Difference between revisions of "2013 AIME II Problems"
(17 intermediate revisions by 6 users not shown) | |||
Line 2: | Line 2: | ||
==Problem 1== | ==Problem 1== | ||
+ | Suppose that the measurement of time during the day is converted to the metric system so that each day has <math>10</math> metric hours, and each metric hour has <math>100</math> metric minutes. Digital clocks would then be produced that would read <math>\text{9:99}</math> just before midnight, <math>\text{0:00}</math> at midnight, <math>\text{1:25}</math> at the former <math>\text{3:00}</math> AM, and <math>\text{7:50}</math> at the former <math>\text{6:00}</math> PM. After the conversion, a person who wanted to wake up at the equivalent of the former <math>\text{6:36}</math> AM would set his new digital alarm clock for <math>\text{A:BC}</math>, where <math>\text{A}</math>, <math>\text{B}</math>, and <math>\text{C}</math> are digits. Find <math>100\text{A}+10\text{B}+\text{C}</math>. | ||
+ | |||
+ | [[2013 AIME II Problems/Problem 1|Solution]] | ||
+ | |||
+ | ==Problem 2== | ||
+ | Positive integers <math>a</math> and <math>b</math> satisfy the condition | ||
+ | <cmath>\log_2(\log_{2^a}(\log_{2^b}(2^{1000}))) = 0.</cmath> | ||
+ | Find the sum of all possible values of <math>a+b</math>. | ||
+ | |||
+ | [[2013 AIME II Problems/Problem 2|Solution]] | ||
+ | |||
+ | ==Problem 3== | ||
+ | A large candle is <math>119</math> centimeters tall. It is designed to burn down more quickly when it is first lit and more slowly as it approaches its bottom. Specifically, the candle takes <math>10</math> seconds to burn down the first centimeter from the top, <math>20</math> seconds to burn down the second centimeter, and <math>10k</math> seconds to burn down the <math>k</math>-th centimeter. Suppose it takes <math>T</math> seconds for the candle to burn down completely. Then <math>\tfrac{T}{2}</math> seconds after it is lit, the candle's height in centimeters will be <math>h</math>. Find <math>10h</math>. | ||
+ | |||
+ | [[2013 AIME II Problems/Problem 3|Solution]] | ||
+ | |||
+ | ==Problem 4== | ||
+ | In the Cartesian plane let <math>A = (1,0)</math> and <math>B = \left( 2, 2\sqrt{3} \right)</math>. Equilateral triangle <math>ABC</math> is constructed so that <math>C</math> lies in the first quadrant. Let <math>P=(x,y)</math> be the center of <math>\triangle ABC</math>. Then <math>x \cdot y</math> can be written as <math>\tfrac{p\sqrt{q}}{r}</math>, where <math>p</math> and <math>r</math> are relatively prime positive integers and <math>q</math> is an integer that is not divisible by the square of any prime. Find <math>p+q+r</math>. | ||
+ | |||
+ | [[2013 AIME II Problems/Problem 4|Solution]] | ||
+ | |||
+ | ==Problem 5== | ||
+ | In equilateral <math>\triangle ABC</math> let points <math>D</math> and <math>E</math> trisect <math>\overline{BC}</math>. Then <math>\sin(\angle DAE)</math> can be expressed in the form <math>\frac{a\sqrt{b}}{c}</math>, where <math>a</math> and <math>c</math> are relatively prime positive integers, and <math>b</math> is an integer that is not divisible by the square of any prime. Find <math>a+b+c</math>. | ||
+ | |||
+ | [[2013 AIME II Problems/Problem 5|Solution]] | ||
+ | |||
+ | ==Problem 6== | ||
+ | Find the least positive integer <math>N</math> such that the set of <math>1000</math> consecutive integers beginning with <math>1000\cdot N</math> contains no square of an integer. | ||
+ | |||
+ | [[2013 AIME II Problems/Problem 6|Solution]] | ||
+ | |||
+ | ==Problem 7== | ||
+ | A group of clerks is assigned the task of sorting <math>1775</math> files. Each clerk sorts at a constant rate of <math>30</math> files per hour. At the end of the first hour, some of the clerks are reassigned to another task; at the end of the second hour, the same number of the remaining clerks are also reassigned to another task, and a similar assignment occurs at the end of the third hour. The group finishes the sorting in <math>3</math> hours and <math>10</math> minutes. Find the number of files sorted during the first one and a half hours of sorting. | ||
+ | |||
+ | [[2013 AIME II Problems/Problem 7|Solution]] | ||
+ | |||
+ | ==Problem 8== | ||
+ | A hexagon that is inscribed in a circle has side lengths <math>22</math>, <math>22</math>, <math>20</math>, <math>22</math>, <math>22</math>, and <math>20</math> in that order. The radius of the circle can be written as <math>p+\sqrt{q}</math>, where <math>p</math> and <math>q</math> are positive integers. Find <math>p+q</math>. | ||
