 2\sqrt{3} \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 4\sqrt{3} \qquad \textbf{(D)... | [
"The formula for the area of a circle is $\\pi r^2$ so the radius of this circle is $\\sqrt{156}.$\nBecause $OA=4\\sqrt{3} < \\sqrt{156}, A$ must be in the interior of circle $O.$\nLet $s$ be the unknown value, the sidelength of the triangle, and let $X$ be the point on $BC$ where $OX \\perp BC.$ Since $\\triangle ... |
https://artofproblemsolving.com/wiki/index.php/1997_AHSME_Problems/Problem_19 | D | 2.37 | A circle with center $O$ is tangent to the coordinate axes and to the hypotenuse of the $30^\circ$ $60^\circ$ $90^\circ$ triangle $ABC$ as shown, where $AB=1$ . To the nearest hundredth, what is the radius of the circle?
[asy] defaultpen(linewidth(.8pt)); dotfactor=3; pair A = origin; pair B = (1,0); pair C = (0,sqrt(3... | [
"\nDraw radii $OE$ and $OD$ to the axes, and label the point of tangency to triangle $ABC$ point $F$ . Let the radius of the circle $O$ be $r$ . Square $OEAD$ has side length $r$\nBecause $BD$ and $BF$ are tangents from a common point $B$ $BD = BF$\n$AD = AB + BD$\n$r = 1 + BD$\n$r = 1 + BF$\nSimilarly, $CF = CE$... |
https://artofproblemsolving.com/wiki/index.php/1994_AIME_Problems/Problem_2 | null | 312 | A circle with diameter $\overline{PQ}$ of length 10 is internally tangent at $P$ to a circle of radius 20. Square $ABCD$ is constructed with $A$ and $B$ on the larger circle, $\overline{CD}$ tangent at $Q$ to the smaller circle, and the smaller circle outside $ABCD$ . The length of $\overline{AB}$ can be written in the... | [
"1994 AIME Problem 2 - Solution.png\nCall the center of the larger circle $O$ . Extend the diameter $\\overline{PQ}$ to the other side of the square (at point $E$ ), and draw $\\overline{AO}$ . We now have a right triangle , with hypotenuse of length $20$ . Since $OQ = OP - PQ = 20 - 10 = 10$ , we know that $OE = A... |
https://artofproblemsolving.com/wiki/index.php/2022_AMC_12A_Problems/Problem_25 | E | 17 | A circle with integer radius $r$ is centered at $(r, r)$ . Distinct line segments of length $c_i$ connect points $(0, a_i)$ to $(b_i, 0)$ for $1 \le i \le 14$ and are tangent to the circle, where $a_i$ $b_i$ , and $c_i$ are all positive integers and $c_1 \le c_2 \le \cdots \le c_{14}$ . What is the ratio $\frac{c_{14}}... | [
"Case 1: The tangent and the origin are on the opposite sides of the circle.\nIn this case, $a, b > 2r$\nWe can easily prove that \\[a + b - 2 r = c . \\hspace{1cm} (1)\\]\nRecall that $c = \\sqrt{a^2 + b^2}$\nTaking square of (1) and reorganizing all terms, (1) is converted as \\[\\left( a - 2 r \\right) \\left( b... |
https://artofproblemsolving.com/wiki/index.php/2022_AMC_12A_Problems/Problem_25 | null | 17 | A circle with integer radius $r$ is centered at $(r, r)$ . Distinct line segments of length $c_i$ connect points $(0, a_i)$ to $(b_i, 0)$ for $1 \le i \le 14$ and are tangent to the circle, where $a_i$ $b_i$ , and $c_i$ are all positive integers and $c_1 \le c_2 \le \cdots \le c_{14}$ . What is the ratio $\frac{c_{14}}... | [
"Suppose that with a pair $(a_i,b_i)$ the circle is an excircle. Then notice that the hypotenuse must be $(r-x)+(r-y)$ , so it must be the case that \\[a_i^2+b_i^2=(2r-a_i-b_i)^2.\\] Similarly, if with a pair $(a_i,b_i)$ the circle is an incircle, the hypotenuse must be $(x-r)+(y-r)$ , leading to the same equation.... |
https://artofproblemsolving.com/wiki/index.php/2011_AMC_8_Problems/Problem_25 | A | 12 | A circle with radius $1$ is inscribed in a square and circumscribed about another square as shown. Which fraction is closest to the ratio of the circle's shaded area to the area between the two squares?
[asy] filldraw((-1,-1)--(-1,1)--(1,1)--(1,-1)--cycle,gray,black); filldraw(Circle((0,0),1), mediumgray,black); filldr... | [
"The area of the smaller square is one half of the product of its diagonals. Note that the distance from a corner of the smaller square to the center is equivalent to the circle's radius so the diagonal is equal to the diameter: $2 \\cdot 2 \\cdot \\frac{1}{2}=2.$\nThe circle's shaded area is the area of the small... |
https://artofproblemsolving.com/wiki/index.php/2022_AIME_II_Problems/Problem_7 | null | 192 | A circle with radius $6$ is externally tangent to a circle with radius $24$ . Find the area of the triangular region bounded by the three common tangent lines of these two circles. | [
"\n$r_1 = O_1A = 24$ $r_2 = O_2B = 6$ $AG = BO_2 = r_2 = 6$ $O_1G = r_1 - r_2 = 24 - 6 = 18$ $O_1O_2 = r_1 + r_2 = 30$\n$\\triangle O_2BD \\sim \\triangle O_1GO_2$ $\\frac{O_2D}{O_1O_2} = \\frac{BO_2}{GO_1}$ $\\frac{O_2D}{30} = \\frac{6}{18}$ $O_2D = 10$\n$CD = O_2D + r_2 = 10 + 6 = 16$\n$EF = 2EC = EA + EB = AB = ... |
https://artofproblemsolving.com/wiki/index.php/1972_AHSME_Problems/Problem_28 | null | 32 | A circular disc with diameter $D$ is placed on an $8\times 8$ checkerboard with width $D$ so that the centers coincide. The number of checkerboard squares which are completely covered by the disc is
$\textbf{(A) }48\qquad \textbf{(B) }44\qquad \textbf{(C) }40\qquad \textbf{(D) }36\qquad \textbf{(E) }32$ | [
"Consider the upper right half of the grid, which consists of a $4\\times4$ section of the checkerboard and a quarter-circle of radius $4$ . We can draw this as a coordinate grid and shade in the complete squares. There are $8$ squares in the upper right corner, so there are $8 \\cdot 4 = \\boxed{32}$ whole squares... |
https://artofproblemsolving.com/wiki/index.php/2008_AMC_12B_Problems/Problem_5 | C | 9 | A class collects $50$ dollars to buy flowers for a classmate who is in the hospital. Roses cost $3$ dollars each, and carnations cost $2$ dollars each. No other flowers are to be used. How many different bouquets could be purchased for exactly $50$ dollars?
