Difference between revisions of "2008 AIME II Problems"

Latest revision as of 20:56, 28 June 2024

2008 AIME II (Answer Key) Printable version \| AoPS Contest Collections • PDF
Instructions 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. 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

Let $N = 100^2 + 99^2 - 98^2 - 97^2 + 96^2 + \cdots + 4^2 + 3^2 - 2^2 - 1^2$ , where the additions and subtractions alternate in pairs. Find the remainder when $N$ is divided by $1000$ .

Solution

Problem 2

Rudolph bikes at a constant rate and stops for a five-minute break at the end of every mile. Jennifer bikes at a constant rate which is three-quarters the rate that Rudolph bikes, but Jennifer takes a five-minute break at the end of every two miles. Jennifer and Rudolph begin biking at the same time and arrive at the $50$ -mile mark at exactly the same time. How many minutes has it taken them?

Solution

Problem 3

A block of cheese in the shape of a rectangular solid measures $10$ cm by $13$ cm by $14$ cm. Ten slices are cut from the cheese. Each slice has a width of $1$ cm and is cut parallel to one face of the cheese. The individual slices are not necessarily parallel to each other. What is the maximum possible volume in cubic cm of the remaining block of cheese after ten slices have been cut off?

Solution

Problem 4

There exist $r$ unique nonnegative integers $n_1 > n_2 > \cdots > n_r$ and $r$ integers $a_k$ ( $1\le k\le r$ ) with each $a_k$ either $1$ or $- 1$ such that $a_13^{n_1} + a_23^{n_2} + \cdots + a_r3^{n_r} = 2008.$ Find $n_1 + n_2 + \cdots + n_r$ .

Solution

Problem 5

In trapezoid $ABCD$ with $\overline{BC}\parallel\overline{AD}$ , let $BC = 1000$ and $AD = 2008$ . Let $\angle A = 37^\circ$ , $\angle D = 53^\circ$ , and $M$ and $N$ be the midpoints of $\overline{BC}$ and $\overline{AD}$ , respectively. Find the length $MN$ .

Solution

Problem 6

The sequence $\{a_n\}$ is defined by $a_0 = 1,a_1 = 1, \text{ and } a_n = a_{n - 1} + \frac {a_{n - 1}^2}{a_{n - 2}}\text{ for }n\ge2.$ The sequence $\{b_n\}$ is defined by $b_0 = 1,b_1 = 3, \text{ and } b_n = b_{n - 1} + \frac {b_{n - 1}^2}{b_{n - 2}}\text{ for }n\ge2.$ Find $\frac {b_{32}}{a_{32}}$ .

Solution

Problem 7

Let $r$ , $s$ , and $t$ be the three roots of the equation $8x^3 + 1001x + 2008 = 0.$ Find $(r + s)^3 + (s + t)^3 + (t + r)^3$ .

Solution

Problem 8

Let $a = \pi/2008$ . Find the smallest positive integer $n$ such that $2[\cos(a)\sin(a) + \cos(4a)\sin(2a) + \cos(9a)\sin(3a) + \cdots + \cos(n^2a)\sin(na)]$ is an integer.

Solution

Problem 9

A particle is located on the coordinate plane at $(5,0)$ . Define a move for the particle as a counterclockwise rotation of $\pi/4$ radians about the origin followed by a translation of $10$ units in the positive $x$ -direction. Given that the particle's position after $150$ moves is $(p,q)$ , find the greatest integer less than or equal to $|p| + |q|$ .

Solution

Problem 10

The diagram below shows a $4\times4$ rectangular array of points, each of which is $1$ unit away from its nearest neighbors.

$[asy] unitsize(0.25inch); defaultpen(linewidth(0.7)); int i, j; for(i = 0; i < 4; ++i) for(j = 0; j < 4; ++j) dot(((real)i, (real)j)); [/asy]$

Define a growing path to be a sequence of distinct points of the array with the property that the distance between consecutive points of the sequence is strictly increasing. Let $m$ be the maximum possible number of points in a growing path, and let $r$ be the number of growing paths consisting of exactly $m$ points. Find $mr$ .

Solution

Problem 11

In triangle $ABC$ , $AB = AC = 100$ , and $BC = 56$ . Circle $P$ has radius $16$ and is tangent to $\overline{AC}$ and $\overline{BC}$ . Circle $Q$ is externally tangent to circle $P$ and is tangent to $\overline{AB}$ and $\overline{BC}$ . No point of circle $Q$ lies outside of $\bigtriangleup\overline{ABC}$ . The radius of circle $Q$ can be expressed in the form $m - n\sqrt{k}$ ,where $m$ , $n$ , and $k$ are positive integers and $k$ is the product of distinct primes. Find $m +nk$ .

