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/2014_AMC_10A_Problems/Problem_24 | A | 996,506 | A sequence of natural numbers is constructed by listing the first $4$ , then skipping one, listing the next $5$ , skipping $2$ , listing $6$ , skipping $3$ , and on the $n$ th iteration, listing $n+3$ and skipping $n$ . The sequence begins $1,2,3,4,6,7,8,9,10,13$ . What is the $500,\!000$ th number in the sequence?
$\textbf{(A)}\ 996,\!506\qquad\textbf{(B)}\ 996,\!507\qquad\textbf{(C)}\ 996,\!508\qquad\textbf{(D)}\ 996,\!509\qquad\textbf{(E)}\ 996,\!510$ | [
"If we list the rows by iterations, then we get\n$1,2,3,4$\n$6,7,8,9,10$\n$13,14,15,16,17,18$ etc.\nso that the $500,000$ th number is the $506$ th number on the $997$ th row because $4+5+6+7......+999 = 499,494$ . The last number of the $996$ th row (when including the numbers skipped) is $499,494 + (1+2+3+4.....+... |
https://artofproblemsolving.com/wiki/index.php/2009_AMC_8_Problems/Problem_5 | D | 68 | A sequence of numbers starts with $1$ $2$ , and $3$ . The fourth number of the sequence is the sum of the previous three numbers in the sequence: $1+2+3=6$ . In the same way, every number after the fourth is the sum of the previous three numbers. What is the eighth number in the sequence?
$\textbf{(A)}\ 11 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 37 \qquad \textbf{(D)}\ 68 \qquad \textbf{(E)}\ 99$ | [
"List them out, adding the three previous numbers to get the next number,\n\\[1,2,3,6,11,20,37,\\boxed{68}\\]"
] |
https://artofproblemsolving.com/wiki/index.php/2002_AMC_8_Problems/Problem_11 | C | 13 | A sequence of squares is made of identical square tiles. The edge of each square is one tile length longer than the edge of the previous square. The first three squares are shown. How many more tiles does the seventh square require than the sixth?
[asy] path p=origin--(1,0)--(1,1)--(0,1)--cycle; draw(p); draw(shift(3,0)*p); draw(shift(3,1)*p); draw(shift(4,0)*p); draw(shift(4,1)*p); draw(shift(7,0)*p); draw(shift(7,1)*p); draw(shift(7,2)*p); draw(shift(8,0)*p); draw(shift(8,1)*p); draw(shift(8,2)*p); draw(shift(9,0)*p); draw(shift(9,1)*p); draw(shift(9,2)*p);[/asy]
$\text{(A)}\ 11 \qquad \text{(B)}\ 12 \qquad \text{(C)}\ 13 \qquad \text{(D)}\ 14 \qquad \text{(E)}\ 15$ | [
"The first square has a sidelength of $1$ , the second square $2$ , and so on. The seventh square has $7$ and is made of $7^2=49$ unit tiles. The sixth square has $6$ and is made of $6^2=36$ unit tiles. The seventh square has $49-36=\\boxed{13}$ more tiles than the sixth square.",
"The edge of each square is one ... |
https://artofproblemsolving.com/wiki/index.php/2017_AMC_12A_Problems/Problem_21 | D | 9 | A set $S$ is constructed as follows. To begin, $S = \{0,10\}$ . Repeatedly, as long as possible, if $x$ is an integer root of some polynomial $a_{n}x^n + a_{n-1}x^{n-1} + \dots + a_{1}x + a_0$ for some $n\geq{1}$ , all of whose coefficients $a_i$ are elements of $S$ , then $x$ is put into $S$ . When no more elements can be added to $S$ , how many elements does $S$ have?
$\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 9 \qquad\textbf{(E)}\ 11$ | [
"At first, $S=\\{0,10\\}$\n\\[\\begin{tabular}{r c l c l} \\(10x+10\\) & has root & \\(x=-1\\) & so now & \\(S=\\{-1,0,10\\}\\) \\\\ \\(-x^{10}-x^9-x^8-x^7-x^6-x^5-x^4-x^3-x^2-x+10\\) & has root & \\(x=1\\) & so now & \\(S=\\{-1,0,1,10\\}\\) \\\\ \\(x+10\\) & has root & \\(x=-10\\) & so now & \\(S=\\{-10,-1,0,1,10\... |
https://artofproblemsolving.com/wiki/index.php/2003_AMC_12A_Problems/Problem_9 | D | 8 | A set $S$ of points in the $xy$ -plane is symmetric about the origin, both coordinate axes, and the line $y=x$ . If $(2,3)$ is in $S$ , what is the smallest number of points in $S$
$\mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 8\qquad \mathrm{(E) \ } 16$ | [
"If $(2,3)$ is in $S$ , then $(3,2)$ is also, and quickly we see that every point of the form $(\\pm 2, \\pm 3)$ or $(\\pm 3, \\pm 2)$ must be in $S$ . Now note that these $8$ points satisfy all of the symmetry conditions. Thus the answer is $\\boxed{8}$"
] |
https://artofproblemsolving.com/wiki/index.php/2014_AMC_12B_Problems/Problem_12 | B | 9 | A set S consists of triangles whose sides have integer lengths less than 5, and no two elements of S are congruent or similar. What is the largest number of elements that S can have?
$\textbf{(A)}\ 8\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 12$ | [
"Define $T$ to be the set of all integral triples $(a, b, c)$ such that $a \\ge b \\ge c$ $b+c > a$ , and $a, b, c < 5$ . Now we enumerate the elements of $T$\n$(4, 4, 4)$\n$(4, 4, 3)$\n$(4, 4, 2)$\n$(4, 4, 1)$\n$(4, 3, 3)$\n$(4, 3, 2)$\n$(3, 3, 3)$\n$(3, 3, 2)$\n$(3, 3, 1)$\n$(3, 2, 2)$\n$(2, 2, 2)$\n$(2, 2, 1)$\n... |
https://artofproblemsolving.com/wiki/index.php/2017_AIME_II_Problems/Problem_5 | null | 791 | A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are $189$ $320$ $287$ $234$ $x$ , and $y$ . Find the greatest possible value of $x+y$ | [
"Let these four numbers be $a$ $b$ $c$ , and $d$ , where $a>b>c>d$ $x+y$ needs to be maximized, so let $x=a+b$ and $y=a+c$ because these are the two largest pairwise sums. Now $x+y=2a+b+c$ needs to be maximized. Notice that $2a+b+c=3(a+b+c+d)-(a+2b+2c+3d)=3((a+c)+(b+d))-((a+d)+(b+c)+(b+d)+(c+d))$ . No matter how th... |
https://artofproblemsolving.com/wiki/index.php/2007_AMC_10B_Problems/Problem_20 | C | 600 | A set of $25$ square blocks is arranged into a $5 \times 5$ square. How many different combinations of $3$ blocks can be selected from that set so that no two are in the same row or column?
$\textbf{(A) } 100 \qquad\textbf{(B) } 125 \qquad\textbf{(C) } 600 \qquad\textbf{(D) } 2300 \qquad\textbf{(E) } 3600$ | [
"There are $25$ ways to choose the first square. The four remaining squares in its row and column and the square you chose exclude nine squares from being chosen next time.\nThere are $16$ remaining blocks to be chosen for the second square. The three remaining spaces in its row and column and the square you chose ... |
https://artofproblemsolving.com/wiki/index.php/2017_AMC_12B_Problems/Problem_25 | D | 557 | A set of $n$ people participate in an online video basketball tournament. Each person may be a member of any number of $5$ -player teams, but no two teams may have exactly the same $5$ members. The site statistics show a curious fact: The average, over all subsets of size $9$ of the set of $n$ participants, of the number of complete teams whose members are among those $9$ people is equal to the reciprocal of the average, over all subsets of size $8$ of the set of $n$ participants, of the number of complete teams whose members are among those $8$ people. How many values $n$ $9\leq n\leq 2017$ , can be the number of participants?
$\textbf{(A) } 477 \qquad \textbf{(B) } 482 \qquad \textbf{(C) } 487 \qquad \textbf{(D) } 557 \qquad \textbf{(E) } 562$ | [
"Let there be $T$ teams. For each team, there are ${n-5\\choose 4}$ different subsets of $9$ players that includes a given full team, so the total number of team-(group of 9) pairs is\n\\[T{n-5\\choose 4}.\\]\nThus, the expected value of the number of full teams in a random set of $9$ players is\n\\[\\frac{T{n-5\\c... |
https://artofproblemsolving.com/wiki/index.php/2016_AMC_10B_Problems/Problem_22 | A | 385 | A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won $10$ games and lost $10$ games; there were no ties. How many sets of three teams $\{A, B, C\}$ were there in which $A$ beat $B$ $B$ beat $C$ , and $C$ beat $A?$
$\textbf{(A)}\ 385 \qquad \textbf{(B)}\ 665 \qquad \textbf{(C)}\ 945 \qquad \textbf{(D)}\ 1140 \qquad \textbf{(E)}\ 1330$ | [
"There are $10 \\cdot 2+1=21$ teams. Any of the $\\tbinom{21}3=1330$ sets of three teams must either be a fork (in which one team beat both the others) or a cycle:\n But we know that every team beat exactly $10$ other teams, so for each possible $A$ at the head of a fork, there are always exactly $\\tbinom{10}2$ ch... |
https://artofproblemsolving.com/wiki/index.php/2016_AMC_12B_Problems/Problem_20 | A | 385 | A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won $10$ games and lost $10$ games; there were no ties. How many sets of three teams $\{A, B, C\}$ were there in which $A$ beat $B$ $B$ beat $C$ , and $C$ beat $A?$
$\textbf{(A)}\ 385 \qquad \textbf{(B)}\ 665 \qquad \textbf{(C)}\ 945 \qquad \textbf{(D)}\ 1140 \qquad \textbf{(E)}\ 1330$ | [
"We use complementary counting. First, because each team played $20$ other teams, there are $21$ teams total. All sets that do not have $A$ beat $B$ $B$ beat $C$ , and $C$ beat $A$ have one team that beats both the other teams. Thus we must count the number of sets of three teams such that one team beats the two ot... |
https://artofproblemsolving.com/wiki/index.php/2016_AMC_12B_Problems/Problem_20 | null | 385 | A set of teams held a round-robin tournament in which every team played every other team exactly once. Every team won $10$ games and lost $10$ games; there were no ties. How many sets of three teams $\{A, B, C\}$ were there in which $A$ beat $B$ $B$ beat $C$ , and $C$ beat $A?$
$\textbf{(A)}\ 385 \qquad \textbf{(B)}\ 665 \qquad \textbf{(C)}\ 945 \qquad \textbf{(D)}\ 1140 \qquad \textbf{(E)}\ 1330$ | [
"As above, note that there are 21 teams, and call them A, B, C, ... T, U. WLOG, assume that A beat teams B-L and lost to teams M-U. We will count the number of sets satisfying the “cycle-win” condition—e.g. here, A beats a team in X which beats a team in Y which beats A. The first and third part of the condition ar... |
https://artofproblemsolving.com/wiki/index.php/2002_AMC_10A_Problems/Problem_22 | C | 18 | A set of tiles numbered 1 through 100 is modified repeatedly by the following operation: remove all tiles numbered with a perfect square , and renumber the remaining tiles consecutively starting with 1. How many times must the operation be performed to reduce the number of tiles in the set to one?