$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 9... | [
"The class could send $25$ carnations and no roses, $22$ carnations and $2$ roses, $19$ carnations and $4$ roses, and so on, down to $1$ carnation and $16$ roses. There are 9 total possibilities (from 0 to 16 roses, incrementing by 2 at each step), $\\Rightarrow \\boxed{9}$"
] |
https://artofproblemsolving.com/wiki/index.php/2008_AMC_10B_Problems/Problem_8 | C | 9 | A class collects 50 dollars to buy flowers for a classmate who is in the hospital. Roses cost 3 dollars each, and carnations cost 2 dollars each. No other flowers are to be used. How many different bouquets could be purchased for exactly 50 dollars?
$\mathrm{(A)}\ 1 \qquad \mathrm{(B)}\ 7 \qquad \mathrm{(C)}\ 9 \qquad ... | [
"The cost of a rose is odd, hence we need an even number of roses. Let there be $2r$ roses for some $r\\geq 0$ . Then we have $50-3\\cdot 2r = 50-6r$ dollars left. We can always reach the sum exactly $50$ by buying $(50-6r)/2 = 25-3r$ carnations. Of course, the number of roses must be such that the number of carnat... |
https://artofproblemsolving.com/wiki/index.php/2003_AMC_10B_Problems/Problem_22 | B | 9 | A clock chimes once at $30$ minutes past each hour and chimes on the hour according to the hour. For example, at $1 \text{PM}$ there is one chime and at noon and midnight there are twelve chimes. Starting at $11:15 \text{AM}$ on $\text{February 26, 2003},$ on what date will the $2003^{\text{rd}}$ chime occur?
$\textbf{... | [
"First, find how many chimes will have already happened before midnight (the beginning of the day) of $\\text{February 27, 2003}.$ $13$ half-hours have passed, and the number of chimes according to the hour is $1+2+3+\\cdots+12.$ The total number of chimes is $13+78=91.$\nEvery day, there will be $24$ half-hours an... |
https://artofproblemsolving.com/wiki/index.php/2020_AIME_I_Problems/Problem_7 | null | 81 | A club consisting of $11$ men and $12$ women needs to choose a committee from among its members so that the number of women on the committee is one more than the number of men on the committee. The committee could have as few as $1$ member or as many as $23$ members. Let $N$ be the number of such committees that can be... | [
"Let $k$ be the number of women selected. Then, the number of men not selected is $11-(k-1)=12-k$ .\nNote that the sum of the number of women selected and the number of men not selected is constant at $12$ . Each combination of women selected and men not selected corresponds to a committee selection. Since choosing... |
https://artofproblemsolving.com/wiki/index.php/2009_AIME_I_Problems/Problem_3 | null | 11 | A coin that comes up heads with probability $p > 0$ and tails with probability $1 - p > 0$ independently on each flip is flipped $8$ times. Suppose that the probability of three heads and five tails is equal to $\frac {1}{25}$ of the probability of five heads and three tails. Let $p = \frac {m}{n}$ , where $m$ and $n$ ... | [
"The probability of three heads and five tails is $\\binom {8}{3}p^3(1-p)^5$ and the probability of five heads and three tails is $\\binom {8}{3}p^5(1-p)^3$\n\\begin{align*} 25\\binom {8}{3}p^3(1-p)^5&=\\binom {8}{3}p^5(1-p)^3 \\\\ 25(1-p)^2&=p^2 \\\\ 25p^2-50p+25&=p^2 \\\\ 24p^2-50p+25&=0 \\\\ p&=\\frac {5}{6}\\en... |
https://artofproblemsolving.com/wiki/index.php/2006_AIME_I_Problems/Problem_11 | null | 458 | A collection of 8 cubes consists of one cube with edge length $k$ for each integer $k, 1 \le k \le 8.$ A tower is to be built using all 8 cubes according to the rules:
Let $T$ be the number of different towers than can be constructed. What is the remainder when $T$ is divided by 1000? | [
"We proceed recursively . Suppose we can build $T_m$ towers using blocks of size $1, 2, \\ldots, m$ . How many towers can we build using blocks of size $1, 2, \\ldots, m, m + 1$ ? If we remove the block of size $m + 1$ from such a tower (keeping all other blocks in order), we get a valid tower using blocks $1, 2... |
https://artofproblemsolving.com/wiki/index.php/2001_AMC_8_Problems/Problem_10 | A | 20 | A collector offers to buy state quarters for 2000% of their face value. At that rate how much will Bryden get for his four state quarters?
$\text{(A)}\ 20\text{ dollars} \qquad \text{(B)}\ 50\text{ dollars} \qquad \text{(C)}\ 200\text{ dollars} \qquad \text{(D)}\ 500\text{ dollars} \qquad \text{(E)}\ 2000\text{ dollars... | [
"$2000\\%$ is equivalent to $20\\times100\\%$ . Therefore, $2000\\%$ of a number is the same as $20$ times that number. $4$ quarters is $1$ dollar, so Bryden will get $20\\times1={20}$ dollars, $\\boxed{20}$",
"Since $2000\\%$ is just $\\frac{2000}{100}$ , we can multiply that by $100$ , because four quarters is ... |
https://artofproblemsolving.com/wiki/index.php/2007_AMC_10B_Problems/Problem_3 | B | 24 | A college student drove his compact car $120$ miles home for the weekend and averaged $30$ miles per gallon. On the return trip the student drove his parents' SUV and averaged only $20$ miles per gallon. What was the average gas mileage, in miles per gallon, for the round trip?
$\textbf{(A) } 22 \qquad\textbf{(B) } 24 ... | [
"The trip was $240$ miles long and took $\\dfrac{120}{30}+\\dfrac{120}{20}=4+6=10$ gallons. Therefore, the average mileage was $\\dfrac{240}{10}= \\boxed{24}$",
"Alternatively, we can use the harmonic mean to get $\\frac{2}{\\frac{1}{20} + \\frac{1}{30}} = \\frac{2}{\\frac{1}{12}} = \\boxed{24}$"
] |
https://artofproblemsolving.com/wiki/index.php/2007_AMC_12B_Problems/Problem_2 | B | 24 | A college student drove his compact car $120$ miles home for the weekend and averaged $30$ miles per gallon. On the return trip the student drove his parents' SUV and averaged only $20$ miles per gallon. What was the average gas mileage, in miles per gallon, for the round trip?
$\textbf{(A) } 22 \qquad\textbf{(B) } 24 ... | [
"The trip was $240$ miles long and took $\\dfrac{120}{30}+\\dfrac{120}{20}=4+6=10$ gallons. Therefore, the average mileage was $\\dfrac{240}{10}= \\boxed{24}$",
"Alternatively, we can use the harmonic mean to get $\\frac{2}{\\frac{1}{20} + \\frac{1}{30}} = \\frac{2}{\\frac{1}{12}} = \\boxed{24}$"
] |
https://artofproblemsolving.com/wiki/index.php/2004_AMC_10A_Problems/Problem_11 | C | 36 | A company sells peanut butter in cylindrical jars. Marketing research suggests that using wider jars will increase sales. If the diameter of the jars is increased by $25\%$ without altering the volume , by what percent must the height be decreased?