Solution

Problem 12

There are two distinguishable flagpoles, and there are $19$ flags, of which $10$ are identical blue flags, and $9$ are identical green flags. Let $N$ be the number of distinguishable arrangements using all of the flags in which each flagpole has at least one flag and no two green flags on either pole are adjacent. Find the remainder when $N$ is divided by $1000$ .

Solution

Problem 13

A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let $R$ be the region outside the hexagon, and let $S = \left\lbrace\frac{1}{z}|z \in R\right\rbrace$ . Then the area of $S$ has the form $a\pi + \sqrt{b}$ , where $a$ and $b$ are positive integers. Find $a + b$ .

Solution

Problem 14

Let $a$ and $b$ be positive real numbers with $a \ge b$ . Let $\rho$ be the maximum possible value of $\dfrac{a}{b}$ for which the system of equations $a^2 + y^2 = b^2 + x^2 = (a-x)^2 + (b-y)^2$ has a solution $(x,y)$ satisfying $0 \le x < a$ and $0 \le y < b$ . Then $\rho^2$ can be expressed as a fraction $\dfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ .

Solution

Problem 15

Find the largest integer $n$ satisfying the following conditions:

(i) $n^2$ can be expressed as the difference of two consecutive cubes;

(ii) $2n + 79$ is a perfect square.

