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/2016_AMC_8_Problems/Problem_14 | A | 525 | Karl's car uses a gallon of gas every $35$ miles, and his gas tank holds $14$ gallons when it is full. One day, Karl started with a full tank of gas, drove $350$ miles, bought $8$ gallons of gas, and continued driving to his destination. When he arrived, his gas tank was half full. How many miles did Karl drive that da... | [
"Since he uses a gallon of gas every $35$ miles, he had used $\\frac{350}{35} = 10$ gallons after $350$ miles. Therefore, after the first leg of his trip he had $14 - 10 = 4$ gallons of gas left. Then, he bought $8$ gallons of gas, which brought him up to $12$ gallons of gas in his gas tank. When he arrived, he had... |
https://artofproblemsolving.com/wiki/index.php/2011_AMC_8_Problems/Problem_2 | E | 100 | Karl's rectangular vegetable garden is $20$ feet by $45$ feet, and Makenna's is $25$ feet by $40$ feet. Whose garden is larger in area?
$\textbf{(A)}\ \text{Karl's garden is larger by 100 square feet.}$
$\textbf{(B)}\ \text{Karl's garden is larger by 25 square feet.}$
$\textbf{(C)}\ \text{The gardens are the same size.... | [
"The area of a rectangle is given by the formula length times width. Karl's garden is $20 \\times 45 = 900$ square feet and Makenna's garden is $25 \\times 40 = 1000$ square feet. Since $1000 > 900,$ Makenna's garden is larger by $1000-900=100$ square feet. $\\Rightarrow \\boxed{100}$"
] |
https://artofproblemsolving.com/wiki/index.php/2018_AMC_10B_Problems/Problem_1 | A | 90 | Kate bakes a $20$ -inch by $18$ -inch pan of cornbread. The cornbread is cut into pieces that measure $2$ inches by $2$ inches. How many pieces of cornbread does the pan contain?
$\textbf{(A) } 90 \qquad \textbf{(B) } 100 \qquad \textbf{(C) } 180 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 360$ | [
"The area of the pan is $20\\cdot18=360$ . Since the area of each piece is $2\\cdot2=4$ , there are $\\frac{360}{4} = \\boxed{90}$ pieces.",
"By dividing each of the dimensions by $2$ , we get a $10\\times9$ grid that makes $\\boxed{90}$ pieces."
] |
https://artofproblemsolving.com/wiki/index.php/2018_AMC_12B_Problems/Problem_1 | A | 90 | Kate bakes a $20$ -inch by $18$ -inch pan of cornbread. The cornbread is cut into pieces that measure $2$ inches by $2$ inches. How many pieces of cornbread does the pan contain?
$\textbf{(A) } 90 \qquad \textbf{(B) } 100 \qquad \textbf{(C) } 180 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 360$ | [
"The area of the pan is $20\\cdot18=360$ . Since the area of each piece is $2\\cdot2=4$ , there are $\\frac{360}{4} = \\boxed{90}$ pieces.",
"By dividing each of the dimensions by $2$ , we get a $10\\times9$ grid that makes $\\boxed{90}$ pieces."
] |
https://artofproblemsolving.com/wiki/index.php/2007_AMC_12A_Problems/Problem_4 | A | 7 | Kate rode her bicycle for 30 minutes at a speed of 16 mph, then walked for 90 minutes at a speed of 4 mph. What was her overall average speed in miles per hour?
$\mathrm{(A)}\ 7\qquad \mathrm{(B)}\ 9\qquad \mathrm{(C)}\ 10\qquad \mathrm{(D)}\ 12\qquad \mathrm{(E)}\ 14$ | [
"\\[16 \\cdot \\frac{30}{60}+4\\cdot\\frac{90}{60}=14\\] \\[\\frac{14}2=7\\Rightarrow\\boxed{7}\\]"
] |
https://artofproblemsolving.com/wiki/index.php/2018_AIME_I_Problems/Problem_3 | null | 157 | Kathy has $5$ red cards and $5$ green cards. She shuffles the $10$ cards and lays out $5$ of the cards in a row in a random order. She will be happy if and only if all the red cards laid out are adjacent and all the green cards laid out are adjacent. For example, card orders RRGGG, GGGGR, or RRRRR will make Kathy happy... | [
"We have $2+4\\cdot 2$ cases total.\nThe two are all red and all green. Then, you have 4 of one, 1 of other. 3 of one, 2 of other. 2 of one, 3 of other. 1 of one, 4 of other. Then flip the order, so times two.\nObviously the denominator is $10\\cdot 9\\cdot 8\\cdot 7\\cdot 6$ , since we are choosing a card without ... |
https://artofproblemsolving.com/wiki/index.php/2000_AMC_8_Problems/Problem_21 | B | 38 | Keiko tosses one penny and Ephraim tosses two pennies. The probability that Ephraim gets the same number of heads that Keiko gets is
$\text{(A)}\ \frac{1}{4}\qquad\text{(B)}\ \frac{3}{8}\qquad\text{(C)}\ \frac{1}{2}\qquad\text{(D)}\ \frac{2}{3}\qquad\text{(E)}\ \frac{3}{4}$ | [
"Divide it into $2$ cases:\n1) Keiko and Ephriam both get $0$ heads:\nThis means that they both roll all tails, so there is only $1$ way for this to happen.\n2) Keiko and Ephriam both get $1$ head:\nFor Keiko, there is only $1$ way for this to happen because he is only flipping 1 penny, but for Ephriam, there are 2... |
https://artofproblemsolving.com/wiki/index.php/2011_AMC_10B_Problems/Problem_12 | A | 3 | Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has a width of $6$ meters, and it takes her $36$ seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in me... | [
"Let $s$ be Keiko's speed in meters per second, $a$ be the length of the straight parts of the track, $b$ be the radius of the smaller circles, and $b+6$ be the radius of the larger circles. The length of the inner edge will be $2a+2b \\pi$ and the length of the outer edge will be $2a+2\\pi (b+6).$ Since it takes $... |
https://artofproblemsolving.com/wiki/index.php/2011_AMC_12B_Problems/Problem_8 | A | 3 | Keiko walks once around a track at the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has a width of $6$ meters, and it takes her $36$ seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters ... | [
"To find Keiko's speed, all we need to find is the difference between the distance around the inside edge of the track and the distance around the outside edge of the track, and divide it by the difference in the time it takes her for each distance. We are given the difference in time, so all we need to find is th... |
https://artofproblemsolving.com/wiki/index.php/2009_AMC_10B_Problems/Problem_6 | D | 18 | Kiana has two older twin brothers. The product of their three ages is 128. What is the sum of their three ages?
$\mathrm{(A)}\ 10\qquad \mathrm{(B)}\ 12\qquad \mathrm{(C)}\ 16\qquad \mathrm{(D)}\ 18\qquad \mathrm{(E)}\ 24$ | [
"The age of each person is a factor of $128 = 2^7$ . So the twins could be $2^0 = 1, 2^1 = 2, 2^2 = 4, 2^3 = 8$ years of age and, consequently Kiana could be $128$ $32$ $8$ , or $2$ years old, respectively. Because Kiana is younger than her brothers, she must be $2$ years old. So the sum of their ages is $2 + 8 +... |
https://artofproblemsolving.com/wiki/index.php/2009_AMC_12B_Problems/Problem_5 | D | 18 | Kiana has two older twin brothers. The product of their three ages is 128. What is the sum of their three ages?
$\mathrm{(A)}\ 10\qquad \mathrm{(B)}\ 12\qquad \mathrm{(C)}\ 16\qquad \mathrm{(D)}\ 18\qquad \mathrm{(E)}\ 24$ | [
"The age of each person is a factor of $128 = 2^7$ . So the twins could be $2^0 = 1, 2^1 = 2, 2^2 = 4, 2^3 = 8$ years of age and, consequently Kiana could be $128$ $32$ $8$ , or $2$ years old, respectively. Because Kiana is younger than her brothers, she must be $2$ years old. So the sum of their ages is $2 + 8 +... |
https://artofproblemsolving.com/wiki/index.php/1985_AJHSME_Problems/Problem_23 | E | 50 | King Middle School has $1200$ students. Each student takes $5$ classes a day. Each teacher teaches $4$ classes. Each class has $30$ students and $1$ teacher. How many teachers are there at King Middle School?
$\text{(A)}\ 30 \qquad \text{(B)}\ 32 \qquad \text{(C)}\ 40 \qquad \text{(D)}\ 45 \qquad \text{(E)}\ 50$ | [
"If each student has $5$ classes, and there are $1200$ students, then they have a total of $5\\times 1200=6000$ classes among them.\nEach class has $30$ students, so there must be $\\frac{6000}{30}=200$ classes. Each class has $1$ teacher, so the teachers have a total of $200$ classes among them.\nEach teacher teac... |
https://artofproblemsolving.com/wiki/index.php/2017_AMC_12B_Problems/Problem_1 | E | 25 | Kymbrea's comic book collection currently has $30$ comic books in it, and she is adding to her collection at the rate of $2$ comic books per month. LaShawn's collection currently has $10$ comic books in it, and he is adding to his collection at the rate of $6$ comic books per month. After how many months will LaShawn's... | [
"Kymbrea has $30$ comic books initially and every month, she adds two. This can be represented as $30 + 2x$ where x is the number of months elapsed. LaShawn's collection, similarly, is $10 + 6x$ . To find when LaShawn will have twice the number of comic books as Kymbrea, we solve for x with the equation $2(2x + 30)... |
https://artofproblemsolving.com/wiki/index.php/1994_AHSME_Problems/Problem_19 | null | 415 | Label one disk " $1$ ", two disks " $2$ ", three disks " $3$ $, ...,$ fifty disks " $50$ ". Put these $1+2+3+ \cdots+50=1275$ labeled disks in a box. Disks are then drawn from the box at random without replacement. The minimum number of disks that must be drawn to guarantee drawing at least ten disks with the same labe... | [
"We can solve this problem by thinking of the worst case scenario, essentially an adaptation of the Pigeon-hole principle. \nWe can start by picking up all the disks numbered 1 to 9 since even if we have all those disks we won't have 10 of any one disk. This gives us 45 disks.\nFrom disks numbered from 10 to 50, we... |
https://artofproblemsolving.com/wiki/index.php/2018_AMC_8_Problems/Problem_13 | A | 4 | Laila took five math tests, each worth a maximum of 100 points. Laila's score on each test was an integer between 0 and 100, inclusive. Laila received the same score on the first four tests, and she received a higher score on the last test. Her average score on the five tests was 82. How many values are possible for La... | [
"Say Laila gets a value of $x$ on her first 4 tests, and a value of $y$ on her last test. Thus, $4x+y=410.$\nThe value $y$ has to be greater than $82$ , because otherwise she would receive the same score on her last test. Additionally, the greatest value for $y$ is $98$ (as $y=100$ would make $x$ as a decimal), so ... |
https://artofproblemsolving.com/wiki/index.php/2023_AMC_12B_Problems/Problem_18 | A | 22 | Last academic year Yolanda and Zelda took different courses that did not necessarily administer the same number of quizzes during each of the two semesters. Yolanda's average on all the quizzes she took during the first semester was $3$ points higher than Zelda's average on all the quizzes she took during the first sem... | [
"Denote by $A_i$ the average of person with initial $A$ in semester $i \\in \\left\\{1, 2 \\right\\}$ Thus, $Y_1 = Z_1 + 3$ $Y_2 = Y_1 + 18$ $Y_2 = Z_2 + 3$\nDenote by $A_{12}$ the average of person with initial $A$ in the full year.\nThus, $Y_{12}$ can be any number in $\\left( Y_1 , Y_2 \\right)$ and $Z_{12}$ can... |
https://artofproblemsolving.com/wiki/index.php/1994_AJHSME_Problems/Problem_11 | B | 32 | Last summer $100$ students attended basketball camp. Of those attending, $52$ were boys and $48$ were girls. Also, $40$ students were from Jonas Middle School and $60$ were from Clay Middle School. Twenty of the girls were from Jonas Middle School. How many of the boys were from Clay Middle School?