$\text{(A)}\ 10 \qquad \text{(B)}\ 11 \qquad \text{(C)}\ 18 \qquad \text{(D)}\ 19 \qquad \text{(E)}\ 20$ | [
"The pattern is quite simple to see after listing a couple of terms.\n\\[\\begin{tabular}{|r|r|r|} \\hline \\#&\\text{Removed}&\\text{Left}\\\\ \\hline 1&10&90\\\\ 2&9&81\\\\ 3&9&72\\\\ 4&8&64\\\\ 5&8&56\\\\ 6&7&49\\\\ 7&7&42\\\\ 8&6&36\\\\ 9&6&30\\\\ 10&5&25\\\\ 11&5&20\\\\ 12&4&16\\\\ 13&4&12\\\\ 14&3&9\\\\ 15&3&... |
https://artofproblemsolving.com/wiki/index.php/2002_AMC_10A_Problems/Problem_22 | null | 18 | A set of tiles numbered 1 through 100 is modified repeatedly by the following operation: remove all tiles numbered with a perfect square , and renumber the remaining tiles consecutively starting with 1. How many times must the operation be performed to reduce the number of tiles in the set to one?
$\text{(A)}\ 10 \qquad \text{(B)}\ 11 \qquad \text{(C)}\ 18 \qquad \text{(D)}\ 19 \qquad \text{(E)}\ 20$ | [
"We start of with $100 = 10 \\cdot 10$ numbers. When we use the certain operation, call if $P(x)$ , have $100 - 10 = 90 = 10 \\cdot 9$ .\nThen we do $P(x)$ again, to subtract $9$ numbers and get $9 \\cdot 9$ . In the end, we will want $1 = 1 \\cdot 1$ . We can say we have to use $P(x)$ once to make $n \\cdot n$ int... |
https://artofproblemsolving.com/wiki/index.php/2009_AMC_12A_Problems/Problem_13 | D | 700,800 | A ship sails $10$ miles in a straight line from $A$ to $B$ , turns through an angle between $45^{\circ}$ and $60^{\circ}$ , and then sails another $20$ miles to $C$ . Let $AC$ be measured in miles. Which of the following intervals contains $AC^2$ [asy] unitsize(2mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair B=(0,0), A=(-10,0), C=20*dir(50); draw(A--B--C); draw(A--C,linetype("4 4")); dot(A); dot(B); dot(C); label("$10$",midpoint(A--B),S); label("$20$",midpoint(B--C),SE); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,NE); [/asy]
$\textbf{(A)}\ [400,500] \qquad \textbf{(B)}\ [500,600] \qquad \textbf{(C)}\ [600,700] \qquad \textbf{(D)}\ [700,800]$ $\textbf{(E)}\ [800,900]$ | [
"Let $C_1$ be the point the ship would reach if it turned $45^\\circ$ , and $C_2$ the point it would reach if it turned $60^\\circ$ . Obviously, $C_1$ is the furthest possible point from $A$ , and $C_2$ is the closest possible point to $A$\nHence the interval of possible values for $AC^2$ is $[AC_2^2,AC_1^2]$\n\nWe... |
https://artofproblemsolving.com/wiki/index.php/2012_AMC_8_Problems/Problem_8 | D | 60 | A shop advertises everything is "half price in today's sale." In addition, a coupon gives a 20% discount on sale prices. Using the coupon, the price today represents what percentage off the original price?
$\textbf{(A)}\hspace{.05in}10\qquad\textbf{(B)}\hspace{.05in}33\qquad\textbf{(C)}\hspace{.05in}40\qquad\textbf{(D)}\hspace{.05in}60\qquad\textbf{(E)}\hspace{.05in}70$ | [
"Let the original price of an item be $x$\nFirst, everything is half-off, so the price is now $\\frac{x}{2} = 0.5x$\nNext, the extra coupon applies 20% off on the sale price , so the price after this discount will be $100\\% - 20\\% = 80\\%$ of what it was before. (Notice how this is not applied to the original pri... |
https://artofproblemsolving.com/wiki/index.php/1994_AJHSME_Problems/Problem_9 | A | 81 | A shopper buys a $100$ dollar coat on sale for $20\%$ off. An additional $5$ dollars are taken off the sale price by using a discount coupon. A sales tax of $8\%$ is paid on the final selling price. The total amount the shopper pays for the coat is
$\text{(A)}\ \text{81.00 dollars} \qquad \text{(B)}\ \text{81.40 dollars} \qquad \text{(C)}\ \text{82.00 dollars} \qquad \text{(D)}\ \text{82.08 dollars} \qquad \text{(E)}\ \text{82.40 dollars}$ | [
"After the $20\\%$ sale, the coat costs $100(0.8)=80$ dollars. Then $5$ dollars are taken off for a cost of $80-5=75$ . Adding on the sales tax, the final amount is $(75)(1.08)=\\boxed{81.00}$"
] |
https://artofproblemsolving.com/wiki/index.php/2010_AMC_10B_Problems/Problem_11 | A | 50 | A shopper plans to purchase an item that has a listed price greater than $\textdollar 100$ and can use any one of the three coupons. Coupon A gives $15\%$ off the listed price, Coupon B gives $\textdollar 30$ off the listed price, and Coupon C gives $25\%$ off the amount by which the listed price exceeds $\textdollar 100$ Let $x$ and $y$ be the smallest and largest prices, respectively, for which Coupon A saves at least as many dollars as Coupon B or C. What is $y - x$
$\textbf{(A)}\ 50 \qquad \textbf{(B)}\ 60 \qquad \textbf{(C)}\ 75 \qquad \textbf{(D)}\ 80 \qquad \textbf{(E)}\ 100$ | [
"Let the listed price be $(100 + p)$ , where $p > 0$\nCoupon A saves us: $0.15(100+p) = (0.15p + 15)$\nCoupon B saves us: $30$\nCoupon C saves us: $0.25p$\nNow, the condition is that A has to be greater than or equal to either B or C which gives us the following inequalities:\n$A \\geq B \\Rightarrow 0.15p + 15 \\g... |
https://artofproblemsolving.com/wiki/index.php/2013_AMC_8_Problems/Problem_2 | D | 12 | A sign at the fish market says, "50 $\%$ off, today only: half-pound packages for just $3 per package." What is the regular price for a full pound of fish, in dollars?
$\textbf{(A)}\ 6 \qquad \textbf{(B)}\ 9 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 15$ | [
"50% off the price of half a pound of fish is $3, so 100%, the regular price, of a half pound of fish is $6. If half a pound of fish costs $6, then a whole pound of fish is $\\boxed{12}$ dollars.",
"Suppose a full pound at normal price costs $x$ dollars. Then, with the 50% off deal, the full pound would cost $x/2... |
https://artofproblemsolving.com/wiki/index.php/2020_AMC_10A_Problems/Problem_9 | B | 18 | A single bench section at a school event can hold either $7$ adults or $11$ children. When $N$ bench sections are connected end to end, an equal number of adults and children seated together will occupy all the bench space. What is the least possible positive integer value of $N?$
$\textbf{(A) } 9 \qquad \textbf{(B) } 18 \qquad \textbf{(C) } 27 \qquad \textbf{(D) } 36 \qquad \textbf{(E) } 77$ | [
"The least common multiple of $7$ and $11$ is $77$ . Therefore, there must be $77$ adults and $77$ children. The total number of benches is $\\frac{77}{7}+\\frac{77}{11}=11+7=\\boxed{18}$ .~taarunganesh",
"Let $x$ denote how many adults there are. Since the number of adults is equal to the number of children we c... |
https://artofproblemsolving.com/wiki/index.php/2006_AMC_8_Problems/Problem_20 | C | 2 | A singles tournament had six players. Each player played every other player only once, with no ties. If Helen won 4 games, Ines won 3 games, Janet won 2 games, Kendra won 2 games and Lara won 2 games, how many games did Monica win?
$\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 4$ | [
"Since there are 6 players, a total of $\\frac{6(6-1)}{2}=15$ games are played. So far, $4+3+2+2+2=13$ games finished (one person won from each game), so Monica needs to win $15-13 = \\boxed{2}$"
] |
https://artofproblemsolving.com/wiki/index.php/2011_AMC_10A_Problems/Problem_2 | E | 15 | A small bottle of shampoo can hold $35$ milliliters of shampoo, whereas a large bottle can hold $500$ milliliters of shampoo. Jasmine wants to buy the minimum number of small bottles necessary to completely fill a large bottle. How many bottles must she buy?
$\textbf{(A)}\ 11 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 15$ | [
"To find how many small bottles we need, we can simply divide $500$ by $35$ . This simplifies to $\\frac{100}{7}=14 \\frac{2}{7}.$ Since the answer must be an integer greater than $14$ , we have to round up to $15$ bottles, or $\\boxed{15}$",
"We double $35$ to get $70.$ We see that $70\\cdot7=490,$ which is very... |
https://artofproblemsolving.com/wiki/index.php/2011_AMC_12A_Problems/Problem_3 | E | 15 | A small bottle of shampoo can hold $35$ milliliters of shampoo, whereas a large bottle can hold $500$ milliliters of shampoo. Jasmine wants to buy the minimum number of small bottles necessary to completely fill a large bottle. How many bottles must she buy?
$\textbf{(A)}\ 11 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 15$ | [
"To find how many small bottles we need, we can simply divide $500$ by $35$ . This simplifies to $\\frac{100}{7}=14 \\frac{2}{7}.$ Since the answer must be an integer greater than $14$ , we have to round up to $15$ bottles, or $\\boxed{15}$",
"We double $35$ to get $70.$ We see that $70\\cdot7=490,$ which is very... |
https://artofproblemsolving.com/wiki/index.php/1985_AIME_Problems/Problem_4 | null | 32 | A small square is constructed inside a square of area 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of $n$ if the the area of the small square is exactly $\frac1{1985}$
AIME 1985 Problem 4.png | [
"The lines passing through $A$ and $C$ divide the square into three parts, two right triangles and a parallelogram . Using the smaller side of the parallelogram, $1/n$ , as the base, where the height is 1, we find that the area of the parallelogram is $A = \\frac{1}{n}$ . By the Pythagorean Theorem , the longer b... |
https://artofproblemsolving.com/wiki/index.php/2019_AIME_I_Problems/Problem_4 | null | 122 | A soccer team has $22$ available players. A fixed set of $11$ players starts the game, while the other $11$ are available as substitutes. During the game, the coach may make as many as $3$ substitutions, where any one of the $11$ players in the game is replaced by one of the substitutes. No player removed from the game may reenter the game, although a substitute entering the game may be replaced later. No two substitutions can happen at the same time. The players involved and the order of the substitutions matter. Let $n$ be the number of ways the coach can make substitutions during the game (including the possibility of making no substitutions). Find the remainder when $n$ is divided by $1000$ | [
"There are $0-3$ substitutions. The number of ways to sub any number of times must be multiplied by the previous number. This is defined recursively. The case for $0$ subs is $1$ , and the ways to reorganize after $n$ subs is the product of the number of new subs ( $12-n$ ) and the players that can be ejected ( $11... |
https://artofproblemsolving.com/wiki/index.php/2013_AMC_10A_Problems/Problem_4 | C | 45 | A softball team played ten games, scoring $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ , and $10$ runs. They lost by one run in exactly five games. In each of their other games, they scored twice as many runs as their opponent. How many total runs did their opponents score?
