link stringlengths 75 84 | letter stringclasses 5
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https://artofproblemsolving.com/wiki/index.php/2002_AMC_10A_Problems/Problem_3 | B | 1 | According to the standard convention for exponentiation, \[2^{2^{2^{2}}} = 2^{(2^{(2^2)})} = 2^{16} = 65536.\]
If the order in which the exponentiations are performed is changed, how many other values are possible?
$\textbf{(A) } 0\qquad \textbf{(B) } 1\qquad \textbf{(C) } 2\qquad \textbf{(D) } 3\qquad \textbf{(E) } 4$ | [
"The best way to solve this problem is by simple brute force.\nIt is convenient to drop the usual way how exponentiation is denoted, and to write the formula as $2\\uparrow 2\\uparrow 2\\uparrow 2$ , where $\\uparrow$ denotes exponentiation. We are now examining all ways to add parentheses to this expression. There... |
https://artofproblemsolving.com/wiki/index.php/2002_AMC_12A_Problems/Problem_3 | B | 1 | According to the standard convention for exponentiation, \[2^{2^{2^{2}}} = 2^{(2^{(2^2)})} = 2^{16} = 65536.\]
If the order in which the exponentiations are performed is changed, how many other values are possible?
$\textbf{(A) } 0\qquad \textbf{(B) } 1\qquad \textbf{(C) } 2\qquad \textbf{(D) } 3\qquad \textbf{(E) } 4$ | [
"The best way to solve this problem is by simple brute force.\nIt is convenient to drop the usual way how exponentiation is denoted, and to write the formula as $2\\uparrow 2\\uparrow 2\\uparrow 2$ , where $\\uparrow$ denotes exponentiation. We are now examining all ways to add parentheses to this expression. There... |
https://artofproblemsolving.com/wiki/index.php/2012_AMC_10A_Problems/Problem_23 | B | 170 | Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways can this happen?
$\text{(A)}\ 60\qquad\... | [
"Note that if $n$ is the number of friends each person has, then $n$ can be any integer from $1$ to $4$ , inclusive.\nOne person can have at most 4 friends since they cannot be all friends (stated in the problem).\nAlso note that the cases of $n=1$ and $n=4$ are the same, since a map showing a solution for $n=1$ ca... |
https://artofproblemsolving.com/wiki/index.php/2012_AMC_12A_Problems/Problem_19 | B | 170 | Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways can this happen?
$\text{(A)}\ 60\qquad\... | [
"Note that if $n$ is the number of friends each person has, then $n$ can be any integer from $1$ to $4$ , inclusive.\nOne person can have at most 4 friends since they cannot be all friends (stated in the problem).\nAlso note that the cases of $n=1$ and $n=4$ are the same, since a map showing a solution for $n=1$ ca... |
https://artofproblemsolving.com/wiki/index.php/2022_AIME_II_Problems/Problem_1 | null | 154 | Adults made up $\frac5{12}$ of the crowd of people at a concert. After a bus carrying $50$ more people arrived, adults made up $\frac{11}{25}$ of the people at the concert. Find the minimum number of adults who could have been at the concert after the bus arrived. | [
"Let $x$ be the number of people at the party before the bus arrives. We know that $x\\equiv 0\\pmod {12}$ , as $\\frac{5}{12}$ of people at the party before the bus arrives are adults. Similarly, we know that $x + 50 \\equiv 0 \\pmod{25}$ , as $\\frac{11}{25}$ of the people at the party are adults after the bus ar... |
https://artofproblemsolving.com/wiki/index.php/2019_AMC_8_Problems/Problem_23 | B | 11 | After Euclid High School's last basketball game, it was determined that $\frac{1}{4}$ of the team's points were scored by Alexa and $\frac{2}{7}$ were scored by Brittany. Chelsea scored $15$ points. None of the other $7$ team members scored more than $2$ points. What was the total number of points scored by the other $... | [
"Given the information above, we start with the equation $\\frac{t}{4}+\\frac{2t}{7} + 15 + x = t$ ,where $t$ is the total number of points scored and $x\\le 14$ is the number of points scored by the remaining 7 team members, we can simplify to obtain the Diophantine equation $x+15 = \\frac{13}{28}t$ , or $28x+28\\... |
https://artofproblemsolving.com/wiki/index.php/2004_AMC_8_Problems/Problem_6 | C | 3 | After Sally takes $20$ shots, she has made $55\%$ of her shots. After she takes $5$ more shots, she raises her percentage to $56\%$ . How many of the last $5$ shots did she make?
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$ | [
"Sally made $0.55*20=11$ shots originally. Letting $x$ be the number of shots she made, we have $\\frac{11+x}{25}=0.56$ . Solving for $x$ gives us $x=\\boxed{3}$"
] |
https://artofproblemsolving.com/wiki/index.php/2020_AMC_8_Problems/Problem_11 | E | 24 | After school, Maya and Naomi headed to the beach, $6$ miles away. Maya decided to bike while Naomi took a bus. The graph below shows their journeys, indicating the time and distance traveled. What was the difference, in miles per hour, between Naomi's and Maya's average speeds?
[asy] unitsize(1.25cm); dotfactor = 10; p... | [
"Naomi travels $6$ miles in a time of $10$ minutes, which is equivalent to $\\dfrac{1}{6}$ of an hour. Since $\\text{speed} = \\frac{\\text{distance}}{\\text{time}}$ , her speed is $\\frac{6}{\\left(\\frac{1}{6}\\right)} = 36$ mph. By a similar calculation, Maya's speed is $12$ mph, so the answer is $36-12 = \\boxe... |
https://artofproblemsolving.com/wiki/index.php/1997_AJHSME_Problems/Problem_2 | D | 380 | Ahn chooses a two-digit integer, subtracts it from 200, and doubles the result. What is the largest number Ahn can get?
$\text{(A)}\ 200 \qquad \text{(B)}\ 202 \qquad \text{(C)}\ 220 \qquad \text{(D)}\ 380 \qquad \text{(E)}\ 398$ | [
"The smallest two-digit integer he can subtract from $200$ is $10$ . This will give the largest result for that first operation, and doubling it will keep it as the largest number possible.\n\\[200-10=190\\] \\[190\\times2=380\\]\n$\\boxed{380}$"
] |
https://artofproblemsolving.com/wiki/index.php/2018_AMC_12B_Problems/Problem_23 | C | 120 | Ajay is standing at point $A$ near Pontianak, Indonesia, $0^\circ$ latitude and $110^\circ \text{ E}$ longitude. Billy is standing at point $B$ near Big Baldy Mountain, Idaho, USA, $45^\circ \text{ N}$ latitude and $115^\circ \text{ W}$ longitude. Assume that Earth is a perfect sphere with center $C.$ What is the degre... | [
"This solution refers to the Diagram section.\nLet $D$ be the orthogonal projection of $B$ onto the equator. Note that $\\angle BDA = \\angle BDC = 90^\\circ$ and $\\angle BCD = 45^\\circ.$ Recall that $115^\\circ \\text{ W}$ longitude is the same as $245^\\circ \\text{ E}$ longitude, so $\\angle ACD=135^\\circ.$\n... |
https://artofproblemsolving.com/wiki/index.php/2003_AMC_10B_Problems/Problem_2 | D | 20 | Al gets the disease algebritis and must take one green pill and one pink pill each day for two weeks. A green pill costs $$ $1$ more than a pink pill, and Al's pills cost a total of $\textdollar 546$ for the two weeks. How much does one green pill cost?
$\textbf{(A)}\ \textdollar 7 \qquad\textbf{(B) }\textdollar 14 \qq... | [
"Because there are $14$ days in two weeks, Al spends $546/14 = 39$ dollars per day for the cost of a green pill and a pink pill. If the green pill costs $x$ dollars and the pink pill $x-1$ dollars, the sum of the two costs $2x-1$ should equal $39$ dollars. Then the cost of the green pill $x$ is $\\boxed{20}$"
] |
https://artofproblemsolving.com/wiki/index.php/2003_AMC_12B_Problems/Problem_2 | D | 20 | Al gets the disease algebritis and must take one green pill and one pink pill each day for two weeks. A green pill costs $$ $1$ more than a pink pill, and Al's pills cost a total of $\textdollar 546$ for the two weeks. How much does one green pill cost?
$\textbf{(A)}\ \textdollar 7 \qquad\textbf{(B) }\textdollar 14 \qq... | [
"Because there are $14$ days in two weeks, Al spends $546/14 = 39$ dollars per day for the cost of a green pill and a pink pill. If the green pill costs $x$ dollars and the pink pill $x-1$ dollars, the sum of the two costs $2x-1$ should equal $39$ dollars. Then the cost of the green pill $x$ is $\\boxed{20}$"
] |
https://artofproblemsolving.com/wiki/index.php/1987_AIME_Problems/Problem_10 | null | 120 | Al walks down to the bottom of an escalator that is moving up and he counts 150 steps. His friend, Bob, walks up to the top of the escalator and counts 75 steps. If Al's speed of walking (in steps per unit time) is three times Bob's walking speed, how many steps are visible on the escalator at a given time? (Assume tha... | [
"Let the total number of steps be $x$ , the speed of the escalator be $e$ and the speed of Bob be $b$\nIn the time it took Bob to climb up the escalator he saw 75 steps and also climbed the entire escalator. Thus the contribution of the escalator must have been an additional $x - 75$ steps. Since Bob and the esca... |
https://artofproblemsolving.com/wiki/index.php/1977_AHSME_Problems/Problem_12 | D | 102 | Al's age is $16$ more than the sum of Bob's age and Carl's age, and the square of Al's age is $1632$ more than the square of the sum of
Bob's age and Carl's age. What is the sum of the ages of Al, Bob, and Carl?