$\mathrm{(A) \ } 10 \qquad \mathrm{(B) \ } 25 \qquad \mathrm{(C) \ } ... | [
"When the diameter is increased by $25\\%$ , it is increased by $\\dfrac{5}{4}$ , so the area of the base is increased by $\\left(\\dfrac54\\right)^2=\\dfrac{25}{16}$\nTo keep the volume the same, the height must be $\\dfrac{1}{\\frac{25}{16}}=\\dfrac{16}{25}$ of the original height, which is a $36\\%$ reduction. $... |
https://artofproblemsolving.com/wiki/index.php/2004_AMC_12A_Problems/Problem_9 | C | 36 | A company sells peanut butter in cylindrical jars. Marketing research suggests that using wider jars will increase sales. If the diameter of the jars is increased by $25\%$ without altering the volume , by what percent must the height be decreased?
$\mathrm{(A) \ } 10 \qquad \mathrm{(B) \ } 25 \qquad \mathrm{(C) \ } ... | [
"When the diameter is increased by $25\\%$ , it is increased by $\\dfrac{5}{4}$ , so the area of the base is increased by $\\left(\\dfrac54\\right)^2=\\dfrac{25}{16}$\nTo keep the volume the same, the height must be $\\dfrac{1}{\\frac{25}{16}}=\\dfrac{16}{25}$ of the original height, which is a $36\\%$ reduction. $... |
https://artofproblemsolving.com/wiki/index.php/1987_AJHSME_Problems/Problem_14 | B | 36 | A computer can do $10,000$ additions per second. How many additions can it do in one hour?
$\text{(A)}\ 6\text{ million} \qquad \text{(B)}\ 36\text{ million} \qquad \text{(C)}\ 60\text{ million} \qquad \text{(D)}\ 216\text{ million} \qquad \text{(E)}\ 360\text{ million}$ | [
"There are $3600$ seconds per hour, so we have \\begin{align*} \\frac{3600\\text{ seconds}}{\\text{hour}}\\cdot \\frac{10,000\\text{ additions}}{\\text{second}} &= \\frac{36,000,000\\text{ additions}}{\\text{hour}} \\\\ &= 36\\text{ million additions per hour} \\end{align*}\n$\\boxed{36}$"
] |
https://artofproblemsolving.com/wiki/index.php/2000_AIME_I_Problems/Problem_8 | null | 52 | A container in the shape of a right circular cone is $12$ inches tall and its base has a $5$ -inch radius . The liquid that is sealed inside is $9$ inches deep when the cone is held with its point down and its base horizontal. When the liquid is held with its point up and its base horizontal, the height of the liquid i... | [
"The scale factor is uniform in all dimensions, so the volume of the liquid is $\\left(\\frac{3}{4}\\right)^{3}$ of the container. The remaining section of the volume is $\\frac{1-\\left(\\frac{3}{4}\\right)^{3}}{1}$ of the volume, and therefore $\\frac{\\left(1-\\left(\\frac{3}{4}\\right)^{3}\\right)^{1/3}}{1}$ of... |
https://artofproblemsolving.com/wiki/index.php/2009_AMC_12B_Problems/Problem_20 | C | 300 | A convex polyhedron $Q$ has vertices $V_1,V_2,\ldots,V_n$ , and $100$ edges. The polyhedron is cut by planes $P_1,P_2,\ldots,P_n$ in such a way that plane $P_k$ cuts only those edges that meet at vertex $V_k$ . In addition, no two planes intersect inside or on $Q$ . The cuts produce $n$ pyramids and a new polyhedron $R... | [
"Euler's Polyhedron Formula applied to $Q$ gives $n - 100 + F = 2$ , where F is the number of faces of $Q$ . Each edge of $Q$ is cut by two planes, so $R$ has $200$ vertices. Each cut by a plane $P_k$ creates an additional face on $R$ , so Euler's Polyhedron Formula applied to $R$ gives $200 - E + (F+n) = 2$ , wh... |
https://artofproblemsolving.com/wiki/index.php/2009_AMC_12B_Problems/Problem_20 | null | 300 | A convex polyhedron $Q$ has vertices $V_1,V_2,\ldots,V_n$ , and $100$ edges. The polyhedron is cut by planes $P_1,P_2,\ldots,P_n$ in such a way that plane $P_k$ cuts only those edges that meet at vertex $V_k$ . In addition, no two planes intersect inside or on $Q$ . The cuts produce $n$ pyramids and a new polyhedron $R... | [
"Each edge of $Q$ is cut by two planes, so $R$ has $200$ vertices. Three edges of $R$ meet at each vertex, so $R$ has $\\frac 12 \\cdot 3 \\cdot 200 = \\boxed{300}$ edges.",
"At each vertex, as many new edges are created by this process as there are original edges meeting at that vertex. Thus the total number of... |
https://artofproblemsolving.com/wiki/index.php/2021_AIME_II_Problems/Problem_12 | null | 47 | A convex quadrilateral has area $30$ and side lengths $5, 6, 9,$ and $7,$ in that order. Denote by $\theta$ the measure of the acute angle formed by the diagonals of the quadrilateral. Then $\tan \theta$ can be written in the form $\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ | [
"Since we are asked to find $\\tan \\theta$ , we can find $\\sin \\theta$ and $\\cos \\theta$ separately and use their values to get $\\tan \\theta$ . We can start by drawing a diagram. Let the vertices of the quadrilateral be $A$ $B$ $C$ , and $D$ . Let $AB = 5$ $BC = 6$ $CD = 9$ , and $DA = 7$ . Let $AX = a$ $BX ... |
https://artofproblemsolving.com/wiki/index.php/2002_AMC_8_Problems/Problem_23 | B | 49 | A corner of a tiled floor is shown. If the entire floor is tiled in this way and each of the four corners looks like this one, then what fraction of the tiled floor is made of darker tiles?
[asy] /* AMC8 2002 #23 Problem */ fill((0,2)--(1,3)--(2,3)--(2,4)--(3,5)--(4,4)--(4,3)--(5,3)--(6,2)--(5,1)--(4,1)--(4,0)--(2,0)--... | [
"The same pattern is repeated for every $6 \\times 6$ tile. Looking closer, there is also symmetry of the top $3 \\times 3$ square, so the fraction of the entire floor in dark tiles is the same as the fraction in the square. Counting the tiles, there are $4$ dark tiles, and $9$ total tiles, giving a fraction of $\\... |
https://artofproblemsolving.com/wiki/index.php/1987_AHSME_Problems/Problem_16 | D | 108 | A cryptographer devises the following method for encoding positive integers. First, the integer is expressed in base $5$ .