$\textbf{(A)}\ 35 \qquad\textbf{(B)}\ 40 \qquad\textbf{(C)}\ 45 \qquad\textbf{(D)}\ 50 \qquad\textbf{(E)}\ 55$ | [
"We know that, for the games where they scored an odd number of runs, they cannot have scored twice as many runs as their opponents, as odd numbers are not divisible by $2$ . Thus, from this, we know that the five games where they lost by one run were when they scored $1$ $3$ $5$ $7$ , and $9$ runs, and the others... |
https://artofproblemsolving.com/wiki/index.php/2003_AMC_10A_Problems/Problem_3 | D | 18 | A solid box is $15$ cm by $10$ cm by $8$ cm. A new solid is formed by removing a cube $3$ cm on a side from each corner of this box. What percent of the original volume is removed?
$\mathrm{(A) \ } 4.5\%\qquad \mathrm{(B) \ } 9\%\qquad \mathrm{(C) \ } 12\%\qquad \mathrm{(D) \ } 18\%\qquad \mathrm{(E) \ } 24\%$ | [
"The volume of the original box is $15\\cdot10\\cdot8=1200.$\nThe volume of each cube that is removed is $3\\cdot3\\cdot3=27.$\nSince there are $8$ corners on the box, $8$ cubes are removed.\nSo the total volume removed is $8\\cdot27=216$\nTherefore, the desired percentage is $\\frac{216}{1200}\\cdot100 = \\boxed{1... |
https://artofproblemsolving.com/wiki/index.php/2003_AMC_12A_Problems/Problem_3 | D | 18 | A solid box is $15$ cm by $10$ cm by $8$ cm. A new solid is formed by removing a cube $3$ cm on a side from each corner of this box. What percent of the original volume is removed?
$\mathrm{(A) \ } 4.5\%\qquad \mathrm{(B) \ } 9\%\qquad \mathrm{(C) \ } 12\%\qquad \mathrm{(D) \ } 18\%\qquad \mathrm{(E) \ } 24\%$ | [
"The volume of the original box is $15\\cdot10\\cdot8=1200.$\nThe volume of each cube that is removed is $3\\cdot3\\cdot3=27.$\nSince there are $8$ corners on the box, $8$ cubes are removed.\nSo the total volume removed is $8\\cdot27=216$\nTherefore, the desired percentage is $\\frac{216}{1200}\\cdot100 = \\boxed{1... |
https://artofproblemsolving.com/wiki/index.php/2010_AMC_10A_Problems/Problem_17 | A | 7 | A solid cube has side length 3 inches. A 2-inch by 2-inch square hole is cut into the center of each face. The edges of each cut are parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid?
$\textbf{(A)}\ 7 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 15$ | [
"Imagine making the cuts one at a time. The first cut removes a box $2\\times 2\\times 3$ . The second cut removes two boxes, each of dimensions $2\\times 2\\times 0.5$ , and the third cut does the same as the second cut, on the last two faces. Hence the total volume of all cuts is $12 + 4 + 4 = 20$\nTherefore the ... |
https://artofproblemsolving.com/wiki/index.php/2010_AMC_12A_Problems/Problem_9 | A | 7 | A solid cube has side length 3 inches. A 2-inch by 2-inch square hole is cut into the center of each face. The edges of each cut are parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid?
$\textbf{(A)}\ 7 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 15$ | [
"Imagine making the cuts one at a time. The first cut removes a box $2\\times 2\\times 3$ . The second cut removes two boxes, each of dimensions $2\\times 2\\times 0.5$ , and the third cut does the same as the second cut, on the last two faces. Hence the total volume of all cuts is $12 + 4 + 4 = 20$\nTherefore the ... |
https://artofproblemsolving.com/wiki/index.php/2013_AMC_10A_Problems/Problem_14 | D | 84 | A solid cube of side length $1$ is removed from each corner of a solid cube of side length $3$ . How many edges does the remaining solid have?
$\textbf{(A) }36\qquad\textbf{(B) }60\qquad\textbf{(C) }72\qquad\textbf{(D) }84\qquad\textbf{(E) }108\qquad$ | [
"We can use Euler's polyhedron formula that says that $F+V=E+2$ . We know that there are originally $6$ faces on the cube, and each corner cube creates $3$ more. $6+8(3) = 30$ . In addition, each cube creates $7$ new vertices while taking away the original $8$ , yielding $8(7) = 56$ vertices. Thus $E+2=56+30$ , ... |
https://artofproblemsolving.com/wiki/index.php/2004_AIME_II_Problems/Problem_3 | null | 384 | A solid rectangular block is formed by gluing together $N$ congruent 1-cm cubes face to face. When the block is viewed so that three of its faces are visible, exactly $231$ of the 1-cm cubes cannot be seen. Find the smallest possible value of $N.$ | [
"The $231$ cubes which are not visible must lie below exactly one layer of cubes. Thus, they form a rectangular solid which is one unit shorter in each dimension. If the original block has dimensions $l \\times m \\times n$ , we must have $(l - 1)\\times(m-1) \\times(n - 1) = 231$ . The prime factorization of $2... |
https://artofproblemsolving.com/wiki/index.php/1994_AIME_Problems/Problem_9 | null | 394 | A solitaire game is played as follows. Six distinct pairs of matched tiles are placed in a bag. The player randomly draws tiles one at a time from the bag and retains them, except that matching tiles are put aside as soon as they appear in the player's hand. The game ends if the player ever holds three tiles, no two of which match; otherwise the drawing continues until the bag is empty. The probability that the bag will be emptied is $p/q,\,$ where $p\,$ and $q\,$ are relatively prime positive integers. Find $p+q.\,$ | [
"Let $P_k$ be the probability of emptying the bag when it has $k$ pairs in it. Let's consider the possible draws for the first three cards:\nTherefore, we obtain the recursion $P_k = \\frac {3}{2k - 1}P_{k - 1}$ . Iterating this for $k = 6,5,4,3,2$ (obviously $P_1 = 1$ ), we get $\\frac {3^5}{11*9*7*5*3} = \\frac {... |
https://artofproblemsolving.com/wiki/index.php/1998_AHSME_Problems/Problem_9 | D | 33 | A speaker talked for sixty minutes to a full auditorium. Twenty percent of the audience heard the entire talk and ten percent slept through the entire talk. Half of the remainder heard one third of the talk and the other half heard two thirds of the talk. What was the average number of minutes of the talk heard by members of the audience?
$\mathrm{(A) \ } 24 \qquad \mathrm{(B) \ } 27\qquad \mathrm{(C) \ }30 \qquad \mathrm{(D) \ }33 \qquad \mathrm{(E) \ }36$ | [
"Assume that there are $100$ people in the audience.\n$20$ people heard $60$ minutes of the talk, for a total of $20\\cdot 60 = 1200$ minutes heard.\n$10$ people heard $0$ minutes.\n$\\frac{70}{2} = 35$ people heard $20$ minutes of the talk, for a total of $35\\cdot 20 = 700$ minutes.\n$35$ people heard $40$ minute... |
https://artofproblemsolving.com/wiki/index.php/2017_AIME_II_Problems/Problem_9 | null | 13 | A special deck of cards contains $49$ cards, each labeled with a number from $1$ to $7$ and colored with one of seven colors. Each number-color combination appears on exactly one card. Sharon will select a set of eight cards from the deck at random. Given that she gets at least one card of each color and at least one card with each number, the probability that Sharon can discard one of her cards and $\textit{still}$ have at least one card of each color and at least one card with each number is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p+q$ | [
"Without loss of generality, assume that the $8$ numbers on Sharon's cards are $1$ $1$ $2$ $3$ $4$ $5$ $6$ , and $7$ , in that order, and assume the $8$ colors are red, red, and six different arbitrary colors. There are ${8\\choose2}-1$ ways of assigning the two red cards to the $8$ numbers; we subtract $1$ because... |
https://artofproblemsolving.com/wiki/index.php/2005_AMC_8_Problems/Problem_4 | C | 36 | A square and a triangle have equal perimeters. The lengths of the three sides of the triangle are 6.1 cm, 8.2 cm and 9.7 cm. What is the area of the square in square centimeters?
$\textbf{(A)}\ 24\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 64$ | [
"The perimeter of the triangle is $6.1+8.2+9.7=24$ cm. A square's perimeter is four times its sidelength, since all its sidelengths are equal. If the square's perimeter is $24$ , the sidelength is $24/4=6$ , and the area is $6^2=\\boxed{36}$"
] |
https://artofproblemsolving.com/wiki/index.php/1989_AHSME_Problems/Problem_21 | null | 2 | A square flag has a red cross of uniform width with a blue square in the center on a white background as shown. (The cross is symmetric with respect to each of the diagonals of the square.) If the entire cross (both the red arms and the blue center) takes up 36% of the area of the flag, what percent of the area of the flag is blue?
[asy] unitsize(2.5 cm); pair[] A, B, C; real t = 0.2; A[1] = (0,0); A[2] = (1,0); A[3] = (1,1); A[4] = (0,1); B[1] = (t,0); B[2] = (1 - t,0); B[3] = (1,t); B[4] = (1,1 - t); B[5] = (1 - t,1); B[6] = (t,1); B[7] = (0,1 - t); B[8] = (0,t); C[1] = extension(B[1],B[4],B[7],B[2]); C[2] = extension(B[3],B[6],B[1],B[4]); C[3] = extension(B[5],B[8],B[3],B[6]); C[4] = extension(B[7],B[2],B[5],B[8]); fill(C[1]--C[2]--C[3]--C[4]--cycle,blue); fill(A[1]--B[1]--C[1]--C[4]--B[8]--cycle,red); fill(A[2]--B[3]--C[2]--C[1]--B[2]--cycle,red); fill(A[3]--B[5]--C[3]--C[2]--B[4]--cycle,red); fill(A[4]--B[7]--C[4]--C[3]--B[6]--cycle,red); draw(A[1]--A[2]--A[3]--A[4]--cycle); draw(B[1]--B[4]); draw(B[2]--B[7]); draw(B[3]--B[6]); draw(B[5]--B[8]); [/asy]
$\text{(A)}\ 0.5\qquad\text{(B)}\ 1\qquad\text{(C)}\ 2\qquad\text{(D)}\ 3\qquad\text{(E)}\ 6$ | [
"The diagram can be quartered as shown: and reassembled into two smaller squares of side $k$ , each of which looks like this: The border in this figure is the former cross, which still occupies 36% of the area. Therefore the inner square occupies 64% of the area, from which we deduce that it is $0.8k \\times 0.8k... |
https://artofproblemsolving.com/wiki/index.php/2014_AMC_10A_Problems/Problem_18 | B | 17 | A square in the coordinate plane has vertices whose $y$ -coordinates are $0$ $1$ $4$ , and $5$ . What is the area of the square?