$\text{(A)}\ 64 \qquad \text{(B)}\ 94 \qquad \text{(C)}\ 96 \qquad \text{(D)}\ 102 \qquad \text{(E)}\ 1... | [
"Solution by e_power_pi_times_i\nDenote Al's age, Bob's age, and Carl's age by $a$ $b$ , and $c$ , respectively. Then, $a = 16 + b + c$ and $a^2 = 1632 + b^2 + c^2$ . Substituting the first equation into the second, $(16 + b + c)^2 = b^2 + c^2 + 2bc + 32b + 32c + 256 = b^2 + c^2 + 1632$ . Thus, $bc + 16b + 16c = 68... |
https://artofproblemsolving.com/wiki/index.php/2003_AMC_10B_Problems/Problem_12 | C | 400 | Al, Betty, and Clare split $\textdollar 1000$ among them to be invested in different ways. Each begins with a different amount. At the end of one year, they have a total of $\textdollar 1500$ dollars. Betty and Clare have both doubled their money, whereas Al has managed to lose $\textdollar100$ dollars. What was Al's o... | [
"For this problem, we will have to write a three-variable equation, but not necessarily solve it. Let $a, b,$ and $c$ represent the original portions of Al, Betty, and Clare, respectively. At the end of one year, they each have $a-100, 2b,$ and $2c$ . From this, we can write two equations, marked by (1) and (2).\n\... |
https://artofproblemsolving.com/wiki/index.php/2013_AMC_10B_Problems/Problem_17 | E | 103 | Alex has $75$ red tokens and $75$ blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges... | [
"If Alex goes to the red booth 3 times, then goes to the blue booth once, Alex can exchange 6 red tokens for 4 silver tokens and one red token. Similarly, if Alex goes to the blue booth 2 times, then goes to the red booth once, Alex can exchange 6 blue tokens for 3 silver tokens and one blue token. Let's call the f... |
https://artofproblemsolving.com/wiki/index.php/2013_AMC_12B_Problems/Problem_10 | E | 103 | Alex has $75$ red tokens and $75$ blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges... | [
"If Alex goes to the red booth 3 times, then goes to the blue booth once, Alex can exchange 6 red tokens for 4 silver tokens and one red token. Similarly, if Alex goes to the blue booth 2 times, then goes to the red booth once, Alex can exchange 6 blue tokens for 3 silver tokens and one blue token. Let's call the f... |
https://artofproblemsolving.com/wiki/index.php/1993_AIME_Problems/Problem_11 | null | 93 | Alfred and Bonnie play a game in which they take turns tossing a fair coin. The winner of a game is the first person to obtain a head. Alfred and Bonnie play this game several times with the stipulation that the loser of a game goes first in the next game. Suppose that Alfred goes first in the first game, and that the ... | [
"In order to begin this problem, we need to calculate the probability that Alfred will win on the first round.\nBecause he goes first, Alfred has a $\\frac{1}{2}$ chance of winning (getting heads) on his first flip.\nThen, Bonnie, who goes second, has a $\\frac{1}{2} \\times \\frac{1}{2} = \\frac{1}{4}$ , chance of... |
https://artofproblemsolving.com/wiki/index.php/2003_AMC_8_Problems/Problem_16 | D | 12 | Ali, Bonnie, Carlo, and Dianna are going to drive together to a nearby theme park. The car they are using has $4$ seats: $1$ Driver seat, $1$ front passenger seat, and $2$ back passenger seat. Bonnie and Carlo are the only ones who know how to drive the car. How many possible seating arrangements are there?
$\textbf{(A... | [
"There are only $2$ people who can go in the driver's seat--Bonnie and Carlo. Any of the $3$ remaining people can go in the front passenger seat. There are $2$ people who can go in the first back passenger seat, and the remaining person must go in the last seat. Thus, there are $2\\cdot3\\cdot2$ or $12$ ways. The a... |
https://artofproblemsolving.com/wiki/index.php/2016_AMC_12B_Problems/Problem_13 | E | 5.5 | Alice and Bob live $10$ miles apart. One day Alice looks due north from her house and sees an airplane. At the same time Bob looks due west from his house and sees the same airplane. The angle of elevation of the airplane is $30^\circ$ from Alice's position and $60^\circ$ from Bob's position. Which of the following is ... | [
"We have two 30-60-90 triangles $ABC$ and $DBC$ that are perpendicular and share leg $BC$ (the altitude of the plane $p$ ). $AD=10$ The shared leg is the shortest leg of one triangle and the longest leg of the other. $A$ and $B$ are Bob and Alice respectively.\nFind $AC$ and $DC$ in terms of $p$ . Use Pythagorean T... |
https://artofproblemsolving.com/wiki/index.php/2005_AMC_8_Problems/Problem_20 | A | 6 | Alice and Bob play a game involving a circle whose circumference is divided by 12 equally-spaced points. The points are numbered clockwise, from 1 to 12. Both start on point 12. Alice moves clockwise and Bob, counterclockwise.
In a turn of the game, Alice moves 5 points clockwise and Bob moves 9 points counterclockwise... | [
"Alice moves $5k$ steps and Bob moves $9k$ steps, where $k$ is the turn they are on. Alice and Bob coincide when the number of steps they move collectively, $14k$ , is a multiple of $12$ . Since this number must be a multiple of $12$ , as stated in the previous sentence, $14$ has a factor $2$ $k$ must have a factor... |
https://artofproblemsolving.com/wiki/index.php/2024_AIME_I_Problems/Problem_3 | null | 809 | Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either $1$ token or $4$ tokens from the stack. Whoever removes the last token wins. Find the number of positive integers $n$ less than or equal to $2024$ for whi... | [
"Let's first try some experimentation. Alice obviously wins if there is one coin. She will just take it and win. If there are 2 remaining, then Alice will take one and then Bob will take one, so Bob wins. If there are $3$ , Alice will take $1$ , Bob will take one, and Alice will take the final one. If there are $4$... |
https://artofproblemsolving.com/wiki/index.php/2024_AIME_II_Problems/Problem_6 | null | 55 | Alice chooses a set $A$ of positive integers. Then Bob lists all finite nonempty sets $B$ of positive integers with the property that the maximum element of $B$ belongs to $A$ . Bob's list has 2024 sets. Find the sum of the elements of A. | [
"Let $k$ be one of the elements in Alices set $A$ of positive integers. The number of sets that Bob lists with the property that their maximum element is k is $2^{k-1}$ , since every positive integer less than k can be in the set or out. Thus, for the number of sets bob have listed to be 2024, we want to find a sum... |
https://artofproblemsolving.com/wiki/index.php/2019_AMC_8_Problems/Problem_25 | C | 190 | Alice has $24$ apples. In how many ways can she share them with Becky and Chris so that each of the three people has at least two apples? $\textbf{(A) }105\qquad\textbf{(B) }114\qquad\textbf{(C) }190\qquad\textbf{(D) }210\qquad\textbf{(E) }380$ | [
"Note: This solution uses the non-negative version for stars and bars. A solution using the positive version of stars is similar (first removing an apple from each person instead of 2).\nThis method uses the counting method of stars and bars (non-negative version). Since each person must have at least $2$ apples, w... |
https://artofproblemsolving.com/wiki/index.php/2013_AMC_10A_Problems/Problem_2 | B | 10 | Alice is making a batch of cookies and needs $2\frac{1}{2}$ cups of sugar. Unfortunately, her measuring cup holds only $\frac{1}{4}$ cup of sugar. How many times must she fill that cup to get the correct amount of sugar?
$\textbf{(A)}\ 8 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 12 \qquad\textbf{(D)}\ 16 \qquad\tex... | [
"To get how many cups we need, we realize that we simply need to divide the number of cups needed by the number of cups collected in her measuring cup each time. Thus, we need to evaluate the fraction $\\frac{2\\frac{1}{2}}{\\frac{1}{4}}$ . Simplifying, this is equal to $\\frac{5}{2}(4) = \\boxed{10}$"
] |
https://artofproblemsolving.com/wiki/index.php/2023_AIME_I_Problems/Problem_6 | null | 51 | Alice knows that $3$ red cards and $3$ black cards will be revealed to her one at a time in random order. Before each card is revealed, Alice must guess its color. If Alice plays optimally, the expected number of cards she will guess correctly is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. ... | [
"At any point in the game, Alice should guess whichever color has come up less frequently thus far (although if both colors have come up equally often, she may guess whichever she likes); using this strategy, her probability of guessing correctly is at least $\\frac{1}{2}$ on any given card, as desired.\nThere are ... |
https://artofproblemsolving.com/wiki/index.php/2010_AMC_8_Problems/Problem_5 | B | 34 | Alice needs to replace a light bulb located $10$ centimeters below the ceiling in her kitchen. The ceiling is $2.4$ meters above the floor. Alice is $1.5$ meters tall and can reach $46$ centimeters above the top of her head. Standing on a stool, she can just reach the light bulb. What is the height of the stool, in cen... | [
"Convert everything to the same unit. Since the answer is in centimeters, change meters to centimeters by moving the decimal place two places to the right.\nThe ceiling is $240$ centimeters above the floor. The combined height of Alice and the light bulb when she reaches for it is $10+150+46=206$ centimeters. That ... |
https://artofproblemsolving.com/wiki/index.php/2017_AMC_12A_Problems/Problem_14 | C | 28 | Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of $5$ chairs under these conditions?
