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Difference between revisions of "2024 AIME II Problems"

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==Problem 14==
 
==Problem 14==
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Let <math>b \geq 2</math> be an integer. Call a positive integer <math>n</math> <math>b</math>-eautiful if it has exactly two digits when expressed in base <math>b</math>, and these two digits sum to <math>\sqrt{n}</math>. For example, <math>81</math> is <math>13</math>-eautiful because <math>81=\underline{6}\ \underline{3}_{13}</math> and <math>6+3=\sqrt{81}</math>. Find the least integer <math>b\geq 2</math> for which there are more than then <math>b</math>-eautiful integers   
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Let <math>b \geq 2</math> be an integer. Call a positive integer <math>n</math> <math>b\textit{-eautiful}</math> if it has exactly two digits when expressed in base <math>b</math>, and these two digits sum to <math>\sqrt{n}</math>. For example, <math>81</math> is <math>13</math>-eautiful because <math>81=\underline{6}\ \underline{3}_{13}</math> and <math>6+3=\sqrt{81}</math>. Find the least integer <math>b\geq 2</math> for which there are more than ten <math>b</math>-eautiful integers.  
  
 
[[2024 AIME II Problems/Problem 14|Solution]]
 
[[2024 AIME II Problems/Problem 14|Solution]]

Revision as of 23:53, 8 February 2024

2024 AIME II (Answer Key)
Printable version | AoPS Contest Collections • PDF

Instructions

  1. This is a 15-question, 3-hour examination. All answers are integers ranging from $000$ to $999$, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.
  2. No aids other than scratch paper, graph paper, ruler, compass, and protractor are permitted. In particular, calculators and computers are not permitted.
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15

Problem 1

Among the $900$ residents of Aimeville, there are $195$ who own a diamond ring, $367$ who own a set of golf clubs, and $562$ who own a garden spade. In addition, each of the $900$ residents owns a bag of candy hearts. There are $437$ residents who own exactly two of these things, and $234$ residents who own exactly three of these things. Find the number of residents of Aimeville who own all four of these things.

Solution

Problem 2

A list of positive integers has the following properties:

$\bullet$ The sum of the items in the list is $30$.

$\bullet$ The unique mode of the list is $9$.

$\bullet$ The median of the list is a positive integer that does not appear in the list itself.

Find the sum of the squares of all the items in the list.

Solution

Problem 3

Find the number of ways to place a digit in each cell of a 2x3 grid so that the sum of the two numbers formed by reading left to right is $999$, and the sum of the three numbers formed by reading top to bottom is $99$. The grid below is an example of such an arrangement because $8+991=999$ and $9+9+81=99$.

\[\begin{array}{|c|c|c|} \hline 0 & 0 & 8 \\ \hline 9 & 9 & 1 \\ \hline \end{array}\]

Solution

Problem 4

Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations: \[\log_2\left({x \over yz}\right) = {1 \over 2}\]\[\log_2\left({y \over xz}\right) = {1 \over 3}\]\[\log_2\left({z \over xy}\right) = {1 \over 4}\] Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is ${m \over n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.


Solution

Problem 5

Let $ABCDEF$ be a convex equilateral hexagon in which all pairs of opposite sides are parallel. The triangle whose sides are extensions of segments $\overline{AB}$, $\overline{CD}$, and $\overline{EF}$ has side lengths $200, 240,$ and $300$. Find the side length of the hexagon.

Solution

Problem 6

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 $A$.

Solution

Problem 7

Let $N$ be the greatest four-digit integer with the property that whenever one of its digits is changed to $1$, the resulting number is divisible by $7$. Let $Q$ and $R$ be the quotient and remainder, respectively, when $N$ is divided by $1000$. Find $Q+R$.

Solution

Problem 8

Solution

Problem 9

Solution

Problem 10

Solution

Problem 11

Find the number of triples of nonnegative integers $(a, b, c)$ satisfying $a + b + c = 300$ and

\[a^2 b + a^2 c + b^2 a + b^2 c + c^2 a + c^2 b = 6,000,000.\]

Solution

Problem 12

Solution

Problem 13

Let $\omega\neq 1$ be a 13th root of unity. Find the remainder when \[\prod_{k=0}^{12}(2-2\omega^k+\omega^{2k})\] is divided by 1000.

Solution

Problem 14

Let $b \geq 2$ be an integer. Call a positive integer $n$ $b\textit{-eautiful}$ if it has exactly two digits when expressed in base $b$, and these two digits sum to $\sqrt{n}$. For example, $81$ is $13$-eautiful because $81=\underline{6}\ \underline{3}_{13}$ and $6+3=\sqrt{81}$. Find the least integer $b\geq 2$ for which there are more than ten $b$-eautiful integers.

Solution

Problem 15

Find the number of rectangles inside a fixed regular dodecagon ($12$-gon) where each side of the rectangle lies on a side or on a diagonal of the dodecagon. The diagram below shows three of those rectangles.

[asy] unitsize(0.6 inch); for(int i=0; i<360; i+=30) { dot(dir(i), 4+black); draw(dir(i)--dir(i+30)); } draw(dir(120)--dir(330)); filldraw(dir(210)--dir(240)--dir(30)--dir(60)--cycle, mediumgray, linewidth(1.5)); draw((0,0.366)--(0.366,0), linewidth(1.5)); [/asy]

Solution

See also

2024 AIME II (Problems • Answer Key • Resources)
Preceded by
2024 AIME I 		Followed by
2025 AIME I
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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