Second, a 1-to-1 correspondence is established between the digits that appear in the expressions in base $5$ and the elements of the set $\{V, W, X, Y, Z\}$ . Using this correspondence, the crypto... | [
"Since $VYX + 1 = VVW$ , i.e. adding $1$ causes the \"fives\" digit to change, we must have $X = 4$ and $W = 0$ . Now since $VYZ + 1 = VYX$ , we have $X = Z + 1 \\implies Z = 4 - 1 = 3$ . Finally, note that in $VYX + 1 = VVW$ , adding $1$ will cause the \"fives\" digit to change by $1$ if it changes at all, so $V =... |
https://artofproblemsolving.com/wiki/index.php/2000_AMC_8_Problems/Problem_22 | C | 17 | A cube has edge length $2$ . Suppose that we glue a cube of edge length $1$ on top of the big cube so that one of its faces rests entirely on the top face of the larger cube. The percent increase in the surface area (sides, top, and bottom) from the original cube to the new solid formed is closest to
[asy] draw((0,0)--... | [
"The original cube has $6$ faces, each with an area of $2\\cdot 2 = 4$ square units. Thus the original figure had a total surface area of $24$ square units.\nThe new figure has the original surface, with $6$ new faces that each have an area of $1$ square unit, for a total surface area of of $6$ additional square u... |
https://artofproblemsolving.com/wiki/index.php/1997_AJHSME_Problems/Problem_17 | E | 16 | A cube has eight vertices (corners) and twelve edges. A segment, such as $x$ , which joins two vertices not joined by an edge is called a diagonal. Segment $y$ is also a diagonal. How many diagonals does a cube have?
[asy] draw((0,3)--(0,0)--(3,0)--(5.5,1)--(5.5,4)--(3,3)--(0,3)--(2.5,4)--(5.5,4)); draw((3,0)--(3,3)... | [
"On each face, there are $2$ diagonals like $x$ . There are $6$ faces on a cube. Thus, there are $2\\times 6 = 12$ diagonals that are \"x-like\".\nEvery \"y-like\" diagonal must connect the bottom of the cube to the top of the cube. Thus, for each of the $4$ bottom vertices of the cube, there is a different \"y-... |
https://artofproblemsolving.com/wiki/index.php/2021_Fall_AMC_10B_Problems/Problem_24 | A | 7 | A cube is constructed from $4$ white unit cubes and $4$ blue unit cubes. How many different ways are there to construct the $2 \times 2 \times 2$ cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.)
$\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C... | [
"This problem is about the relationships between the white unit cubes and the blue unit cubes, which can be solved by Graph Theory . We use a Planar Graph to represent the larger cube. Each vertex of the planar graph represents a unit cube. Each edge of the planar graph represents a shared face between $2$ neighbor... |
https://artofproblemsolving.com/wiki/index.php/2021_Fall_AMC_10B_Problems/Problem_24 | null | 7 | A cube is constructed from $4$ white unit cubes and $4$ blue unit cubes. How many different ways are there to construct the $2 \times 2 \times 2$ cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.)
$\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C... | [
"Burnside lemma is used to counting number of orbit where the element on the same orbit can be achieved by the defined operator, naming rotation, reflection and etc.\nThe fact for Burnside lemma are\n1. the sum of stablizer on the same orbit equals to the # of operators;\n2. the sum of stablizer can be counted as $... |
https://artofproblemsolving.com/wiki/index.php/2021_Fall_AMC_12B_Problems/Problem_20 | A | 7 | A cube is constructed from $4$ white unit cubes and $4$ blue unit cubes. How many different ways are there to construct the $2 \times 2 \times 2$ cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.)
$\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C... | [
"This problem is about the relationships between the white unit cubes and the blue unit cubes, which can be solved by Graph Theory . We use a Planar Graph to represent the larger cube. Each vertex of the planar graph represents a unit cube. Each edge of the planar graph represents a shared face between $2$ neighbor... |
https://artofproblemsolving.com/wiki/index.php/2021_Fall_AMC_12B_Problems/Problem_20 | null | 7 | A cube is constructed from $4$ white unit cubes and $4$ blue unit cubes. How many different ways are there to construct the $2 \times 2 \times 2$ cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.)
$\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C... | [
"Burnside lemma is used to counting number of orbit where the element on the same orbit can be achieved by the defined operator, naming rotation, reflection and etc.\nThe fact for Burnside lemma are\n1. the sum of stablizer on the same orbit equals to the # of operators;\n2. the sum of stablizer can be counted as $... |
https://artofproblemsolving.com/wiki/index.php/1987_AHSME_Problems/Problem_27 | B | 6 | A cube of cheese $C=\{(x, y, z)| 0 \le x, y, z \le 1\}$ is cut along the planes $x=y, y=z$ and $z=x$ . How many pieces are there?
(No cheese is moved until all three cuts are made.)
$\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$ | [
"The cut $x = y$ separates the cube into points with $x < y$ and points with $x > y$ , and analogous results apply for the other cuts. Thus, which piece a particular point is in depends only on the relative sizes of its coordinates $x$ $y$ , and $z$ - for example, all points with the ordering $x < y < z$ are in the... |
https://artofproblemsolving.com/wiki/index.php/1991_AJHSME_Problems/Problem_24 | E | 20 | A cube of edge $3$ cm is cut into $N$ smaller cubes, not all the same size. If the edge of each of the smaller cubes is a whole number of centimeters, then $N=$
$\text{(A)}\ 4 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 16 \qquad \text{(E)}\ 20$ | [
"If none of the cubes have edge length $2$ , then all of the cubes have edge length $1$ , meaning they all are the same size, a contradiction.\nIt is clearly impossible to split a cube of edge $3$ into two or more cubes of edge $2$ with extra unit cubes, so there is one $2\\times 2\\times 2$ cube and $3^3-2^3=19$ u... |
https://artofproblemsolving.com/wiki/index.php/2011_AIME_I_Problems/Problem_13 | null | 330 | A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labeled $A$ . The three vertices adjacent to vertex $A$ are at heights 10, 11, and 12 above the plane. The distance from vertex $A$ to the plane can be expressed as $\frac{r-\sqrt{s}}{t}$ , where $r$ $s$ , and $t$ are positive int... | [
"Set the cube at the origin with the three vertices along the axes and the plane equal to $ax+by+cz+d=0$ , where $a^2+b^2+c^2=1$ . The distance from a point $(X,Y,Z)$ to a plane with equation $Ax+By+Cz+D=0$ is \\[\\frac{AX+BY+CZ+D}{\\sqrt{A^2+B^2+C^2}},\\] so the (directed) distance from any point $(x,y,z)$ to the... |
https://artofproblemsolving.com/wiki/index.php/2023_AIME_II_Problems/Problem_14 | null | 751 | A cube-shaped container has vertices $A,$ $B,$ $C,$ and $D,$ where $\overline{AB}$ and $\overline{CD}$ are parallel edges of the cube, and $\overline{AC}$ and $\overline{BD}$ are diagonals of faces of the cube, as shown. Vertex $A$ of the cube is set on a horizontal plane $\mathcal{P}$ so that the plane of the rectangl... | [
"\nLet's first view the cube from a direction perpendicular to $ABDC$ , as illustrated above. Let $x$ be the cube's side length. Since $\\triangle CHA \\sim \\triangle AGB$ , we have \\[\\frac{CA}{CH} = \\frac{AB}{AG}.\\] We know $AB = x$ $AG = \\sqrt{x^2-2^2}$ $AC = \\sqrt{2}x$ $CH = 8$ . Plug them into the above ... |
https://artofproblemsolving.com/wiki/index.php/2022_AMC_8_Problems/Problem_9 | B | 86 | A cup of boiling water ( $212^{\circ}\text{F}$ ) is placed to cool in a room whose temperature remains constant at $68^{\circ}\text{F}$ . Suppose the difference between the water temperature and the room temperature is halved every $5$ minutes. What is the water temperature, in degrees Fahrenheit, after $15$ minutes?
