Full text
... Then X is a complete partition of w. Proof: Suppose not. Then there must be some numbers between 1 and n that cannot be expressed as a sum of elements of Xx,..., Xk. Let m be the least such number. Then we have Xl+-- + Xi1. We claim that m
... Then X is a complete partition of w. Proof: Suppose not. Then there must be some numbers between 1 and n that cannot be expressed as a sum of elements of Xx,..., Xk. Let m be the least such number. Then we have Xl+-- + Xi
CHAP02 Linear Congruences
... The following Theorem is known as Fermat's “Little” Theorem. This is to distinguish it from his celebrated “Last Theorem”. Fermat's Last Theorem states that for all integers n ≥ 3 there are no solutions to the equation xn + yn = zn for non-zero integers x, y and z. We all know that 32 + 42 = 52 and ...
... The following Theorem is known as Fermat's “Little” Theorem. This is to distinguish it from his celebrated “Last Theorem”. Fermat's Last Theorem states that for all integers n ≥ 3 there are no solutions to the equation xn + yn = zn for non-zero integers x, y and z. We all know that 32 + 42 = 52 and ...
Problems short list - International Mathematical Olympiad
... C6. We are given an infinite deck of cards, each with a real number on it. For every real number x, there is exactly one card in the deck that has x written on it. Now two players draw disjoint sets A and B of 100 cards each from this deck. We would like to define a rule that declares one of them a ...
... C6. We are given an infinite deck of cards, each with a real number on it. For every real number x, there is exactly one card in the deck that has x written on it. Now two players draw disjoint sets A and B of 100 cards each from this deck. We would like to define a rule that declares one of them a ...
Document
... numerator into the denominator Step 2. Divide the remainder in Step 1 into the divisor of Step 1 Step 3. Divide the remainder of Step 2 into the divisor of Step 2. Continue until the remainder is 0 ...
... numerator into the denominator Step 2. Divide the remainder in Step 1 into the divisor of Step 1 Step 3. Divide the remainder of Step 2 into the divisor of Step 2. Continue until the remainder is 0 ...
Collatz conjecture
The Collatz conjecture is a conjecture in mathematics named after Lothar Collatz, who first proposed it in 1937. The conjecture is also known as the 3n + 1 conjecture, the Ulam conjecture (after Stanisław Ulam), Kakutani's problem (after Shizuo Kakutani), the Thwaites conjecture (after Sir Bryan Thwaites), Hasse's algorithm (after Helmut Hasse), or the Syracuse problem; the sequence of numbers involved is referred to as the hailstone sequence or hailstone numbers (because the values are usually subject to multiple descents and ascents like hailstones in a cloud), or as wondrous numbers.Take any natural number n. If n is even, divide it by 2 to get n / 2. If n is odd, multiply it by 3 and add 1 to obtain 3n + 1. Repeat the process (which has been called ""Half Or Triple Plus One"", or HOTPO) indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach 1. The property has also been called oneness.Paul Erdős said about the Collatz conjecture: ""Mathematics may not be ready for such problems."" He also offered $500 for its solution.