Difference between revisions of "2008 AIME II Problems"
(→Problem 9) |
JoetheFixer (talk | contribs) (→Problem 4) |
||
Line 17: | Line 17: | ||
== 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> | + | 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> | <cmath> | ||
a_13^{n_1} + a_23^{n_2} + \cdots + a_r3^{n_r} = 2008. | a_13^{n_1} + a_23^{n_2} + \cdots + a_r3^{n_r} = 2008. |
Revision as of 14:52, 31 March 2013
2008 AIME II (Answer Key) | AoPS Contest Collections | ||
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.
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.