$... | [
"Initially, the difference between the water temperature and the room temperature is $212-68=144$ degrees Fahrenheit.\nAfter $5$ minutes, the difference between the temperatures is $144\\div2=72$ degrees Fahrenheit.\nAfter $10$ minutes, the difference between the temperatures is $72\\div2=36$ degrees Fahrenheit.\nA... |
https://artofproblemsolving.com/wiki/index.php/2015_AIME_II_Problems/Problem_9 | null | 384 | A cylindrical barrel with radius $4$ feet and height $10$ feet is full of water. A solid cube with side length $8$ feet is set into the barrel so that the diagonal of the cube is vertical. The volume of water thus displaced is $v$ cubic feet. Find $v^2$
[asy] import three; import solids; size(5cm); currentprojection=or... | [
"Our aim is to find the volume of the part of the cube submerged in the cylinder. \nIn the problem, since three edges emanate from each vertex, the boundary of the cylinder touches the cube at three points. Because the space diagonal of the cube is vertical, by the symmetry of the cube, the three points form an eq... |
https://artofproblemsolving.com/wiki/index.php/2003_AIME_II_Problems/Problem_5 | null | 216 | A cylindrical log has diameter $12$ inches. A wedge is cut from the log by making two planar cuts that go entirely through the log. The first is perpendicular to the axis of the cylinder, and the plane of the second cut forms a $45^\circ$ angle with the plane of the first cut. The intersection of these two planes has e... | [
"The volume of the wedge is half the volume of a cylinder with height $12$ and radius $6$ . (Imagine taking another identical wedge and sticking it to the existing one). Thus, $V=\\dfrac{6^2\\cdot 12\\pi}{2}=216\\pi$ , so $n=\\boxed{216}$"
] |
https://artofproblemsolving.com/wiki/index.php/2022_AMC_10A_Problems/Problem_8 | D | 36 | A data set consists of $6$ (not distinct) positive integers: $1$ $7$ $5$ $2$ $5$ , and $X$ . The average (arithmetic mean) of the $6$ numbers equals a value in the data set. What is the sum of all possible values of $X$
$\textbf{(A) } 10 \qquad \textbf{(B) } 26 \qquad \textbf{(C) } 32 \qquad \textbf{(D) } 36 \qquad \te... | [
"First, note that $1+7+5+2+5=20$ . There are $3$ possible cases:\nCase 1: the mean is $5$\n$X = 5 \\cdot 6 - 20 = 10$\nCase 2: the mean is $7$\n$X = 7 \\cdot 6 - 20 = 22$\nCase 3: the mean is $X$\n$X= \\frac{20+X}{6} \\Rightarrow X=4$\nTherefore, the answer is $10+22+4=\\boxed{36}$"
] |
https://artofproblemsolving.com/wiki/index.php/2022_AMC_12A_Problems/Problem_6 | D | 36 | A data set consists of $6$ (not distinct) positive integers: $1$ $7$ $5$ $2$ $5$ , and $X$ . The average (arithmetic mean) of the $6$ numbers equals a value in the data set. What is the sum of all possible values of $X$
$\textbf{(A) } 10 \qquad \textbf{(B) } 26 \qquad \textbf{(C) } 32 \qquad \textbf{(D) } 36 \qquad \te... | [
"First, note that $1+7+5+2+5=20$ . There are $3$ possible cases:\nCase 1: the mean is $5$\n$X = 5 \\cdot 6 - 20 = 10$\nCase 2: the mean is $7$\n$X = 7 \\cdot 6 - 20 = 22$\nCase 3: the mean is $X$\n$X= \\frac{20+X}{6} \\Rightarrow X=4$\nTherefore, the answer is $10+22+4=\\boxed{36}$"
] |
https://artofproblemsolving.com/wiki/index.php/2021_AMC_12A_Problems/Problem_6 | C | 12 | A deck of cards has only red cards and black cards. The probability of a randomly chosen card being red is $\frac13$ . When $4$ black cards are added to the deck, the probability of choosing red becomes $\frac14$ . How many cards were in the deck originally?
$\textbf{(A) }6 \qquad \textbf{(B) }9 \qquad \textbf{(C) }12 ... | [
"If the probability of choosing a red card is $\\frac{1}{3}$ , the red and black cards are in ratio $1:2$ . This means at the beginning there are $x$ red cards and $2x$ black cards.\nAfter $4$ black cards are added, there are $2x+4$ black cards. This time, the probability of choosing a red card is $\\frac{1}{4}$ so... |
https://artofproblemsolving.com/wiki/index.php/2000_AIME_II_Problems/Problem_3 | null | 758 | A deck of forty cards consists of four $1$ 's, four $2$ 's,..., and four $10$ 's. A matching pair (two cards with the same number) is removed from the deck. Given that these cards are not returned to the deck, let $m/n$ be the probability that two randomly selected cards also form a pair, where $m$ and $n$ are relati... | [
"There are ${38 \\choose 2} = 703$ ways we can draw two cards from the reduced deck. The two cards will form a pair if both are one of the nine numbers that were not removed, which can happen in $9{4 \\choose 2} = 54$ ways, or if the two cards are the remaining two cards of the number that was removed, which can ha... |
https://artofproblemsolving.com/wiki/index.php/2010_AMC_8_Problems/Problem_18 | C | 6 | A decorative window is made up of a rectangle with semicircles at either end. The ratio of $AD$ to $AB$ is $3:2$ . And $AB$ is 30 inches. What is the ratio of the area of the rectangle to the combined area of the semicircles?
[asy] import graph; size(5cm); real lsf=0; pen dps=linewidth(0.7)+fontsize(8); defaultpen(dps)... | [
"We can set a proportion:\n\\[\\dfrac{AD}{AB}=\\dfrac{3}{2}\\]\nWe substitute $AB$ with 30 and solve for $AD$\n\\[\\dfrac{AD}{30}=\\dfrac{3}{2}\\]\n\\[AD=45\\]\nWe calculate the combined area of semicircle by putting together semicircle $AB$ and $CD$ to get a circle with radius $15$ . Thus, the area is $225\\pi$ . ... |
https://artofproblemsolving.com/wiki/index.php/2012_AMC_10B_Problems/Problem_11 | A | 729 | A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different dessert menus for the week are possible?