$\textbf{(A)}\ 16\qquad\textbf{(B)}\ 17\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 26\qquad\textbf{(E)}\ 27$ | [
"Let the points be $A=(x_1,0)$ $B=(x_2,1)$ $C=(x_3,5)$ , and $D=(x_4,4)$\nNote that the difference in $y$ value of $B$ and $C$ is $4$ . By rotational symmetry of the square, the difference in $x$ value of $A$ and $B$ is also $4$ . Note that the difference in $y$ value of $A$ and $B$ is $1$ . We now know that $AB$ ,... |
https://artofproblemsolving.com/wiki/index.php/2017_AMC_12A_Problems/Problem_22 | E | 39 | A square is drawn in the Cartesian coordinate plane with vertices at $(2, 2)$ $(-2, 2)$ $(-2, -2)$ $(2, -2)$ . A particle starts at $(0,0)$ . Every second it moves with equal probability to one of the eight lattice points (points with integer coordinates) closest to its current position, independently of its previous moves. In other words, the probability is $1/8$ that the particle will move from $(x, y)$ to each of $(x, y + 1)$ $(x + 1, y + 1)$ $(x + 1, y)$ $(x + 1, y - 1)$ $(x, y - 1)$ $(x - 1, y - 1)$ $(x - 1, y)$ , or $(x - 1, y + 1)$ . The particle will eventually hit the square for the first time, either at one of the 4 corners of the square or at one of the 12 lattice points in the interior of one of the sides of the square. The probability that it will hit at a corner rather than at an interior point of a side is $m/n$ , where $m$ and $n$ are relatively prime positive integers. What is $m + n$
$\textbf{(A) } 4 \qquad\textbf{(B) } 5 \qquad\textbf{(C) } 7 \qquad\textbf{(D) } 15 \qquad\textbf{(E) } 39$ | [
"We let $c, e,$ and $m$ be the probability of reaching a corner before an edge when starting at an \"inside corner\" (e.g. $(1, 1)$ ), an \"inside edge\" (e.g. $(1, 0)$ ), and the middle respectively.\nStarting in the middle, there is a $\\frac{4}{8}$ chance of moving to an inside edge and a $\\frac{4}{8}$ chance o... |
https://artofproblemsolving.com/wiki/index.php/2006_AMC_10B_Problems/Problem_8 | B | 25 | A square of area 40 is inscribed in a semicircle as shown. What is the area of the semicircle?
[asy] defaultpen(linewidth(0.8)); size(100); real r=sqrt(50), s=sqrt(10); draw(Arc(origin, r, 0, 180)); draw((r,0)--(-r,0), dashed); draw((-s,0)--(s,0)--(s,2*s)--(-s,2*s)--cycle); [/asy]
$\textbf{(A) } 20\pi\qquad \textbf{(B) } 25\pi\qquad \textbf{(C) } 30\pi\qquad \textbf{(D) } 40\pi\qquad \textbf{(E) } 50\pi$ | [
"Since the area of the square is $40$ , the length of a side is $\\sqrt{40}=2\\sqrt{10}$ . The distance between the center of the semicircle and one of the bottom vertices of the square is half the length of the side, which is $\\sqrt{10}$\nUsing the Pythagorean Theorem to find the radius $r$ of the semicircle, $r^... |
https://artofproblemsolving.com/wiki/index.php/2021_AMC_10B_Problems/Problem_21 | A | 2 | A square piece of paper has side length $1$ and vertices $A,B,C,$ and $D$ in that order. As shown in the figure, the paper is folded so that vertex $C$ meets edge $\overline{AD}$ at point $C'$ , and edge $\overline{BC}$ intersects edge $\overline{AB}$ at point $E$ . Suppose that $C'D = \frac{1}{3}$ . What is the perimeter of triangle $\bigtriangleup AEC' ?$
$\textbf{(A)} ~2 \qquad\textbf{(B)} ~1+\frac{2}{3}\sqrt{3} \qquad\textbf{(C)} ~\frac{13}{6} \qquad\textbf{(D)} ~1 + \frac{3}{4}\sqrt{3} \qquad\textbf{(E)} ~\frac{7}{3}$ [asy] /* Made by samrocksnature */ pair A=(0,1); pair CC=(0.666666666666,1); pair D=(1,1); pair F=(1,0.440062); pair C=(1,0); pair B=(0,0); pair G=(0,0.22005); pair H=(-0.13,0.41); pair E=(0,0.5); dot(A^^CC^^D^^C^^B^^E); draw(E--A--D--F); draw(G--B--C--F, dashed); fill(E--CC--F--G--H--E--CC--cycle, gray); draw(E--CC--F--G--H--E--CC); label("A",A,NW); label("B",B,SW); label("C",C,SE); label("D",D,NE); label("E",E,NW); label("C'",CC,N); label("F",F,NE); [/asy] | [
"We can set the point on $CD$ where the fold occurs as point $F$ . Then, we can set $FD$ as $x$ , and $CF$ as $1-x$ because of symmetry due to the fold. It can be recognized that this is a right triangle, and solving for $x$ , we get,\n\\[x^2 + \\left(\\frac{1}{3}\\right)^2 = (1-x)^2 \\rightarrow x^2 + \\frac{1}{9}... |
https://artofproblemsolving.com/wiki/index.php/2008_AIME_I_Problems/Problem_15 | null | 871 | A square piece of paper has sides of length $100$ . From each corner a wedge is cut in the following manner: at each corner, the two cuts for the wedge each start at a distance $\sqrt{17}$ from the corner, and they meet on the diagonal at an angle of $60^{\circ}$ (see the figure below). The paper is then folded up along the lines joining the vertices of adjacent cuts. When the two edges of a cut meet, they are taped together. The result is a paper tray whose sides are not at right angles to the base. The height of the tray, that is, the perpendicular distance between the plane of the base and the plane formed by the upper edges, can be written in the form $\sqrt[n]{m}$ , where $m$ and $n$ are positive integers, $m<1000$ , and $m$ is not divisible by the $n$ th power of any prime. Find $m+n$ | [
"In the original picture, let $P$ be the corner, and $M$ and $N$ be the two points whose distance is $\\sqrt{17}$ from $P$ . Also, let $R$ be the point where the two cuts intersect.\nUsing $\\triangle{MNP}$ (a 45-45-90 triangle), $MN=MP\\sqrt{2}\\quad\\Longrightarrow\\quad MN=\\sqrt{34}$ $\\triangle{MNR}$ is equila... |
https://artofproblemsolving.com/wiki/index.php/2007_AIME_I_Problems/Problem_13 | null | 80 | A square pyramid with base $ABCD$ and vertex $E$ has eight edges of length $4$ . A plane passes through the midpoints of $AE$ $BC$ , and $CD$ . The plane's intersection with the pyramid has an area that can be expressed as $\sqrt{p}$ . Find $p$
AIME I 2007-13.png | [
"Note first that the intersection is a pentagon\nUse 3D analytical geometry, setting the origin as the center of the square base and the pyramid’s points oriented as shown above. $A(-2,2,0),\\ B(2,2,0),\\ C(2,-2,0),\\ D(-2,-2,0),\\ E(0,2\\sqrt{2})$ . Using the coordinates of the three points of intersection $(-1,1,... |
https://artofproblemsolving.com/wiki/index.php/2012_AMC_8_Problems/Problem_17 | B | 4 | A square with integer side length is cut into 10 squares, all of which have integer side length and at least 8 of which have area 1. What is the smallest possible value of the length of the side of the original square?
$\text{(A)}\hspace{.05in}3\qquad\text{(B)}\hspace{.05in}4\qquad\text{(C)}\hspace{.05in}5\qquad\text{(D)}\hspace{.05in}6\qquad\text{(E)}\hspace{.05in}7$ | [
"The first answer choice ${\\textbf{(A)}\\ 3}$ , can be eliminated since there must be $10$ squares with integer side lengths. We then test the next smallest sidelength which is $4$ . The square with area $16$ can be partitioned into $8$ squares with area $1$ and two squares with area $4$ , which satisfies all the ... |
https://artofproblemsolving.com/wiki/index.php/2021_AMC_10B_Problems/Problem_23 | C | 68 | A square with side length $8$ is colored white except for $4$ black isosceles right triangular regions with legs of length $2$ in each corner of the square and a black diamond with side length $2\sqrt{2}$ in the center of the square, as shown in the diagram. A circular coin with diameter $1$ is dropped onto the square and lands in a random location where the coin is completely contained within the square. The probability that the coin will cover part of the black region of the square can be written as $\frac{1}{196}\left(a+b\sqrt{2}+\pi\right)$ , where $a$ and $b$ are positive integers. What is $a+b$ [asy] /* Made by samrocksnature */ draw((0,0)--(8,0)--(8,8)--(0,8)--(0,0)); fill((2,0)--(0,2)--(0,0)--cycle, black); fill((6,0)--(8,0)--(8,2)--cycle, black); fill((8,6)--(8,8)--(6,8)--cycle, black); fill((0,6)--(2,8)--(0,8)--cycle, black); fill((4,6)--(2,4)--(4,2)--(6,4)--cycle, black); filldraw(circle((2.6,3.31),0.5),gray); [/asy]
$\textbf{(A)} ~64 \qquad\textbf{(B)} ~66 \qquad\textbf{(C)} ~68 \qquad\textbf{(D)} ~70 \qquad\textbf{(E)} ~72$ | [
"To find the probability, we look at the $\\frac{\\text{success region}}{\\text{total possible region}}$ . For the coin to be completely contained within the square, we must have the distance from the center of the coin to a side of the square to be at least $\\frac{1}{2}$ , as it's the radius of the coin. This imp... |
https://artofproblemsolving.com/wiki/index.php/2005_AMC_8_Problems/Problem_25 | A | 2 | A square with side length 2 and a circle share the same center. The total area of the regions that are inside the circle and outside the square is equal to the total area of the regions that are outside the circle and inside the square. What is the radius of the circle?
[asy] pair a=(4,4), b=(0,0), c=(0,4), d=(4,0), o=(2,2); draw(a--d--b--c--cycle); draw(circle(o, 2.5)); [/asy]
$\textbf{(A)}\ \frac{2}{\sqrt{\pi}} \qquad \textbf{(B)}\ \frac{1+\sqrt{2}}{2} \qquad \textbf{(C)}\ \frac{3}{2} \qquad \textbf{(D)}\ \sqrt{3} \qquad \textbf{(E)}\ \sqrt{\pi}$ | [
"Let the region within the circle and square be $a$ . In other words, it is the area inside the circle $\\textbf{and}$ the square. Let $r$ be the radius. We know that the area of the circle minus $a$ is equal to the area of the square, minus $a$\nWe get:\n$\\pi r^2 -a=4-a$\n$r^2=\\frac{4}{\\pi}$\n$r=\\frac{2}{\\sq... |
https://artofproblemsolving.com/wiki/index.php/1998_AHSME_Problems/Problem_8 | D | 56 | A square with sides of length $1$ is divided into two congruent trapezoids and a pentagon, which have equal areas, by joining the center of the square with points on three of the sides, as shown. Find $x$ , the length of the longer parallel side of each trapezoid.
$\mathrm{(A) \ } \frac 35 \qquad \mathrm{(B) \ } \frac 23 \qquad \mathrm{(C) \ } \frac 34 \qquad \mathrm{(D) \ } \frac 56 \qquad \mathrm{(E) \ } \frac 78$ | [
"The area of the trapezoid is $\\frac{1}{3}$ , and the shorter base and height are both $\\frac{1}{2}$ . Therefore, \\[\\frac{1}{3}=\\frac{1}{2}\\cdot \\frac{1}{2}\\cdot \\left(\\frac{1}{2}+x\\right) \\Rightarrow x=\\frac{5}{6}\\rightarrow \\boxed{56}\\]",
"Divide the pentagon into 2 small congruent trapezoids b... |
https://artofproblemsolving.com/wiki/index.php/2017_AMC_8_Problems/Problem_11 | C | 361 | A square-shaped floor is covered with congruent square tiles. If the total number of tiles that lie on the two diagonals is 37, how many tiles cover the floor?
