Difference between revisions of "1999 AIME Problems"
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== Problem 5 == | == Problem 5 == | ||
| − | For any positive integer <math>x_{}</math>, let <math>S(x)</math> be the sum of the digits of <math>x_{}</math>, and let <math>T(x)</math> be <math>|S(x+2)-S(x)|.</math> For example, <math>T(199)=|S(201)-S(199)|=|3-19|=16.</math> How many values <math>T(x)</math> do not exceed 1999? | + | For any positive integer <math>x_{}</math>, let <math>S(x)</math> be the sum of the digits of <math>x_{}</math>, and let <math>T(x)</math> be <math>|S(x+2)-S(x)|.</math> For example, <math>T(199)=|S(201)-S(199)|=|3-19|=16.</math> How many values of <math>T(x)</math> do not exceed 1999? |
[[1999 AIME Problems/Problem 5|Solution]] | [[1999 AIME Problems/Problem 5|Solution]] | ||
Revision as of 14:33, 17 June 2011
| 1999 AIME (Answer Key) | AoPS Contest Collections • PDF | ||
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Instructions
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| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
to 
, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.Contents
Problem 1
Find the smallest prime that is the fifth term of an increasing arithmetic sequence, all four preceding terms also being prime.
Problem 2
Consider the parallelogram with vertices 
and 
A line through the origin cuts this figure into two congruent polygons. The slope of the line is 
where 
and 
are relatively prime positive integers. Find 
Problem 3
Find the sum of all positive integers 
for which 
is a perfect square.
Problem 4
The two squares shown share the same center 
and have sides of length 1. The length of 
is 
and the area of octagon 
is where and are relatively prime positive integers. Find
Problem 5
For any positive integer 
, let 
be the sum of the digits of , and let 
be 
For example, 
How many values of do not exceed 1999?
Problem 6
A transformation of the first quadrant of the coordinate plane maps each point 
to the point 
The vertices of quadrilateral 
are 
and 
Let 
be the area of the region enclosed by the image of quadrilateral 
Find the greatest integer that does not exceed 
Problem 7
There is a set of 1000 switches, each of which has four positions, called 
, and 
. When the position of any switch changes, it is only from 
to 
, from to 
, from to , or from to . Initially each switch is in position . The switches are labeled with the 1000 different integers 
, where 
, and 
take on the values 
. At step i of a 1000-step process, the 
-th switch is advanced one step, and so are all the other switches whose labels divide the label on the -th switch. After step 1000 has been completed, how many switches will be in position ?
Problem 8
Let 
be the set of ordered triples 
of nonnegative real numbers that lie in the plane 
Let us say that supports 
when exactly two of the following are true: 
Let 
consist of those triples in that support 
The area of divided by the area of is where and are relatively prime positive integers, find
Problem 9
A function 
is defined on the complex numbers by 
where 
and 
are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. Given that 
and that 
where and are relatively prime positive integers. Find
Problem 10
Ten points in the plane are given, with no three collinear. Four distinct segments joining pairs of these points are chosen at random, all such segments being equally likely. The probability that some three of the segments form a triangle whose vertices are among the ten given points is where and are relatively prime positive integers. Find
Problem 11
Given that 
where angles are measured in degrees, and and are relatively prime positive integers that satisfy 
find
Problem 12
The inscribed circle of triangle 
is tangent to at 
and its radius is 21. Given that 
and 
find the perimeter of the triangle.
Problem 13
Forty teams play a tournament in which every team plays every other team exactly once. No ties occur, and each team has a 
chance of winning any game it plays. The probability that no two teams win the same number of games is where and are relatively prime positive integers. Find 
Problem 14
Point 
is located inside triangle so that angles 
and 
are all congruent. The sides of the triangle have lengths 
and 
and the tangent of angle 
is where and are relatively prime positive integers. Find
Problem 15
Consider the paper triangle whose vertices are 
and 
The vertices of its midpoint triangle are the midpoints of its sides. A triangular pyramid is formed by folding the triangle along the sides of its midpoint triangle. What is the volume of this pyramid?
