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6
votes
1answer
131 views

IMO 2015 warm up problem

I get this problem from IMO 2015 facebook page. Let $x_i$ be positive integers for $i=1,2,...,11$. If $x_i+x_{i+1}\geq 100$, $|x_i-x_{i+1}|\geq 20$ for $i=1,2,...,10$. And $x_{11}+x_{1}\geq 100$, ...
1
vote
0answers
72 views

Integers Placed On A Circle [on hold]

My problem is such: On a circle there are $9$ distinct positive integers aranced in such a way that the product of two non-adjacent numbers in the circle is a multiple of $n$ and the product of any ...
3
votes
1answer
20 views

Boundedness of $\gcd(|x-y|,|a_x-a_y|)$ in sequence

Let $a_1,a_2,\ldots$ be an infinite sequence of distinct positive integers, and let $n$ be a positive integer. Does there always exist integers $x,y$ such that $\gcd(|x-y|,|a_x-a_y|)>n$? When ...
24
votes
3answers
487 views

Arithmetic Progressions in slowly oscillating sequences

An infinite sequence ($a_0$, $a_1$, ...) is such that the absolute value of the difference between any 2 consecutive terms is equal to $1$. Is there a length-8 subsequence such that the terms are ...
1
vote
3answers
147 views

this is a conjecture or a result? every arithmetic progression contains a sequence of $k$ “consecutive” primes for possibly all natural numbers $k$?

Writing a little better the previous question: is it true that if we let $a$ and $b$ be coprime integers, then the arithmetic progression : $a + bh: h\in {\mathbb Z}$, contains a sequence of $k$ ...
6
votes
2answers
197 views

Number of sudokus with no consecutive arithmetic progression of length 3 in any row or column.

How many such Sudokus are there? Any reference to papers, books, articles or any insight into the problem will be greatly appreciated. I've tried several search engines, scholarly and not, with no ...
1
vote
1answer
56 views

Circular variation with repetition

I would like to know formula for circular variation with repetition. What I mean is : You have round table with n-spots. On every spot there can be number from 1 to k. So for n = 4 and k = 3 ...
2
votes
0answers
54 views

Efficient way to count number of arithmetic progression on coloring of $\mathbb{N}$.

Consider a coloring of $\mathbb{N}$ with two colors. How many monochromatic arithmetic progressions of a fixed length $m$ (i.e. numbers of the form $a+nd$ are colored the same) are there in the subset ...
1
vote
1answer
44 views

Parity of Partition Function

Let $T(n)$ denote the number of partitions of $n$ into parts not congruent to $3$ mod $6$. Deduce that $T(n)$ is also the number of partitions of $n $ in which odd parts appear at most twice (even ...
10
votes
2answers
134 views

What are Green's almost primes?

In a general-audience talk, Ben Green explains his famous proof with Terence Tao as an application of Szemer├ędi's theorem, but placing the primes within a smaller set of almost-primes in which they ...
13
votes
1answer
426 views

Narcissistic numbers in other bases

It is well known that $153$ is a narcissistic number; that is, it is equal to the sum of the cubes of its digits since $153=1^3+5^3+3^3$. Other bases have similar numbers. For example, in base $3$, ...
2
votes
1answer
81 views

Numbers not of the form $x^2+My^2$

Why are there only 436 numbers not of the form $x^2+My^2$ for $x>0$, $y>1$ and $M>0$? This is A074885 from OEIS. The last number is 1875902. Can the following argument be fixed up? I ...
0
votes
1answer
55 views

Combinations problem with 2 contradicted answers!

If I have the digits from $0$ up till $9$: $0,1,2,3,\cdots,9$. How many 3-digit number can be made from these set of digits if the number is greater than $600$? My solution was as follows:There are ...
2
votes
3answers
422 views

Multiset Combination in Combinatorics

I want to buy a $k$-combination of doughnuts, where $k$ is any amount less than or equal to the total doughnuts available. At the bakery there are $n$ different types of doughnuts but there are ...
8
votes
1answer
257 views

Expectation of a Random Subset of the Roots of Unity.

Let $p$ be a prime. If $1_A(x)$ denotes the indicator function of the set $A\subset\mathbb{Z}/p\mathbb{Z}$ and $$\hat{1}_A(t):=\frac{1}{p}\sum_{n=1}^p 1_A(n)e^{2\pi i \frac{nt}{p}}$$ denotes the ...