$\textbf{(A) }148\qquad\textbf{(B) }324\qquad\textbf{(C) }361\qquad\textbf{(D) }1296\qquad\textbf{(E) }1369$ | [
"Since the number of tiles lying on both diagonals is $37$ , counting one tile twice, there are $37=2x-1\\implies x=19$ tiles on each side. Therefore, our answer is $19^2=361=\\boxed{361}$",
"Visualize it as 4 separate diagonals connecting to one square in the middle. Each square on the diagonal corresponds to on... |
https://artofproblemsolving.com/wiki/index.php/2000_AIME_I_Problems/Problem_15 | null | 927 | A stack of $2000$ cards is labelled with the integers from $1$ to $2000,$ with different integers on different cards. The cards in the stack are not in numerical order. The top card is removed from the stack and placed on the table, and the next card is moved to the bottom of the stack. The new top card is removed from the stack and placed on the table, to the right of the card already there, and the next card in the stack is moved to the bottom of the stack. The process - placing the top card to the right of the cards already on the table and moving the next card in the stack to the bottom of the stack - is repeated until all cards are on the table. It is found that, reading from left to right, the labels on the cards are now in ascending order: $1,2,3,\ldots,1999,2000.$ In the original stack of cards, how many cards were above the card labeled $1999$ | [
"We try to work backwards from when there are 2 cards left, since this is when the 1999 card is laid onto the table. When there are 2 cards left, the 1999 card is on the top of the deck. In order for this to occur, it must be 2nd on the deck when there are 4 cards remaining, and this means it must be the 4th card w... |
https://artofproblemsolving.com/wiki/index.php/2004_AMC_10B_Problems/Problem_4 | B | 12 | A standard six-sided die is rolled, and $P$ is the product of the five numbers that are visible. What is the largest number that is certain to divide $P$
$\mathrm{(A) \ } 6 \qquad \mathrm{(B) \ } 12 \qquad \mathrm{(C) \ } 24 \qquad \mathrm{(D) \ } 144\qquad \mathrm{(E) \ } 720$ | [
"The product of all six numbers is $6!=720$ . The products of numbers that can be visible are $720/1$ $720/2$ , ..., $720/6$ .\nThe answer to this problem is their greatest common divisor -- which is $720/L$ , where $L$ is the least common multiple of $\\{1,2,3,4,5,6\\}$ .\nClearly $L=60$ and the answer is $720/60=... |
https://artofproblemsolving.com/wiki/index.php/2019_AIME_II_Problems/Problem_4 | null | 187 | A standard six-sided fair die is rolled four times. The probability that the product of all four numbers rolled is a perfect square is $\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ | [
"Notice that, other than the number 5, the remaining numbers 1, 2, 3, 4, 6 are only divisible by 2 and/or 3. We can do some cases on the number of 5's rolled (note that there are $6^4 = 1296$ outcomes).\nCase 1 (easy): Four 5's are rolled. This has probability $\\frac{1}{6^4}$ of occurring.\nCase 2: Two 5's are rol... |
https://artofproblemsolving.com/wiki/index.php/2019_AMC_8_Problems/Problem_22 | E | 40 | A store increased the original price of a shirt by a certain percent and then lowered the new price by the same amount. Given that the resulting price was $84\%$ of the original price, by what percent was the price increased and decreased $?$
$\textbf{(A) }16\qquad\textbf{(B) }20\qquad\textbf{(C) }28\qquad\textbf{(D) }36\qquad\textbf{(E) }40$ | [
"Suppose the fraction of discount is $x$ . That means $(1-x)(1+x)=0.84$ ; so, $1-x^{2}=0.84$ , and $(x^{2})=0.16$ , obtaining $x=0.4$ . Therefore, the price was increased and decreased by $40$ %, or $\\boxed{40}$",
"After the first increase by $p$ percent, the shirt price became $(1+p)$ times greater than the ori... |
https://artofproblemsolving.com/wiki/index.php/2019_AMC_8_Problems/Problem_22 | null | 40 | A store increased the original price of a shirt by a certain percent and then lowered the new price by the same amount. Given that the resulting price was $84\%$ of the original price, by what percent was the price increased and decreased $?$
$\textbf{(A) }16\qquad\textbf{(B) }20\qquad\textbf{(C) }28\qquad\textbf{(D) }36\qquad\textbf{(E) }40$ | [
"Let our original cost be $$ 100.$ We are looking for a result of $$ 84,$ then. We try 16% and see it gets us higher than 84. We try 20% and see it gets us lower than 16 but still higher than 84. We know that the higher the percent, the less the value. We try 36, as we are not progressing much, and we are close! We... |
https://artofproblemsolving.com/wiki/index.php/2005_AMC_10A_Problems/Problem_5 | A | 100 | A store normally sells windows at $$100$ each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately?
$\textbf{(A) } 100\qquad \textbf{(B) } 200\qquad \textbf{(C) } 300\qquad \textbf{(D) } 400\qquad \textbf{(E) } 500$ | [
"The store's offer means that every $5$ th window is free.\nDave would get $\\left\\lfloor\\frac{7}{5}\\right\\rfloor=1$ free window.\nDoug would get $\\left\\lfloor\\frac{8}{5}\\right\\rfloor=1$ free window.\nThis is a total of $2$ free windows.\nTogether, they would get $\\left\\lfloor\\frac{8+7}{5}\\right\\rfloo... |
https://artofproblemsolving.com/wiki/index.php/1992_AJHSME_Problems/Problem_8 | C | 1,000 | A store owner bought $1500$ pencils at $$ 0.10$ each. If he sells them for $$ 0.25$ each, how many of them must he sell to make a profit of exactly $$ 100.00$
$\text{(A)}\ 400 \qquad \text{(B)}\ 667 \qquad \text{(C)}\ 1000 \qquad \text{(D)}\ 1500 \qquad \text{(E)}\ 1900$ | [
"$1500\\times 0.1=150$ , so the store owner is $$150$ below profit. Therefore he needs to sell $150+100= 250$ dollars worth of pencils. Selling them at $$0.25$ each gives $250/0.25= \\boxed{1000}$"
] |
https://artofproblemsolving.com/wiki/index.php/1990_AJHSME_Problems/Problem_17 | A | 2 | A straight concrete sidewalk is to be $3$ feet wide, $60$ feet long, and $3$ inches thick. How many cubic yards of concrete must a contractor order for the sidewalk if concrete must be ordered in a whole number of cubic yards?
$\text{(A)}\ 2 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 20 \qquad \text{(E)}\ \text{more than 20}$ | [
"This is a $1$ yard by $20$ yard by $1/12$ yard sidewalk, so its volume in yards is \\[1\\times 20\\times \\frac{1}{12} = 1.\\overline{6}.\\] Since concrete must be ordered in a whole number of cubic yards, we need $2\\rightarrow \\boxed{2}$"
] |
https://artofproblemsolving.com/wiki/index.php/2022_AIME_I_Problems/Problem_5 | null | 550 | A straight river that is $264$ meters wide flows from west to east at a rate of $14$ meters per minute. Melanie and Sherry sit on the south bank of the river with Melanie a distance of $D$ meters downstream from Sherry. Relative to the water, Melanie swims at $80$ meters per minute, and Sherry swims at $60$ meters per minute. At the same time, Melanie and Sherry begin swimming in straight lines to a point on the north bank of the river that is equidistant from their starting positions. The two women arrive at this point simultaneously. Find $D$ | [
"Define $m$ as the number of minutes they swim for.\nLet their meeting point be $A$ . Melanie is swimming against the current, so she must aim upstream from point $A$ , to compensate for this; in particular, since she is swimming for $m$ minutes, the current will push her $14m$ meters downstream in that time, so sh... |
https://artofproblemsolving.com/wiki/index.php/2001_AMC_10_Problems/Problem_15 | C | 12 | A street has parallel curbs $40$ feet apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The length of the curb between the stripes is $15$ feet and each stripe is $50$ feet long. Find the distance, in feet, between the stripes.
$\textbf{(A)}\ 9 \qquad \textbf{(B)}\ 10 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 15 \qquad \textbf{(E)}\ 25$ | [
"Drawing the problem out, we see we get a parallelogram with a height of $40$ and a base of $15$ , giving an area of $600$\nIf we look at it the other way, we see the distance between the stripes is the height and the base is $50$\n\nThe area is still the same, so the distance between the stripes is $600/50 = \\box... |
https://artofproblemsolving.com/wiki/index.php/2016_AIME_I_Problems/Problem_10 | null | 504 | A strictly increasing sequence of positive integers $a_1$ $a_2$ $a_3$ $\cdots$ has the property that for every positive integer $k$ , the subsequence $a_{2k-1}$ $a_{2k}$ $a_{2k+1}$ is geometric and the subsequence $a_{2k}$ $a_{2k+1}$ $a_{2k+2}$ is arithmetic. Suppose that $a_{13} = 2016$ . Find $a_1$ | [
"Instead of setting $a_1$ equal to something and $a_2$ equal to something, note that it is rather easier to set $a_1=x^2$ and $a_3=y^2$ so that $a_2=xy,a_4=y(2y-x),a_5=(2y-x)^2$ and so on until you reached $a_{13}=(6y-5x)^2$ (Or simply notice the pattern), so $6y-5x=\\sqrt{2016}=12\\sqrt{14}$ . Note that since each... |
https://artofproblemsolving.com/wiki/index.php/2013_AMC_10A_Problems/Problem_11 | A | 10 | A student council must select a two-person welcoming committee and a three-person planning committee from among its members. There are exactly $10$ ways to select a two-person team for the welcoming committee. It is possible for students to serve on both committees. In how many different ways can a three-person planning committee be selected?
$\textbf{(A)}\ 10\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 25$ | [
"Let the number of students on the council be $x$ . To select a two-person committee, we can select a \"first person\" and a \"second person.\" There are $x$ choices to select a first person; subsequently, there are $x-1$ choices for the second person. This gives a preliminary count of $x(x-1)$ ways to choose a tw... |
https://artofproblemsolving.com/wiki/index.php/2013_AMC_10A_Problems/Problem_7 | C | 9 | A student must choose a program of four courses from a menu of courses consisting of English, Algebra, Geometry, History, Art, and Latin. This program must contain English and at least one mathematics course. In how many ways can this program be chosen?
$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 16$ | [
"Let us split this up into two cases.\nCase $1$ : The student chooses both algebra and geometry.\nThis means that $3$ courses have already been chosen. We have $3$ more options for the last course, so there are $3$ possibilities here.\nCase $2$ : The student chooses one or the other.\nHere, we simply count how man... |
https://artofproblemsolving.com/wiki/index.php/1987_AHSME_Problems/Problem_5 | B | 8 | A student recorded the exact percentage frequency distribution for a set of measurements, as shown below.
However, the student neglected to indicate $N$ , the total number of measurements. What is the smallest possible value of $N$
\[\begin{tabular}{c c}\text{measured value}&\text{percent frequency}\\ \hline 0 & 12.5\\ 1 & 0\\ 2 & 50\\ 3 & 25\\ 4 & 12.5\\ \hline\ & 100\\ \end{tabular}\]
$\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 16 \qquad \textbf{(D)}\ 25 \qquad \textbf{(E)}\ 50$ | [
"Note that $12.5\\% = \\frac{1}{8}$ $25\\% = \\frac{1}{4}$ , and $50\\% = \\frac{1}{2}$ . Thus, since the frequencies must be integers, $N$ must be divisible by $2$ $4$ , and $8$ (so that $\\frac{N}{8}$ etc. are integers), or in other words, $N$ is divisible by $8$ . Thus the smallest possible value of $N$ is the s... |
https://artofproblemsolving.com/wiki/index.php/1976_AHSME_Problems/Problem_12 | C | 6 | A supermarket has $128$ crates of apples. Each crate contains at least $120$ apples and at most $144$ apples.
What is the largest integer $n$ such that there must be at least $n$ crates containing the same number of apples?