$\textbf{(A)}\ 12 \qquad \textbf{(B)}\ 16 \qquad\textbf{(C)}\ 28 \qquad\textbf{(D)}\ 32 \qquad\textbf{(E)}\ 40$ | [
"Alice may sit in the center chair, in an end chair, or in a next-to-end chair. Suppose she sits in the center chair. The 2nd and 4th chairs (next to her) must be occupied by Derek and Eric, in either order, leaving the end chairs for Bob and Carla in either order; this yields $2! * 2! = 4$ ways to seat the group.\... |
https://artofproblemsolving.com/wiki/index.php/2017_AMC_10A_Problems/Problem_19 | C | 28 | Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of $5$ chairs under these conditions?
$\textbf{(A)}\ 12\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 32\qquad\textbf{(E)}\ 40$ | [
"Let Alice be A, Bob be B, Carla be C, Derek be D, and Eric be E.\nWe can split this problem up into two cases:\n$\\textbf{Case 1: }$ A sits on an edge seat.\nSince B and C can't sit next to A, that must mean either D or E sits next to A. After we pick either D or E, then either B or C must sit next to D/E. Then, w... |
https://artofproblemsolving.com/wiki/index.php/1983_AHSME_Problems/Problem_7 | B | 30 | Alice sells an item at $$10$ less than the list price and receives $10\%$ of her selling price as her commission.
Bob sells the same item at $$20$ less than the list price and receives $20\%$ of his selling price as his commission.
If they both get the same commission, then the list price is
$\textbf{(A) } $20\qquad ... | [
"If $x$ is the list price, then $10\\%(x-10)=20\\%(x-20)$ . Solving this equation gives $x=30$ , so the answer is $\\boxed{30}$"
] |
https://artofproblemsolving.com/wiki/index.php/2018_AMC_10A_Problems/Problem_5 | D | 56 | Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, "We are at least $6$ miles away," Bob replied, "We are at most $5$ miles away." Charlie then remarked, "Actually the nearest town is at most $4$ miles away." It turned out that none of the three statements were... | [
"For each of the false statements, we identify its corresponding true statement. Note that:\nWe construct the following table: \\[\\begin{array}{c||c|c} & & \\\\ [-2.5ex] \\textbf{Hiker} & \\textbf{False Statement} & \\textbf{True Statement} \\\\ [0.5ex] \\hline & & \\\\ [-2ex] \\textbf{Alice} & [6,\\infty) & [0,6... |
https://artofproblemsolving.com/wiki/index.php/2018_AMC_12A_Problems/Problem_4 | D | 56 | Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, "We are at least $6$ miles away," Bob replied, "We are at most $5$ miles away." Charlie then remarked, "Actually the nearest town is at most $4$ miles away." It turned out that none of the three statements were... | [
"For each of the false statements, we identify its corresponding true statement. Note that:\nWe construct the following table: \\[\\begin{array}{c||c|c} & & \\\\ [-2.5ex] \\textbf{Hiker} & \\textbf{False Statement} & \\textbf{True Statement} \\\\ [0.5ex] \\hline & & \\\\ [-2ex] \\textbf{Alice} & [6,\\infty) & [0,6... |
https://artofproblemsolving.com/wiki/index.php/2004_AMC_10A_Problems/Problem_3 | E | 29 | Alicia earns 20 dollars per hour, of which $1.45\%$ is deducted to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes?
$\mathrm{(A) \ } 0.0029 \qquad \mathrm{(B) \ } 0.029 \qquad \mathrm{(C) \ } 0.29 \qquad \mathrm{(D) \ } 2.9 \qquad \mathrm{(E) \ } 29$ | [
"$20$ dollars is the same as $2000$ cents, and $1.45\\%$ of $2000$ is $0.0145\\times2000=29$ cents. $\\Rightarrow\\boxed{29}$",
"Since there can't be decimal values of cents, the answer must be $\\Rightarrow\\boxed{29}$"
] |
https://artofproblemsolving.com/wiki/index.php/2004_AMC_12A_Problems/Problem_1 | E | 29 | Alicia earns 20 dollars per hour, of which $1.45\%$ is deducted to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes?
$\mathrm{(A) \ } 0.0029 \qquad \mathrm{(B) \ } 0.029 \qquad \mathrm{(C) \ } 0.29 \qquad \mathrm{(D) \ } 2.9 \qquad \mathrm{(E) \ } 29$ | [
"$20$ dollars is the same as $2000$ cents, and $1.45\\%$ of $2000$ is $0.0145\\times2000=29$ cents. $\\Rightarrow\\boxed{29}$",
"Since there can't be decimal values of cents, the answer must be $\\Rightarrow\\boxed{29}$"
] |
https://artofproblemsolving.com/wiki/index.php/2017_AMC_8_Problems/Problem_2 | E | 120 | Alicia, Brenda, and Colby were the candidates in a recent election for student president. The pie chart below shows how the votes were distributed among the three candidates. If Brenda received $36$ votes, then how many votes were cast all together?
[asy] draw((-1,0)--(0,0)--(0,1)); draw((0,0)--(0.309, -0.951)); filldr... | [
"Let $x$ be the total amount of votes casted. From the chart, Brenda received $30\\%$ of the votes and had $36$ votes. We can express this relationship as $\\frac{30}{100}x=36$ . Solving for $x$ , we get $x=\\boxed{120}.$",
"We're being asked for the total number of votes cast -- that represents $100\\%$ of the t... |
https://artofproblemsolving.com/wiki/index.php/2023_AMC_8_Problems/Problem_21 | C | 2 | Alina writes the numbers $1, 2, \dots , 9$ on separate cards, one number per card. She wishes to divide the cards into $3$ groups of $3$ cards so that the sum of the numbers in each group will be the same. In how many ways can this be done?
$\textbf{(A) } 0 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } 2 \qquad \textbf{(... | [
"First, we need to find the sum of each group when split. This is the total sum of all the elements divided by the # of groups. $1 + 2 \\cdots + 9 = \\frac{9(10)}{2} = 45$ . Then, dividing by $3$ , we have $\\frac{45}{3} = 15$ , so each group of $3$ must have a sum of 15. To make the counting easier, we will just s... |
https://artofproblemsolving.com/wiki/index.php/2013_AMC_10A_Problems/Problem_25 | A | 49 | All $20$ diagonals are drawn in a regular octagon. At how many distinct points in the interior
of the octagon (not on the boundary) do two or more diagonals intersect?
$\textbf{(A)}\ 49\qquad\textbf{(B)}\ 65\qquad\textbf{(C)}\ 70\qquad\textbf{(D)}\ 96\qquad\textbf{(E)}\ 128$ | [
"If you draw a clear diagram like the one below, it is easy to see that there are $\\boxed{49}$ points.",
"Let the number of intersections be $x$ . We know that $x\\le \\dbinom{8}{4} = 70$ , as every $4$ vertices on the octagon forms a quadrilateral with intersecting diagonals which is an intersection point. Ho... |
https://artofproblemsolving.com/wiki/index.php/2019_AMC_10B_Problems/Problem_4 | A | 12 | All lines with equation $ax+by=c$ such that $a,b,c$ form an arithmetic progression pass through a common point. What are the coordinates of that point?
$\textbf{(A) } (-1,2) \qquad\textbf{(B) } (0,1) \qquad\textbf{(C) } (1,-2) \qquad\textbf{(D) } (1,0) \qquad\textbf{(E) } (1,2)$ | [
"If all lines satisfy the condition, then we can just plug in values for $a$ $b$ , and $c$ that form an arithmetic progression. Let's use $a=1$ $b=2$ $c=3$ , and $a=1$ $b=3$ $c=5$ . Then the two lines we get are: \\[x+2y=3\\] \\[x+3y=5\\] Use elimination to deduce \\[y = 2\\] and plug this into one of the previous ... |
https://artofproblemsolving.com/wiki/index.php/2005_AMC_10B_Problems/Problem_18 | D | 8 | All of David's telephone numbers have the form $555-abc-defg$ , where $a$ $b$ $c$ $d$ $e$ $f$ , and $g$ are distinct digits and in increasing order, and none is either $0$ or $1$ . How many different telephone numbers can David have?
$\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 7 \qquad \textbf{(D) } 8 ... | [
"The only digits available to use in the phone number are $2$ $3$ $4$ $5$ $6$ $7$ $8$ , and $9$ . There are only $7$ spots left among the $8$ numbers, so we need to find the number of ways to choose $7$ numbers from $8$ . The answer is then $\\dbinom{8}{7}=\\dfrac{8!}{7!\\,(8-7)!}=\\boxed{8}$"
] |
https://artofproblemsolving.com/wiki/index.php/2017_AMC_8_Problems/Problem_9 | D | 4 | All of Marcy's marbles are blue, red, green, or yellow. One third of her marbles are blue, one fourth of them are red, and six of them are green. What is the smallest number of yellow marbles that Macy could have?
