Tagged Questions
30
votes
1answer
550 views
Proving that $x$ is an integer, if the differences between any two of $x^{1919}$, $x^{1960}$, and $x^{2100}$ are integers
For a specific real number $x$, the difference between any two of $x^{1919}$, $x^{1960}$ , and $x^{2100}$ is always an integer. How would one prove that $x$ is an integer?
4
votes
2answers
53 views
Find $x,y$ such that $x=4y$ and $1$-$9$ occur in $x$ or $y$ exactly once.
$x$ is a $5$-digits number, while $y$ is $4$-digits number. $x=4y$, and they used up all numbers from 1 to 9. Find $x,y$.
Can someone give me some ideas please? Thank you.
2
votes
1answer
39 views
Imperfect digit-to-digit invariants in Base $10$
$3435 = 3^3 + 4^4 + 3^3 + 5^5$ is an example of a perfect digit-to-digit invariant.
Fact: The number of PDDIs is finite for any given base; in particular, for base $10$.
Question: Working over base ...
8
votes
2answers
48 views
Having $A_1=a+b+c$,$A_2=a^2+b^2+c^2$, $A_3=a^3+b^3+c^3$ - how to get $a,b,c$?
Perhaps I'm just a bit dense at the moment - I've re-read some of my notes from monthes ago concerning elementary symmetric polynomials, and I find that I've no idea how to approach the "inverse" ...
8
votes
1answer
97 views
Proving a number defined by a sequence is a square number
I found this problem in a math magazine:
Given the sequence $(x_n)_{n \in \mathbb{N}}$ defined by:
$$
x_0 = 0\\
x_1 = 1\\
x_{n+2}+x_{n+1}+2x_{n}=0
$$
Prove that $s_n = 2^{n+1}-7x_{n-1}^2, n ...
5
votes
1answer
163 views
How did Euler solve the 4-whole-numbers-adding-up-to-a-perfect-square problem?
So I was watching a video on Leonhard Euler about how he amazingly solved so many difficult problems and one of the many problems that he solved was this:
...
4
votes
4answers
301 views
Mathematical proof for long-term behavior of a sequence of integer vectors
There are some children sitting around a round table. Each child is given an even amount of $1$-cent coins ($0$ is even) by their teacher, all the children at once. A child will give half his money to ...
1
vote
1answer
235 views
Multiplication Table with a frame and picture of equal sum
Is there an $n \times n$ multiplication table such that if you form a border of width $k$ ("the frame") and sum its elements, the total will equal the sum of the remaining elements ("the picture")?
...
0
votes
1answer
152 views
Can this crate have even numbers in all rows and columns?
A milk crate holds 24 bottles in four rows and six columns. Can you put 18 bottles of milk in the crate so that each row and each column of the crate have an even number of bottles in it?
0
votes
3answers
180 views
Which is the biggest integer that divides all integers that are the product of three consecutive odd numbers?
I read this problem from a high-school-math-problems-calendar, and I'm solving them in my spare time just for the fun of it (what in math is not about the fun? =) ), but this little one it's been hard ...
0
votes
3answers
242 views
Smallest multiple whose digits are only ones and zeros
I have a collection of typewritten pages that formed the basis of a third year problem solving course offered about 25 years ago at U. Waterloo. I've been slowly working through the problems and have ...
15
votes
1answer
563 views
What was Ramanujan's solution?
The wikipedia entry on Ramanujan contains the following passage:
One of his remarkable capabilities was the rapid solution for
problems. He was sharing a room with P. C. Mahalanobis who had a
...
4
votes
2answers
163 views
A game of numbers: When can we have 2011?
Two friends are playing a game. In every turn, after one of them says a number $k$, the other one has to say a number in form $a\cdot b$ where $a,b\in \mathbb{N}$ such that $a+b=k$ holds. The game ...
13
votes
2answers
854 views
Asking 2011 Putnam B6
I wish to ask today's Putnam problem B6:
Suppose $p$ is an odd prime. Prove that for $n\in \{0,1,2...p-1\}$, at least $\frac{p+1}{2}$ number of $\sum^{p-1}_{k=0} k! n^{k}$ is not divisble by $p$.
...
