Algebra, Sequences
A sequence is an ordered list of numbers (or other items) that often follows a specific rule or pattern. This topic covers identifying patterns, finding specific terms, determining general formulas (`n`-th term), and understanding different types of sequences (arithmetic, geometric, recursive).
Arithmetic Progression / Arithmetic Sequence Complete/Continue the Sequence Recurrence Relations-
Question
Given a real number `a` such that `a+1/a` is an integer. Prove that `a^n+1/a^n` is also an integer for every natural number `n`.
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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? -
Numbers on the Board
The following numbers are written on the board:`1/3,1/2,1,2,3`. In each step, you are allowed to choose any two numbers written on the board and replace each of them with their product.
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Can you reach a quintet of numbers in this way whose sum is `4 1/4`? -
Sums of Factorials
For every positive integer n, let us denote `n! = 1*2*3*...*n`
Find all integers n such that the sum `1! + 2! + 3! + ... + n!` is the square of an integer.
(Solution format: a,b,c,... i.e., the three smallest numbers separated by commas with no spaces, and three dots after if there are more solutions)
Sources:- Grossman Math Olympiad, 2022, Seniors Question 1
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Consecutive Numbers and Deletion
Yossi wrote 10 consecutive natural numbers on the board. Danny erased one of the numbers. The sum of the remaining numbers on the board is 2020.
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Which number was erased? -
Numbers Squared
Ayala took all the numbers divisible by 3 from 1 to 99, inclusive, squared each of them, summed the results, and multiplied the resulting number by 2.
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The duck took all the numbers not divisible by 3 from 1 to 100, inclusive, squared each of them, and summed the results.
What is the difference between the number the duck obtained and the number Ayala obtained? -
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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The Number Circle
The numbers 1 through 6 are written in a circle in order, as shown in the figure.
In each step, Lior chooses a number a in the circle whose neighbors are b and c, and replaces
it with the number `(bc)/a`.
Can Lior reach a state where the product of the numbers in the circle is greater than `10^100`
(a) in 100 steps.
(b) in 110 steps
Sources:- Gillis Mathematical Olympiad, 2019-2020 Question 6
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The Confrontation
Ehud and Benjamin are participating in a public debate. Each one, in turn, presents a question to their opponent. Ehud is chosen to be the first to present a question. A "trick question" is a question to which the opponent has no answer. A contestant who manages to ask a trick question immediately wins the debate. The probability of each of the two contestants finding a trick question (in their turn) is exactly 1/2. It is also known that there is no dependence between the questions. What is the probability of Ehud winning the debate?