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }5$ | [
"The $6$ green marbles and yellow marbles form $1 - \\frac{1}{3} - \\frac{1}{4} = \\frac{5}{12}$ of the total marbles. Now, suppose the total number of marbles is $x$ . We know the number of yellow marbles is $\\frac{5}{12}x - 6$ and a positive integer. Therefore, $12$ must divide $x$ . Trying the smallest multiple... |
https://artofproblemsolving.com/wiki/index.php/1997_AJHSME_Problems/Problem_25 | D | 6 | All of the even numbers from 2 to 98 inclusive, excluding those ending in 0, are multiplied together. What is the rightmost digit (the units digit) of the product?
$\text{(A)}\ 0 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 8$ | [
"All the tens digits of the product will be irrelevant to finding the units digit. Thus, we are searching for the units digit of $(2\\cdot 4\\cdot 6 \\cdot 8) \\cdot (2 \\cdot 4 \\cdot 6 \\cdot 8) \\cdot (2\\cdot 4\\cdot 6 \\cdot 8) \\cdot ...$\nThere will be $10$ groups of $4$ numbers. The number now can be rewr... |
https://artofproblemsolving.com/wiki/index.php/2018_AMC_10A_Problems/Problem_9 | D | 24 | All of the triangles in the diagram below are similar to isosceles triangle $ABC$ , in which $AB=AC$ . Each of the $7$ smallest triangles has area $1,$ and $\triangle ABC$ has area $40$ . What is the area of trapezoid $DBCE$
[asy] unitsize(5); dot((0,0)); dot((60,0)); dot((50,10)); dot((10,10)); dot((30,30)); draw((0,0... | [
"Note that the area of an isosceles triangle is equivalent to the square of its height. Using this information, the height of the smallest isosceles triangle is $1$ , and thus its base is $2.$\nLet $h$ be the height of the top triangle. We can set up a height-to-base similarity ratio, using the top triangle and $\\... |
https://artofproblemsolving.com/wiki/index.php/2018_AMC_10A_Problems/Problem_9 | E | 24 | All of the triangles in the diagram below are similar to isosceles triangle $ABC$ , in which $AB=AC$ . Each of the $7$ smallest triangles has area $1,$ and $\triangle ABC$ has area $40$ . What is the area of trapezoid $DBCE$
[asy] unitsize(5); dot((0,0)); dot((60,0)); dot((50,10)); dot((10,10)); dot((30,30)); draw((0,0... | [
"Let $x$ be the area of $ADE$ . Note that $x$ is comprised of the $7$ small isosceles triangles and a triangle similar to $ADE$ with side length ratio $3:4$ (so an area ratio of $9:16$ ). Thus, we have \\[x=7+\\dfrac{9}{16}x.\\] This gives $x=16$ , so the area of $DBCE=40-x=\\boxed{24}$"
] |
https://artofproblemsolving.com/wiki/index.php/2018_AMC_10A_Problems/Problem_9 | null | 24 | All of the triangles in the diagram below are similar to isosceles triangle $ABC$ , in which $AB=AC$ . Each of the $7$ smallest triangles has area $1,$ and $\triangle ABC$ has area $40$ . What is the area of trapezoid $DBCE$
[asy] unitsize(5); dot((0,0)); dot((60,0)); dot((50,10)); dot((10,10)); dot((30,30)); draw((0,0... | [
"Let the base length of the small triangle be $x$ . Then, there is a triangle $ADE$ encompassing the 7 small triangles and sharing the top angle with a base length of $4x$ . Because the area is proportional to the square of the side, let the base $BC$ be $\\sqrt{40}x$ . The ratio of the area of triangle $ADE$ to tr... |
https://artofproblemsolving.com/wiki/index.php/2018_AMC_12A_Problems/Problem_8 | D | 24 | All of the triangles in the diagram below are similar to isosceles triangle $ABC$ , in which $AB=AC$ . Each of the $7$ smallest triangles has area $1,$ and $\triangle ABC$ has area $40$ . What is the area of trapezoid $DBCE$
[asy] unitsize(5); dot((0,0)); dot((60,0)); dot((50,10)); dot((10,10)); dot((30,30)); draw((0,0... | [
"Note that the area of an isosceles triangle is equivalent to the square of its height. Using this information, the height of the smallest isosceles triangle is $1$ , and thus its base is $2.$\nLet $h$ be the height of the top triangle. We can set up a height-to-base similarity ratio, using the top triangle and $\\... |
https://artofproblemsolving.com/wiki/index.php/2018_AMC_12A_Problems/Problem_8 | E | 24 | All of the triangles in the diagram below are similar to isosceles triangle $ABC$ , in which $AB=AC$ . Each of the $7$ smallest triangles has area $1,$ and $\triangle ABC$ has area $40$ . What is the area of trapezoid $DBCE$
[asy] unitsize(5); dot((0,0)); dot((60,0)); dot((50,10)); dot((10,10)); dot((30,30)); draw((0,0... | [
"Let $x$ be the area of $ADE$ . Note that $x$ is comprised of the $7$ small isosceles triangles and a triangle similar to $ADE$ with side length ratio $3:4$ (so an area ratio of $9:16$ ). Thus, we have \\[x=7+\\dfrac{9}{16}x.\\] This gives $x=16$ , so the area of $DBCE=40-x=\\boxed{24}$"
] |
https://artofproblemsolving.com/wiki/index.php/2018_AMC_12A_Problems/Problem_8 | null | 24 | All of the triangles in the diagram below are similar to isosceles triangle $ABC$ , in which $AB=AC$ . Each of the $7$ smallest triangles has area $1,$ and $\triangle ABC$ has area $40$ . What is the area of trapezoid $DBCE$
[asy] unitsize(5); dot((0,0)); dot((60,0)); dot((50,10)); dot((10,10)); dot((30,30)); draw((0,0... | [
"Let the base length of the small triangle be $x$ . Then, there is a triangle $ADE$ encompassing the 7 small triangles and sharing the top angle with a base length of $4x$ . Because the area is proportional to the square of the side, let the base $BC$ be $\\sqrt{40}x$ . The ratio of the area of triangle $ADE$ to tr... |
https://artofproblemsolving.com/wiki/index.php/2007_AMC_10B_Problems/Problem_7 | E | 150 | All sides of the convex pentagon $ABCDE$ are of equal length, and $\angle A= \angle B = 90^\circ.$ What is the degree measure of $\angle E?$
$\textbf{(A) } 90 \qquad\textbf{(B) } 108 \qquad\textbf{(C) } 120 \qquad\textbf{(D) } 144 \qquad\textbf{(E) } 150$ | [
"$AB = EC$ because they are opposite sides of a square. Also, $ED = DC = AB$ because all sides of the convex pentagon are of equal length. Since $ABCE$ is a square and $\\triangle CED$ is an equilateral triangle, $\\angle AEC = 90$ and $\\angle CED = 60.$ Use angle addition: \\[\\angle E = \\angle AEC + \\angle CED... |
https://artofproblemsolving.com/wiki/index.php/2024_AMC_8_Problems/Problem_9 | E | 28 | All the marbles in Maria's collection are red, green, or blue. Maria has half as many red marbles as green marbles and twice as many blue marbles as green marbles. Which of the following could be the total number of marbles in Maria's collection?
$\textbf{(A) } 24\qquad\textbf{(B) } 25\qquad\textbf{(C) } 26\qquad\textb... | [
"Since she has half as many red marbles as green, we can call the number of red marbles $x$ , and the number of green marbles $2x$ .\nSince she has half as many green marbles as blue, we can call the number of blue marbles $4x$ . \nAdding them up, we have $7x$ marbles. The number of marbles therefore must be a mult... |
https://artofproblemsolving.com/wiki/index.php/2016_AMC_10B_Problems/Problem_15 | C | 7 | All the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ are written in a $3\times3$ array of squares, one number in each square, in such a way that if two numbers are consecutive then they occupy squares that share an edge. The numbers in the four corners add up to $18$ . What is the number in the center?