$\text{(A)}\ 20 ... | [
"Make a table with the information given.\n\\[\\begin{tabular}{c|ccc} & \\text{Jonas} & \\text{Clay} & \\text{total} \\\\ \\hline \\text{boys} & & & 52 \\\\ \\text{girls} & 20 & & 48 \\\\ \\text{total} & 40 & 60 & 100 \\end{tabular}\\]\nBecause the first two columns must add up to the third column, and the same wi... |
https://artofproblemsolving.com/wiki/index.php/2011_AMC_12A_Problems/Problem_5 | C | 40 | Last summer $30\%$ of the birds living on Town Lake were geese, $25\%$ were swans, $10\%$ were herons, and $35\%$ were ducks. What percent of the birds that were not swans were geese?
$\textbf{(A)}\ 20 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 40 \qquad \textbf{(D)}\ 50 \qquad \textbf{(E)}\ 60$ | [
"To simplify the problem, WLOG, let us say that there were a total of $100$ birds. The number of birds that are not swans is $75$ . The number of geese is $30$ . Therefore the percentage is just $\\frac{30}{75} \\times 100 = 40 \\Rightarrow \\boxed{40}$"
] |
https://artofproblemsolving.com/wiki/index.php/2011_AMC_10A_Problems/Problem_8 | C | 40 | Last summer 30% of the birds living on Town Lake were geese, 25% were swans, 10% were herons, and 35% were ducks. What percent of the birds that were not swans were geese?
$\textbf{(A)}\ 20\qquad\textbf{(B)}\ 30\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 50\qquad\textbf{(E)}\ 60$ | [
"75% of the total birds were not swans. Out of that 75%, there was $30\\% / 75\\% = \\boxed{40}$ of the birds that were not swans that were geese.",
"WLOG, suppose there were 100 birds in total living on Town Lake, then 30 were geese, 25 were swans, 10 were herons, and 35 were ducks. $100-25 = 75$ of the birds ar... |
https://artofproblemsolving.com/wiki/index.php/2012_AMC_10A_Problems/Problem_5 | B | 200 | Last year 100 adult cats, half of whom were female, were brought into the Smallville Animal Shelter. Half of the adult female cats were accompanied by a litter of kittens. The average number of kittens per litter was 4. What was the total number of cats and kittens received by the shelter last year?
$\textbf{(A)}\ 150\... | [
"Half of the 100 adult cats are female, so there are $\\frac{100}{2}$ $50$ female cats. Half of those female adult cats have a litter of kittens, so there would be $\\frac{50}{2}$ $25$ litters. Since the average number of kittens per litter is 4, this implies that there are $25 \\times 4$ $100$ kittens. So the t... |
https://artofproblemsolving.com/wiki/index.php/2017_AMC_10B_Problems/Problem_25 | E | 100 | Last year Isabella took $7$ math tests and received $7$ different scores, each an integer between $91$ and $100$ , inclusive. After each test she noticed that the average of her test scores was an integer. Her score on the seventh test was $95$ . What was her score on the sixth test?
$\textbf{(A)}\ 92\qquad\textbf{(B)}... | [
"Let the sum of the scores of Isabella's first $6$ tests be $S$ . Since the mean of her first $7$ scores is an integer, then $S + 95 \\equiv 0 \\text{ (mod 7)}$ , or $S \\equiv3 \\text{ (mod 7)}$ . Also, $S \\equiv 0 \\text{ (mod 6)}$ , so by the CRT $S \\equiv 24 \\text{ (mod 42)}$ . We also know that $91 \\cdot 6... |
https://artofproblemsolving.com/wiki/index.php/2017_AMC_12B_Problems/Problem_21 | E | 100 | Last year, Isabella took 7 math tests and received 7 different scores, each an integer between 91 and 100, inclusive. After each test she noticed that the average of her test scores was an integer. Her score on the seventh test was 95. What was her score on the sixth test?
$\textbf{(A)}\ 92\qquad\textbf{(B)}\ 94\qquad\... | [
"First, remove all the 90s, since they make no impact. So, we have numbers from $1$ to $10$ . Then, $5$ is the 7th number. Let the sum of the first $6$ numbers be $k$ . Then, $k\\equiv 0 \\mod 6$ and $k\\equiv 3 \\mod 7$ . We easily solve this as $k \\equiv 24 \\mod 42$ . Clearly, the sum of the first $6$ numbers m... |
https://artofproblemsolving.com/wiki/index.php/2022_AMC_8_Problems/Problem_15 | C | 3 | Laszlo went online to shop for black pepper and found thirty different black pepper options varying in weight and price, shown in the scatter plot below. In ounces, what is the weight of the pepper that offers the lowest price per ounce?
[asy] //diagram by pog size(5.5cm); usepackage("mathptmx"); defaultpen(mediumgray*... | [
"\nWe are looking for a black point, that when connected to the origin, yields the lowest slope. The slope represents the price per ounce. We can visually find that the point with the lowest slope is the blue point. Furthermore, it is the only one with a price per ounce significantly less than $1$ . Finally, we see... |
https://artofproblemsolving.com/wiki/index.php/2016_AMC_10B_Problems/Problem_6 | B | 4 | Laura added two three-digit positive integers. All six digits in these numbers are different. Laura's sum is a three-digit number $S$ . What is the smallest possible value for the sum of the digits of $S$
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 20$ | [
"Let the two three-digit numbers she added be $a$ and $b$ with $a+b=S$ and $a<b$ . The hundreds digits of these numbers must be at least $1$ and $2$ , so $a\\ge 100$ and $b\\ge 200$\nSay $a=100+p$ and $b=200+q$ ; then we just need $p+q=100$ with $p$ and $q$ having different digits which aren't $1$ or $2$ .There are... |
https://artofproblemsolving.com/wiki/index.php/2011_AMC_10B_Problems/Problem_4 | C | 2 | LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid $A$ dollars and Bernardo had paid $B$ dollars, where $A < B$ . How many dollar... | [
"The difference in how much LeRoy and Bernardo paid is $B-A$ . To share the costs equally, LeRoy must give Bernardo half of the difference, which is $\\boxed{2}$",
"Since there are no restrictions on cost paid besides $A<B$ , we can use an example where $A = 40$ and $B = 50$ . Quickly, we realize the only way the... |
https://artofproblemsolving.com/wiki/index.php/2011_AMC_12B_Problems/Problem_3 | C | 2 | LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid A dollars and Bernardo had paid B dollars, where $A < B.$ How many dollars mus... | [
"The total amount of money that was spent during the trip was \\[A + B\\] So each person should pay \\[\\frac{A+B}{2}\\] if they were to share the costs equally. Because LeRoy has already paid $A$ dollars of his part, he still has to pay \\[\\frac{A+B}{2} - A =\\] \\[= \\boxed{2}\\]"
] |
https://artofproblemsolving.com/wiki/index.php/2014_AMC_10B_Problems/Problem_1 | C | 37 | Leah has $13$ coins, all of which are pennies and nickels. If she had one more nickel than she has now, then she would have the same number of pennies and nickels. In cents, how much are Leah's coins worth?
$\textbf {(A) } 33 \qquad \textbf {(B) } 35 \qquad \textbf {(C) } 37 \qquad \textbf {(D) } 39 \qquad \textbf {(E)... | [
"If Leah has $1$ more nickel, she has $14$ total coins. Because she has the same number of nickels and pennies, she has $7$ nickels and $7$ pennies. This is after the nickel has been added, so we must subtract $1$ nickel to get $6$ nickels and $7$ pennies. Therefore, Leah has $6\\cdot5+7=\\boxed{37}$ cents."
] |
https://artofproblemsolving.com/wiki/index.php/2014_AMC_12B_Problems/Problem_1 | C | 37 | Leah has $13$ coins, all of which are pennies and nickels. If she had one more nickel than she has now, then she would have the same number of pennies and nickels. In cents, how much are Leah's coins worth?
$\textbf{(A)}\ 33\qquad\textbf{(B)}\ 35\qquad\textbf{(C)}\ 37\qquad\textbf{(D)}\ 39\qquad\textbf{(E)}\ 41$ | [
"She has $p$ pennies and $n$ nickels, where $n + p = 13$ . If she had $n+1$ nickels then $n+1 = p$ , so $2n+ 1 = 13$ and $n=6$ . So she has 6 nickels and 7 pennies, which clearly have a value of $\\boxed{37}$ cents."