$\textb... | [
"Desserts must be chosen for $7$ days: Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday.\nThere are $3$ choices for dessert on Saturday: pie, ice cream, or pudding, as there must be cake on Friday and the same dessert may not be served two days in a row. Likewise, there are $3$ choices for dessert on ... |
https://artofproblemsolving.com/wiki/index.php/2012_AMC_10B_Problems/Problem_11 | null | 729 | A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different dessert menus for the week are possible?
$\textb... | [
"There are $4 \\cdot 3^6$ ways for the desserts to be chosen. By symmetry, any of the desserts that are chosen on Friday share $\\frac{1}{4}$ of the total arrangements. Therefore our answer is $\\frac{4\\cdot3^6}{4} = 3^6 = \\boxed{729}.$"
] |
https://artofproblemsolving.com/wiki/index.php/2012_AMC_12B_Problems/Problem_8 | A | 729 | A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different dessert menus for the week are possible?
$\textb... | [
"We can count the number of possible foods for each day and then multiply to enumerate the number of combinations.\nOn Friday, we have one possibility: cake.\nOn Saturday, we have three possibilities: pie, ice cream, or pudding. This is the end of the week.\nOn Thursday, we have three possibilities: pie, ice cream,... |
https://artofproblemsolving.com/wiki/index.php/2023_AMC_10A_Problems/Problem_9 | E | 9 | A digital display shows the current date as an $8$ -digit integer consisting of a $4$ -digit year, followed by a $2$ -digit month, followed by a $2$ -digit date within the month. For example, Arbor Day this year is displayed as 20230428. For how many dates in $2023$ will each digit appear an even number of times in the... | [
"Do careful casework by each month. In the month and the date, we need a $0$ , a $3$ , and two digits repeated (which has to be $1$ and $2$ after consideration). After the casework, we get $\\boxed{9}$ .\nFor curious readers, the numbers (in chronological order) are:",
"There is one $3$ , so we need one more (thr... |
https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_7 | E | 9 | A digital display shows the current date as an $8$ -digit integer consisting of a $4$ -digit year, followed by a $2$ -digit month, followed by a $2$ -digit date within the month. For example, Arbor Day this year is displayed as 20230428. For how many dates in $2023$ will each digit appear an even number of times in the... | [
"Do careful casework by each month. In the month and the date, we need a $0$ , a $3$ , and two digits repeated (which has to be $1$ and $2$ after consideration). After the casework, we get $\\boxed{9}$ .\nFor curious readers, the numbers (in chronological order) are:",
"There is one $3$ , so we need one more (thr... |
https://artofproblemsolving.com/wiki/index.php/2006_AMC_10A_Problems/Problem_4 | E | 23 | A digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display?
$\textbf{(A)}\ 17\qquad\textbf{(B)}\ 19\qquad\textbf{(C)}\ 21\qquad\textbf{(D)}\ 22\qquad\textbf{(E)}\ 23$ | [
"From the greedy algorithm , we have $9$ in the hours section and $59$ in the minutes section. $9+5+9=\\boxed{23}$",
"With a matrix, we can see $\\begin{bmatrix} 1+2&9&6&3\\\\ 1+1&8&5&2\\\\ 1+0&7&4&1 \\end{bmatrix}$ The largest single digit sum we can get is $9$ .\nFor the minutes digits, we can combine the large... |
https://artofproblemsolving.com/wiki/index.php/2006_AMC_12A_Problems/Problem_4 | E | 23 | A digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display?
$\textbf{(A)}\ 17\qquad\textbf{(B)}\ 19\qquad\textbf{(C)}\ 21\qquad\textbf{(D)}\ 22\qquad\textbf{(E)}\ 23$ | [
"From the greedy algorithm , we have $9$ in the hours section and $59$ in the minutes section. $9+5+9=\\boxed{23}$",
"With a matrix, we can see $\\begin{bmatrix} 1+2&9&6&3\\\\ 1+1&8&5&2\\\\ 1+0&7&4&1 \\end{bmatrix}$ The largest single digit sum we can get is $9$ .\nFor the minutes digits, we can combine the large... |
https://artofproblemsolving.com/wiki/index.php/2016_AMC_10B_Problems/Problem_20 | null | 13 | A dilation of the plane—that is, a size transformation with a positive scale factor—sends the circle of radius $2$ centered at $A(2,2)$ to the circle of radius $3$ centered at $A’(5,6)$ . What distance does the origin $O(0,0)$ , move under this transformation?
$\textbf{(A)}\ 0\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ \s... | [
"The center of dilation must lie on the line $A A'$ , which can be expressed as $y = \\dfrac{4x}{3} - \\dfrac{2}{3}$ . Note that the center of dilation must have an $x$ -coordinate less than $2$ ; if the $x$ -coordinate were otherwise, then the circle under the transformation would not have an increased $x$ -coordi... |
https://artofproblemsolving.com/wiki/index.php/2021_Fall_AMC_10A_Problems/Problem_19 | A | 10 | A disk of radius $1$ rolls all the way around the inside of a square of side length $s>4$ and sweeps out a region of area $A$ . A second disk of radius $1$ rolls all the way around the outside of the same square and sweeps out a region of area $2A$ . The value of $s$ can be written as $a+\frac{b\pi}{c}$ , where $a,b$ ,... | [
"The side length of the inner square traced out by the disk with radius $1$ is $s-4.$ However, there is a piece at each corner (bounded by two line segments and one $90^\\circ$ arc) where the disk never sweeps out. The combined area of these four pieces is $(1+1)^2-\\pi\\cdot1^2=4-\\pi.$ As a result, we have \\[A=s... |
https://artofproblemsolving.com/wiki/index.php/2014_AIME_I_Problems/Problem_10 | null | 58 | A disk with radius $1$ is externally tangent to a disk with radius $5$ . Let $A$ be the point where the disks are tangent, $C$ be the center of the smaller disk, and $E$ be the center of the larger disk. While the larger disk remains fixed, the smaller disk is allowed to roll along the outside of the larger disk until ... | [
"\nLet $F$ be the new tangency point of the two disks. The smaller disk rolled along minor arc $\\overarc{AF}$ on the larger disk.\nLet $\\alpha = \\angle AEF$ , in radians. The smaller disk must then have rolled along an arc of length $5\\alpha$ , since the larger disk has a radius of $5$ . Since all of the point... |
https://artofproblemsolving.com/wiki/index.php/1991_AIME_Problems/Problem_13 | null | 990 | A drawer contains a mixture of red socks and blue socks, at most $1991$ in all. It so happens that, when two socks are selected randomly without replacement, there is a probability of exactly $\frac{1}{2}$ that both are red or both are blue. What is the largest possible number of red socks in the drawer that is consist... | [
"Let $r$ and $b$ denote the number of red and blue socks, respectively. Also, let $t=r+b$ . The probability $P$ that when two socks are drawn randomly, without replacement, both are red or both are blue is given by\n\\[\\frac{r(r-1)}{(r+b)(r+b-1)}+\\frac{b(b-1)}{(r+b)(r+b-1)}=\\frac{r(r-1)+(t-r)(t-r-1)}{t(t-1)}=\\f... |
https://artofproblemsolving.com/wiki/index.php/2010_AMC_10B_Problems/Problem_3 | C | 5 | A drawer contains red, green, blue, and white socks with at least 2 of each color. What is
the minimum number of socks that must be pulled from the drawer to guarantee a matching
pair?