$\textbf{(A)}\ 5\qquad\te... | [
"Consecutive numbers share an edge. That means that it is possible to walk from $1$ to $9$ by single steps north, south, east, or west. Consequently, the squares in the diagram with different shades have different parity: But since there are only four even numbers in the set, the five darker squares must contain t... |
https://artofproblemsolving.com/wiki/index.php/2016_AMC_10B_Problems/Problem_17 | D | 729 | All the numbers $2, 3, 4, 5, 6, 7$ are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of th... | [
"Let us call the six faces of our cube $a,b,c,d,e,$ and $f$ (where $a$ is opposite $d$ $c$ is opposite $e$ , and $b$ is opposite $f$ .\nThus, for the eight vertices, we have the following products: $abc,abe,bcd,bde,acf,cdf,aef,$ and $def$ .\nLet us find the sum of these products: \\[abc+abe+bcd+bde+acf+cdf+aef+def\... |
https://artofproblemsolving.com/wiki/index.php/2016_AMC_12B_Problems/Problem_15 | D | 729 | All the numbers $2, 3, 4, 5, 6, 7$ are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of th... | [
"First assign each face the letters $a,b,c,d,e,f$ . The sum of the product of the faces is $abc+acd+ade+aeb+fbc+fcd+fde+feb$ . We can factor this into $(a+f)(b+c)(d+e)$ which is the product of the sum of each pair of opposite faces. In order to maximize $(a+f)(b+c)(d+e)$ we use the numbers $(7+2)(6+3)(5+4)$ or $\\b... |
https://artofproblemsolving.com/wiki/index.php/2021_AMC_10A_Problems/Problem_14 | A | 88 | All the roots of the polynomial $z^6-10z^5+Az^4+Bz^3+Cz^2+Dz+16$ are positive integers, possibly repeated. What is the value of $B$
$\textbf{(A) }{-}88 \qquad \textbf{(B) }{-}80 \qquad \textbf{(C) }{-}64 \qquad \textbf{(D) }{-}41\qquad \textbf{(E) }{-}40$ | [
"By Vieta's formulas, the sum of the six roots is $10$ and the product of the six roots is $16$ . By inspection, we see the roots are $1, 1, 2, 2, 2,$ and $2$ , so the function is $(z-1)^2(z-2)^4=(z^2-2z+1)(z^4-8z^3+24z^2-32z+16)$ . Therefore, calculating just the $z^3$ terms, we get $B = -32 - 48 - 8 = \\boxed{88}... |
https://artofproblemsolving.com/wiki/index.php/2021_AMC_12A_Problems/Problem_12 | A | 88 | All the roots of the polynomial $z^6-10z^5+Az^4+Bz^3+Cz^2+Dz+16$ are positive integers, possibly repeated. What is the value of $B$
$\textbf{(A) }{-}88 \qquad \textbf{(B) }{-}80 \qquad \textbf{(C) }{-}64 \qquad \textbf{(D) }{-}41\qquad \textbf{(E) }{-}40$ | [
"By Vieta's formulas, the sum of the six roots is $10$ and the product of the six roots is $16$ . By inspection, we see the roots are $1, 1, 2, 2, 2,$ and $2$ , so the function is $(z-1)^2(z-2)^4=(z^2-2z+1)(z^4-8z^3+24z^2-32z+16)$ . Therefore, calculating just the $z^3$ terms, we get $B = -32 - 48 - 8 = \\boxed{88}... |
https://artofproblemsolving.com/wiki/index.php/2004_AMC_12B_Problems/Problem_11 | D | 13 | All the students in an algebra class took a $100$ -point test. Five students scored $100$ , each student scored at least $60$ , and the mean score was $76$ . What is the smallest possible number of students in the class?
$\mathrm{(A)}\ 10 \qquad \mathrm{(B)}\ 11 \qquad \mathrm{(C)}\ 12 \qquad \mathrm{(D)}\ 13 \qquad \m... | [
"Let the number of students be $n\\geq 5$ . Then the sum of their scores is at least $5\\cdot 100 + (n-5)\\cdot 60$ . At the same time, we need to achieve the mean $76$ , which is equivalent to achieving the sum $76n$\nHence we get a necessary condition on $n$ : we must have $5\\cdot 100 + (n-5)\\cdot 60 \\leq 76n$... |
https://artofproblemsolving.com/wiki/index.php/2016_AMC_10B_Problems/Problem_9 | C | 8 | All three vertices of $\bigtriangleup ABC$ lie on the parabola defined by $y=x^2$ , with $A$ at the origin and $\overline{BC}$ parallel to the $x$ -axis. The area of the triangle is $64$ . What is the length of $BC$
$\textbf{(A)}\ 4\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 16$ | [
"Let the point where the height of the triangle intersects with the base be $D$ . Now we can guess what $x$ is and find $y$ . If $x$ is $3$ , then $y$ is $9$ . The cords of $B$ and $C$ would be $(-3,9)$ and $(3,9)$ , respectively. The distance between $B$ and $C$ is $6$ , meaning the area would be $\\frac{6 \\cdot ... |
https://artofproblemsolving.com/wiki/index.php/2016_AMC_12B_Problems/Problem_6 | C | 8 | All three vertices of $\bigtriangleup ABC$ lie on the parabola defined by $y=x^2$ , with $A$ at the origin and $\overline{BC}$ parallel to the $x$ -axis. The area of the triangle is $64$ . What is the length of $BC$
$\textbf{(A)}\ 4\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 16$ | [
"By: Albert471\nPlotting points $B$ and $C$ on the graph shows that they are at $\\left( -x,x^2\\right)$ and $\\left( x,x^2\\right)$ , which is isosceles. By setting up the triangle area formula you get: $64=\\frac{1}{2}*2x*x^2 = 64=x^3$ Making x=4, and the length of $BC$ is $2x$ , so the answer is $\\boxed{8}$"... |
https://artofproblemsolving.com/wiki/index.php/2005_AMC_12B_Problems/Problem_24 | A | 14 | All three vertices of an equilateral triangle are on the parabola $y = x^2$ , and one of its sides has a slope of $2$ . The $x$ -coordinates of the three vertices have a sum of $m/n$ , where $m$ and $n$ are relatively prime positive integers. What is the value of $m + n$
$\mathrm{(A)}\ {{{14}}}\qquad\mathrm{(B)}\ {{{15... | [
"Let the three points be at $A = (x_1, x_1^2)$ $B = (x_2, x_2^2)$ , and $C = (x_3, x_3^2)$ , such that the slope between the first two is $2$ , and $A$ is the point with the least $y$ -coordinate.\nThen, we have $\\textrm{Slope of }AC = \\frac{x_1^2 - x_3^2}{x_1 - x_3} = x_1 + x_3$ . Similarly, the slope of $BC$ is... |
https://artofproblemsolving.com/wiki/index.php/2023_AMC_8_Problems/Problem_13 | D | 48 | Along the route of a bicycle race, $7$ water stations are evenly spaced between the start and finish lines,
as shown in the figure below. There are also $2$ repair stations evenly spaced between the start and
finish lines. The $3$ rd water station is located $2$ miles after the $1$ st repair station. How long is the ra... | [
"Suppose that the race is $d$ miles long. The water stations are located at \\[\\frac{d}{8}, \\frac{2d}{8}, \\ldots, \\frac{7d}{8}\\] miles from the start, and the repair stations are located at \\[\\frac{d}{3}, \\frac{2d}{3}\\] miles from the start.\nWe are given that $\\frac{3d}{8}=\\frac{d}{3}+2,$ from which \\b... |
https://artofproblemsolving.com/wiki/index.php/2004_AIME_I_Problems/Problem_5 | null | 849 | Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of 500 points. Alpha scored 160 points out of 300 points attempted on the first day, and scored 140 points out of 200 points attempted on the second day. Beta who did not atte... | [
"Let $q$ be the number of questions Beta takes on day 1 and $a$ be the number he gets right. Let $b$ be the number he gets right on day 2.\nThese inequalities follow: \\[\\frac{a}{q} < \\frac{160}{300} = \\frac{8}{15}\\] \\[\\frac{b}{500-q} < \\frac{140}{200} = \\frac{7}{10}\\] Solving for a and b and adding the tw... |
https://artofproblemsolving.com/wiki/index.php/2017_AMC_10A_Problems/Problem_18 | D | 4 | Amelia has a coin that lands heads with probability $\frac{1}{3}\,$ , and Blaine has a coin that lands on heads with probability $\frac{2}{5}$ . Amelia and Blaine alternately toss their coins until someone gets a head; the first one to get a head wins. All coin tosses are independent. Amelia goes first. The probability... | [
"Let $P$ be the probability Amelia wins. Note that $P = \\text{chance she wins on her first turn} + \\text{chance she gets to her turn again}\\cdot P$ , since if she gets to her turn again, she is back where she started with probability of winning $P$ . The chance she wins on her first turn is $\\frac{1}{3}$ . The ... |
https://artofproblemsolving.com/wiki/index.php/2017_AMC_10A_Problems/Problem_18 | null | 59 | Amelia has a coin that lands heads with probability $\frac{1}{3}\,$ , and Blaine has a coin that lands on heads with probability $\frac{2}{5}$ . Amelia and Blaine alternately toss their coins until someone gets a head; the first one to get a head wins. All coin tosses are independent. Amelia goes first. The probability... | [
"We can solve this by using 'casework,' the cases being:\nCase 1: Amelia wins on her first turn.\nCase 2 Amelia wins on her second turn.\nand so on.\nThe probability of her winning on her first turn is $\\dfrac13$ . The probability of all the other cases is determined by the probability that Amelia and Blaine all l... |
https://artofproblemsolving.com/wiki/index.php/2012_AMC_10B_Problems/Problem_24 | B | 132 | Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those two girls but disliked by the third. In how many different ways is this possible?
$\textbf{(A)}\ 108\qquad\te... | [
"Let the ordered triple $(a,b,c)$ denote that $a$ songs are liked by Amy and Beth, $b$ songs by Beth and Jo, and $c$ songs by Jo and Amy. The only possible triples are $(1,1,1), (2,1,1), (1,2,1)(1,1,2)$\nTo show this, observe these are all valid conditions. Second, note that none of $a,b,c$ can be bigger than 3. Su... |
https://artofproblemsolving.com/wiki/index.php/2012_AMC_12B_Problems/Problem_16 | B | 132 | Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those two girls but disliked by the third. In how many different ways is this possible?