+ | |||
+ | [[2013 AIME II Problems/Problem 8|Solution]] | ||
+ | ==Problem 9== | ||
+ | A <math>7\times 1</math> board is completely covered by <math>m\times 1</math> tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the board. Each tile is either red, blue, or green. Let <math>N</math> be the number of tilings of the <math>7\times 1</math> board in which all three colors are used at least once. For example, a <math>1\times 1</math> red tile followed by a <math>2\times 1</math> green tile, a <math>1\times 1</math> green tile, a <math>2\times 1</math> blue tile, and a <math>1\times 1</math> green tile is a valid tiling. Note that if the <math>2\times 1</math> blue tile is replaced by two <math>1\times 1</math> blue tiles, this results in a different tiling. Find the remainder when <math>N</math> is divided by <math>1000</math>. | ||
+ | |||
+ | [[2013 AIME II Problems/Problem 9|Solution]] | ||
+ | |||
+ | ==Problem 10== | ||
+ | Given a circle of radius <math>\sqrt{13}</math>, let <math>A</math> be a point at a distance <math>4 + \sqrt{13}</math> from the center <math>O</math> of the circle. Let <math>B</math> be the point on the circle nearest to point <math>A</math>. A line passing through the point <math>A</math> intersects the circle at points <math>K</math> and <math>L</math>. The maximum possible area for <math>\triangle BKL</math> can be written in the form <math>\frac{a - b\sqrt{c}}{d}</math>, where <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> are positive integers, <math>a</math> and <math>d</math> are relatively prime, and <math>c</math> is not divisible by the square of any prime. Find <math>a+b+c+d</math>. | ||
+ | |||
+ | [[2013 AIME II Problems/Problem 10|Solution]] | ||
+ | |||
+ | ==Problem 11== | ||
+ | Let <math>A = \{1, 2, 3, 4, 5, 6, 7\}</math>, and let <math>N</math> be the number of functions <math>f</math> from set <math>A</math> to set <math>A</math> such that <math>f(f(x))</math> is a constant function. Find the remainder when <math>N</math> is divided by <math>1000</math>. | ||
+ | |||
+ | [[2013 AIME II Problems/Problem 11|Solution]] | ||
+ | |||
+ | ==Problem 12== | ||
+ | Let <math>S</math> be the set of all polynomials of the form <math>z^3 + az^2 + bz + c</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are integers. Find the number of polynomials in <math>S</math> such that each of its roots <math>z</math> satisfies either <math>|z| = 20</math> or <math>|z| = 13</math>. | ||
+ | |||
+ | [[2013 AIME II Problems/Problem 12|Solution]] | ||
+ | |||
+ | ==Problem 13== | ||
+ | In <math>\triangle ABC</math>, <math>AC = BC</math>, and point <math>D</math> is on <math>\overline{BC}</math> so that <math>CD = 3\cdot BD</math>. Let <math>E</math> be the midpoint of <math>\overline{AD}</math>. Given that <math>CE = \sqrt{7}</math> and <math>BE = 3</math>, the area of <math>\triangle ABC</math> can be expressed in the form <math>m\sqrt{n}</math>, where <math>m</math> and <math>n</math> are positive integers and <math>n</math> is not divisible by the square of any prime. Find <math>m+n</math>. | ||
+ | |||
+ | [[2013 AIME II Problems/Problem 13|Solution]] | ||
+ | |||
+ | ==Problem 14== | ||
+ | For positive integers <math>n</math> and <math>k</math>, let <math>f(n, k)</math> be the remainder when <math>n</math> is divided by <math>k</math>, and for <math>n > 1</math> let <math>F(n) = \max_{\substack{1\le k\le \frac{n}{2}}} f(n, k)</math>. Find the remainder when <math>\sum\limits_{n=20}^{100} F(n)</math> is divided by <math>1000</math>. | ||
+ | |||
+ | [[2013 AIME II Problems/Problem 14|Solution]] | ||
+ | |||
+ | ==Problem 15== | ||
+ | Let <math>A,B,C</math> be angles of an acute triangle with | ||
+ | <cmath> \begin{align*} | ||
+ | \cos^2 A + \cos^2 B + 2 \sin A \sin B \cos C &= \frac{15}{8} \text{ and} \\ | ||
+ | \cos^2 B + \cos^2 C + 2 \sin B \sin C \cos A &= \frac{14}{9} | ||
+ | \end{align*} </cmath> | ||
+ | There are positive integers <math>p</math>, <math>q</math>, <math>r</math>, and <math>s</math> for which <cmath> \cos^2 C + \cos^2 A + 2 \sin C \sin A \cos B = \frac{p-q\sqrt{r}}{s}, </cmath> where <math>p+q</math> and <math>s</math> are relatively prime and <math>r</math> is not divisible by the square of any prime. Find <math>p+q+r+s</math>. | ||
+ | |||
+ | [[2013 AIME II Problems/Problem 15|Solution]] | ||
+ | |||