$\textbf{(A) }4\qquad \textbf{(B) }5\qquad \textbf{(C) }6\qquad \textbf{(D) }24\qquad \textbf{(E) }25$ | [
"To find the largest number of \"repeated\" crates necessary, we must account for all the possibilities of the number of apples in each crate. Since each crate contains a minimum of $120$ apples and a maximum of $144$ apples, there are $144 - 120 + 1 = 25$ different amounts possible for the number of apples per cra... |
https://artofproblemsolving.com/wiki/index.php/2013_AMC_10A_Problems/Problem_1 | C | 2.75 | A taxi ride costs $$1.50$ plus $$0.25$ per mile traveled. How much does a $5$ -mile taxi ride cost?
$\textbf{(A)}\ 2.25 \qquad\textbf{(B)}\ 2.50 \qquad\textbf{(C)}\ 2.75 \qquad\textbf{(D)}\ 3.00 \qquad\textbf{(E)}\ 3.75$ | [
"There are five miles which need to be traveled. The cost of these five miles is $(0.25\\cdot5) = 1.25$ . Adding this to $1.50$ , we get $\\boxed{2.75}$"
] |
https://artofproblemsolving.com/wiki/index.php/2007_AMC_10B_Problems/Problem_16 | C | 93 | A teacher gave a test to a class in which $10\%$ of the students are juniors and $90\%$ are seniors. The average score on the test was $84.$ The juniors all received the same score, and the average score of the seniors was $83.$ What score did each of the juniors receive on the test?
$\textbf{(A) } 85 \qquad\textbf{(B) } 88 \qquad\textbf{(C) } 93 \qquad\textbf{(D) } 94 \qquad\textbf{(E) } 98$ | [
"We can assume there are $10$ people in the class. Then there will be $1$ junior and $9$ seniors. The sum of everyone's scores is $10 \\cdot 84 = 840$ . Since the average score of the seniors was $83$ , the sum of all the senior's scores is $9 \\cdot 83 = 747$ . The only score that has not been added to that is the... |
https://artofproblemsolving.com/wiki/index.php/2007_AMC_12B_Problems/Problem_12 | C | 93 | A teacher gave a test to a class in which $10\%$ of the students are juniors and $90\%$ are seniors. The average score on the test was $84.$ The juniors all received the same score, and the average score of the seniors was $83.$ What score did each of the juniors receive on the test?
$\textbf{(A) } 85 \qquad\textbf{(B) } 88 \qquad\textbf{(C) } 93 \qquad\textbf{(D) } 94 \qquad\textbf{(E) } 98$ | [
"We can assume there are $10$ people in the class. Then there will be $1$ junior and $9$ seniors. The sum of everyone's scores is $10 \\cdot 84 = 840$ . Since the average score of the seniors was $83$ , the sum of all the senior's scores is $9 \\cdot 83 = 747$ . The only score that has not been added to that is the... |
https://artofproblemsolving.com/wiki/index.php/1995_AJHSME_Problems/Problem_4 | C | 26 | A teacher tells the class,
Ben thinks of $6$ , and gives his answer to Sue. What should Sue's answer be?
$\text{(A)}\ 18 \qquad \text{(B)}\ 24 \qquad \text{(C)}\ 26 \qquad \text{(D)}\ 27 \qquad \text{(E)}\ 30$ | [
"Ben adds $1$ to $6$ , getting $1 + 6 = 7$ . Then Ben doubles it, to get $7\\cdot 2 = 14$ . He gives this number to Sue. Then Sue subtracts $1$ from the number, to get $14 - 1 = 13$ . Finally, Sue doubles the result to get $13\\cdot 2 = 26$ , and the answer is $\\boxed{26}$"
] |
https://artofproblemsolving.com/wiki/index.php/2021_AIME_II_Problems/Problem_11 | null | 258 | A teacher was leading a class of four perfectly logical students. The teacher chose a set $S$ of four integers and gave a different number in $S$ to each student. Then the teacher announced to the class that the numbers in $S$ were four consecutive two-digit positive integers, that some number in $S$ was divisible by $6$ , and a different number in $S$ was divisible by $7$ . The teacher then asked if any of the students could deduce what $S$ is, but in unison, all of the students replied no.
However, upon hearing that all four students replied no, each student was able to determine the elements of $S$ . Find the sum of all possible values of the greatest element of $S$ | [
"Note that $\\operatorname{lcm}(6,7)=42.$ It is clear that $42\\not\\in S$ and $84\\not\\in S,$ otherwise the three other elements in $S$ are divisible by neither $6$ nor $7.$\nIn the table below, the multiples of $6$ are colored in yellow, and the multiples of $7$ are colored in green. By the least common multiple... |
https://artofproblemsolving.com/wiki/index.php/1995_AJHSME_Problems/Problem_14 | B | 23 | A team won $40$ of its first $50$ games. How many of the remaining $40$ games must this team win so it will have won exactly $70 \%$ of its games for the season?
$\text{(A)}\ 20 \qquad \text{(B)}\ 23 \qquad \text{(C)}\ 28 \qquad \text{(D)}\ 30 \qquad \text{(E)}\ 35$ | [
"Noting that 70% is the same as $\\frac{70}{100}=\\frac{7}{10}$ , and that, when x is the amount of wins in the last 40 games, the fraction of games won is $\\frac{40+x}{50+40}=\\frac{40+x}{90}$ , all we have to do is set them equal: \\[\\frac{40+x}{90}=\\frac{7}{10}\\] \\[40+x=63\\] \\[x=\\boxed{23}\\]",
"Altern... |
https://artofproblemsolving.com/wiki/index.php/1992_AIME_Problems/Problem_3 | null | 164 | A tennis player computes her win ratio by dividing the number of matches she has won by the total number of matches she has played. At the start of a weekend, her win ratio is exactly $.500$ . During the weekend, she plays four matches, winning three and losing one. At the end of the weekend, her win ratio is greater than $.503$ . What's the largest number of matches she could've won before the weekend began? | [
"Let $n$ be the number of matches won, so that $\\frac{n}{2n}=\\frac{1}{2}$ , and $\\frac{n+3}{2n+4}>\\frac{503}{1000}$\nCross multiplying $1000n+3000>1006n+2012$ , so $n<\\frac{988}{6}=164 \\dfrac {4}{6}=164 \\dfrac{2}{3}$ . Thus, the answer is $\\boxed{164}$",
"Let $n$ be the number of matches she won before th... |
https://artofproblemsolving.com/wiki/index.php/1999_AHSME_Problems/Problem_29 | C | 0.2 | A tetrahedron with four equilateral triangular faces has a sphere inscribed within it and a sphere circumscribed about it. For each of the four faces, there is a sphere tangent externally to the face at its center and to the circumscribed sphere. A point $P$ is selected at random inside the circumscribed sphere. The probability that $P$ lies inside one of the five small spheres is closest to
$\mathrm{(A) \ }0 \qquad \mathrm{(B) \ }0.1 \qquad \mathrm{(C) \ }0.2 \qquad \mathrm{(D) \ }0.3 \qquad \mathrm{(E) \ }0.4$ | [
"Let the side length of the tetrahedron be $1$\nThe altitude of the equilateral triangle base is $\\frac{\\sqrt{3}}{2}$ . Thus, the distance from the center of the equilateral triangle to its vertex is $\\frac{\\sqrt{3}}{2} \\cdot \\frac 23 = \\frac{\\sqrt{3}}{3}$ . Therefore, the altitude of the tetrahedron is $\\... |
https://artofproblemsolving.com/wiki/index.php/2016_AMC_10B_Problems/Problem_10 | D | 33.3 | A thin piece of wood of uniform density in the shape of an equilateral triangle with side length $3$ inches weighs $12$ ounces. A second piece of the same type of wood, with the same thickness, also in the shape of an equilateral triangle, has side length of $5$ inches. Which of the following is closest to the weight, in ounces, of the second piece?
$\textbf{(A)}\ 14.0\qquad\textbf{(B)}\ 16.0\qquad\textbf{(C)}\ 20.0\qquad\textbf{(D)}\ 33.3\qquad\textbf{(E)}\ 55.6$ | [
"We can solve this problem by using similar triangles, since two equilateral triangles are always similar. We can then use\n$\\left(\\frac{3}{5}\\right)^2=\\frac{12}{x}$\nWe can then solve the equation to get $x=\\frac{100}{3}$ which is closest to $\\boxed{33.3}$",
"Also recall that the area of an equilateral tri... |
https://artofproblemsolving.com/wiki/index.php/2016_AMC_12B_Problems/Problem_8 | D | 33.3 | A thin piece of wood of uniform density in the shape of an equilateral triangle with side length $3$ inches weighs $12$ ounces. A second piece of the same type of wood, with the same thickness, also in the shape of an equilateral triangle, has side length of $5$ inches. Which of the following is closest to the weight, in ounces, of the second piece?
