Number Theory, Division, Parity (Even/Odd)
Parity refers to whether an integer is even (divisible by 2) or odd (not divisible by 2). Many problems, especially in number theory and combinatorics, can be solved or simplified by considering the parity of the numbers involved. Questions often require analyzing how operations affect parity.
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Pumbaa and the Candies
Pumbaa has 11 chocolate candies and 13 toffee candies. Each time he can eat either two candies of different types,
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or three candies of the same type. What is the largest number of candies that Pumbaa can eat according to these rules? -
Circle of Liars - The Truth Claim
In a circle, `n` people are seated, each of whom is either a liar or a truth-teller.
The people are looking towards the center of the circle. A liar always lies, and a truth-teller always tells the truth.
Each of the people knows exactly who is a liar and who is a truth-teller.
Each of the people says that the person sitting two places to their left (that is, next to the person sitting next to them), is a truth-teller.
It is known that in the circle there is at least one liar, and at least one truth-teller.
a. Is it possible that `n = 2017`?
b. Is it possible that `n = 5778`?
(Solution format: "word, word" for example "cat, puppy")
Sources:Topics:Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Proof and Example -> Proof by Contradiction Logic -> Truth-tellers and Liars Problems- Gillis Mathematical Olympiad, 2017-2018 Question 1
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6 on the Board
The number 6 is written on the board. At each step, you can add the digit 6 to the end of the number (so that it is the units digit,) or replace the number with the sum of its digits.
Which numbers can be obtained in this way? Describe the entire set of numbers and explain why there are no moreSources:Topics:Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Combinatorics -> Induction (Mathematical Induction) Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Algebra -> Sequences Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures -
Consecutive Numbers
a. Avi wants to find 10 consecutive numbers whose sum is divisible by 90. Will he succeed?
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b. Benny wants to find 11 consecutive numbers such that their sum is divisible by 90. Will he succeed? -
Four Numbers
Given four distinct positive integers. The sum of these numbers is equal to 18.
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Additionally, it is known that the product of these four numbers is odd. Calculate this product. -
Five Numbers
Given five distinct positive integers. The sum of these numbers is 27. Additionally, it is known that the product of these five numbers is odd. Calculate this product.
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Numbers on a Board
At the beginning of the day, four integers are written on the board (`a_0,b_0,c_0,d_0`). Every minute, Danny replaces the four numbers on the board with a new set of four numbers according to the following rule: If the numbers written on the board are (a,b,c,d), Danny first generates the numbers
`a'=a+4b+16c+64d`
`b'=b+4c+16d+64a`
`c'=c+4d+16a+64b`
`d'=d+4a+16b+64c`
Then he erases the numbers (a,b,c,d) and writes in their place the numbers (a',d',c',b'). For which initial sets (`a_0,b_0,c_0,d_0`) will Danny eventually write a set of four numbers that are all divisible by `5780^5780`Sources:Topics:Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Number Theory -> Division -> Parity (Even/Odd) Algebra -> Sequences- Gillis Mathematical Olympiad, 2019-2020 Question 4
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Finite Division
Find all integers x, y, z, w that satisfy `x^2+y^2=3z^2+3w^2`.
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Integer Coefficients?
Given real numbers a, b, c such that for every integer x, the number `ax^2+bx+c` is an integer. Does this necessarily imply that a, b, c are all integers? Prove it, or provide a counterexample.
Sources:Topics:Algebra Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Constructing an Example / Counterexample- Grossman Math Olympiad, 2017, Juniors Question 2
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Equality in Stages
The numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 are written on the board, and David is supposed to change them in stages. At each stage, David is allowed to choose two numbers and change them by 1, that is, to add 1 to both, subtract 1 from both, or add 1 to one and subtract 1 from the other.
Can David, after a number of stages, reach a situation where all the numbers on the board are equal? If so, show an example, and if not, explain your answer in detail.
Sources:Topics:Combinatorics -> Invariants Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures- Grossman Math Olympiad, 2017, Juniors Question 3