] |
https://artofproblemsolving.com/wiki/index.php/2008_AIME_I_Problems/Problem_13 | null | 40 | Let
\[p(x,y) = a_0 + a_1x + a_2y + a_3x^2 + a_4xy + a_5y^2 + a_6x^3 + a_7x^2y + a_8xy^2 + a_9y^3.\]
Suppose that
\[p(0,0) = p(1,0) = p( - 1,0) = p(0,1) = p(0, - 1)\\ = p(1,1) = p(1, - 1) = p(2,2) = 0.\]
There is a point $\left(\frac {a}{c},\frac {b}{c}\right)$ for which $p\left(\frac {a}{c},\frac {b}{c}\right) = 0$ for... | [
"\\begin{align*} p(0,0) &= a_0 \\\\ &= 0 \\\\ p(1,0) &= a_0 + a_1 + a_3 + a_6 \\\\ &= a_1 + a_3 + a_6 \\\\ &= 0 \\\\ p(-1,0) &= -a_1 + a_3 - a_6 \\\\ &= 0 \\end{align*}\nAdding the above two equations gives $a_3 = 0$ , and so we can deduce that $a_6 = -a_1$\nSimilarly, plugging in $(0,1)$ and $(0,-1)$ gives $a_5 = ... |
https://artofproblemsolving.com/wiki/index.php/2010_AIME_I_Problems/Problem_9 | null | 158 | Let $(a,b,c)$ be the real solution of the system of equations $x^3 - xyz = 2$ $y^3 - xyz = 6$ $z^3 - xyz = 20$ . The greatest possible value of $a^3 + b^3 + c^3$ can be written in the form $\frac {m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ | [
"This is almost the same as solution 1. Note $a^3 + b^3 + c^3 = 28 + 3abc$ . Next, let $k = a^3$ . Note that $b = \\sqrt [3]{k + 4}$ and $c = \\sqrt [3]{k + 18}$ , so we have $28 + 3\\sqrt [3]{k(k+4)(k+18)} = 28+3abc=a^3+b^3+c^3=3k+22$ . Move 28 over, divide both sides by 3, then cube to get $k^3-6k^2+12k-8 = k^3+2... |
https://artofproblemsolving.com/wiki/index.php/2020_AMC_10A_Problems/Problem_18 | C | 96 | Let $(a,b,c,d)$ be an ordered quadruple of not necessarily distinct integers, each one of them in the set $\{0,1,2,3\}.$ For how many such quadruples is it true that $a\cdot d-b\cdot c$ is odd? (For example, $(0,3,1,1)$ is one such quadruple, because $0\cdot 1-3\cdot 1 = -3$ is odd.)
$\textbf{(A) } 48 \qquad \textbf{(B... | [
"In order for $a\\cdot d-b\\cdot c$ to be odd, consider parity. We must have (even)-(odd) or (odd)-(even). There are $2(2 + 4) = 12$ ways to pick numbers to obtain an even product. There are $2 \\cdot 2 = 4$ ways to obtain an odd product. Therefore, the total amount of ways to make $a\\cdot d-b\\cdot c$ odd is $2 \... |
https://artofproblemsolving.com/wiki/index.php/2012_AMC_12B_Problems/Problem_18 | B | 512 | Let $(a_1,a_2, \dots ,a_{10})$ be a list of the first 10 positive integers such that for each $2 \le i \le 10$ either $a_i+1$ or $a_i-1$ or both appear somewhere before $a_i$ in the list. How many such lists are there?
$\textbf{(A)}\ 120\qquad\textbf{(B)}\ 512\qquad\textbf{(C)}\ 1024\qquad\textbf{(D)}\ 181,440\qquad\te... | [
"This problem is worded awkwardly. More simply, it asks: “How many ways can you order numbers 1-10 so that each number is one above or below some previous term?”\nThen, the method becomes clear. For some initial number, WLOG examine the numbers greater than it. They always must appear in ascending order later in th... |
https://artofproblemsolving.com/wiki/index.php/2006_AIME_II_Problems/Problem_4 | null | 462 | Let $(a_1,a_2,a_3,\ldots,a_{12})$ be a permutation of $(1,2,3,\ldots,12)$ for which
An example of such a permutation is $(6,5,4,3,2,1,7,8,9,10,11,12).$ Find the number of such permutations. | [
"Clearly, $a_6=1$ . Now, consider selecting $5$ of the remaining $11$ values. Sort these values in descending order, and sort the other $6$ values in ascending order. Now, let the $5$ selected values be $a_1$ through $a_5$ , and let the remaining $6$ be $a_7$ through ${a_{12}}$ . It is now clear that there is a... |
https://artofproblemsolving.com/wiki/index.php/2004_AMC_10B_Problems/Problem_21 | A | 3,722 | Let $1$ $4$ $\ldots$ and $9$ $16$ $\ldots$ be two arithmetic progressions. The set $S$ is the union of the first $2004$ terms of each sequence. How many distinct numbers are in $S$
$\mathrm{(A) \ } 3722 \qquad \mathrm{(B) \ } 3732 \qquad \mathrm{(C) \ } 3914 \qquad \mathrm{(D) \ } 3924 \qquad \mathrm{(E) \ } 4007$ | [
"The two sets of terms are $A=\\{ 3k+1 : 0\\leq k < 2004 \\}$ and $B=\\{ 7l+9 : 0\\leq l<2004\\}$\nNow $S=A\\cup B$ . We can compute $|S|=|A\\cup B|=|A|+|B|-|A\\cap B|=4008-|A\\cap B|$ . We will now find $|A\\cap B|$\nConsider the numbers in $B$ . We want to find out how many of them lie in $A$ . In other words, we... |
https://artofproblemsolving.com/wiki/index.php/2003_AIME_II_Problems/Problem_14 | null | 51 | Let $A = (0,0)$ and $B = (b,2)$ be points on the coordinate plane. Let $ABCDEF$ be a convex equilateral hexagon such that $\angle FAB = 120^\circ,$ $\overline{AB}\parallel \overline{DE},$ $\overline{BC}\parallel \overline{EF,}$ $\overline{CD}\parallel \overline{FA},$ and the y-coordinates of its vertices are distinct e... | [
"The y-coordinate of $F$ must be $4$ . All other cases yield non-convex and/or degenerate hexagons, which violate the problem statement.\nLetting $F = (f,4)$ , and knowing that $\\angle FAB = 120^\\circ$ , we can use rewrite $F$ using complex numbers: $f + 4 i = (b + 2 i)\\left(e^{i(2 \\pi / 3)}\\right) = (b + 2 i... |
https://artofproblemsolving.com/wiki/index.php/2013_AIME_II_Problems/Problem_11 | null | 399 | Let $A = \{1, 2, 3, 4, 5, 6, 7\}$ , and let $N$ be the number of functions $f$ from set $A$ to set $A$ such that $f(f(x))$ is a constant function. Find the remainder when $N$ is divided by $1000$ | [
"Any such function can be constructed by distributing the elements of $A$ on three tiers.\nThe bottom tier contains the constant value, $c=f(f(x))$ for any $x$ . (Obviously $f(c)=c$ .)\nThe middle tier contains $k$ elements $x\\ne c$ such that $f(x)=c$ , where $1\\le k\\le 6$\nThe top tier contains $6-k$ elements s... |
https://artofproblemsolving.com/wiki/index.php/2024_AIME_I_Problems/Problem_9 | null | 480 | Let $A$ $B$ $C$ , and $D$ be point on the hyperbola $\frac{x^2}{20}- \frac{y^2}{24} = 1$ such that $ABCD$ is a rhombus whose diagonals intersect at the origin. Find the greatest real number that is less than $BD^2$ for all such rhombi. | [
"A quadrilateral is a rhombus if and only if its two diagonals bisect each other and are perpendicular to each other. The first condition is automatically satisfied because of the hyperbola's symmetry about the origin. To satisfy the second condition, we set $BD$ as the line $y = mx$ and $AC$ as $y = -\\frac{1}{m}x... |
https://artofproblemsolving.com/wiki/index.php/1985_AIME_Problems/Problem_12 | null | 182 | Let $A$ $B$ $C$ and $D$ be the vertices of a regular tetrahedron, each of whose edges measures $1$ meter. A bug, starting from vertex $A$ , observes the following rule: at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the... | [
"We evaluate $P(7)$ recursively: \\begin{alignat*}{6} P(0)&=1, \\\\ P(1)&=\\frac13(1-P(0))&&=0, \\\\ P(2)&=\\frac13(1-P(1))&&=\\frac13, \\\\ P(3)&=\\frac13(1-P(2))&&=\\frac29, \\\\ P(4)&=\\frac13(1-P(3))&&=\\frac{7}{27}, \\\\ P(5)&=\\frac13(1-P(4))&&=\\frac{20}{81}, \\\\ P(6)&=\\frac13(1-P(5))&&=\\frac{61}{243},\\\... |
https://artofproblemsolving.com/wiki/index.php/2000_AMC_10_Problems/Problem_20 | C | 69 | Let $A$ $M$ , and $C$ be nonnegative integers such that $A+M+C=10$ . What is the maximum value of $A\cdot M\cdot C+A\cdot M+M\cdot C+C\cdot A$
$\textbf{(A)}\ 49 \qquad\textbf{(B)}\ 59 \qquad\textbf{(C)}\ 69 \qquad\textbf{(D)}\ 79 \qquad\textbf{(E)}\ 89$ | [
"The trick is to realize that the sum $AMC+AM+MC+CA$ is similar to the product $(A+1)(M+1)(C+1)$ . If we multiply $(A+1)(M+1)(C+1)$ , we get \\[(A+1)(M+1)(C+1) = AMC + AM + AC + MC + A + M + C + 1.\\] We know that $A+M+C=10$ , therefore $(A+1)(M+1)(C+1) = (AMC + AM + MC + CA) + 11$ and \\[AMC + AM + MC + CA = (A+1)... |
https://artofproblemsolving.com/wiki/index.php/2009_AIME_II_Problems/Problem_13 | null | 672 | Let $A$ and $B$ be the endpoints of a semicircular arc of radius $2$ . The arc is divided into seven congruent arcs by six equally spaced points $C_1$ $C_2$ $\dots$ $C_6$ . All chords of the form $\overline {AC_i}$ or $\overline {BC_i}$ are drawn. Let $n$ be the product of the lengths of these twelve chords. Find the r... | [
"Let the radius be 1 instead. All lengths will be halved so we will multiply by $2^{12}$ at the end. Place the semicircle on the complex plane, with the center of the circle being 0 and the diameter being the real axis. Then $C_1,\\ldots, C_6$ are 6 of the 14th roots of unity. Let $\\omega=\\text{cis}\\frac{360^{\\... |