$\textbf{(A)}\ 14.0\qquad\textbf{(B)}\ 16.0\qquad\textbf{(C)}\ 20.0\qquad\textbf{(D)}\ 33.3\qquad\textbf{(E)}\ 55.6$ | [
"By: dragonfly\nWe can solve this problem by using similar triangles, since two equilateral triangles are always similar. We can then use\n$\\left(\\frac{3}{5}\\right)^2=\\frac{12}{x}$\nWe can then solve the equation to get $x=\\frac{100}{3}$ which is closest to $\\boxed{33.3}$",
"Another approach to this problem... |
https://artofproblemsolving.com/wiki/index.php/2009_AMC_8_Problems/Problem_13 | B | 13 | A three-digit integer contains one of each of the digits $1$ $3$ , and $5$ . What is the probability that the integer is divisible by $5$
$\textbf{(A)}\ \frac{1}{6} \qquad \textbf{(B)}\ \frac{1}{3} \qquad \textbf{(C)}\ \frac{1}{2} \qquad \textbf{(D)}\ \frac{2}{3} \qquad \textbf{(E)}\ \frac{5}{6}$ | [
"The three digit numbers are $135,153,351,315,513,531$ . The numbers that end in $5$ are divisible are $5$ , and the probability of choosing those numbers is $\\boxed{13}$",
"The number is divisible by 5 if and only if the number ends in $5$ (also $0$ , but that case can be ignored, as none of the digits are $0$ ... |
https://artofproblemsolving.com/wiki/index.php/2014_AIME_I_Problems/Problem_11 | null | 391 | A token starts at the point $(0,0)$ of an $xy$ -coordinate grid and then makes a sequence of six moves. Each move is 1 unit in a direction parallel to one of the coordinate axes. Each move is selected randomly from the four possible directions and independently of the other moves. The probability the token ends at a point on the graph of $|y|=|x|$ is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ | [
"Perform the coordinate transformation $(x, y)\\rightarrow (x+y, x-y)$ . Then we can see that a movement up, right, left, or down in the old coordinates adds the vectors $\\langle 1, -1 \\rangle$ $\\langle 1, 1 \\rangle$ $\\langle -1, -1 \\rangle$ $\\langle -1, 1 \\rangle$ respectively. Moreover, the transformation... |
https://artofproblemsolving.com/wiki/index.php/1954_AHSME_Problems/Problem_48 | C | 600 | A train, an hour after starting, meets with an accident which detains it a half hour, after which it proceeds at $\frac{3}{4}$ of its former rate and arrives $3\tfrac{1}{2}$ hours late. Had the accident happened $90$ miles farther along the line, it would have arrived only $3$ hours late. The length of the trip in miles was:
$\textbf{(A)}\ 400 \qquad \textbf{(B)}\ 465 \qquad \textbf{(C)}\ 600 \qquad \textbf{(D)}\ 640 \qquad \textbf{(E)}\ 550$ | [
"Let the speed of the train be $x$ miles per hour, and let $D$ miles be the total distance of the trip, where $x$ and $D$ are unit-less quantities. Then for the trip that actually occurred, the train travelled 1 hour before the crash, and then travelled $D-x$ miles after the crash. In other words, the train travell... |
https://artofproblemsolving.com/wiki/index.php/1999_AIME_Problems/Problem_6 | null | 314 | A transformation of the first quadrant of the coordinate plane maps each point $(x,y)$ to the point $(\sqrt{x},\sqrt{y}).$ The vertices of quadrilateral $ABCD$ are $A=(900,300), B=(1800,600), C=(600,1800),$ and $D=(300,900).$ Let $k_{}$ be the area of the region enclosed by the image of quadrilateral $ABCD.$ Find the greatest integer that does not exceed $k_{}.$ | [
"\\begin{eqnarray*}A' = & (\\sqrt {900}, \\sqrt {300})\\\\ B' = & (\\sqrt {1800}, \\sqrt {600})\\\\ C' = & (\\sqrt {600}, \\sqrt {1800})\\\\ D' = & (\\sqrt {300}, \\sqrt {900}) \\end{eqnarray*}\nFirst we see that lines passing through $AB$ and $CD$ have equations $y = \\frac {1}{3}x$ and $y = 3x$ , respectively. Lo... |
https://artofproblemsolving.com/wiki/index.php/2012_AMC_12B_Problems/Problem_20 | D | 63 | A trapezoid has side lengths 3, 5, 7, and 11. The sum of all the possible areas of the trapezoid can be written in the form of $r_1\sqrt{n_1}+r_2\sqrt{n_2}+r_3$ , where $r_1$ $r_2$ , and $r_3$ are rational numbers and $n_1$ and $n_2$ are positive integers not divisible by the square of any prime. What is the greatest integer less than or equal to $r_1+r_2+r_3+n_1+n_2$
$\textbf{(A)}\ 57\qquad\textbf{(B)}\ 59\qquad\textbf{(C)}\ 61\qquad\textbf{(D)}\ 63\qquad\textbf{(E)}\ 65$ | [
"Name the trapezoid $ABCD$ , where $AB$ is parallel to $CD$ $AB<CD$ , and $AD<BC$ . Draw a line through $B$ parallel to $AD$ , crossing the side $CD$ at $E$ . Then $BE=AD$ $EC=DC-DE=DC-AB$ . One needs to guarantee that $BE+EC>BC$ , so there are only three possible trapezoids:\n\\[AB=3, BC=7, CD=11, DA=5, CE=8\\] \\... |
https://artofproblemsolving.com/wiki/index.php/2012_AMC_12A_Problems/Problem_10 | D | 23 | A triangle has area $30$ , one side of length $10$ , and the median to that side of length $9$ . Let $\theta$ be the acute angle formed by that side and the median. What is $\sin{\theta}$
$\textbf{(A)}\ \frac{3}{10}\qquad\textbf{(B)}\ \frac{1}{3}\qquad\textbf{(C)}\ \frac{9}{20}\qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{9}{10}$ | [
"$AB$ is the side of length $10$ , and $CD$ is the median of length $9$ . The altitude of $C$ to $AB$ is $6$ because the 0.5(altitude)(base)=Area of the triangle. $\\theta$ is $\\angle CDE$ . To find $\\sin{\\theta}$ , just use opposite over hypotenuse with the right triangle $\\triangle DCE$ . This is equal to $\\... |
https://artofproblemsolving.com/wiki/index.php/2010_AMC_10B_Problems/Problem_7 | D | 32 | A triangle has side lengths $10$ $10$ , and $12$ . A rectangle has width $4$ and area equal to the
area of the triangle. What is the perimeter of this rectangle?
$\textbf{(A)}\ 16 \qquad \textbf{(B)}\ 24 \qquad \textbf{(C)}\ 28 \qquad \textbf{(D)}\ 32 \qquad \textbf{(E)}\ 36$ | [
"The triangle is isosceles. The height of the triangle is therefore given by $h = \\sqrt{10^2 - \\left( \\dfrac{12}{2} \\right)^2} = \\sqrt{64} = 8$\nNow, the area of the triangle is $\\dfrac{bh}{2} = \\dfrac{12*8}{2} = \\dfrac{96}{2} = 48$\nWe have that the area of the rectangle is the same as the area of the tri... |
https://artofproblemsolving.com/wiki/index.php/2017_AIME_II_Problems/Problem_3 | null | 409 | A triangle has vertices $A(0,0)$ $B(12,0)$ , and $C(8,10)$ . The probability that a randomly chosen point inside the triangle is closer to vertex $B$ than to either vertex $A$ or vertex $C$ can be written as $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p+q$ | [
"\nThe set of all points closer to point $B$ than to point $A$ lie to the right of the perpendicular bisector of $AB$ (line $PZ$ in the diagram), and the set of all points closer to point $B$ than to point $C$ lie below the perpendicular bisector of $BC$ (line $PX$ in the diagram). Therefore, the set of points insi... |
https://artofproblemsolving.com/wiki/index.php/1967_AHSME_Problems/Problem_5 | C | 2 | A triangle is circumscribed about a circle of radius $r$ inches. If the perimeter of the triangle is $P$ inches and the area is $K$ square inches, then $\frac{P}{K}$ is:
$\text{(A)}\text{ independent of the value of} \; r\qquad\text{(B)}\ \frac{\sqrt{2}}{r}\qquad\text{(C)}\ \frac{2}{\sqrt{r}}\qquad\text{(D)}\ \frac{2}{r}\qquad\text{(E)}\ \frac{r}{2}$ | [
"The area $K$ of the triangle can be expressed in terms of its inradius $r$ and its semiperimeter $s$ as: \\[K = r \\times s = r \\times \\frac{P}{2}\\]\nSo, $\\frac{P}{K} = \\boxed{2}$"
] |
https://artofproblemsolving.com/wiki/index.php/2006_AMC_10B_Problems/Problem_23 | D | 18 | A triangle is partitioned into three triangles and a quadrilateral by drawing two lines from vertices to their opposite sides. The areas of the three triangles are 3, 7, and 7, as shown. What is the area of the shaded quadrilateral?
[asy] unitsize(1.5cm); defaultpen(.8); pair A = (0,0), B = (3,0), C = (1.4, 2), D = B + 0.4*(C-B), Ep = A + 0.3*(C-A); pair F = intersectionpoint( A--D, B--Ep ); draw( A -- B -- C -- cycle ); draw( A -- D ); draw( B -- Ep ); filldraw( D -- F -- Ep -- C -- cycle, mediumgray, black ); label("$7$",(1.25,0.2)); label("$7$",(2.2,0.45)); label("$3$",(0.45,0.35)); [/asy]
$\mathrm{(A) \ } 15\qquad \mathrm{(B) \ } 17\qquad \mathrm{(C) \ } \frac{35}{2}\qquad \mathrm{(D) \ } 18\qquad \mathrm{(E) \ } \frac{55}{3}$ | [
"Label the points in the figure as shown below, and draw the segment $CF$ . This segment divides the quadrilateral into two triangles, let their areas be $x$ and $y$\n\nSince triangles $AFB$ and $DFB$ share an altitude from $B$ and have equal area, their bases must be equal, hence $AF=DF$\nSince triangles $AFC$ and... |
https://artofproblemsolving.com/wiki/index.php/2016_AMC_10A_Problems/Problem_9 | D | 9 | A triangular array of $2016$ coins has $1$ coin in the first row, $2$ coins in the second row, $3$ coins in the third row, and so on up to $N$ coins in the $N$ th row. What is the sum of the digits of $N$
$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10$ | [
"We are trying to find the value of $N$ such that \\[1+2+3\\cdots+(N-1)+N=\\frac{N(N+1)}{2}=2016.\\] Noticing that $\\frac{63\\cdot 64}{2}=2016,$ we have $N=63,$ so our answer is $\\boxed{9}.$",
"Knowing that each row number can stand for the number of coins there are in the row, we can just add until we get $201... |
https://artofproblemsolving.com/wiki/index.php/2016_AMC_12A_Problems/Problem_6 | D | 9 | A triangular array of $2016$ coins has $1$ coin in the first row, $2$ coins in the second row, $3$ coins in the third row, and so on up to $N$ coins in the $N$ th row. What is the sum of the digits of $N$
$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10$ | [
"We are trying to find the value of $N$ such that \\[1+2+3\\cdots+(N-1)+N=\\frac{N(N+1)}{2}=2016.\\] Noticing that $\\frac{63\\cdot 64}{2}=2016,$ we have $N=63,$ so our answer is $\\boxed{9}.$",
"Knowing that each row number can stand for the number of coins there are in the row, we can just add until we get $201... |
https://artofproblemsolving.com/wiki/index.php/2008_AMC_10A_Problems/Problem_6 | D | 6 | A triathlete competes in a triathlon in which the swimming, biking, and running segments are all of the same length. The triathlete swims at a rate of 3 kilometers per hour, bikes at a rate of 20 kilometers per hour, and runs at a rate of 10 kilometers per hour. Which of the following is closest to the triathlete's average speed, in kilometers per hour, for the entire race?
$\mathrm{(A)}\ 3\qquad\mathrm{(B)}\ 4\qquad\mathrm{(C)}\ 5\qquad\mathrm{(D)}\ 6\qquad\mathrm{(E)}\ 7$ | [
"Since the three segments are all the same length, the triathlete's average speed is the harmonic mean of the three given rates. Therefore, the average speed is \\[\\frac{3}{\\frac{1}{3}+\\frac{1}{20}+\\frac{1}{10}}=\\frac{3}{\\frac{29}{60}}=\\frac{180}{29}\\approx6\\Rightarrow\\boxed{6}\\]"
] |
https://artofproblemsolving.com/wiki/index.php/2006_AIME_I_Problems/Problem_14 | null | 183 | A tripod has three legs each of length $5$ feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is $4$ feet from the ground. In setting up the tripod, the lower 1 foot of one leg breaks off. Let $h$ be the height in feet of the top of the tripod from the ground when the broken tripod is set up. Then $h$ can be written in the form $\frac m{\sqrt{n}},$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $\lfloor m+\sqrt{n}\rfloor.$ (The notation $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x.$ | [
"Diagram borrowed from Solution 1\nApply Pythagorean Theorem on $\\bigtriangleup TOB$ yields \\[BO=\\sqrt{TB^2-TO^2}=3\\] Since $\\bigtriangleup ABC$ is equilateral, we have $\\angle MOB=60^{\\circ}$ and \\[BC=2BM=2(OB\\sin MOB)=3\\sqrt{3}\\] Apply Pythagorean Theorem on $\\bigtriangleup TMB$ yields \\[TM=\\sqrt{TB... |
https://artofproblemsolving.com/wiki/index.php/2014_AMC_10B_Problems/Problem_8 | E | 10 | A truck travels $\dfrac{b}{6}$ feet every $t$ seconds. There are $3$ feet in a yard. How many yards does the truck travel in $3$ minutes?
