Difference between revisions of "1992 AIME Problems"
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== Problem 8 == | == Problem 8 == | ||
− | For any sequence of real numbers <math>A=(a_1,a_2,a_3,\ldots)</math>, define <math>\Delta A^{}_{}</math> to be the sequence <math>(a_2-a_1,a_3-a_2,a_4-a_3,\ldots)</math>, whose <math>n^\mbox{th | + | For any sequence of real numbers <math>A=(a_1,a_2,a_3,\ldots)</math>, define <math>\Delta A^{}_{}</math> to be the sequence <math>(a_2-a_1,a_3-a_2,a_4-a_3,\ldots)</math>, whose <math>n^\mbox{th}</math> term is <math>a_{n+1}-a_n^{}</math>. Suppose that all of the terms of the sequence <math>\Delta(\Delta A^{}_{})</math> are <math>1^{}_{}</math>, and that <math>a_{19}=a_{92}^{}=0</math>. Find <math>a_1^{}</math>. |
[[1992 AIME Problems/Problem 8|Solution]] | [[1992 AIME Problems/Problem 8|Solution]] |
Revision as of 22:15, 2 March 2015
1992 AIME (Answer Key) | AoPS Contest Collections | ||
Instructions
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Contents
Problem 1
Find the sum of all positive rational numbers that are less than 10 and that have denominator 30 when written in lowest terms.
Problem 2
A positive integer is called ascending if, in its decimal representation, there are at least two digits and each digit is less than any digit to its right. How many ascending positive integers are there?
Problem 3
A tennis player computes her win ratio by dividing the number of matches she has won by the total number of matches she has played. At the start of a weekend, her win ratio is exactly . During the weekend, she plays four matches, winning three and losing one. At the end of the weekend, her win ratio is greater than . What's the largest number of matches she could've won before the weekend began?
Problem 4
In Pascal's Triangle, each entry is the sum of the two entries above it. In which row of Pascal's Triangle do three consecutive entries occur that are in the ratio ?
Problem 5
Let be the set of all rational numbers , , that have a repeating decimal expansion in the form , where the digits , , and are not necessarily distinct. To write the elements of as fractions in lowest terms, how many different numerators are required?
Problem 6
For how many pairs of consecutive integers in is no carrying required when the two integers are added?
Problem 7
Faces and of tetrahedron meet at an angle of . The area of face is , the area of face is , and . Find the volume of the tetrahedron.
Problem 8
For any sequence of real numbers , define to be the sequence , whose $n^\mbox{th}$ (Error compiling LaTeX. ! Missing { inserted.) term is . Suppose that all of the terms of the sequence are , and that . Find .
Problem 9
Trapezoid has sides , , , and , with parallel to . A circle with center on is drawn tangent to and . Given that , where and are relatively prime positive integers, find .
Problem 10
Consider the region in the complex plane that consists of all points such that both and have real and imaginary parts between and , inclusive. What is the integer that is nearest the area of ?
Problem 11
Lines and both pass through the origin and make first-quadrant angles of and radians, respectively, with the positive x-axis. For any line , the transformation produces another line as follows: is reflected in , and the resulting line is reflected in . Let and . Given that is the line , find the smallest positive integer for which .
Problem 12
In a game of Chomp, two players alternately take bites from a 5-by-7 grid of unit squares. To take a bite, a player chooses one of the remaining squares, then removes ("eats") all squares in the quadrant defined by the left edge (extended upward) and the lower edge (extended rightward) of the chosen square. For example, the bite determined by the shaded square in the diagram would remove the shaded square and the four squares marked by (The squares with two or more dotted edges have been removed form the original board in previous moves.)
The object of the game is to make one's opponent take the last bite. The diagram shows one of the many subsets of the set of 35 unit squares that can occur during the game of Chomp. How many different subsets are there in all? Include the full board and empty board in your count.
Problem 13
Triangle has and . What's the largest area that this triangle can have?
Problem 14
In triangle , , , and are on the sides , , and , respectively. Given that , , and are concurrent at the point , and that , find .
Problem 15
Define a positive integer to be a factorial tail if there is some positive integer such that the decimal representation of ends with exactly zeroes. How many positive integers less than are not factorial tails?
See also
- 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.