Tagged Questions
9
votes
0answers
131 views
A contest question
$p$ is an odd prime,denote $$f(x)=\sum_{k=0}^{p-1}\binom{2k}{k}^2x^k.$$
Prove that for every $x\in Z$,$$(-1)^\frac{p-1}2f(x)\equiv f(\frac{1}{16}-x)\pmod{p^2}.$$
This is a contest question,I do not ...
1
vote
1answer
75 views
Number array divided into several parts, genelize $a>b>c>d>0$ so $ab+cd>ac+bd$ to more numbers
Now, we have an original number array: $$a_1 > a_2 > a_3 > ... > a_{mn} > 0$$, I wonder whether the following inequality is the truth, if so, could you give me the proof or some ...
0
votes
1answer
68 views
How to prove $p(n\mathrel{;} \{1, 2, 4\}) = p(n - 4\mathrel{;}\{1, 2, 4\}) + p(n\mathrel{;} \{1, 2\})$?
Let $n_1,...,n_k$ be distinct natural numbers and let $p(n\mathrel{;} \{n_1,...,n_k\})$ denote the number of partitions of $n$ into parts, each of which is equal to one of $n_1,...,n_k$. Show that ...
2
votes
1answer
22 views
Partitioning of subsets
This is a previous exam question.
Let $S$ be a subset of $\{10,11,...,99\}$ containing 10 elements. Show that there will exist two disjoint subsets $A$ and $B$ of $S$ such that sum of the elements of ...
3
votes
1answer
36 views
A combinatorial number theory question (pigeonhole principle)
Let $n$ be a positive integer such that $n$ and $10$ are coprime. Prove that $n|11\cdots11$ for some $11\cdots11$ in base 10 representation.
This problem is about pigeonhole principle, I have a great ...
1
vote
0answers
35 views
Upper bound on degree of coefficients required to write polynomials as a linear combination of $f_1,…,f_n$
All polynomials will be elements of $\mathbb{Q}[x]$. Suppose $f_1,...,f_n$ are polynomials of degree at most $d$ which are coprime. What is a (hopefully sharp) upper bound on the degree of ...
0
votes
1answer
47 views
How i could find this three-digits
How I find this :
Find the least three-digits number that is equal to :
the sum of its digits plus twice the product of its digit?
In how many ways i can write 12 as an ordered sum of integers ...
0
votes
3answers
31 views
Representing positions on a grid as a base-3 number
I have an assignment that presents some math I don't understand.
Consider the following grid.
...
4
votes
4answers
70 views
How to calculate the number of pieces in the border of a puzzle?
Is there any way to calculate how many border-pieces a puzzle has, without knowing it's width-height ratio? I guess it's not even possible, but I am trying to be sure about it.
Thanks for your help!
...
2
votes
0answers
33 views
Characterization of $\lambda,\mu\vdash n$ for which $\displaystyle{n\choose\mu}\mid{n\choose\lambda}$
In view of this question, I was wondering about general characterizations of $\mu,\lambda\vdash n$ for which
$${n\choose\mu}\,\left|\,{n\choose\lambda}\right.,$$
(see multinomial coefficient and ...
1
vote
2answers
216 views
Number of non-negative integer solutions of an equation using generating functions
I have question. Can anyone able to solve this problem or figure out the idea
how non negative integer solution $x_1+5x_2=n$ and $x_1+5x_2=60$
Using the convolution
thank you so much
-1
votes
1answer
74 views
$10\times10$ table of numbers with differences of adjacent entries no more than 5
In each square in a table $10\times 10$ we write a whole number so the difference between any two numbers by the neighboring squares must be no more than $5$ (two squares are neighboring if they have ...
3
votes
1answer
57 views
Number of roots for quadratic residues
Let $n \in Z$ and define $QR_n=\{x \in Z_n|\exists y \in Z_n :y^2=x (mod\ n)\}$.
How can I show that $\forall x \in QR_n$ it hold that $|\{ y \in Z_n:y^2=x (mod\ n)\}|=\frac{n}{|QR_n|}$ ?
Why am I ...
3
votes
4answers
97 views
How many zeros are there at $1000!$ in the base $24$
I know $1000!$ has $\frac{1000}{5}+\frac{1000}{25}+\frac{1000}{125}+\frac{1000}{625}=249$ terminal zeros in decimal notation, but what if we write $1000!$ in base $24$, how many terminal zeros would ...
5
votes
1answer
66 views
Given $A$ and $B$, how many positive integers $N$ such that $N\times B$ has at least one divisior $D$ that lies in $N \lt D \le A$?
For two integers $A$ and $B$, how can we find the number of positive integers $N$ such that $N\times B$ has at least one divisior $D$ that lies in $N \lt D \le A$?
For example, if $A = 100$ and $B = ...