$\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$ | [
"After you draw $4$ socks, you can have one of each color, so (according to the pigeonhole principle ), if you pull $\\boxed{5}$ then you will be guaranteed a matching pair."
] |
https://artofproblemsolving.com/wiki/index.php/1986_AHSME_Problems/Problem_17 | B | 23 | A drawer in a darkened room contains $100$ red socks, $80$ green socks, $60$ blue socks and $40$ black socks.
A youngster selects socks one at a time from the drawer but is unable to see the color of the socks drawn.
What is the smallest number of socks that must be selected to guarantee that the selection contains a... | [
"Solution by e_power_pi_times_i\nSuppose that you wish to draw one pair of socks from the drawer. Then you would pick $5$ socks (one of each kind, plus one). Notice that in the worst possible situation, you will continue to draw the same sock, until you get $10$ pairs. This is because drawing the same sock results ... |
https://artofproblemsolving.com/wiki/index.php/1990_AJHSME_Problems/Problem_8 | D | 66 | A dress originally priced at $80$ dollars was put on sale for $25\%$ off. If $10\%$ tax was added to the sale price, then the total selling price (in dollars) of the dress was
$\text{(A)}\ \text{45 dollars} \qquad \text{(B)}\ \text{52 dollars} \qquad \text{(C)}\ \text{54 dollars} \qquad \text{(D)}\ \text{66 dollars} \... | [
"After the price reduction, the sale price is $80-.25\\times 80 = 60$ dollars. The tax makes the final price $60+.1\\times 60 = 66$ dollars $\\rightarrow \\boxed{66}$"
] |
https://artofproblemsolving.com/wiki/index.php/2020_AMC_10A_Problems/Problem_4 | E | 26 | A driver travels for $2$ hours at $60$ miles per hour, during which her car gets $30$ miles per gallon of gasoline. She is paid $$0.50$ per mile, and her only expense is gasoline at $$2.00$ per gallon. What is her net rate of pay, in dollars per hour, after this expense?
$\textbf{(A)}\ 20\qquad\textbf{(B)}\ 22\qquad\te... | [
"Since the driver travels $60$ miles per hour and each hour she uses $2$ gallons of gasoline, she spends $$4$ per hour on gas. If she gets $$0.50$ per mile, then she gets $$30$ per hour of driving. Subtracting the gas cost, her net rate of money earned per hour is $\\boxed{26}$ .\n~mathsmiley",
"The driver is dri... |
https://artofproblemsolving.com/wiki/index.php/2020_AMC_12A_Problems/Problem_3 | E | 26 | A driver travels for $2$ hours at $60$ miles per hour, during which her car gets $30$ miles per gallon of gasoline. She is paid $$0.50$ per mile, and her only expense is gasoline at $$2.00$ per gallon. What is her net rate of pay, in dollars per hour, after this expense?
$\textbf{(A)}\ 20\qquad\textbf{(B)}\ 22\qquad\te... | [
"Since the driver travels $60$ miles per hour and each hour she uses $2$ gallons of gasoline, she spends $$4$ per hour on gas. If she gets $$0.50$ per mile, then she gets $$30$ per hour of driving. Subtracting the gas cost, her net rate of money earned per hour is $\\boxed{26}$ .\n~mathsmiley",
"The driver is dri... |
https://artofproblemsolving.com/wiki/index.php/2001_AIME_I_Problems/Problem_6 | null | 79 | A fair die is rolled four times. The probability that each of the final three rolls is at least as large as the roll preceding it may be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers . Find $m + n$ | [
"Recast the problem entirely as a block-walking problem. Call the respective dice $a, b, c, d$ . In the diagram below, the lowest $y$ -coordinate at each of $a$ $b$ $c$ , and $d$ corresponds to the value of the roll.\nAIME01IN6.png\nThe red path corresponds to the sequence of rolls $2, 3, 5, 5$ . This establishe... |
https://artofproblemsolving.com/wiki/index.php/2014_AMC_12A_Problems/Problem_13 | null | 2,220 | A fancy bed and breakfast inn has $5$ rooms, each with a distinctive color-coded decor. One day $5$ friends arrive to spend the night. There are no other guests that night. The friends can room in any combination they wish, but with no more than $2$ friends per room. In how many ways can the innkeeper assign the gu... | [
"We can work in reverse by first determining the number of combinations in which there are more than $2$ friends in at least one room. There are three cases:\nCase 1: Three friends are in one room. Since there are $5$ possible rooms in which this can occur, we are choosing three friends from the five, and the other... |
https://artofproblemsolving.com/wiki/index.php/1964_AHSME_Problems/Problem_14 | C | 7 | A farmer bought $749$ sheep. He sold $700$ of them for the price paid for the $749$ sheep.
The remaining $49$ sheep were sold at the same price per head as the other $700$ .
Based on the cost, the percent gain on the entire transaction is:
$\textbf{(A)}\ 6.5 \qquad \textbf{(B)}\ 6.75 \qquad \textbf{(C)}\ 7 \qquad \te... | [
"Let us say each sheep cost $x$ dollars. The farmer paid $749x$ for the sheep. He sold $700$ of them for $749x$ , so each sheep sold for $\\frac{749}{700} = 1.07x$\nSince every sheep sold for the same price per head, and since every sheep cost $x$ and sold for $1.07x$ , there is an increase of $\\frac{1.07x - 1x}... |
https://artofproblemsolving.com/wiki/index.php/1959_AHSME_Problems/Problem_9 | C | 140 | A farmer divides his herd of $n$ cows among his four sons so that one son gets one-half the herd, a second son, one-fourth, a third son, one-fifth, and the fourth son, $7$ cows. Then $n$ is: $\textbf{(A)}\ 80 \qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 140\qquad\textbf{(D)}\ 180\qquad\textbf{(E)}\ 240$ | [
"The first three sons get $\\frac{1}{2}+\\frac{1}{4}+\\frac{1}{5}=\\frac{19}{20}$ of the herd, so that the fourth son should get $\\frac{1}{20}$ of it. But the fourth son gets $7$ cows, so the size of the herd is $n=\\frac{7}{\\frac{1}{20}} = 140$ . Then our answer is $\\boxed{140}$ , and we are done."