$\textbf{(A)}\ 108\qquad\te... | [
"Let the ordered triple $(a,b,c)$ denote that $a$ songs are liked by Amy and Beth, $b$ songs by Beth and Jo, and $c$ songs by Jo and Amy. The only possible triples are $(1,1,1), (2,1,1), (1,2,1)(1,1,2)$\nTo show this, observe these are all valid conditions. Second, note that none of $a,b,c$ can be bigger than 3. Su... |
https://artofproblemsolving.com/wiki/index.php/2016_AIME_II_Problems/Problem_4 | null | 180 | An $a \times b \times c$ rectangular box is built from $a \cdot b \cdot c$ unit cubes. Each unit cube is colored red, green, or yellow. Each of the $a$ layers of size $1 \times b \times c$ parallel to the $(b \times c)$ faces of the box contains exactly $9$ red cubes, exactly $12$ green cubes, and some yellow cubes. Ea... | [
"By counting the number of green cubes $2$ different ways, we have $12a=20b$ , or $a=\\dfrac{5}{3} b$ . Notice that there are only $3$ possible colors for unit cubes, so for each of the $1 \\times b \\times c$ layers, there are $bc-21$ yellow cubes, and similarly there are $ac-45$ red cubes in each of the $1 \\time... |
https://artofproblemsolving.com/wiki/index.php/1998_AIME_Problems/Problem_14 | null | 130 | An $m\times n\times p$ rectangular box has half the volume of an $(m + 2)\times(n + 2)\times(p + 2)$ rectangular box, where $m, n,$ and $p$ are integers, and $m\le n\le p.$ What is the largest possible value of $p$ | [
"\\[2mnp = (m+2)(n+2)(p+2)\\]\nLet’s solve for $p$\n\\[(2mn)p = p(m+2)(n+2) + 2(m+2)(n+2)\\] \\[[2mn - (m+2)(n+2)]p = 2(m+2)(n+2)\\] \\[p = \\frac{2(m+2)(n+2)}{mn - 2n - 2m - 4} = \\frac{2(m+2)(n+2)}{(m-2)(n-2) - 8}\\]\nClearly, we want to minimize the denominator, so we test $(m-2)(n-2) - 8 = 1 \\Longrightarrow (m... |
https://artofproblemsolving.com/wiki/index.php/2016_AMC_8_Problems/Problem_17 | D | 9,990 | An ATM password at Fred's Bank is composed of four digits from $0$ to $9$ , with repeated digits allowable. If no password may begin with the sequence $9,1,1,$ then how many passwords are possible?
$\textbf{(A)}\mbox{ }30\qquad\textbf{(B)}\mbox{ }7290\qquad\textbf{(C)}\mbox{ }9000\qquad\textbf{(D)}\mbox{ }9990\qquad\te... | [
"For the first three digits, there are $10^3-1=999$ combinations since $911$ is not allowed. For the final digit, any of the $10$ numbers are allowed. $999 \\cdot 10 = 9990 \\rightarrow \\boxed{9990}$",
"Counting the prohibited cases, we find that there are 10 of them. This is because, when we start with 9,1, and... |
https://artofproblemsolving.com/wiki/index.php/1995_AJHSME_Problems/Problem_8 | D | 1,875 | An American traveling in Italy wishes to exchange American money (dollars) for Italian money (lire). If 3000 lire = 1.60, how much lire will the traveler receive in exchange for 1.00?
$\text{(A)}\ 180 \qquad \text{(B)}\ 480 \qquad \text{(C)}\ 1800 \qquad \text{(D)}\ 1875 \qquad \text{(E)}\ 4875$ | [
"$\\frac{3000 \\text{ lire}}{1.6 \\text{ dollars}} = \\frac{x \\text{ lire}}{1 \\text{ dollar}}$\n$x = 1875$ , and the answer is $\\boxed{1875}$"
] |
https://artofproblemsolving.com/wiki/index.php/1998_AJHSME_Problems/Problem_14 | E | 40 | An Annville Junior High School, $30\%$ of the students in the Math Club are in the Science Club, and $80\%$ of the students in the Science Club are in the Math Club. There are 15 students in the Science Club. How many students are in the Math Club?
$\text{(A)}\ 12 \qquad \text{(B)}\ 15 \qquad \text{(C)}\ 30 \qquad \t... | [
"If $80\\%$ of the people in the science club of $15$ people are in the Math Club, $\\frac{4}{5}\\times15=12$ people are in the both the Math Club and the Science Club.\nThese $12$ people make up $30\\%$ of the Math Club.\nSetting up a proportion:\n$12\\cdot 1.00 =0.30\\cdot x$\n$\\frac{12}{0.3} = 40=x$\nThere are ... |
https://artofproblemsolving.com/wiki/index.php/2018_AMC_8_Problems/Problem_1 | A | 14 | An amusement park has a collection of scale models, with a ratio of $1: 20$ , of buildings and other sights from around the country. The height of the United States Capitol is $289$ feet. What is the height in feet of its duplicate to the nearest whole number?
$\textbf{(A) }14\qquad\textbf{(B) }15\qquad\textbf{(C) }16\... | [
"You can see that since the ratio of real building's heights to the model building's height is $1:20$ . We also know that the U.S Capitol is $289$ feet in real life, so to find the height of the model, we divide by 20. That gives us $14.45$ which rounds to 14. Therefore, to the nearest whole number, the duplicate i... |
https://artofproblemsolving.com/wiki/index.php/2003_AIME_I_Problems/Problem_11 | null | 92 | An angle $x$ is chosen at random from the interval $0^\circ < x < 90^\circ.$ Let $p$ be the probability that the numbers $\sin^2 x, \cos^2 x,$ and $\sin x \cos x$ are not the lengths of the sides of a triangle. Given that $p = d/n,$ where $d$ is the number of degrees in $\text{arctan}$ $m$ and $m$ and $n$ are positive ... | [
"Note that the three expressions are symmetric with respect to interchanging $\\sin$ and $\\cos$ , and so the probability is symmetric around $45^\\circ$ . Thus, take $0 < x < 45$ so that $\\sin x < \\cos x$ . Then $\\cos^2 x$ is the largest of the three given expressions and those three lengths not forming a tri... |
https://artofproblemsolving.com/wiki/index.php/2006_AIME_I_Problems/Problem_7 | null | 408 | An angle is drawn on a set of equally spaced parallel lines as shown. The ratio of the area of shaded region $C$ to the area of shaded region $B$ is 11/5. Find the ratio of shaded region $D$ to the area of shaded region $A.$
[asy] defaultpen(linewidth(0.7)+fontsize(10)); for(int i=0; i<4; i=i+1) { fill((2*i,0)--(2*i+1,... | [
"Note that the apex of the angle is not on the parallel lines. Set up a coordinate proof\nLet the set of parallel lines be perpendicular to the x-axis , such that they cross it at $0, 1, 2 \\ldots$ . The base of region $\\mathcal{A}$ is on the line $x = 1$ . The bigger base of region $\\mathcal{D}$ is on the line $... |
https://artofproblemsolving.com/wiki/index.php/2004_AMC_10B_Problems/Problem_12 | A | 2 | An annulus is the region between two concentric circles. The concentric circles in the figure have radii $b$ and $c$ , with $b>c$ . Let $OX$ be a radius of the larger circle, let $XZ$ be tangent to the smaller circle at $Z$ , and let $OY$ be the radius of the larger circle that contains $Z$ . Let $a=XZ$ $d=YZ$ , and $e... | [
"The area of the large circle is $\\pi b^2$ , the area of the small one is $\\pi c^2$ , hence the shaded area is $\\pi(b^2-c^2)$\nFrom the Pythagorean Theorem for the right triangle $OXZ$ we have $a^2 + c^2 = b^2$ , hence $b^2-c^2=a^2$ and thus the shaded area is $\\boxed{2}$",
"Set $c=0,$ then the shaded area is... |
https://artofproblemsolving.com/wiki/index.php/2004_AMC_12B_Problems/Problem_10 | A | 2 | An annulus is the region between two concentric circles. The concentric circles in the figure have radii $b$ and $c$ , with $b>c$ . Let $OX$ be a radius of the larger circle, let $XZ$ be tangent to the smaller circle at $Z$ , and let $OY$ be the radius of the larger circle that contains $Z$ . Let $a=XZ$ $d=YZ$ , and $e... | [
"The area of the large circle is $\\pi b^2$ , the area of the small one is $\\pi c^2$ , hence the shaded area is $\\pi(b^2-c^2)$\nFrom the Pythagorean Theorem for the right triangle $OXZ$ we have $a^2 + c^2 = b^2$ , hence $b^2-c^2=a^2$ and thus the shaded area is $\\boxed{2}$",
"Set $c=0,$ then the shaded area is... |
https://artofproblemsolving.com/wiki/index.php/2021_AIME_II_Problems/Problem_8 | null | 49 | An ant makes a sequence of moves on a cube where a move consists of walking from one vertex to an adjacent vertex along an edge of the cube. Initially the ant is at a vertex of the bottom face of the cube and chooses one of the three adjacent vertices to move to as its first move. For all moves after the first move, th... | [
"For all positive integers $k,$ let\nThe base case occurs at $k=1,$ from which $\\left(N(1,\\mathrm{BB}),N(1,\\mathrm{BT}),N(1,\\mathrm{TB}),N(1,\\mathrm{TT})\\right)=(2,1,0,0).$\nSuppose the ant makes exactly $k$ moves for some $k\\geq2.$ We perform casework on its last move:\nAlternatively, this recursion argumen... |
https://artofproblemsolving.com/wiki/index.php/2006_AMC_8_Problems/Problem_21 | A | 0.25 | An aquarium has a rectangular base that measures $100$ cm by $40$ cm and has a height of $50$ cm. The aquarium is filled with water to a depth of $37$ cm. A rock with volume $1000\text{cm}^3$ is then placed in the aquarium and completely submerged. By how many centimeters does the water level rise?