https://artofproblemsolving.com/wiki/index.php/2023_AIME_II_Problems/Problem_13 | null | 167 | Let $A$ be an acute angle such that $\tan A = 2 \cos A.$ Find the number of positive integers $n$ less than or equal to $1000$ such that $\sec^n A + \tan^n A$ is a positive integer whose units digit is $9.$ | [
"Denote $a_n = \\sec^n A + \\tan^n A$ .\nFor any $k$ , we have \\begin{align*} a_n & = \\sec^n A + \\tan^n A \\\\ & = \\left( \\sec^{n-k} A + \\tan^{n-k} A \\right) \\left( \\sec^k A + \\tan^k A \\right) - \\sec^{n-k} A \\tan^k A - \\tan^{n-k} A \\sec^k A \\\\ & = a_{n-k} a_k - 2^k \\sec^{n-k} A \\cos^k A - 2^k \\t... |
https://artofproblemsolving.com/wiki/index.php/2018_AMC_12A_Problems/Problem_19 | C | 19 | Let $A$ be the set of positive integers that have no prime factors other than $2$ $3$ , or $5$ . The infinite sum \[\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} + \frac{1}{12} + \frac{1}{15} + \frac{1}{16} + \frac{1}{18} + \frac{1}{20} + \c... | [
"Note that the fractions of the form $\\frac{1}{2^a3^b5^c},$ where $a,b,$ and $c$ are nonnegative integers, span all terms of the infinite sum.\nTherefore, the infinite sum becomes \\begin{align*} \\sum_{a=0}^{\\infty}\\sum_{b=0}^{\\infty}\\sum_{c=0}^{\\infty}\\frac{1}{2^a3^b5^c} &= \\left(\\sum_{a=0}^{\\infty}\\fr... |
https://artofproblemsolving.com/wiki/index.php/2005_AMC_12B_Problems/Problem_18 | C | 51 | Let $A(2,2)$ and $B(7,7)$ be points in the plane. Define $R$ as the region in the first quadrant consisting of those points $C$ such that $\triangle ABC$ is an acute triangle. What is the closest integer to the area of the region $R$
$\mathrm{(A)}\ 25 \qquad \mathrm{(B)}\ 39 \qquad \mathrm{(C)}\ 51 \qquad \... | [
"\nFor angle $A$ and $B$ to be acute, $C$ must be between the two lines that are perpendicular to $\\overline{AB}$ and contain points $A$ and $B$ . For angle $C$ to be acute, first draw a $45-45-90$ triangle with $\\overline{AB}$ as the hypotenuse. Note $C$ cannot be inside this triangle's circumscribed circle or e... |
https://artofproblemsolving.com/wiki/index.php/2013_AIME_II_Problems/Problem_15 | null | 222 | Let $A,B,C$ be angles of a triangle with \begin{align*} \cos^2 A + \cos^2 B + 2 \sin A \sin B \cos C &= \frac{15}{8} \text{ and} \\ \cos^2 B + \cos^2 C + 2 \sin B \sin C \cos A &= \frac{14}{9} \end{align*} There are positive integers $p$ $q$ $r$ , and $s$ for which \[\cos^2 C + \cos^2 A + 2 \sin C \sin A \cos B = \frac... | [
"Let's draw the triangle. Since the problem only deals with angles, we can go ahead and set one of the sides to a convenient value. Let $BC = \\sin{A}$\nBy the Law of Sines, we must have $CA = \\sin{B}$ and $AB = \\sin{C}$\nNow let us analyze the given:\n\\begin{align*} \\cos^2A + \\cos^2B + 2\\sin A\\sin B\\cos C ... |
https://artofproblemsolving.com/wiki/index.php/2014_AIME_I_Problems/Problem_12 | null | 453 | Let $A=\{1,2,3,4\}$ , and $f$ and $g$ be randomly chosen (not necessarily distinct) functions from $A$ to $A$ . The probability that the range of $f$ and the range of $g$ are disjoint is $\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m$ | [
"The natural way to go is casework. And the natural process is to sort $f$ and $g$ based on range size! Via Pigeonhole Principle, we see that the only real possibilities are: $f 1 g 1; f 1 g 2; f 1 g 3; f 2 g 2$ . Note that the $1, 2$ and $1, 3$ cases are symmetrical and we need just a $*2$ . Note also that the tot... |
https://artofproblemsolving.com/wiki/index.php/2024_AIME_I_Problems/Problem_10 | null | 113 | Let $ABC$ be a triangle inscribed in circle $\omega$ . Let the tangents to $\omega$ at $B$ and $C$ intersect at point $D$ , and let $\overline{AD}$ intersect $\omega$ at $P$ . If $AB=5$ $BC=9$ , and $AC=10$ $AP$ can be written as the form $\frac{m}{n}$ , where $m$ and $n$ are relatively prime integers. Find $m + n$ | [
"From the tangency condition we have $\\let\\angle BCD = \\let\\angle CBD = \\let\\angle A$ . With LoC we have $\\cos(A) = \\frac{25+100-81}{2*5*10} = \\frac{11}{25}$ and $\\cos(B) = \\frac{81+25-100}{2*9*5} = \\frac{1}{15}$ . Then, $CD = \\frac{\\frac{9}{2}}{\\cos(A)} = \\frac{225}{22}$ . Using LoC we can find $AD... |
https://artofproblemsolving.com/wiki/index.php/2004_AIME_I_Problems/Problem_9 | null | 35 | Let $ABC$ be a triangle with sides 3, 4, and 5, and $DEFG$ be a 6-by-7 rectangle . A segment is drawn to divide triangle $ABC$ into a triangle $U_1$ and a trapezoid $V_1$ and another segment is drawn to divide rectangle $DEFG$ into a triangle $U_2$ and a trapezoid $V_2$ such that $U_1$ is similar to $U_2$ and $V_1$ is ... | [
"We let $AB=3, AC=4, DE=6, DG=7$ for the purpose of labeling. Clearly, the dividing segment in $DEFG$ must go through one of its vertices, without loss of generality $D$ . The other endpoint ( $D'$ ) of the segment can either lie on $\\overline{EF}$ or $\\overline{FG}$ $V_2$ is a trapezoid with a right angle then, ... |
https://artofproblemsolving.com/wiki/index.php/2007_AIME_I_Problems/Problem_15 | null | 989 | Let $ABC$ be an equilateral triangle , and let $D$ and $F$ be points on sides $BC$ and $AB$ , respectively, with $FA = 5$ and $CD = 2$ . Point $E$ lies on side $CA$ such that angle $DEF = 60^{\circ}$ . The area of triangle $DEF$ is $14\sqrt{3}$ . The two possible values of the length of side $AB$ are $p \pm q \sqrt{... | [
"AIME I 2007-15.png\nDenote the length of a side of the triangle $x$ , and of $\\overline{AE}$ as $y$ . The area of the entire equilateral triangle is $\\frac{x^2\\sqrt{3}}{4}$ . Add up the areas of the triangles using the $\\frac{1}{2}ab\\sin C$ formula (notice that for the three outside triangles, $\\sin 60 = \\f... |
https://artofproblemsolving.com/wiki/index.php/2017_AMC_12B_Problems/Problem_15 | E | 37 | Let $ABC$ be an equilateral triangle. Extend side $\overline{AB}$ beyond $B$ to a point $B'$ so that $BB'=3 \cdot AB$ . Similarly, extend side $\overline{BC}$ beyond $C$ to a point $C'$ so that $CC'=3 \cdot BC$ , and extend side $\overline{CA}$ beyond $A$ to a point $A'$ so that $AA'=3 \cdot CA$ . What is the ratio of ... | [
"Note that by symmetry, $\\triangle A'B'C'$ is also equilateral. Therefore, we only need to find one of the sides of $A'B'C'$ to determine the area ratio. WLOG, let $AB = BC = CA = 1$ . Therefore, $BB' = 3$ and $BC' = 4$ . Also, $\\angle B'BC' = 120^{\\circ}$ , so by the Law of Cosines, $B'C' = \\sqrt{37}$ . Theref... |
https://artofproblemsolving.com/wiki/index.php/1998_AIME_Problems/Problem_12 | null | 83 | Let $ABC$ be equilateral , and $D, E,$ and $F$ be the midpoints of $\overline{BC}, \overline{CA},$ and $\overline{AB},$ respectively. There exist points $P, Q,$ and $R$ on $\overline{DE}, \overline{EF},$ and $\overline{FD},$ respectively, with the property that $P$ is on $\overline{CQ}, Q$ is on $\overline{AR},$ and $... | [
"1998 AIME-12.png\nWe let $x = EP = FQ$ $y = EQ$ $k = PQ$ . Since $AE = \\frac {1}{2}AB$ and $AD = \\frac {1}{2}AC$ $\\triangle AED \\sim \\triangle ABC$ and $ED \\parallel BC$\nBy alternate interior angles, we have $\\angle PEQ = \\angle BFQ$ and $\\angle EPQ = \\angle FBQ$ . By vertical angles, $\\angle EQP = \\a... |
https://artofproblemsolving.com/wiki/index.php/2002_AIME_I_Problems/Problem_11 | null | 230 | Let $ABCD$ and $BCFG$ be two faces of a cube with $AB=12$ . A beam of light emanates from vertex $A$ and reflects off face $BCFG$ at point $P$ , which is 7 units from $\overline{BG}$ and 5 units from $\overline{BC}$ . The beam continues to be reflected off the faces of the cube. The length of the light path from the ti... | [
"When a light beam reflects off a surface, the path is like that of a ball bouncing. Picture that, and also imagine X, Y, and Z coordinates for the cube vertices. The coordinates will all involve 0's and 12's only, so that means that the X, Y, and Z distance traveled by the light must all be divisible by 12. Since ... |
https://artofproblemsolving.com/wiki/index.php/2018_AIME_II_Problems/Problem_12 | null | 112 | Let $ABCD$ be a convex quadrilateral with $AB = CD = 10$ $BC = 14$ , and $AD = 2\sqrt{65}$ . Assume that the diagonals of $ABCD$ intersect at point $P$ , and that the sum of the areas of triangles $APB$ and $CPD$ equals the sum of the areas of triangles $BPC$ and $APD$ . Find the area of quadrilateral $ABCD$ | [
"For reference, $2\\sqrt{65} \\approx 16$ , so $\\overline{AD}$ is the longest of the four sides of $ABCD$ . Let $h_1$ be the length of the altitude from $B$ to $\\overline{AC}$ , and let $h_2$ be the length of the altitude from $D$ to $\\overline{AC}$ . Then, the triangle area equation becomes\n\\[\\frac{h_1}{2}AP... |
