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Number theory

AHSME 1992

Subject: 
Algebra
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Tags: 
Mathematics
Abstract algebra
Integer sequences
Number theory
Triangles
Triangle geometry
Combinatorics
Number
Prime number
Triangle
Coprime
Integer
sports

USA AIME 1992 1 Find the sum of all positive rational numbers that are less than 10 and that have denominator 30 when written in lowest terms. 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? 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 .500. During the weekend, she plays four matches, winning three and losing one. At the end of the weekend, her win ratio is greater than .503. What?s the largest number of matches she could?ve won before the weekend began?

AHSME 1990

Subject: 
Algebra
Rating: 
0
No votes yet
Tags: 
Mathematics
Mathematical analysis
Number theory
Complex analysis
pi
Factorial

USA AIME 1990 1 The increasing sequence 2, 3, 5, 6, 7, 10, 11, . . . consists of all positive integers that are neither the square nor the cube of a positive integer. Find the 500th term of this sequence. 2 Find the value of (52 + 6 ? 43)3/2 ? (52? 6 ? 43)3/2. 3 Let P1 be a regular r-gon and P2 be a regular s-gon (r ? s ? 3) such that each interior angle of P1 is 5958 as large as each interior angle of P2. What?s the largest possible value of s? 4 Find the positive solution to 1 x2 ? 10x? 29 + 1 x2 ? 10x? 45 ? 2 x2 ? 10x? 69 = 0 5 Let n be the smallest positive integer that is a multiple of 75 and has exactly 75 positive integral divisors, including 1 and itself. Find n/75. 6 A biologist wants to calculate the number of fish in a lake. On May 1 she catches a random
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