$\textbf{(A)}\ 0.25... | [
"The water level will rise $1$ cm for every $100 \\cdot 40 = 4000\\text{cm}^2$ . Since $1000$ is $\\frac{1}{4}$ of $4000$ , the water will rise $\\frac{1}{4}\\cdot1 = \\boxed{0.25}$"
] |
https://artofproblemsolving.com/wiki/index.php/1985_AHSME_Problems/Problem_10 | E | 16 | An arbitrary circle can intersect the graph of $y = \sin x$ in
$\mathrm{(A) \ } \text{at most }2\text{ points} \qquad \mathrm{(B) \ }\text{at most }4\text{ points} \qquad \mathrm{(C) \ } \text{at most }6\text{ points} \qquad \mathrm{(D) \ } \text{at most }8\text{ points}$ $\mathrm{(E) \ }\text{more than }16\text{ poi... | [
"Consider a circle whose center lies on the positive $y$ -axis and which passes through the origin. As the radius of this circle becomes arbitrarily large, its curvature near the $x$ -axis becomes almost flat, and so it can intersect the curve $y = \\sin x$ arbitrarily many times (since the $x$ -axis itself interse... |
https://artofproblemsolving.com/wiki/index.php/2021_Fall_AMC_10A_Problems/Problem_17 | D | 17 | An architect is building a structure that will place vertical pillars at the vertices of regular hexagon $ABCDEF$ , which is lying horizontally on the ground. The six pillars will hold up a flat solar panel that will not be parallel to the ground. The heights of pillars at $A$ $B$ , and $C$ are $12$ $9$ , and $10$ mete... | [
"The pillar at $B$ has height $9$ and the pillar at $A$ has height $12.$ Since the solar panel is flat, the inclination from pillar $B$ to pillar $A$ is $3.$ Call the center of the hexagon $G.$ Since $\\overrightarrow{CG}\\parallel\\overrightarrow{BA},$ it follows that the solar panel has height $13$ at $G.$ Since ... |
https://artofproblemsolving.com/wiki/index.php/2015_AMC_8_Problems/Problem_18 | B | 31 | An arithmetic sequence is a sequence in which each term after the first is obtained by adding a constant to the previous term. For example, $2,5,8,11,14$ is an arithmetic sequence with five terms, in which the first term is $2$ and the constant added is $3$ . Each row and each column in this $5\times5$ array is an arit... | [
"We begin filling in the table. The top row has a first term $1$ and a fifth term $25$ , so we have the common difference is $\\frac{25-1}4=6$ . This means we can fill in the first row of the table: \nThe fifth row has a first term of $17$ and a fifth term of $81$ , so the common difference is $\\frac{81-17}4=16$... |
https://artofproblemsolving.com/wiki/index.php/2023_AMC_10B_Problems/Problem_23 | B | 20 | An arithmetic sequence of positive integers has $\text{n} \ge 3$ terms, initial term $a$ , and common difference $d > 1$ . Carl wrote down all the terms in this sequence correctly except for one term, which was off by $1$ . The sum of the terms he wrote was $222$ . What is $a + d + n$
$\textbf{(A) } 24 \qquad \textbf{(... | [
"Since one of the terms was either $1$ more or $1$ less than it should have been, the sum should have been $222-1=221$ or $222+1=223.$\nThe formula for an arithmetic series is $an+d\\left(\\dfrac{(n-1)n}2\\right)=\\dfrac n2\\left(2a+d(n-1)\\right).$ This can quickly be rederived by noticing that the sequence goes $... |
https://artofproblemsolving.com/wiki/index.php/1989_AJHSME_Problems/Problem_23 | C | 33 | An artist has $14$ cubes, each with an edge of $1$ meter. She stands them on the ground to form a sculpture as shown. She then paints the exposed surface of the sculpture. How many square meters does she paint?
$\text{(A)}\ 21 \qquad \text{(B)}\ 24 \qquad \text{(C)}\ 33 \qquad \text{(D)}\ 37 \qquad \text{(E)}\ 42$
[... | [
"We can consider the contributions of the sides of the three layers and the tops of the layers separately.\nLayer $n$ (counting from the top starting at $1$ ) has $4$ side faces each with $n$ unit squares, so the sides of the pyramid contribute $4+8+12=24$ for the surface area.\nThe tops of the layers when combined... |
https://artofproblemsolving.com/wiki/index.php/2004_AMC_8_Problems/Problem_7 | B | 155 | An athlete's target heart rate, in beats per minute, is $80\%$ of the theoretical maximum heart rate. The maximum heart rate is found by subtracting the athlete's age, in years, from $220$ . To the nearest whole number, what is the target heart rate of an athlete who is $26$ years old?
$\textbf{(A)}\ 134\qquad\textbf{(... | [
"The maximum heart rate is $220-26=194$ beats per minute. The target heart rate is then $0.8*194 \\approx \\boxed{155}$ beats per minute."
] |
https://artofproblemsolving.com/wiki/index.php/1991_AJHSME_Problems/Problem_17 | C | 200 | An auditorium with $20$ rows of seats has $10$ seats in the first row. Each successive row has one more seat than the previous row. If students taking an exam are permitted to sit in any row, but not next to another student in that row, then the maximum number of students that can be seated for an exam is
$\text{(A)}... | [
"We first note that if a row has $n$ seats, then the maximum number of students that can be seated in that row is $\\left\\lceil \\frac{n}{2} \\right\\rceil$ , where $\\lceil x \\rceil$ is the smallest integer greater than or equal to $x$ . If a row has $2k$ seats, clearly we can only fit $k$ students in that row.... |
https://artofproblemsolving.com/wiki/index.php/1961_AHSME_Problems/Problem_2 | E | 10 | An automobile travels $a/6$ feet in $r$ seconds. If this rate is maintained for $3$ minutes, how many yards does it travel in $3$ minutes?
$\textbf{(A)}\ \frac{a}{1080r}\qquad \textbf{(B)}\ \frac{30r}{a}\qquad \textbf{(C)}\ \frac{30a}{r}\qquad \textbf{(D)}\ \frac{10r}{a}\qquad \textbf{(E)}\ \frac{10a}{r}$ | [
"Use dimensional analysis. \\[\\frac{a/6 \\text{ feet}}{r \\text{ seconds}} \\cdot \\frac{1 \\text{ yard}}{3 \\text{ feet}} \\cdot \\frac{60 \\text{ seconds}}{1 \\text{ minute}} \\cdot 3 \\text{ minutes}\\] \\[\\frac{10a}{r} \\text{ yards}\\] The answer is $\\boxed{10}$"
] |
https://artofproblemsolving.com/wiki/index.php/1985_AIME_Problems/Problem_11 | null | 85 | An ellipse has foci at $(9,20)$ and $(49,55)$ in the $xy$ -plane and is tangent to the $x$ -axis. What is the length of its major axis | [
"An ellipse is defined to be the locus of points $P$ such that the sum of the distances between $P$ and the two foci is constant. Let $F_1 = (9, 20)$ $F_2 = (49, 55)$ and $X = (x, 0)$ be the point of tangency of the ellipse with the $x$ -axis. Then $X$ must be the point on the axis such that the sum $F_1X + F_2X$... |
https://artofproblemsolving.com/wiki/index.php/2020_USOJMO_Problems/Problem_3 | null | 3,030 | An empty $2020 \times 2020 \times 2020$ cube is given, and a $2020 \times 2020$ grid of square unit cells is drawn on each of its six faces. A beam is a $1 \times 1 \times 2020$ rectangular prism. Several beams are placed inside the cube subject to the following conditions:
What is the smallest positive number of beam... | [
"Place the cube in the xyz-coordinate, with the positive x-axis pointing forward, the positive y-axis pointing right, and the positive z-axis pointing up. Let the position of a unit cube be $(x, y, z)$ if it is centered at $(x, y, z)$ . Place the $2020 \\times 2020 \\times 2020$ cube so that the edges are parallel ... |
https://artofproblemsolving.com/wiki/index.php/2005_AMC_10A_Problems/Problem_20 | A | 72 | An equilangular octagon has four sides of length $1$ and four sides of length $\frac{\sqrt{2}}{2}$ , arranged so that no two consecutive sides have the same length. What is the area of the octagon?
$\textbf{(A) } \frac72\qquad \textbf{(B) } \frac{7\sqrt2}{2}\qquad \textbf{(C) } \frac{5+4\sqrt2}{2}\qquad \textbf{(D) ... | [
"The area of the octagon can be divided up into $5$ squares with side $\\frac{\\sqrt2}2$ and $4$ right triangles, which are half the area of each of the squares.\nTherefore, the area of the octagon is equal to the area of $5+4\\left(\\frac12\\right)=7$ squares.\nThe area of each square is $\\left(\\frac{\\sqrt2}2\\... |
https://artofproblemsolving.com/wiki/index.php/2012_AMC_8_Problems/Problem_23 | C | 6 | An equilateral triangle and a regular hexagon have equal perimeters. If the triangle's area is 4, what is the area of the hexagon?