] |
https://artofproblemsolving.com/wiki/index.php/2021_Fall_AMC_10A_Problems/Problem_18 | C | 84 | A farmer's rectangular field is partitioned into $2$ by $2$ grid of $4$ rectangular sections as shown in the figure. In each section the farmer will plant one crop: corn, wheat, soybeans, or potatoes. The farmer does not want to grow corn and wheat in any two sections that share a border, and the farmer does not want t... | [
"There are $4$ possibilities for the top-left section. It follows that the top-right and bottom-left sections each have $3$ possibilities, so they have $3^2=9$ combinations. We have two cases:\nTogether, the answer is $36+48=\\boxed{84}.$",
"We will do casework on the type of crops in the field.\nCase 1: all of a... |
https://artofproblemsolving.com/wiki/index.php/1996_AHSME_Problems/Problem_7 | B | 2 | A father takes his twins and a younger child out to dinner on the twins' birthday. The restaurant charges $4.95$ for the father and $0.45$ for each year of a child's age, where age is defined as the age at the most recent birthday. If the bill is $9.45$ , which of the following could be the age of the youngest child?
$... | [
"The bill for the three children is $9.45 - 4.95 = 4.50$ . Since the charge is $0.45$ per year for the children, they must have $\\frac{4.50}{0.45} = 10$ years among the three of them.\nThe twins must have an even number of years in total (presuming that they did not dine in the 17 minutes between the time when th... |
https://artofproblemsolving.com/wiki/index.php/2005_AMC_12A_Problems/Problem_19 | null | 1,462 | A faulty car odometer proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. If the odometer now reads 002005 , how many miles has the car actually traveled? $(\mathrm {A}) \ 1404 \qquad (\mathrm {B}) \ 1462 \qquad (\mathrm {C})\ 1604 \qquad (\mathrm {D}) \ 1605 \qquad (\mathrm {E})\ 180... | [
"Alternatively, consider that counting without the number $4$ is almost equivalent to counting in base $9$ ; only, in base $9$ , the number $9$ is not counted. Since $4$ is skipped, the symbol $5$ represents $4$ miles of travel, and we have traveled $2004_9$ miles. By basic conversion, $2004_9=9^3(2)+9^0(4)=729(2)+... |
https://artofproblemsolving.com/wiki/index.php/1994_AIME_Problems/Problem_12 | null | 702 | A fenced, rectangular field measures $24$ meters by $52$ meters. An agricultural researcher has 1994 meters of fence that can be used for internal fencing to partition the field into congruent, square test plots. The entire field must be partitioned, and the sides of the squares must be parallel to the edges of the fie... | [
"Suppose there are $n$ squares in every column of the grid, so there are $\\frac{52}{24}n = \\frac {13}6n$ squares in every row. Then $6|n$ , and our goal is to maximize the value of $n$\nEach vertical fence has length $24$ , and there are $\\frac{13}{6}n - 1$ vertical fences; each horizontal fence has length $52$ ... |
https://artofproblemsolving.com/wiki/index.php/2010_AMC_12A_Problems/Problem_2 | A | 585 | A ferry boat shuttles tourists to an island every hour starting at 10 AM until its last trip, which starts at 3 PM. One day the boat captain notes that on the 10 AM trip there were 100 tourists on the ferry boat, and that on each successive trip, the number of tourists was 1 fewer than on the previous trip. How many to... | [
"It is easy to see that the ferry boat takes $6$ trips total. The total number of people taken to the island is\n\\begin{align*}&100+99+98+97+96+95\\\\ &=6(100)-(1+2+3+4+5)\\\\ &=600 - 15\\\\ &=\\boxed{585}"
] |
https://artofproblemsolving.com/wiki/index.php/1988_AJHSME_Problems/Problem_21 | C | 3 | A fifth number, $n$ , is added to the set $\{ 3,6,9,10 \}$ to make the mean of the set of five numbers equal to its median . The number of possible values of $n$ is
$\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad \text{(E)}\ \text{more than }4$ | [
"The possible medians after $n$ is added are $6$ $n$ , or $9$ . Now we use casework\nCase 1: The median is $6$\nIn this case, $n<6$ and \\[\\frac{3+n+6+9+10}{5}=6 \\Rightarrow n=2\\] so this case contributes $1$\nCase 2: The median is $n$\nWe have $6<n<9$ and \\[\\frac{3+6+n+9+10}{5}=n \\Rightarrow n=7\\] so this ... |
https://artofproblemsolving.com/wiki/index.php/2003_AMC_8_Problems/Problem_15 | B | 4 | A figure is constructed from unit cubes. Each cube shares at least one face with another cube. What is the minimum number of cubes needed to build a figure with the front and side views shown?
[asy] defaultpen(linewidth(0.8)); path p=unitsquare; draw(p^^shift(0,1)*p^^shift(1,0)*p); draw(shift(4,0)*p^^shift(5,0)*p^^shif... | [
"In order to minimize the amount of cubes needed, we must match up as many squares of our given figures with each other to make different sides of the same cube. One example of the solution with $\\boxed{4}$ cubes. Notice the corner cube cannot be removed for a figure of 3 cubes because each face of a cube must be... |
https://artofproblemsolving.com/wiki/index.php/2007_AMC_10A_Problems/Problem_22 | D | 37 | A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might be... | [
"A given digit appears as the hundreds digit, the tens digit, and the units digit of a term the same number of times. Let $k$ be the sum of the units digits in all the terms. Then $S=111k=3 \\cdot 37k$ , so $S$ must be divisible by $37\\ \\mathrm{(D)}$ . To see that it need not be divisible by any larger prime, the... |
https://artofproblemsolving.com/wiki/index.php/2007_AMC_12A_Problems/Problem_11 | D | 37 | A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might be... | [
"A given digit appears as the hundreds digit, the tens digit, and the units digit of a term the same number of times. Let $k$ be the sum of the units digits in all the terms. Then $S=111k=3 \\cdot 37k$ , so $S$ must be divisible by $37\\ \\mathrm{(D)}$ . To see that it need not be divisible by any larger prime, the... |
https://artofproblemsolving.com/wiki/index.php/2001_AIME_I_Problems/Problem_2 | null | 651 | A finite set $\mathcal{S}$ of distinct real numbers has the following properties: the mean of $\mathcal{S}\cup\{1\}$ is $13$ less than the mean of $\mathcal{S}$ , and the mean of $\mathcal{S}\cup\{2001\}$ is $27$ more than the mean of $\mathcal{S}$ . Find the mean of $\mathcal{S}$ | [
"Let $x$ be the mean of $\\mathcal{S}$ . Let $a$ be the number of elements in $\\mathcal{S}$ .\nThen, the given tells us that $\\frac{ax+1}{a+1}=x-13$ and $\\frac{ax+2001}{a+1}=x+27$ . Subtracting, we have \\begin{align*}\\frac{ax+2001}{a+1}-40=\\frac{ax+1}{a+1} \\Longrightarrow \\frac{2000}{a+1}=40 \\Longrightarro... |
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