https://artofproblemsolving.com/wiki/index.php/2022_AIME_II_Problems/Problem_11 | null | 180 | Let $ABCD$ be a convex quadrilateral with $AB=2, AD=7,$ and $CD=3$ such that the bisectors of acute angles $\angle{DAB}$ and $\angle{ADC}$ intersect at the midpoint of $\overline{BC}.$ Find the square of the area of $ABCD.$ | [
"\nAccording to the problem, we have $AB=AB'=2$ $DC=DC'=3$ $MB=MB'$ $MC=MC'$ , and $B'C'=7-2-3=2$\nBecause $M$ is the midpoint of $BC$ , we have $BM=MC$ , so: \\[MB=MB'=MC'=MC.\\]\nThen, we can see that $\\bigtriangleup{MB'C'}$ is an isosceles triangle with $MB'=MC'$\nTherefore, we could start our angle chasing: $\... |
https://artofproblemsolving.com/wiki/index.php/2021_AIME_I_Problems/Problem_11 | null | 301 | Let $ABCD$ be a cyclic quadrilateral with $AB=4,BC=5,CD=6,$ and $DA=7.$ Let $A_1$ and $C_1$ be the feet of the perpendiculars from $A$ and $C,$ respectively, to line $BD,$ and let $B_1$ and $D_1$ be the feet of the perpendiculars from $B$ and $D,$ respectively, to line $AC.$ The perimeter of $A_1B_1C_1D_1$ is $\frac mn... | [
"Note that $\\cos(180^\\circ-\\theta)=-\\cos\\theta$ holds for all $\\theta.$ We apply the Law of Cosines to $\\triangle ABE, \\triangle BCE, \\triangle CDE,$ and $\\triangle DAE,$ respectively: \\begin{alignat*}{12} &&&AE^2+BE^2-2\\cdot AE\\cdot BE\\cdot\\cos\\angle AEB&&=AB^2&&\\quad\\implies\\quad AE^2+BE^2-2\\c... |
https://artofproblemsolving.com/wiki/index.php/1998_AIME_Problems/Problem_6 | null | 308 | Let $ABCD$ be a parallelogram . Extend $\overline{DA}$ through $A$ to a point $P,$ and let $\overline{PC}$ meet $\overline{AB}$ at $Q$ and $\overline{DB}$ at $R.$ Given that $PQ = 735$ and $QR = 112,$ find $RC.$ | [
"We have $\\triangle BRQ\\sim \\triangle DRC$ so $\\frac{112}{RC} = \\frac{BR}{DR}$ . We also have $\\triangle BRC \\sim \\triangle DRP$ so $\\frac{ RC}{847} = \\frac {BR}{DR}$ . Equating the two results gives $\\frac{112}{RC} = \\frac{ RC}{847}$ and so $RC^2=112*847$ which solves to $RC=\\boxed{308}$"
] |
https://artofproblemsolving.com/wiki/index.php/1997_AHSME_Problems/Problem_25 | B | 1 | Let $ABCD$ be a parallelogram and let $\overrightarrow{AA^\prime}$ $\overrightarrow{BB^\prime}$ $\overrightarrow{CC^\prime}$ , and $\overrightarrow{DD^\prime}$ be parallel rays in space on the same side of the plane determined by $ABCD$ . If $AA^{\prime} = 10$ $BB^{\prime}= 8$ $CC^\prime = 18$ , and $DD^\prime = 22$ an... | [
"Let $ABCD$ be a unit square with $A(0,0,0)$ $B(0,1,0)$ $C(1,1,0)$ , and $D(1,0,0)$ . Assume that the rays go in the +z direction. In this case, $A^\\prime(0,0,10)$ $B^\\prime(0,1,8)$ $C^\\prime(1,1,18)$ , and $D^\\prime(1,0,22)$ . Finding the midpoints of $A^\\prime C^\\prime$ and $B^\\prime D^\\prime$ gives $M... |
https://artofproblemsolving.com/wiki/index.php/1992_AHSME_Problems/Problem_24 | null | 5 | Let $ABCD$ be a parallelogram of area $10$ with $AB=3$ and $BC=5$ . Locate $E,F$ and $G$ on segments $\overline{AB},\overline{BC}$ and $\overline{AD}$ , respectively, with $AE=BF=AG=2$ . Let the line through $G$ parallel to $\overline{EF}$ intersect $\overline{CD}$ at $H$ . The area of quadrilateral $EFHG$ is
$\text{(A... | [
"We note that $ABFG$ is a parallelogram because $AG = BF = 2$ and $AG \\parallel BF$ . Using the same reasoning, $GFCD$ is also a parallelogram.\nAssume that the height of parallelogram $ABFG$ with respect to base $AB$ is $x$ . Then, the area of parallelogram $ABFG$ is $AB * x$ . The area of triangle $EFG$ is $\\fr... |
https://artofproblemsolving.com/wiki/index.php/2021_AMC_12B_Problems/Problem_24 | A | 81 | Let $ABCD$ be a parallelogram with area $15$ . Points $P$ and $Q$ are the projections of $A$ and $C,$ respectively, onto the line $BD;$ and points $R$ and $S$ are the projections of $B$ and $D,$ respectively, onto the line $AC.$ See the figure, which also shows the relative locations of these points.
[asy] size(350); d... | [
"Let $X$ denote the intersection point of the diagonals $AC$ and $BD$ . Remark that by symmetry $X$ is the midpoint of both $\\overline{PQ}$ and $\\overline{RS}$ , so $XP = XQ = 3$ and $XR = XS = 4$ . Now note that since $\\angle APB = \\angle ARB = 90^\\circ$ , quadrilateral $ARPB$ is cyclic, and so \\[XR\\cdot XA... |
https://artofproblemsolving.com/wiki/index.php/2003_AMC_12B_Problems/Problem_22 | C | 7 | Let $ABCD$ be a rhombus with $AC = 16$ and $BD = 30$ . Let $N$ be a point on $\overline{AB}$ , and let $P$ and $Q$ be the feet of the perpendiculars from $N$ to $\overline{AC}$ and $\overline{BD}$ , respectively. Which of the following is closest to the minimum possible value of $PQ$
$\mathrm{(A)}\ 6.5 \qquad\mathrm{(B... | [
"Let the intersection of $\\overline{AC}$ and $\\overline{BD}$ be $E$ . Since $ABCD$ is a rhombus, we have $\\overline{AC} \\perp \\overline{BD}$ and $AE = CE = \\dfrac{AC}{2} = 8$ . Since $\\overline{NQ} \\perp \\overline{BD}$ , we have $\\overline{NQ} \\parallel \\overline{AC}$ , so $\\triangle{BNQ} \\sim \\trian... |
https://artofproblemsolving.com/wiki/index.php/2022_AMC_10B_Problems/Problem_20 | D | 113 | Let $ABCD$ be a rhombus with $\angle ADC = 46^\circ$ . Let $E$ be the midpoint of $\overline{CD}$ , and let $F$ be the point
on $\overline{BE}$ such that $\overline{AF}$ is perpendicular to $\overline{BE}$ . What is the degree measure of $\angle BFC$
$\textbf{(A)}\ 110 \qquad\textbf{(B)}\ 111 \qquad\textbf{(C)}\ 112 \q... | [
"Without loss of generality, we assume the length of each side of $ABCD$ is $2$ .\nBecause $E$ is the midpoint of $CD$ $CE = 1$\nBecause $ABCD$ is a rhombus, $\\angle BCE = 180^\\circ - \\angle D$\nIn $\\triangle BCE$ , following from the law of sines, \\[ \\frac{CE}{\\sin \\angle FBC} = \\frac{BC}{\\sin \\angle BE... |
https://artofproblemsolving.com/wiki/index.php/2013_AIME_I_Problems/Problem_3 | null | 18 | Let $ABCD$ be a square, and let $E$ and $F$ be points on $\overline{AB}$ and $\overline{BC},$ respectively. The line through $E$ parallel to $\overline{BC}$ and the line through $F$ parallel to $\overline{AB}$ divide $ABCD$ into two squares and two nonsquare rectangles. The sum of the areas of the two squares is $\frac... | [
"It's important to note that $\\dfrac{AE}{EB} + \\dfrac{EB}{AE}$ is equivalent to $\\dfrac{AE^2 + EB^2}{(AE)(EB)}$\nWe define $a$ as the length of the side of larger inner square, which is also $EB$ $b$ as the length of the side of the smaller inner square which is also $AE$ , and $s$ as the side length of $ABCD$ .... |
https://artofproblemsolving.com/wiki/index.php/2024_AIME_I_Problems/Problem_14 | null | 104 | Let $ABCD$ be a tetrahedron such that $AB=CD= \sqrt{41}$ $AC=BD= \sqrt{80}$ , and $BC=AD= \sqrt{89}$ . There exists a point $I$ inside the tetrahedron such that the distances from $I$ to each of the faces of the tetrahedron are all equal. This distance can be written in the form $\frac{m \sqrt n}{p}$ , where $m$ $n$ , ... | [
"Notice that \\(41=4^2+5^2\\), \\(89=5^2+8^2\\), and \\(80=8^2+4^2\\), let \\(A~(0,0,0)\\), \\(B~(4,5,0)\\), \\(C~(0,5,8)\\), and \\(D~(4,0,8)\\). Then the plane \\(BCD\\) has a normal\n\\begin{equation*}\n\\mathbf n:=\\frac14\\overrightarrow{BC}\\times\\overrightarrow{CD}=\\frac14\\begin{pmatrix}-4\\\\0\\\\8\\end{... |
https://artofproblemsolving.com/wiki/index.php/2004_AIME_II_Problems/Problem_12 | null | 134 | Let $ABCD$ be an isosceles trapezoid , whose dimensions are $AB = 6, BC=5=DA,$ and $CD=4.$ Draw circles of radius 3 centered at $A$ and $B,$ and circles of radius 2 centered at $C$ and $D.$ A circle contained within the trapezoid is tangent to all four of these circles. Its radius is $\frac{-k+m\sqrt{n}}p,$ where $k, m... | [
"Let the radius of the center circle be $r$ and its center be denoted as $O$\nClearly line $AO$ passes through the point of tangency of circle $A$ and circle $O$ . Let $y$ be the height from the base of the trapezoid to $O$ . From the Pythagorean Theorem \\[3^2 + y^2 = (r + 3)^2 \\Longrightarrow y = \\sqrt {r^2 + 6... |
https://artofproblemsolving.com/wiki/index.php/2021_AIME_I_Problems/Problem_9 | null | 567 | Let $ABCD$ be an isosceles trapezoid with $AD=BC$ and $AB<CD.$ Suppose that the distances from $A$ to the lines $BC,CD,$ and $BD$ are $15,18,$ and $10,$ respectively. Let $K$ be the area of $ABCD.$ Find $\sqrt2 \cdot K.$ | [
"Let $\\overline{AE}, \\overline{AF},$ and $\\overline{AG}$ be the perpendiculars from $A$ to $\\overleftrightarrow{BC}, \\overleftrightarrow{CD},$ and $\\overleftrightarrow{BD},$ respectively. Next, let $H$ be the intersection of $\\overline{AF}$ and $\\overline{BD}.$\nWe set $AB=x$ and $AH=y,$ as shown below. Fr... |