$\textbf {(A) } \frac{b}{1080t} \qquad \textbf {(B) } \frac{30t}{b} \qquad \textbf {(C) } \frac{30b}{t}\qquad \textbf {(D) } \frac{10t}{b} \qquad \textbf {(E) } \frac{10b}{t}$ | [
"Converting feet to yards and minutes to second, we see that the truck travels $\\dfrac{b}{18}$ yards every $t$ seconds for $180$ seconds. We see that he does $\\dfrac{180}{t}$ cycles of $\\dfrac{b}{18}$ yards. Multiplying, we get $\\dfrac{180b}{18t}$ , or $\\dfrac{10b}{t}$ , or $\\boxed{10}$",
"We set a proporti... |
https://artofproblemsolving.com/wiki/index.php/2004_AMC_12B_Problems/Problem_19 | A | 6 | A truncated cone has horizontal bases with radii $18$ and $2$ . A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere?
$\mathrm{(A)}\ 6 \qquad\mathrm{(B)}\ 4\sqrt{5} \qquad\mathrm{(C)}\ 9 \qquad\mathrm{(D)}\ 10 \qquad\mathrm{(E)}\ 6\sqrt{3}$ | [
"Consider a trapezoid (label it $ABCD$ as follows) cross-section of the truncate cone along a diameter of the bases:\nAbove, $E,F,$ and $G$ are points of tangency . By the Two Tangent Theorem, $BF = BE = 18$ and $CF = CG = 2$ , so $BC = 20$ . We draw $H$ such that it is the foot of the altitude $\\overline{HD}$ to ... |
https://artofproblemsolving.com/wiki/index.php/2021_Fall_AMC_10A_Problems/Problem_8 | B | 1 | A two-digit positive integer is said to be $\emph{cuddly}$ if it is equal to the sum of its nonzero tens digit and the square of its units digit. How many two-digit positive integers are cuddly?
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4$ | [
"Note that the number $\\underline{xy} = 10x + y.$ By the problem statement, \\[10x + y = x + y^2 \\implies 9x = y^2 - y \\implies 9x = y(y-1).\\] From this we see that $y(y-1)$ must be divisible by $9.$ This only happens when $y=9.$ Then, $x=8.$ Thus, there is only $\\boxed{1}$ cuddly number, which is $89.$",
"I... |
https://artofproblemsolving.com/wiki/index.php/1997_AJHSME_Problems/Problem_22 | E | 675 | A two-inch cube $(2\times 2\times 2)$ of silver weighs 3 pounds and is worth 200 dollars. How much is a three-inch cube of silver worth?
$\textbf{(A) }\text{300 dollars} \qquad \textbf{(B) }\text{375 dollars} \qquad \textbf{(C) }\text{450 dollars} \qquad \textbf{(D) }\text{560 dollars}\qquad \textbf{(E) }\text{675 dollars}$ | [
"The two-inch cube has a volume of $8$ cubic inches, and the three-inch cube has a volume of $27$ cubic inches. Thus, the three-inch cube has a weight that is $\\frac{27}{8}$ times that of the two-inch cube. Then its value is $\\frac{27}{8} \\cdot 200 = \\boxed{675}$"
] |
https://artofproblemsolving.com/wiki/index.php/2004_AIME_I_Problems/Problem_14 | null | 813 | A unicorn is tethered by a $20$ -foot silver rope to the base of a magician's cylindrical tower whose radius is $8$ feet. The rope is attached to the tower at ground level and to the unicorn at a height of $4$ feet. The unicorn has pulled the rope taut, the end of the rope is $4$ feet from the nearest point on the tower, and the length of the rope that is touching the tower is $\frac{a-\sqrt{b}}c$ feet, where $a, b,$ and $c$ are positive integers , and $c$ is prime. Find $a+b+c.$ | [
"Looking from an overhead view, call the center of the circle $O$ , the tether point to the unicorn $A$ and the last point where the rope touches the tower $B$ $\\triangle OAB$ is a right triangle because $OB$ is a radius and $BA$ is a tangent line at point $B$ . We use the Pythagorean Theorem to find the horizont... |
https://artofproblemsolving.com/wiki/index.php/2018_AMC_10A_Problems/Problem_3 | E | 12 | A unit of blood expires after $10!=10\cdot 9 \cdot 8 \cdots 1$ seconds. Yasin donates a unit of blood at noon of January 1. On what day does his unit of blood expire?
$\textbf{(A) }\text{January 2}\qquad\textbf{(B) }\text{January 12}\qquad\textbf{(C) }\text{January 22}\qquad\textbf{(D) }\text{February 11}\qquad\textbf{(E) }\text{February 12}$ | [
"There are $60 \\cdot 60 \\cdot 24 = 86400$ seconds in a day, which means that Yasin's blood expires in $10! \\div 86400 = 42$ days. Since there are $31$ days in January (consult a calendar), then $42-31+1$ (Jan 1 doesn't count) is $12$ days into February, so $\\boxed{12}$"
] |
https://artofproblemsolving.com/wiki/index.php/2004_AMC_10A_Problems/Problem_19 | C | 240 | A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet?
[asy] size(250);defaultpen(linewidth(0.8)); draw(ellipse(origin, 3, 1)); fill((3,0)--(3,2)--(-3,2)--(-3,0)--cycle, white); draw((3,0)--(3,16)^^(-3,0)--(-3,16)); draw((0, 15)--(3, 12)^^(0, 16)--(3, 13)); filldraw(ellipse((0, 16), 3, 1), white, black); draw((-3,11)--(3, 5)^^(-3,10)--(3, 4)); draw((-3,2)--(0,-1)^^(-3,1)--(-1,-0.89)); draw((0,-1)--(0,15), dashed); draw((3,-2)--(3,-4)^^(-3,-2)--(-3,-4)); draw((-7,0)--(-5,0)^^(-7,16)--(-5,16)); draw((3,-3)--(-3,-3), Arrows(6)); draw((-6,0)--(-6,16), Arrows(6)); draw((-2,9)--(-1,9), Arrows(3)); label("$3$", (-1.375,9.05), dir(260), fontsize(7)); label("$A$", (0,15), N); label("$B$", (0,-1), NE); label("$30$", (0, -3), S); label("$80$", (-6, 8), W);[/asy]
$\mathrm{(A) \ } 120 \qquad \mathrm{(B) \ } 180 \qquad \mathrm{(C) \ } 240 \qquad \mathrm{(D) \ } 360 \qquad \mathrm{(E) \ } 480$ | [
"The cylinder can be \"unwrapped\" into a rectangle, and we see that the stripe is a parallelogram with base $3$ and height $80$ . Thus, we get $3\\times80=240\\Rightarrow\\boxed{240}$"
] |
https://artofproblemsolving.com/wiki/index.php/2005_AMC_10A_Problems/Problem_11 | B | 4 | A wooden cube $n$ units on a side is painted red on all six faces and then cut into $n^3$ unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is $n$
$\textbf{(A) } 3\qquad \textbf{(B) } 4\qquad \textbf{(C) } 5\qquad \textbf{(D) } 6\qquad \textbf{(E) } 7$ | [
"Since there are $n^2$ little faces on each face of the big wooden cube , there are $6n^2$ little faces painted red.\nSince each unit cube has $6$ faces, there are $6n^3$ little faces total.\nSince one-fourth of the little faces are painted red,\n$\\frac{6n^2}{6n^3}=\\frac{1}{4}$\n$\\frac{1}{n}=\\frac{1}{4}$\n$n=\\... |
https://artofproblemsolving.com/wiki/index.php/1996_AIME_Problems/Problem_4 | null | 166 | A wooden cube , whose edges are one centimeter long, rests on a horizontal surface. Illuminated by a point source of light that is $x$ centimeters directly above an upper vertex , the cube casts a shadow on the horizontal surface. The area of the shadow, which does not include the area beneath the cube is 48 square centimeters. Find the greatest integer that does not exceed $1000x$ | [
" (Figure not to scale) The area of the square shadow base is $48 + 1 = 49$ , and so the sides of the shadow are $7$ . Using the similar triangles in blue, $\\frac {x}{1} = \\frac {1}{6}$ , and $\\left\\lfloor 1000x \\right\\rfloor = \\boxed{166}$"
] |
https://artofproblemsolving.com/wiki/index.php/1985_AHSME_Problems/Problem_20 | D | 8 | A wooden cube with edge length $n$ units (where $n$ is an integer $>2$ ) is painted black all over. By slices parallel to its faces, the cube is cut into $n^3$ smaller cubes each of unit edge length. If the number of smaller cubes with just one face painted black is equal to the number of smaller cubes completely free of paint, what is $n$
$\mathrm{(A)\ } 5 \qquad \mathrm{(B) \ }6 \qquad \mathrm{(C) \ } 7 \qquad \mathrm{(D) \ } 8 \qquad \mathrm{(E) \ }\text{none of these}$ | [
"Observe that if we remove the outer layer of unit cubes from the entire cube, what remains is a smaller cube of side length $(n-2)$ , which contains all of the unpainted cubes and no others. This shows that there are exactly $(n-2)^3$ unpainted cubes. Similarly, taking one face of the cube and removing the outer e... |
https://artofproblemsolving.com/wiki/index.php/2024_AMC_8_Problems/Problem_5 | B | 6 | Aaliyah rolls two standard 6-sided dice. She notices that the product of the two numbers rolled is a multiple of $6$ . Which of the following integers cannot be the sum of the two numbers?
$\textbf{(A) } 5\qquad\textbf{(B) } 6\qquad\textbf{(C) } 7\qquad\textbf{(D) } 8\qquad\textbf{(E) } 9$ | [
"First, figure out all pairs of numbers whose product is 6. Then, using the process of elimination, we can find the following:\n$\\textbf{(A)}$ is possible: $2\\times 3$\n$\\textbf{(C)}$ is possible: $1\\times 6$\n$\\textbf{(D)}$ is possible: $2\\times 6$\n$\\textbf{(E)}$ is possible: $3\\times 6$\nThe only integer... |
https://artofproblemsolving.com/wiki/index.php/2014_AIME_II_Problems/Problem_1 | null | 334 | Abe can paint the room in 15 hours, Bea can paint 50 percent faster than Abe, and Coe can paint twice as fast as Abe. Abe begins to paint the room and works alone for the first hour and a half. Then Bea joins Abe, and they work together until half the room is painted. Then Coe joins Abe and Bea, and they work together until the entire room is painted. Find the number of minutes after Abe begins for the three of them to finish painting the room. | [
"From the given information, we can see that Abe can paint $\\frac{1}{15}$ of the room in an hour, Bea can paint $\\frac{1}{15}\\times\\frac{3}{2} = \\frac{1}{10}$ of the room in an hour, and Coe can paint the room in $\\frac{1}{15}\\times 2 = \\frac{2}{15}$ of the room in an hour. After $90$ minutes, Abe has pain... |
https://artofproblemsolving.com/wiki/index.php/2013_AMC_8_Problems/Problem_14 | C | 38 | Abe holds 1 green and 1 red jelly bean in his hand. Bob holds 1 green, 1 yellow, and 2 red jelly beans in his hand. Each randomly picks a jelly bean to show the other. What is the probability that the colors match?
$\textbf{(A)}\ \frac14 \qquad \textbf{(B)}\ \frac13 \qquad \textbf{(C)}\ \frac38 \qquad \textbf{(D)}\ \frac12 \qquad \textbf{(E)}\ \frac23$ | [
"The favorable responses are either they both show a green bean or they both show a red bean. The probability that both show a green bean is $\\frac{1}{2}\\cdot\\frac{1}{4}=\\frac{1}{8}$ . The probability that both show a red bean is $\\frac{1}{2}\\cdot\\frac{2}{4}=\\frac{1}{4}$ . Therefore the probability is $\\fr... |
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