36
votes
7answers
783 views
Problems regarding $\{x_n \}$ defined by $x_1=1$; $x_n$ is the smallest distinct natural number such that $x_1+…+x_n$ is divisible by $n$.
Let me denote a sequence of distinct natural numbers by $x_n$ whose terms are determined as follows:
$x_1$ is $1$ and $x_2$ is the smallest distinct natural number $n$ such that $x_1+x_2$ is divisible ...
1
vote
1answer
123 views
Number of integer solutions to a system of equations
How many nonnegative integer solutions are there to the pair of equations $x_1 + x_2 + \ldots + x_6 = 20$ and $x_1 + x_2 + x_3 = 7$?
Is my reasoning to the above problem correct?
Take the ...
2
votes
2answers
118 views
Number of subsets of $U$ whose arithmetic mean is integral
Question is :-
$n$ is a positive integer. Call a non-empty subset $S$ of $\{1,2,\dots,n\}$ "good" if the arithmetic mean of
elements of $S$ is also an integer. Further Let $t_n$ denote the ...
0
votes
0answers
69 views
Finding all number combination which XOR results to 0
Let's say I have a fixed list of numbers:
$2, 3, 1, 2$
and I can reduce every number from $n$ to $0$, for instance: $1,3,1,2$ or $0,3,0,1$ etc.
I am looking for all combinations of this sort, where ...
6
votes
2answers
193 views
Combinatorial proof: $p^{r-n}$ divides $\binom{p^{r-2}}{n}$
Let $p$ be an odd prime. Then if $1<n<r$, $$p^{r-n}\,\left|\,\binom{p^{r-2}}{n}\right.$$
Does anyone have a clever combinatorial proof of this fact? There's an easy argument just by counting ...
1
vote
3answers
102 views
bound for the product of numbers
Let $n \in N$. Fix $m \in [-n,n]$. I am curious, how to bound from above the following expression
$$
(n-m)^{\frac{n-m}{2}+1}(n+m)^{\frac{n+m+1}{2}}\leq \quad ?
$$
Thank you.
5
votes
2answers
137 views
How many $n$-digit palindromes are there?
How can one count the number of all $n$-digit palindromes? Is there any recurrence for that?
Thanks.
I'm not sure if my reasoning is right, but I thought that for n=1 we have 10 such numbers ...
6
votes
3answers
218 views
number of multiples of 4 that are multiples of 4 even if you permute their digits
How many 4 digit numbers are multiples of 4 no matter how you permute them?
(base 10)
10
votes
2answers
121 views
Permuting elements of a set around a circle
Given $15$ objects placed around a circle, is it possible to permute their order such that the distance between any two elements is different in the second permutation from that in the original state? ...
0
votes
1answer
74 views
Total number of ways to arrange the prime divisor of a number so it can be written using M digits
How many ways we can arrange all the prime divisor of a number so it can be written using M factors, where M <=T. T is the total number of prime divisor of the give number N.
Example:N=27, its ...
2
votes
2answers
75 views
Counting pairs $(n,n+1)$ where $n$ and $n+1$ are both quadratic residues, etc.?
This is an interesting problem I read that has me stumped.
Let $(RR)$ denote the number of pairs $(n,n+1)$ in the set $\{1,2,\dots,p-1\}$ such that $n$ and $n+1$ are both residues modulo $p$. Let ...
1
vote
2answers
95 views
Counting the number of distinct greatest common divisors for an integer.
What is the expression for the number of distinct greatest common divisors possible for the number $N$? Let us say that $N$ is composed of 4 prime numbers $N = p_3 p_2 p_1 p_0$. Now if $p_i$ are all ...
0
votes
1answer
86 views
Sum of consecutive sub-sequence of 100 natural numbers divides 100 (equals 0 mod 100) [duplicate]
Possible Duplicate:
pigeonhole principle and division
I need a little help in an exercise.
Given 100 natural numbers $a_{1},..,a_{100}$ , prove that there is a consecutive sub-sequence ...
7
votes
2answers
254 views
Fibonacci numbers divisible by $9$
The $n$th Fibonacci number $F_n$ is defined as follows,$$F_1=F_2=1\mbox{ and } F_{n+2}=F_{n+1}+F_{n}\mbox{ for } n\geq 1$$
I want to know how many of the first $1000$ Fibonacci numbers are divisible ...
4
votes
1answer
70 views
inequality with numbers--when its true?
Help me please to understand when the inequality true.
Let $n<N,$ where $n, N$ are natural numbers.
For which $n$ and $N$ the following is true
$$
n^{2n+1}\leq N^{N+1}?
$$
Thank you.
4
votes
2answers
315 views
Change-making problem - counterexample for greedy algorithm
Let D be set of denominations and m the largest element of D. We say c is counterexample if greedy algorithm is giving answer different from optimal one.
I found statement that if for given set ...
10
votes
1answer
241 views
Given a set of digits, what is the biggest number we can make using exponentiation - numberphile noodle quiz
The question is motivated by a question on a can of number noodles.