+ | {{AIME box|year=2013|n=II|before=[[2013 AIME I Problems]]|after=[[2014 AIME I Problems]]}} | ||
+ | |||
+ | {{MAA Notice}} |
Latest revision as of 12:17, 13 March 2020
2013 AIME II (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
Contents
Problem 1
Suppose that the measurement of time during the day is converted to the metric system so that each day has metric hours, and each metric hour has metric minutes. Digital clocks would then be produced that would read just before midnight, at midnight, at the former AM, and at the former PM. After the conversion, a person who wanted to wake up at the equivalent of the former AM would set his new digital alarm clock for , where , , and are digits. Find .
Problem 2
Positive integers and satisfy the condition Find the sum of all possible values of .
Problem 3
A large candle is centimeters tall. It is designed to burn down more quickly when it is first lit and more slowly as it approaches its bottom. Specifically, the candle takes seconds to burn down the first centimeter from the top, seconds to burn down the second centimeter, and seconds to burn down the -th centimeter. Suppose it takes seconds for the candle to burn down completely. Then seconds after it is lit, the candle's height in centimeters will be . Find .
Problem 4
In the Cartesian plane let and . Equilateral triangle is constructed so that lies in the first quadrant. Let be the center of . Then can be written as , where and are relatively prime positive integers and is an integer that is not divisible by the square of any prime. Find .
Problem 5
In equilateral let points and trisect . Then can be expressed in the form , where and are relatively prime positive integers, and is an integer that is not divisible by the square of any prime. Find .
Problem 6
Find the least positive integer such that the set of consecutive integers beginning with contains no square of an integer.
Problem 7
A group of clerks is assigned the task of sorting files. Each clerk sorts at a constant rate of files per hour. At the end of the first hour, some of the clerks are reassigned to another task; at the end of the second hour, the same number of the remaining clerks are also reassigned to another task, and a similar assignment occurs at the end of the third hour. The group finishes the sorting in hours and minutes. Find the number of files sorted during the first one and a half hours of sorting.
Problem 8
A hexagon that is inscribed in a circle has side lengths , , , , , and in that order. The radius of the circle can be written as , where and are positive integers. Find .
Problem 9
A board is completely covered by tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the board. Each tile is either red, blue, or green. Let be the number of tilings of the board in which all three colors are used at least once. For example, a red tile followed by a green tile, a green tile, a blue tile, and a green tile is a valid tiling. Note that if the blue tile is replaced by two blue tiles, this results in a different tiling. Find the remainder when is divided by .
Problem 10
Given a circle of radius , let be a point at a distance from the center of the circle. Let be the point on the circle nearest to point . A line passing through the point intersects the circle at points and . The maximum possible area for can be written in the form , where , , , and are positive integers, and are relatively prime, and is not divisible by the square of any prime. Find .
Problem 11
Let , and let be the number of functions from set to set such that is a constant function. Find the remainder when is divided by .
Problem 12
Let be the set of all polynomials of the form , where , , and are integers. Find the number of polynomials in such that each of its roots satisfies either or .
Problem 13
In , , and point is on so that . Let be the midpoint of . Given that and , the area of can be expressed in the form , where and are positive integers and is not divisible by the square of any prime. Find .
Problem 14
For positive integers and , let be the remainder when is divided by , and for let . Find the remainder when is divided by .
Problem 15
Let be angles of an acute triangle with There are positive integers , , , and for which where and are relatively prime and is not divisible by the square of any prime. Find .
2013 AIME II (Problems • Answer Key • Resources) | ||
Preceded by 2013 AIME I Problems |
Followed by 2014 AIME I Problems | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.