Solution

@@ Line 1: / Line 1: @@
+{{AIME Problems|year=2008|n=II}}
 == Problem 1 ==
 Let <math>N = 100^2 + 99^2 - 98^2 - 97^2 + 96^2 + \cdots + 4^2 + 3^2 - 2^2 - 1^2</math>, where the additions and subtractions alternate in pairs. Find the remainder when <math>N</math> is divided by <math>1000</math>.
@@ Line 5: / Line 7: @@
 == Problem 2 ==
-Rudolph bikes at a [[constant]] rate and stops for a five-minute break at the end of every mile. Jennifer bikes at a constant rate which is three-quarters the rate that Rudolph bikes, but Jennifer takes a five-minute break at the end of every two miles. Jennifer and Rudolph begin biking at the same time and arrive at the <math>50</math>-mile mark at exactly the same time. How many minutes has it taken them?
+Rudolph bikes at a constant rate and stops for a five-minute break at the end of every mile. Jennifer bikes at a constant rate which is three-quarters the rate that Rudolph bikes, but Jennifer takes a five-minute break at the end of every two miles. Jennifer and Rudolph begin biking at the same time and arrive at the <math>50</math>-mile mark at exactly the same time. How many minutes has it taken them?
 [[2008_AIME_II_Problems/Problem_2|Solution]]
@@ Line 15: / Line 17: @@
 == Problem 4 ==
-There exist <math>r</math> unique nonnegative integers <math>n_1 > n_2 > \cdots > n_r</math> and <math>r</math> unique integers <math>a_k</math> (<math>1\le k\le r</math>) with each <math>a_k</math> either <math>1</math> or <math>- 1</math> such that
+There exist <math>r</math> unique nonnegative integers <math>n_1 > n_2 > \cdots > n_r</math> and <math>r</math> integers <math>a_k</math> (<math>1\le k\le r</math>) with each <math>a_k</math> either <math>1</math> or <math>- 1</math> such that
 <cmath>
 a_13^{n_1} + a_23^{n_2} + \cdots + a_r3^{n_r} = 2008.
@@ Line 24: / Line 26: @@
 == Problem 5 ==
-In [[trapezoid]] <math>ABCD</math> with <math>\overline{BC}\parallel\overline{AD}</math>, let <math>BC = 1000</math> and <math>AD = 2008</math>. Let <math>\angle A = 37^\circ</math>, <math>\angle D = 53^\circ</math>, and <math>M</math> and <math>N</math> be the [[midpoint]]s of <math>\overline{BC}</math> and <math>\overline{AD}</math>, respectively. Find the length <math>MN</math>.
+In trapezoid <math>ABCD</math> with <math>\overline{BC}\parallel\overline{AD}</math>, let <math>BC = 1000</math> and <math>AD = 2008</math>. Let <math>\angle A = 37^\circ</math>, <math>\angle D = 53^\circ</math>, and <math>M</math> and <math>N</math> be the midpoints of <math>\overline{BC}</math> and <math>\overline{AD}</math>, respectively. Find the length <math>MN</math>.
 [[2008_AIME_II_Problems/Problem_5|Solution]]
@@ Line 50: / Line 52: @@
 == Problem 8 ==
-Let <math>a = pi/2008</math>. Find the smallest possible integer <math>n</math> such that <math>2[cos(a)sin(a) + cos(4a)sin(2a) + cos(9a)sin(3a) + /cdots + cos(n^2a)sin(na)]</math> is an integer.
+Let <math>a = \pi/2008</math>. Find the smallest positive integer <math>n</math> such that
+<cmath>
+[\cos(a)\sin(a) + \cos(4a)\sin(2a) + \cos(9a)\sin(3a) + \cdots + \cos(n^2a)\sin(na)]
+</cmath>
+is an integer.
 [[2008_AIME_II_Problems/Problem_8|Solution]]
 == Problem 9 ==
-{{problem}}
+A particle is located on the coordinate plane at <math>(5,0)</math>. Define a ''move'' for the particle as a counterclockwise rotation of <math>\pi/4</math> radians about the origin followed by a translation of <math>10</math> units in the positive <math>x</math>-direction. Given that the particle's position after <math>150</math> moves is <math>(p,q)</math>, find the greatest integer less than or equal to <math>|p| + |q|</math>.
 [[2008_AIME_II_Problems/Problem_9|Solution]]
 == Problem 10 ==
-{{problem}}
+The diagram below shows a <math>4\times4</math> rectangular array of points, each of which is <math>1</math> unit away from its nearest neighbors.
+<center><asy>
+unitsize(0.25inch);
+defaultpen(linewidth(0.7));
+int i, j;
+for(i = 0; i < 4; ++i)
+	for(j = 0; j < 4; ++j)
+		dot(((real)i, (real)j));
+</asy></center>
+Define a ''growing path'' to be a sequence of distinct points of the array with the property that the distance between consecutive points of the sequence is strictly increasing. Let <math>m</math> be the maximum possible number of points in a growing path, and let <math>r</math> be the number of growing paths consisting of exactly <math>m</math> points. Find <math>mr</math>.
 [[2008_AIME_II_Problems/Problem_10|Solution]]
 == Problem 11 ==
-{{problem}}
+In triangle <math>ABC</math>, <math>AB = AC = 100</math>, and <math>BC = 56</math>. Circle <math>P</math> has radius <math>16</math> and is tangent to <math>\overline{AC}</math> and <math>\overline{BC}</math>. Circle <math>Q</math> is externally tangent to circle <math>P</math> and is tangent to <math>\overline{AB}</math> and <math>\overline{BC}</math>. No point of circle <math>Q</math> lies outside of <math>\bigtriangleup\overline{ABC}</math>. The radius of circle <math>Q</math> can be expressed in the form <math>m - n\sqrt{k}</math>,where <math>m</math>, <math>n</math>, and <math>k</math> are positive integers and <math>k</math> is the product of distinct primes. Find <math>m +nk</math>.
 [[2008_AIME_II_Problems/Problem_11|Solution]]
 == Problem 12 ==
-{{problem}}
+There are two distinguishable flagpoles, and there are <math>19</math> flags, of which <math>10</math> are identical blue flags, and <math>9</math> are identical green flags. Let <math>N</math> be the number of distinguishable arrangements using all of the flags in which each flagpole has at least one flag and no two green flags on either pole are adjacent. Find the remainder when <math>N</math> is divided by <math>1000</math>.
 [[2008_AIME_II_Problems/Problem_12|Solution]]
 == Problem 13 ==
-{{problem}}
+A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let <math>R</math> be the region outside the hexagon, and let <math>S = \left\lbrace\frac{1}{z}|z \in R\right\rbrace</math>. Then the area of <math>S</math> has the form <math>a\pi + \sqrt{b}</math>, where <math>a</math> and <math>b</math> are positive integers. Find <math>a + b</math>.
 [[2008_AIME_II_Problems/Problem_13|Solution]]
 == Problem 14 ==
-{{problem}}
+Let <math>a</math> and <math>b</math> be positive real numbers with <math>a \ge b</math>. Let <math>\rho</math> be the maximum possible value of <math>\dfrac{a}{b}</math> for which the system of equations
+<cmath>
+a^2 + y^2 = b^2 + x^2 = (a-x)^2 + (b-y)^2
+</cmath>
+has a solution <math>(x,y)</math> satisfying <math>0 \le x < a</math> and <math>0 \le y < b</math>. Then <math>\rho^2</math> can be expressed as a fraction <math>\dfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>.
 [[2008_AIME_II_Problems/Problem_14|Solution]]
 == Problem 15 ==
-{{problem}}
+Find the largest integer <math>n</math> satisfying the following conditions:
+:(i) <math>n^2</math> can be expressed as the difference of two consecutive cubes;
+:(ii) <math>2n + 79</math> is a perfect square.
 [[2008_AIME_II_Problems/Problem_15|Solution]]
 == See also ==
+{{AIME box|year=2008|n=II|before=[[2008 AIME I Problems]]|after=[[2009 AIME I Problems]]}}
 * [[American Invitational Mathematics Examination]]
 * [[AIME Problems and Solutions]]
 * [[Mathematics competition resources]]
+{{MAA Notice}}

2008 AIME II (Problems • Answer Key • Resources)
Preceded by 2008 AIME I Problems	Followed by 2009 AIME I Problems
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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Difference between revisions of "2008 AIME II Problems"

Latest revision as of 20:56, 28 June 2024

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

See also