https://artofproblemsolving.com/wiki/index.php/2008_AIME_I_Problems/Problem_10 | null | 32 | Let $ABCD$ be an isosceles trapezoid with $\overline{AD}||\overline{BC}$ whose angle at the longer base $\overline{AD}$ is $\dfrac{\pi}{3}$ . The diagonals have length $10\sqrt {21}$ , and point $E$ is at distances $10\sqrt {7}$ and $30\sqrt {7}$ from vertices $A$ and $D$ , respectively. Let $F$ be the foot of the alti... | [
"Key observation. $AD = 20\\sqrt{7}$\nProof 1. By the triangle inequality , we can immediately see that $AD \\geq 20\\sqrt{7}$ . However, notice that $10\\sqrt{21} = 20\\sqrt{7}\\cdot\\sin\\frac{\\pi}{3}$ , so by the law of sines, when $AD = 20\\sqrt{7}$ $\\angle ACD$ is right and the circle centered at $A$ with ra... |
https://artofproblemsolving.com/wiki/index.php/2004_AIME_II_Problems/Problem_13 | null | 484 | Let $ABCDE$ be a convex pentagon with $AB \parallel CE, BC \parallel AD, AC \parallel DE, \angle ABC=120^\circ, AB=3, BC=5,$ and $DE = 15.$ Given that the ratio between the area of triangle $ABC$ and the area of triangle $EBD$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$ | [
"Let the intersection of $\\overline{AD}$ and $\\overline{CE}$ be $F$ . Since $AB \\parallel CE, BC \\parallel AD,$ it follows that $ABCF$ is a parallelogram , and so $\\triangle ABC \\cong \\triangle CFA$ . Also, as $AC \\parallel DE$ , it follows that $\\triangle ABC \\sim \\triangle EFD$\nBy the Law of Cosines $... |
https://artofproblemsolving.com/wiki/index.php/2013_AMC_12B_Problems/Problem_16 | A | 0 | Let $ABCDE$ be an equiangular convex pentagon of perimeter $1$ . The pairwise intersections of the lines that extend the sides of the pentagon determine a five-pointed star polygon. Let $s$ be the perimeter of this star. What is the difference between the maximum and the minimum possible values of $s$
$\textbf{(A)}\ 0 ... | [
"The five pointed star can be thought of as five triangles sitting on the five sides of the pentagon. Because the pentagon is equiangular, each of its angles has measure $\\frac{180^\\circ (5-2)}{5}=108^\\circ$ , and so the base angles of the aforementioned triangles (i.e., the angles adjacent to the pentagon) have... |
https://artofproblemsolving.com/wiki/index.php/2010_AIME_II_Problems/Problem_9 | null | 11 | Let $ABCDEF$ be a regular hexagon . Let $G$ $H$ $I$ $J$ $K$ , and $L$ be the midpoints of sides $AB$ $BC$ $CD$ $DE$ $EF$ , and $AF$ , respectively. The segments $\overline{AH}$ $\overline{BI}$ $\overline{CJ}$ $\overline{DK}$ $\overline{EL}$ , and $\overline{FG}$ bound a smaller regular hexagon. Let the ratio of the are... | [
"Without loss of generality, let $BC=2.$\nNote that $\\angle BMH$ is the vertical angle to an angle of the regular hexagon, so it has a measure of $120^\\circ$\nBecause $\\triangle ABH$ and $\\triangle BCI$ are rotational images of one another, we get that $\\angle{MBH}=\\angle{HAB}$ and hence $\\triangle ABH \\sim... |
https://artofproblemsolving.com/wiki/index.php/2021_AMC_10A_Problems/Problem_21 | C | 55 | Let $ABCDEF$ be an equiangular hexagon. The lines $AB, CD,$ and $EF$ determine a triangle with area $192\sqrt{3}$ , and the lines $BC, DE,$ and $FA$ determine a triangle with area $324\sqrt{3}$ . The perimeter of hexagon $ABCDEF$ can be expressed as $m +n\sqrt{p}$ , where $m, n,$ and $p$ are positive integers and $p$ i... | [
"Let $P,Q,R,X,Y,$ and $Z$ be the intersections $\\overleftrightarrow{AB}\\cap\\overleftrightarrow{CD},\\overleftrightarrow{CD}\\cap\\overleftrightarrow{EF},\\overleftrightarrow{EF}\\cap\\overleftrightarrow{AB},\\overleftrightarrow{BC}\\cap\\overleftrightarrow{DE},\\overleftrightarrow{DE}\\cap\\overleftrightarrow{FA... |
https://artofproblemsolving.com/wiki/index.php/2008_AMC_12B_Problems/Problem_24 | C | 17 | Let $A_0=(0,0)$ . Distinct points $A_1,A_2,\dots$ lie on the $x$ -axis, and distinct points $B_1,B_2,\dots$ lie on the graph of $y=\sqrt{x}$ . For every positive integer $n,\ A_{n-1}B_nA_n$ is an equilateral triangle. What is the least $n$ for which the length $A_0A_n\geq100$
$\textbf{(A)}\ 13\qquad \textbf{(B)}\ 15\qq... | [
"We can iteratively calculate out the first few $A_i$ and $B_i$ . We know that $A_0 = (0,0)$ and the line through $A_0$ and $B_1$ needs to make a $60^{\\circ}$ angle with the x-axis (because the triangle is equilateral). The equation of a line that makes an angle $\\theta$ with the x-axis and passes through the ori... |
https://artofproblemsolving.com/wiki/index.php/2008_AMC_12B_Problems/Problem_24 | null | 17 | Let $A_0=(0,0)$ . Distinct points $A_1,A_2,\dots$ lie on the $x$ -axis, and distinct points $B_1,B_2,\dots$ lie on the graph of $y=\sqrt{x}$ . For every positive integer $n,\ A_{n-1}B_nA_n$ is an equilateral triangle. What is the least $n$ for which the length $A_0A_n\geq100$
$\textbf{(A)}\ 13\qquad \textbf{(B)}\ 15\qq... | [
"Let $a_n=|A_{n-1}A_n|$ . We need to rewrite the recursion into something manageable. The two strange conditions, $B$ 's lie on the graph of $y=\\sqrt{x}$ and $A_{n-1}B_nA_n$ is an equilateral triangle, can be compacted as follows: \\[\\left(a_n\\frac{\\sqrt{3}}{2}\\right)^2=\\frac{a_n}{2}+a_{n-1}+a_{n-2}+\\cdots+a... |
https://artofproblemsolving.com/wiki/index.php/2011_AIME_I_Problems/Problem_14 | null | 37 | Let $A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8$ be a regular octagon. Let $M_1$ $M_3$ $M_5$ , and $M_7$ be the midpoints of sides $\overline{A_1 A_2}$ $\overline{A_3 A_4}$ $\overline{A_5 A_6}$ , and $\overline{A_7 A_8}$ , respectively. For $i = 1, 3, 5, 7$ , ray $R_i$ is constructed from $M_i$ towards the interior of the octag... | [
"We use coordinates. Let the octagon have side length $2$ and center $(0, 0)$ . Then all of its vertices have the form $(\\pm 1, \\pm\\left(1+\\sqrt{2}\\right))$ or $(\\pm\\left(1+\\sqrt{2}\\right), \\pm 1)$\nBy symmetry, $B_{1}B_{3}B_{5}B_{7}$ is a square. Thus lines $\\overleftrightarrow{B_{1}B_{3}}$ and $\\overl... |
https://artofproblemsolving.com/wiki/index.php/2002_AIME_I_Problems/Problem_5 | null | 183 | Let $A_1,A_2,A_3,\cdots,A_{12}$ be the vertices of a regular dodecagon. How many distinct squares in the plane of the dodecagon have at least two vertices in the set $\{A_1,A_2,A_3,\cdots,A_{12}\} ?$ | [
"Proceed as above to initially get 198 squares (with overcounting). Then note that any square with all four vertices on the dodecagon has to have three sides \"between\" each vertex, giving us a total of three squares. However, we counted these squares with all four of their sides plus both of their diagonals, mean... |
https://artofproblemsolving.com/wiki/index.php/2021_AIME_I_Problems/Problem_12 | null | 19 | Let $A_1A_2A_3\ldots A_{12}$ be a dodecagon ( $12$ -gon). Three frogs initially sit at $A_4,A_8,$ and $A_{12}$ . At the end of each minute, simultaneously, each of the three frogs jumps to one of the two vertices adjacent to its current position, chosen randomly and independently with both choices being equally likely.... | [
"Define the distance between two frogs as the number of sides between them that do not contain the third frog.\nLet $E(a,b,c)$ denote the expected number of minutes until the frogs stop jumping, such that the distances between the frogs are $a,b,$ and $c$ (in either clockwise or counterclockwise order). Without the... |
https://artofproblemsolving.com/wiki/index.php/2020_AMC_10B_Problems/Problem_20 | B | 19 | Let $B$ be a right rectangular prism (box) with edges lengths $1,$ $3,$ and $4$ , together with its interior. For real $r\geq0$ , let $S(r)$ be the set of points in $3$ -dimensional space that lie within a distance $r$ of some point in $B$ . The volume of $S(r)$ can be expressed as $ar^{3} + br^{2} + cr +d$ , where $a,... | [
"Split $S(r)$ into 4 regions:\n1. The rectangular prism itself\n2. The extensions of the faces of $B$\n3. The quarter cylinders at each edge of $B$\n4. The one-eighth spheres at each corner of $B$\nRegion 1: The volume of $B$ is $1 \\cdot 3 \\cdot 4 = 12$ , so $d=12$\nRegion 2: This volume is equal to the surface a... |
https://artofproblemsolving.com/wiki/index.php/2012_AIME_I_Problems/Problem_5 | null | 330 | Let $B$ be the set of all binary integers that can be written using exactly $5$ zeros and $8$ ones where leading zeros are allowed. If all possible subtractions are performed in which one element of $B$ is subtracted from another, find the number of times the answer $1$ is obtained. | [
