Difference between revisions of "2008 AIME II Problems"
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+ | {{AIME Problems|year=2008|n=II}} | ||
+ | |||
== Problem 1 == | == 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>. | 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 == | == 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? | |
[[2008_AIME_II_Problems/Problem_2|Solution]] | [[2008_AIME_II_Problems/Problem_2|Solution]] | ||
== Problem 3 == | == Problem 3 == | ||
− | + | A block of cheese in the shape of a rectangular solid measures <math>10</math> cm by <math>13</math> cm by <math>14</math> cm. Ten slices are cut from the cheese. Each slice has a width of <math>1</math> 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? | |
[[2008_AIME_II_Problems/Problem_3|Solution]] | [[2008_AIME_II_Problems/Problem_3|Solution]] | ||
== Problem 4 == | == Problem 4 == | ||
− | {{ | + | 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. | ||
+ | </cmath> | ||
+ | Find <math>n_1 + n_2 + \cdots + n_r</math>. | ||
[[2008_AIME_II_Problems/Problem_4|Solution]] | [[2008_AIME_II_Problems/Problem_4|Solution]] | ||
== Problem 5 == | == 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 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]] | [[2008_AIME_II_Problems/Problem_5|Solution]] | ||
== Problem 6 == | == Problem 6 == | ||
− | {{ | + | The sequence <math>\{a_n\}</math> is defined by |
+ | <cmath> | ||
+ | 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. | ||
+ | </cmath> | ||
+ | The sequence <math>\{b_n\}</math> is defined by | ||
+ | <cmath> | ||
+ | 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.</cmath> | ||
+ | Find <math>\frac {b_{32}}{a_{32}}</math>. | ||
[[2008_AIME_II_Problems/Problem_6|Solution]] | [[2008_AIME_II_Problems/Problem_6|Solution]] | ||
== Problem 7 == | == Problem 7 == | ||
− | + | Let <math>r</math>, <math>s</math>, and <math>t</math> be the three roots of the equation | |
+ | <cmath> | ||
+ | 8x^3 + 1001x + 2008 = 0. | ||
+ | </cmath> | ||
+ | Find <math>(r + s)^3 + (s + t)^3 + (t + r)^3</math>. | ||
[[2008_AIME_II_Problems/Problem_7|Solution]] | [[2008_AIME_II_Problems/Problem_7|Solution]] | ||
== Problem 8 == | == Problem 8 == | ||
− | + | Let <math>a = \pi/2008</math>. Find the smallest positive integer <math>n</math> such that | |
+ | <cmath> | ||
+ | 2[\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]] | [[2008_AIME_II_Problems/Problem_8|Solution]] | ||
== Problem 9 == | == Problem 9 == | ||
− | + | 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]] | [[2008_AIME_II_Problems/Problem_9|Solution]] | ||
== Problem 10 == | == Problem 10 == | ||
− | + | 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]] | [[2008_AIME_II_Problems/Problem_10|Solution]] | ||
== Problem 11 == | == Problem 11 == | ||
− | {{ | + | 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]] | [[2008_AIME_II_Problems/Problem_11|Solution]] | ||
== Problem 12 == | == Problem 12 == | ||
− | + | 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]] | [[2008_AIME_II_Problems/Problem_12|Solution]] | ||
== Problem 13 == | == 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 <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]] | [[2008_AIME_II_Problems/Problem_13|Solution]] | ||
== Problem 14 == | == Problem 14 == | ||
− | {{ | + | 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]] | [[2008_AIME_II_Problems/Problem_14|Solution]] | ||
== Problem 15 == | == Problem 15 == | ||
− | + | 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]] | [[2008_AIME_II_Problems/Problem_15|Solution]] | ||
== See also == | == See also == | ||
+ | {{AIME box|year=2008|n=II|before=[[2008 AIME I Problems]]|after=[[2009 AIME I Problems]]}} | ||
* [[American Invitational Mathematics Examination]] | * [[American Invitational Mathematics Examination]] | ||
* [[AIME Problems and Solutions]] | * [[AIME Problems and Solutions]] | ||
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
+ | {{MAA Notice}} |
Latest revision as of 21:56, 28 June 2024
2008 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
Let , where the additions and subtractions alternate in pairs. Find the remainder when
is divided by
.
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 -mile mark at exactly the same time. How many minutes has it taken them?
Problem 3
A block of cheese in the shape of a rectangular solid measures cm by
cm by
cm. Ten slices are cut from the cheese. Each slice has a width of
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?
Problem 4
There exist unique nonnegative integers
and
integers
(
) with each
either
or
such that
Find
.
Problem 5
In trapezoid with
, let
and
. Let
,
, and
and
be the midpoints of
and
, respectively. Find the length
.
Problem 6
The sequence is defined by
The sequence
is defined by
Find
.
Problem 7
Let ,
, and
be the three roots of the equation
Find
.
Problem 8
Let . Find the smallest positive integer
such that
is an integer.
Problem 9
A particle is located on the coordinate plane at . Define a move for the particle as a counterclockwise rotation of
radians about the origin followed by a translation of
units in the positive
-direction. Given that the particle's position after
moves is
, find the greatest integer less than or equal to
.
Problem 10
The diagram below shows a rectangular array of points, each of which is
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]](http://latex.artofproblemsolving.com/1/6/e/16ed1460ee16eabb872eb9645928df7b6cf2f60a.png)
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 be the maximum possible number of points in a growing path, and let
be the number of growing paths consisting of exactly
points. Find
.
Problem 11
In triangle ,
, and
. Circle
has radius
and is tangent to
and
. Circle
is externally tangent to circle
and is tangent to
and
. No point of circle
lies outside of
. The radius of circle
can be expressed in the form
,where
,
, and
are positive integers and
is the product of distinct primes. Find
.
Problem 12
There are two distinguishable flagpoles, and there are flags, of which
are identical blue flags, and
are identical green flags. Let
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
is divided by
.
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 be the region outside the hexagon, and let
. Then the area of
has the form
, where
and
are positive integers. Find
.
Problem 14
Let and
be positive real numbers with
. Let
be the maximum possible value of
for which the system of equations
has a solution
satisfying
and
. Then
can be expressed as a fraction
, where
and
are relatively prime positive integers. Find
.
Problem 15
Find the largest integer satisfying the following conditions:
- (i)
can be expressed as the difference of two consecutive cubes;
- (ii)
is a perfect square.
See also
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 | ||
All AIME Problems and Solutions |
- American Invitational Mathematics Examination
- AIME Problems and Solutions
- Mathematics competition resources
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.