$\textbf{(A)}\hspace{.05in}4\qquad\textbf{(B)}\hspace{.05in}5\qquad\textbf{(C)}\hspace{.05in}6\qquad\textbf{(D)}\hspace{.05in}4\sqrt3\qquad\textbf{(E)}\hspace{.05in}6\sqrt3$ | [
"Let the perimeter of the equilateral triangle be $3s$ . The side length of the equilateral triangle would then be $s$ and the sidelength of the hexagon would be $\\frac{s}{2}$\nA hexagon contains six equilateral triangles. One of these triangles would be similar to the large equilateral triangle in the ratio $1 : ... |
https://artofproblemsolving.com/wiki/index.php/1951_AHSME_Problems/Problem_9 | D | 6 | An equilateral triangle is drawn with a side of length $a$ . A new equilateral triangle is formed by joining the midpoints of the sides of the first one. Then a third equilateral triangle is formed by joining the midpoints of the sides of the second; and so on forever. The limit of the sum of the perimeters of all the ... | [
"The perimeter of the first triangle is $3a$ . The perimeter of the 2nd triangle is half of that, after drawing a picture. The 3rd triangle's perimeter is half the second's, and so on. Therefore, we are computing $3a+\\frac{3a}{2}+\\frac{3a}{4}+\\cdots$\nThe starting term is $3a$ , and the common ratio is $1/2$ . T... |
https://artofproblemsolving.com/wiki/index.php/2001_AIME_I_Problems/Problem_5 | null | 937 | An equilateral triangle is inscribed in the ellipse whose equation is $x^2+4y^2=4$ . One vertex of the triangle is $(0,1)$ , one altitude is contained in the y-axis, and the square of the length of each side is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ | [
"Denote the vertices of the triangle $A,B,$ and $C,$ where $B$ is in quadrant 4 and $C$ is in quadrant $3.$\nNote that the slope of $\\overline{AC}$ is $\\tan 60^\\circ = \\sqrt {3}.$ Hence, the equation of the line containing $\\overline{AC}$ is \\[y = x\\sqrt {3} + 1.\\] This will intersect the ellipse when \\beg... |
https://artofproblemsolving.com/wiki/index.php/2008_AMC_10B_Problems/Problem_7 | C | 100 | An equilateral triangle of side length $10$ is completely filled in by non-overlapping equilateral triangles of side length $1$ . How many small triangles are required?
$\mathrm{(A)}\ 10\qquad\mathrm{(B)}\ 25\qquad\mathrm{(C)}\ 100\qquad\mathrm{(D)}\ 250\qquad\mathrm{(E)}\ 1000$ | [
"The area of the large triangle is $\\frac{10^2\\sqrt3}{4}$ , while the area of each small triangle is $\\frac{1^2\\sqrt3}{4}$ . Dividing these two quantities results in $100$ , therefore $\\boxed{100}$ small triangles can fill the large one without overlap."
] |
https://artofproblemsolving.com/wiki/index.php/2008_AMC_10B_Problems/Problem_7 | null | 100 | An equilateral triangle of side length $10$ is completely filled in by non-overlapping equilateral triangles of side length $1$ . How many small triangles are required?
$\mathrm{(A)}\ 10\qquad\mathrm{(B)}\ 25\qquad\mathrm{(C)}\ 100\qquad\mathrm{(D)}\ 250\qquad\mathrm{(E)}\ 1000$ | [
"\nThe number of triangles is $1+3+\\dots+19 = \\boxed{100}$"
] |
https://artofproblemsolving.com/wiki/index.php/2023_AMC_10A_Problems/Problem_15 | E | 64 | An even number of circles are nested, starting with a radius of $1$ and increasing by $1$ each time, all sharing a common point. The region between every other circle is shaded, starting with the region inside the circle of radius $2$ but outside the circle of radius $1.$ An example showing $8$ circles is displayed bel... | [
"Notice that the area of the shaded region is $(2^2\\pi-1^2\\pi)+(4^2\\pi-3^2\\pi)+(6^2\\pi-5^2\\pi)+ \\cdots + (n^2\\pi-(n-1)^2 \\pi)$ for any even number $n$\nUsing the difference of squares, this simplifies to $(1+2+3+4+\\cdots+n) \\pi$ . So, we are basically finding the smallest $n$ such that $\\frac{n(n+1)}{2}... |
https://artofproblemsolving.com/wiki/index.php/2005_AIME_II_Problems/Problem_3 | null | 802 | An infinite geometric series has sum 2005. A new series, obtained by squaring each term of the original series, has 10 times the sum of the original series. The common ratio of the original series is $\frac mn$ where $m$ and $n$ are relatively prime integers . Find $m+n.$ | [
"Let's call the first term of the original geometric series $a$ and the common ratio $r$ , so $2005 = a + ar + ar^2 + \\ldots$ . Using the sum formula for infinite geometric series, we have $\\;\\;\\frac a{1 -r} = 2005$ . Then we form a new series, $a^2 + a^2 r^2 + a^2 r^4 + \\ldots$ . We know this series has su... |
https://artofproblemsolving.com/wiki/index.php/2001_AMC_12_Problems/Problem_15 | null | 1 | An insect lives on the surface of a regular tetrahedron with edges of length 1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? (Note: Two edges of a tetrahedron are opposite if they have no common en... | [
"Given any path on the surface, we can unfold the surface into a plane to get a path of the same length in the plane. Consider the net of a tetrahedron in the picture below. A pair of opposite points is marked by dots. It is obvious that in the plane the shortest path is just a segment that connects these two point... |
https://artofproblemsolving.com/wiki/index.php/2017_AMC_10B_Problems/Problem_14 | D | 45 | An integer $N$ is selected at random in the range $1\leq N \leq 2020$ . What is the probability that the remainder when $N^{16}$ is divided by $5$ is $1$
$\textbf{(A)}\ \frac{1}{5}\qquad\textbf{(B)}\ \frac{2}{5}\qquad\textbf{(C)}\ \frac{3}{5}\qquad\textbf{(D)}\ \frac{4}{5}\qquad\textbf{(E)}\ 1$ | [
"Notice that we can rewrite $N^{16}$ as $(N^{4})^4$ . By Fermat's Little Theorem , we know that $N^{(5-1)} \\equiv 1 \\pmod {5}$ if $N \\not \\equiv 0 \\pmod {5}$ . Therefore for all $N \\not \\equiv 0 \\pmod {5}$ we have $N^{16} \\equiv (N^{4})^4 \\equiv 1^4 \\equiv 1 \\pmod 5$ . Since $1\\leq N \\leq 2020$ , and ... |
https://artofproblemsolving.com/wiki/index.php/2003_AIME_I_Problems/Problem_9 | null | 615 | An integer between $1000$ and $9999$ , inclusive, is called balanced if the sum of its two leftmost digits equals the sum of its two rightmost digits. How many balanced integers are there? | [
"If the common sum of the first two and last two digits is $n$ , such that $1 \\leq n \\leq 9$ , there are $n$ choices for the first two digits and $n + 1$ choices for the second two digits (since zero may not be the first digit). This gives $\\sum_{n = 1}^9 n(n + 1) = 330$ balanced numbers. If the common sum of ... |
https://artofproblemsolving.com/wiki/index.php/2023_AMC_10A_Problems/Problem_6 | D | 126 | An integer is assigned to each vertex of a cube. The value of an edge is defined to be the sum of the values of the two vertices it touches, and the value of a face is defined to be the sum of the values of the four edges surrounding it. The value of the cube is defined as the sum of the values of its six faces. Suppos... | [
"Each of the vertices is counted $3$ times because each vertex is shared by three different edges. \nEach of the edges is counted $2$ times because each edge is shared by two different faces. \nSince the sum of the integers assigned to all vertices is $21$ , the final answer is $21\\times3\\times2=\\boxed{126}$",
... |
https://artofproblemsolving.com/wiki/index.php/2007_AIME_II_Problems/Problem_6 | null | 640 | An integer is called parity-monotonic if its decimal representation $a_{1}a_{2}a_{3}\cdots a_{k}$ satisfies $a_{i}<a_{i+1}$ if $a_{i}$ is odd , and $a_{i}>a_{i+1}$ if $a_{i}$ is even . How many four-digit parity-monotonic integers are there? | [
"This problem can be solved via recursion since we are \"building a string\" of numbers with some condition. We want to create a new string by adding a new digit at the front so we avoid complications( $0$ can't be at the front and no digit is less than $9$ ). There are $4$ options to add no matter what(try some ex... |
https://artofproblemsolving.com/wiki/index.php/2004_AIME_I_Problems/Problem_6 | null | 882 | An integer is called snakelike if its decimal representation $a_1a_2a_3\cdots a_k$ satisfies $a_i<a_{i+1}$ if $i$ is odd and $a_i>a_{i+1}$ if $i$ is even . How many snakelike integers between 1000 and 9999 have four distinct digits? | [
"We divide the problem into two cases: one in which zero is one of the digits and one in which it is not. In the latter case, suppose we pick digits $x_1,x_2,x_3,x_4$ such that $x_1<x_2<x_3<x_4$ . There are five arrangements of these digits that satisfy the condition of being snakelike: $x_1x_3x_2x_4$ $x_1x_4x_2x_3... |
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