"When $1$ is subtracted from a binary number, the number of digits will remain constant if and only if the original number ended in $10.$ Therefore, every subtraction involving two numbers from $B$ will necessarily involve exactly one number ending in $10.$ To solve the problem, then, we can simply count the instan... |
https://artofproblemsolving.com/wiki/index.php/2004_AIME_I_Problems/Problem_7 | null | 588 | Let $C$ be the coefficient of $x^2$ in the expansion of the product $(1 - x)(1 + 2x)(1 - 3x)\cdots(1 + 14x)(1 - 15x).$ Find $|C|.$ | [
"Let our polynomial be $P(x)$\nIt is clear that the coefficient of $x$ in $P(x)$ is $-1 + 2 - 3 + \\ldots + 14 - 15 = -8$ , so $P(x) = 1 -8x + Cx^2 + Q(x)$ , where $Q(x)$ is some polynomial divisible by $x^3$\nThen $P(-x) = 1 + 8x + Cx^2 + Q(-x)$ and so $P(x)\\cdot P(-x) = 1 + (2C - 64)x^2 + R(x)$ , where $R(x)$ is... |
https://artofproblemsolving.com/wiki/index.php/1988_AIME_Problems/Problem_14 | null | 84 | Let $C$ be the graph of $xy = 1$ , and denote by $C^*$ the reflection of $C$ in the line $y = 2x$ . Let the equation of $C^*$ be written in the form
\[12x^2 + bxy + cy^2 + d = 0.\]
Find the product $bc$ | [
"Given a point $P (x,y)$ on $C$ , we look to find a formula for $P' (x', y')$ on $C^*$ . Both points lie on a line that is perpendicular to $y=2x$ , so the slope of $\\overline{PP'}$ is $\\frac{-1}{2}$ . Thus $\\frac{y' - y}{x' - x} = \\frac{-1}{2} \\Longrightarrow x' + 2y' = x + 2y$ . Also, the midpoint of $\\over... |
https://artofproblemsolving.com/wiki/index.php/2002_AMC_12A_Problems/Problem_18 | null | 20 | Let $C_1$ and $C_2$ be circles defined by $(x-10)^2 + y^2 = 36$ and $(x+15)^2 + y^2 = 81$ respectively. What is the length of the shortest line segment $PQ$ that is tangent to $C_1$ at $P$ and to $C_2$ at $Q$
$\text{(A) }15 \qquad \text{(B) }18 \qquad \text{(C) }20 \qquad \text{(D) }21 \qquad \text{(E) }24$ | [
"First examine the formula $(x-10)^2+y^2=36$ , for the circle $C_1$ . Its center, $D_1$ , is located at (10,0) and it has a radius of $\\sqrt{36}$ = 6. The next circle, using the same pattern, has its center, $D_2$ , at (-15,0) and has a radius of $\\sqrt{81}$ = 9. So we can construct this diagram: Line PQ is tang... |
https://artofproblemsolving.com/wiki/index.php/2020_AMC_10B_Problems/Problem_25 | A | 112 | Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product \[n = f_1\cdot f_2\cdots f_k,\] where $k\ge1$ , the $f_i$ are integers strictly greater than $1$ , and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are co... | [
"Note that $96 = 2^5 \\cdot 3$ . Since there are at most six not necessarily distinct factors $>1$ multiplying to $96$ , we have six cases: $k=1, 2, ..., 6.$ Now we look at each of the six cases.\n$k=1$ : We see that there is $1$ way, merely $96$\n$k=2$ : This way, we have the $3$ in one slot and $2$ in another, an... |
https://artofproblemsolving.com/wiki/index.php/2020_AMC_10B_Problems/Problem_25 | null | 112 | Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product \[n = f_1\cdot f_2\cdots f_k,\] where $k\ge1$ , the $f_i$ are integers strictly greater than $1$ , and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are co... | [
"Ignore the $3$ first and first count $2^{x_1+x_2+...+x^n}=32$ which $x_1+x_2+...+x_n=5$ . This implies that $n$ is less than or equal to $5$ . Now, we can see that $3$ can lie between the two $x_i, x_{i+1}$ , or contribute to one of them. This gives $2k+1$ if $x_1+...+x_k=5$ . Now, just sum up gives $\\binom{5-1}{... |
https://artofproblemsolving.com/wiki/index.php/2020_AMC_12B_Problems/Problem_24 | A | 112 | Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product \[n = f_1\cdot f_2\cdots f_k,\] where $k\ge1$ , the $f_i$ are integers strictly greater than $1$ , and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are co... | [
"Note that $96 = 2^5 \\cdot 3$ . Since there are at most six not necessarily distinct factors $>1$ multiplying to $96$ , we have six cases: $k=1, 2, ..., 6.$ Now we look at each of the six cases.\n$k=1$ : We see that there is $1$ way, merely $96$\n$k=2$ : This way, we have the $3$ in one slot and $2$ in another, an... |
https://artofproblemsolving.com/wiki/index.php/2020_AMC_12B_Problems/Problem_24 | null | 112 | Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product \[n = f_1\cdot f_2\cdots f_k,\] where $k\ge1$ , the $f_i$ are integers strictly greater than $1$ , and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are co... | [
"Ignore the $3$ first and first count $2^{x_1+x_2+...+x^n}=32$ which $x_1+x_2+...+x_n=5$ . This implies that $n$ is less than or equal to $5$ . Now, we can see that $3$ can lie between the two $x_i, x_{i+1}$ , or contribute to one of them. This gives $2k+1$ if $x_1+...+x_k=5$ . Now, just sum up gives $\\binom{5-1}{... |
https://artofproblemsolving.com/wiki/index.php/1996_AHSME_Problems/Problem_14 | C | 400 | Let $E(n)$ denote the sum of the even digits of $n$ . For example, $E(5681) = 6+8 = 14$ . Find $E(1)+E(2)+E(3)+\cdots+E(100)$
$\text{(A)}\ 200\qquad\text{(B)}\ 360\qquad\text{(C)}\ 400\qquad\text{(D)}\ 900\qquad\text{(E)}\ 2250$ | [
"The problem is asking for the sum of all the even digits in the numbers $1$ to $100$ . We can remove $100$ from the list, add $00$ to the list, and tack on some leading zeros to the single digit numbers without changing the sum of the even digits. This gives the list:\n$00, 01, 02, 03, ..., 10, 11, ..., 98, 99$\... |
https://artofproblemsolving.com/wiki/index.php/2001_AIME_II_Problems/Problem_15 | null | 417 | Let $EFGH$ $EFDC$ , and $EHBC$ be three adjacent square faces of a cube , for which $EC = 8$ , and let $A$ be the eighth vertex of the cube. Let $I$ $J$ , and $K$ , be the points on $\overline{EF}$ $\overline{EH}$ , and $\overline{EC}$ , respectively, so that $EI = EJ = EK = 2$ . A solid $S$ is obtained by drilling a t... | [
"Set the coordinate system so that vertex $E$ , where the drilling starts, is at $(8,8,8)$ . Using a little visualization (involving some similar triangles , because we have parallel lines) shows that the tunnel meets the bottom face (the xy plane one) in the line segments joining $(1,0,0)$ to $(2,2,0)$ , and $(0,1... |
https://artofproblemsolving.com/wiki/index.php/2002_AIME_I_Problems/Problem_12 | null | 275 | Let $F(z)=\dfrac{z+i}{z-i}$ for all complex numbers $z\neq i$ , and let $z_n=F(z_{n-1})$ for all positive integers $n$ . Given that $z_0=\dfrac{1}{137}+i$ and $z_{2002}=a+bi$ , where $a$ and $b$ are real numbers, find $a+b$ | [
"Iterating $F$ we get:\n\\begin{align*} F(z) &= \\frac{z+i}{z-i}\\\\ F(F(z)) &= \\frac{\\frac{z+i}{z-i}+i}{\\frac{z+i}{z-i}-i} = \\frac{(z+i)+i(z-i)}{(z+i)-i(z-i)}= \\frac{z+i+zi+1}{z+i-zi-1}= \\frac{(z+1)(i+1)}{(z-1)(1-i)}\\\\ &= \\frac{(z+1)(i+1)^2}{(z-1)(1^2+1^2)}= \\frac{(z+1)(2i)}{(z-1)(2)}= \\frac{z+1}{z-1}i\... |
https://artofproblemsolving.com/wiki/index.php/2013_AMC_12B_Problems/Problem_25 | B | 528 | Let $G$ be the set of polynomials of the form \[P(z)=z^n+c_{n-1}z^{n-1}+\cdots+c_2z^2+c_1z+50,\] where $c_1,c_2,\cdots, c_{n-1}$ are integers and $P(z)$ has distinct roots of the form $a+ib$ with $a$ and $b$ integers. How many polynomials are in $G$
$\textbf{(A)}\ 288\qquad\textbf{(B)}\ 528\qquad\textbf{(C)}\ 576\qquad... | [
"If we factor into irreducible polynomials (in $\\mathbb{Q}[x]$ ), each factor $f_i$ has exponent $1$ in the factorization and degree at most $2$ (since the $a+bi$ with $b\\ne0$ come in conjugate pairs with product $a^2+b^2$ ). Clearly we want the product of constant terms of these polynomials to equal $50$ ; for $... |
https://artofproblemsolving.com/wiki/index.php/2013_AMC_12B_Problems/Problem_25 | null | 528 | Let $G$ be the set of polynomials of the form \[P(z)=z^n+c_{n-1}z^{n-1}+\cdots+c_2z^2+c_1z+50,\] where $c_1,c_2,\cdots, c_{n-1}$ are integers and $P(z)$ has distinct roots of the form $a+ib$ with $a$ and $b$ integers. How many polynomials are in $G$
$\textbf{(A)}\ 288\qquad\textbf{(B)}\ 528\qquad\textbf{(C)}\ 576\qquad... | [
"Disregard sign; we can tack on $x-1$ if the product ends up being negative.\n$1: \\pm i,-1$ (2) (1 is not included)\n$2: \\pm 2, \\pm 1\\pm i$ (4)\n$5: \\pm 2\\pm i, \\pm 1\\pm 2i, \\pm 5$ (6)\n$10: \\pm 3\\pm i, \\pm 1\\pm 3i, \\pm 10$ (6)\n$25: \\pm 25, \\pm 3\\pm 4i, \\pm 4\\pm 3i, \\pm 5i$ (7)\n$50: \\pm 50, \... |