Each item is a digit between $0$ and $9$. Clearly, if you form a string and consider it to represent a base $10$ integer, then ...
5
votes
0answers
101 views
divisibility by powers of $2$ of diagonal sums of infinite latin square
This array is formed by placing integers such that each is the smallest such that the rectangle with it and the top left hand corner as opposite corners does not contain the same integer twice in any ...
2
votes
1answer
99 views
Number combination which brings different result in sum
I am looking for 6 numbers which when added their sum should not be a repeated no like one below for 3 Numbers
Num 1 Num 2 Num 3 Sum
1 3 5 9
1 ...
4
votes
4answers
299 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 ...
0
votes
2answers
93 views
The number of possible factorizations of a positive integer.
Given a positive integer $n>1$ with prime factorization
$$n=\prod_{p_i \text{ prime}}p_i^{k_i}, \space i\ge1, \space k_i \in \mathbb N^*$$
how can I compute the number of factorizations of ...
1
vote
0answers
41 views
Number of ways to write a dyadic rational number into a sum of fixed $n$ terms of negative powers of $2$
As the title describes, I will post here my question clearer:
Let $z=\frac{m}{2^k}$ be a dyadic rational number in $(0,1)$ where $m$ is odd and $k >0$, and also $n$ is a fixed positive integer.
...
1
vote
1answer
43 views
Intervals of Circle Method
I'm trying to understand how to use the circle method to derive an asymptotic formula for Waring's Problem. Do so using the circle method developed by Hardy and Littlewood. In doing this, I want to ...
6
votes
2answers
330 views
Number of permutations which fixes a certain number of point
Given the set $N:=\{1,\cdots,n\}$, let $\pi$ be a permutation on $N$. We say $i \in \{1,\cdots,n\}$ is fixed by $g$ iff $\pi(i)=i.$
Denote the set of all permuations on $N$ by $S_n$.
Define $f :~N ...
1
vote
1answer
151 views
Can you determine a formula for this problem?
Given: A list of integers is there.Now there are 2 buckets -bucket A and bucket B.This step is repeated as long as there are numbers left in the list.Integers from start or end of the list are ...
6
votes
1answer
530 views
In how many ways can we colour $n$ baskets with $r$ colours?
In how many ways can we colour $n$ baskets using up to $r$ colours such that no two consecutive baskets have the same colour and the first and the last baskets also have different colours?
For ...
3
votes
2answers
475 views
Trying to prove that $p$ prime divides $\binom{p-1}{k} + \binom{p-2}{k-1} + \cdots +\binom{p-k}{1} + 1$
So I'm trying to prove that for any natural number $1\leq k<p$, that $p$ prime divides:
$$\binom{p-1}{k} + \binom{p-2}{k-1} + \cdots +\binom{p-k}{1} + 1$$
Writing these choice functions in ...
3
votes
0answers
135 views
How many different ways are there to get the same product from a sequence of integers?
Let's say we have a list of integers from 0 to (n-1), where n > 0:
0, 1, 2, 3, 4, ..., ...
4
votes
1answer
51 views
A harder tournament to schedule
Let us suppose that I have $n$ students in my class, and I break them up into $k$ groups per week. Let's also suppose that I want to repeat this each week, except that I don't want any student to ...
-4
votes
1answer
188 views
What is the number of positive integers?
b. In a system of ternary (Base 3) number, with n digits, how many number can be represented?
Answer:
c. For an n-digit signed 3's complement ternary number (n > 1), what is the number of positive ...
3
votes
1answer
43 views
Can the coefficients of a given term in this family of power series have a common divisor?
Let $g_m(b)$ be the coefficient of $x^m$ the power series
$$\dfrac{1}{1-x-bx^2}$$
(When $b=1$, this is just the generating function of Fibonacci numbers.) Of course, $g_m(b)$ depends on both $m$ and ...
4
votes
3answers
132 views
How to generate equiprobable numbers with a random generator?
Is it possible to emulate a 2 sided coin flip (50/50) with a random number generator which outputs equiprobably the numbers 1, 2 and 3 ?
If yes, how ?
If not why ?
Is there a theorem?
Can it be ...
3
votes
1answer
159 views
Lower bounds for the partition function
In this question we consider the partition function $p(n)$ - that is, the number of ways to express $n$ as a sum of positive integers.
One easy exercise is to show that $$ p(n) \geq 2^{\lfloor ...
1
vote
1answer
109 views
If a number can be expressed as a product of n unique primes…
If a number can be expressed as a product of n unique primes, in how many ways can the number be expressed as a difference of two squares?
4
votes
1answer
71 views
Number of combinations of sets over a function.
Does anyone know if the following question has been solved in general or has any insight in the question.
Let us take for example the sets {0,1} and {1,2} and function multiplication (*) over the ...