https://artofproblemsolving.com/wiki/index.php/1969_AHSME_Problems/Problem_22 | C | 36.5 | Let $K$ be the measure of the area bounded by the $x$ -axis, the line $x=8$ , and the curve defined by
\[f={(x,y)\quad |\quad y=x \text{ when } 0 \le x \le 5, y=2x-5 \text{ when } 5 \le x \le 8}.\]
Then $K$ is:
$\text{(A) } 21.5\quad \text{(B) } 36.4\quad \text{(C) } 36.5\quad \text{(D) } 44\quad \text{(E) less tha... | [
"\nThe shape can be divided into a triangle and a trapezoid. For the triangle, the base is $5$ and the height is $5$ , so the area is $\\frac{5 \\cdot 5}{2} = \\frac{25}{2}$ . For the trapezoid, the two bases are $5$ and $11$ and the height is $3$ , so the area is $\\frac{3(5+11)}{2} = 24$ . Thus, the total area... |
https://artofproblemsolving.com/wiki/index.php/2023_AMC_12A_Problems/Problem_24 | C | 5 | Let $K$ be the number of sequences $A_1$ $A_2$ $\dots$ $A_n$ such that $n$ is a positive integer less than or equal to $10$ , each $A_i$ is a subset of $\{1, 2, 3, \dots, 10\}$ , and $A_{i-1}$ is a subset of $A_i$ for each $i$ between $2$ and $n$ , inclusive. For example, $\{\}$ $\{5, 7\}$ $\{2, 5, 7\}$ $\{2, 5, 7\}$ $... | [
"Consider any sequence with $n$ terms. Every 10 number has such choices: never appear, appear the first time in the first spot, appear the first time in the second spot… and appear the first time in the $n$ th spot, which means every number has $(n+1)$ choices to show up in the sequence. Consequently, for each seq... |
https://artofproblemsolving.com/wiki/index.php/2010_AIME_II_Problems/Problem_3 | null | 150 | Let $K$ be the product of all factors $(b-a)$ (not necessarily distinct) where $a$ and $b$ are integers satisfying $1\le a < b \le 20$ . Find the greatest positive integer $n$ such that $2^n$ divides $K$ | [
"In general, there are $20-n$ pairs of integers $(a, b)$ that differ by $n$ because we can let $b$ be any integer from $n+1$ to $20$ and set $a$ equal to $b-n$ . Thus, the product is $(1^{19})(2^{18})\\cdots(19^1)$ (or alternatively, $19! \\cdot 18! \\cdots 1!$ .)\nWhen we count the number of factors of $2$ , we ha... |
https://artofproblemsolving.com/wiki/index.php/2011_AIME_I_Problems/Problem_3 | null | 31 | Let $L$ be the line with slope $\frac{5}{12}$ that contains the point $A=(24,-1)$ , and let $M$ be the line perpendicular to line $L$ that contains the point $B=(5,6)$ . The original coordinate axes are erased, and line $L$ is made the $x$ -axis and line $M$ the $y$ -axis. In the new coordinate system, point $A$ is o... | [
"Given that $L$ has slope $\\frac{5}{12}$ and contains the point $A=(24,-1)$ , we may write the point-slope equation for $L$ as $y+1=\\frac{5}{12}(x-24)$ .\nSince $M$ is perpendicular to $L$ and contains the point $B=(5,6)$ , we have that the slope of $M$ is $-\\frac{12}{5}$ , and consequently that the point-slope... |
https://artofproblemsolving.com/wiki/index.php/2023_AMC_10B_Problems/Problem_6 | E | 674 | Let $L_{1}=1, L_{2}=3$ , and $L_{n+2}=L_{n+1}+L_{n}$ for $n\geq 1$ . How many terms in the sequence $L_{1}, L_{2}, L_{3},...,L_{2023}$ are even?
$\textbf{(A) }673\qquad\textbf{(B) }1011\qquad\textbf{(C) }675\qquad\textbf{(D) }1010\qquad\textbf{(E) }674$ | [
"We calculate more terms: \\[1,3,4,7,11,18,\\ldots.\\] We find a pattern: if $n$ is a multiple of $3$ , then the term is even, or else it is odd. There are $\\left\\lfloor \\frac{2023}{3} \\right\\rfloor =\\boxed{674}$ multiples of $3$ from $1$ to $2023$",
"Like in the other solution, we find a pattern, except in... |
https://artofproblemsolving.com/wiki/index.php/2021_Fall_AMC_12A_Problems/Problem_8 | D | 74 | Let $M$ be the least common multiple of all the integers $10$ through $30,$ inclusive. Let $N$ be the least common multiple of $M,32,33,34,35,36,37,38,39,$ and $40.$ What is the value of $\frac{N}{M}?$
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 37 \qquad\textbf{(D)}\ 74 \qquad\textbf{(E)}\ 2886$ | [
"By the definition of least common mutiple, we take the greatest powers of the prime numbers of the prime factorization of all the numbers, that we are taking the $\\text{lcm}$ of. In this case, \\[M = 2^4 \\cdot 3^3 \\cdot 5^2 \\cdot 7 \\cdot 11 \\cdot 13 \\cdot 17 \\cdot 19 \\cdot 23 \\cdot 29.\\] Now, using the ... |
https://artofproblemsolving.com/wiki/index.php/2022_AMC_12A_Problems/Problem_12 | B | 13 | Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$ . What is $\cos(\angle CMD)$
$\textbf{(A) } \frac14 \qquad \textbf{(B) } \frac13 \qquad \textbf{(C) } \frac25 \qquad \textbf{(D) } \frac12 \qquad \textbf{(E) } \frac{\sqrt{3}}{2}$ | [
"Without loss of generality, let the edge-length of $ABCD$ be $2.$ It follows that $MC=MD=\\sqrt3.$\nLet $O$ be the center of $\\triangle ABD,$ so $\\overline{CO}\\perp\\overline{MOD}.$ Note that $MO=\\frac13 MD=\\frac{\\sqrt{3}}{3}.$\nIn right $\\triangle CMO,$ we have \\[\\cos(\\angle CMD)=\\frac{MO}{MC}=\\boxed{... |
https://artofproblemsolving.com/wiki/index.php/2011_AIME_II_Problems/Problem_11 | null | 73 | Let $M_n$ be the $n \times n$ matrix with entries as follows: for $1 \le i \le n$ $m_{i,i} = 10$ ; for $1 \le i \le n - 1$ $m_{i+1,i} = m_{i,i+1} = 3$ ; all other entries in $M_n$ are zero. Let $D_n$ be the determinant of matrix $M_n$ . Then $\sum_{n=1}^{\infty} \frac{1}{8D_n+1}$ can be represented as $\frac{p}{q}$ , w... | [
"\\[D_{1}=\\begin{vmatrix} 10 \\end{vmatrix} = 10, \\quad D_{2}=\\begin{vmatrix} 10 & 3 \\\\ 3 & 10 \\\\ \\end{vmatrix} =(10)(10) - (3)(3) = 91, \\quad D_{3}=\\begin{vmatrix} 10 & 3 & 0 \\\\ 3 & 10 & 3 \\\\ 0 & 3 & 10 \\\\ \\end{vmatrix}.\\] Using the expansionary/recursive definition of determinants ... |
https://artofproblemsolving.com/wiki/index.php/2008_AIME_II_Problems/Problem_1 | null | 100 | Let $N = 100^2 + 99^2 - 98^2 - 97^2 + 96^2 + \cdots + 4^2 + 3^2 - 2^2 - 1^2$ , where the additions and subtractions alternate in pairs. Find the remainder when $N$ is divided by $1000$ | [
"Rewriting this sequence with more terms, we have\nFactoring this expression yields\nNext, we get\nThen,\nDividing $10100$ by $1000$ yields a remainder of $\\boxed{100}$",
"By observation, we realize that the sequence \\[(a+3)^2 + (a+2)^2 - (a+1)^2 - a^2\\] alternates every 4 terms. Simplifying, we get \\[(a+3)^2... |
https://artofproblemsolving.com/wiki/index.php/2007_AIME_I_Problems/Problem_7 | null | 477 | Let $N = \sum_{k = 1}^{1000} k ( \lceil \log_{\sqrt{2}} k \rceil - \lfloor \log_{\sqrt{2}} k \rfloor )$
Find the remainder when $N$ is divided by 1000. ( $\lfloor{k}\rfloor$ is the greatest integer less than or equal to $k$ , and $\lceil{k}\rceil$ is the least integer greater than or equal to $k$ .) | [
"The ceiling of a number minus the floor of a number is either equal to zero (if the number is an integer ); otherwise, it is equal to 1. Thus, we need to find when or not $\\log_{\\sqrt{2}} k$ is an integer.\nThe change of base formula shows that $\\frac{\\log k}{\\log \\sqrt{2}} = \\frac{2 \\log k}{\\log 2}$ . Fo... |
https://artofproblemsolving.com/wiki/index.php/2016_AMC_10A_Problems/Problem_17 | A | 12 | Let $N$ be a positive multiple of $5$ . One red ball and $N$ green balls are arranged in a line in random order. Let $P(N)$ be the probability that at least $\tfrac{3}{5}$ of the green balls are on the same side of the red ball. Observe that $P(5)=1$ and that $P(N)$ approaches $\tfrac{4}{5}$ as $N$ grows large. What is... | [
"Let $n = \\frac{N}{5}$ . Then, consider $5$ blocks of $n$ green balls in a line, along with the red ball. Shuffling the line is equivalent to choosing one of the $N + 1$ positions between the green balls to insert the red ball. Less than $\\frac{3}{5}$ of the green balls will be on the same side of the red ball if... |
https://artofproblemsolving.com/wiki/index.php/2016_AMC_12A_Problems/Problem_13 | A | 12 | Let $N$ be a positive multiple of $5$ . One red ball and $N$ green balls are arranged in a line in random order. Let $P(N)$ be the probability that at least $\tfrac{3}{5}$ of the green balls are on the same side of the red ball. Observe that $P(5)=1$ and that $P(N)$ approaches $\tfrac{4}{5}$ as $N$ grows large. What is... | [
"Let $n = \\frac{N}{5}$ . Then, consider $5$ blocks of $n$ green balls in a line, along with the red ball. Shuffling the line is equivalent to choosing one of the $N + 1$ positions between the green balls to insert the red ball. Less than $\\frac{3}{5}$ of the green balls will be on the same side of the red ball if... |
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