0
votes
1answer
14 views

Prove that for a sequence of people sets $S_1,…,S_d$, $\Delta_i \not = 0$ for all people

We have $k$ people $p_1,...,p_k$, and $d$ people sets $S_1,...,S_d$, where the sizes of $S_1,...,S_d$ can vary between $1$ and $k$ (so each $S_1,...,S_d$ is a set of some people from ...
0
votes
1answer
30 views

how many people are at the party with such restriction applied [on hold]

At a party, each person shakes hands with 5 other people. There are a total of 60 handshakes. How many people are at the party? i try firs by dividing the total number of hand shake by the number of ...
-1
votes
0answers
21 views

Ways of representing the product of N numbers as sum of two squares

Given N numbers, we need to tell the number of ways of representing the product of these N numbers as sum of two squares. Example : Let $N=3$ and numbers be $[2,1,2]$ then as $2*1*2=4$ There are 4 ...
0
votes
1answer
52 views

how many people are at the party

At a party, each person shakes hands with 5 other people. There are a total of 60 handshakes. How many people are at the party? i am lost because of the 60 hand shake that is mentioned.
0
votes
0answers
19 views

Calculate sum of distinct pairs [on hold]

Given an array A we need to find the sum of all distinct pairs of indexes from the array and adds the value ⌊$A[i]+A[j]\over A[i]×A[j]$⌋ to the sum Note: ⌊$A\over B$⌋ is the integer division ...
4
votes
1answer
56 views

How many integers could be in such a way that any digits is not bigger than the left digits?

How many 4-digits integers could be in such a way that any digits is not bigger than it's left digits? I Try it with simulation, i get 714. anyone could describe a formula for me? My try:
5
votes
1answer
63 views

Application of Davenport theorem

The Davenport theorem (or Cauchy-Davenport theorem for some authors ) states that for any two nonempty subsets $A$ and $B$ of the prime field $\mathbb Z/p\mathbb Z$ we have $$|A+B| ≥ \min(p, ...
1
vote
1answer
141 views

Count ways to distribute candies

N students sit in a line, and each of them must be given at least one candy. Teacher wants to distribute the candies in such a way that the product of the number of candies any two adjacent students ...
-4
votes
1answer
184 views

Count good numbers in between L and R

Let length(A) denote the count of digits of a number A in its decimal representation. All non-negative numbers of length 1 are Good. Further, a number X with length(X) $≥ 1$ can also be considered ...
0
votes
0answers
34 views

How do you make the lighest change possible?

Suppose you have coin denominations $1 = c_1 < c_2 <... < c_k$ each with associated weight $w_1, ..., w_k$ and that you are trying to make change for $n$ cents. How can you make the ...
1
vote
1answer
28 views

Five digit numbers where each digit can appear up to three times

The question is to determine how many five-digit numbers there are (using the digits 0-9) where each digit can appear up to three times in the number. The total number of numbers that can be made ...
-2
votes
1answer
50 views

Find minimum possible area of brush

A rectangular brush has been moved right and down on the painting. Consider the painting as a $n × m$ rectangular grid. At the beginning an $x × y$ rectangular brush is placed somewhere in the frame, ...
3
votes
1answer
53 views

Number of answers of equation amongs odd natural numbers

How many answer The following Equation has, in set of odd natural numbers? $x_1+x_2+...+x_k=n$, $k \equiv^2 n$ Solution: Comb ( [(n+k)/2]-1, k-1), comb means combination. how we get this?
1
vote
1answer
34 views

Number of different vectors.

Let's say that I have a vector with 6 elements. I put two wedges in the vector, i.e., at position 2 and position 6, for instance. And when I say put a wedge, it means... for every time you traverse ...
0
votes
1answer
63 views

Number of ways to make n digit number?

Given M digits which are between 1 to 9, Find the number of ways to form N digit number, by repeating one or more given digits such that each of M digits are present in N digit number at least once. ...
2
votes
0answers
90 views
+50

Game theoretical approach to other branches of mathematics

Are there some methods and ideas derived from game theory that are successfully applied to better (or more intuitively) understand theorems and proofs or tackling problems from other areas of ...
-1
votes
0answers
55 views

Maximum pairs of men and women

There are shoes of n different colors. We will enumerate the colors from 1 to n. For each i, there are M[i] pairs of men's shoes, W[i] pairs of women's shoes and S[i] pairs of shoelaces of color i. ...
4
votes
2answers
31 views

Can we find an $x, y : x < y$ and $x, y > 0$ and $\lfloor \frac{n}{x}\rfloor$ < $\lfloor \frac{n}{y}\rfloor$ for some integer $n > 0$?

I know there are no solutions when we have just the fraction without the floor, but how do we consider solutions when the floor is there?
0
votes
0answers
27 views

Count ways to paint the grid

Given a rectangular grid of dimension N x M. We need to paint the grid with black or white color such that there is no rectangle of size X x Y having same color in each cell. Find the number of ways ...
0
votes
1answer
39 views

A counting problem on the integer lattice

Let $K$ be a subset of the integer lattice $\mathbb Z^2$such that it contains elements of the form $k=(k_1,k_2) $ where $k_1,k_2$ are integers and $k_2\neq 0$. Find $m$, an integer if possible, such ...
0
votes
0answers
41 views

counting the good numbers

We need to calculate Good Numbers in range from $A$ to $B$ (Both inclusive). A number $N$ is said to be a good Number if it satisfy following conditions : If we extract every $2$-digit number of $N$ ...
0
votes
1answer
24 views

show existence of subsequence $\{a_{i_b}\}_b^{n+1}$

Suppose $\{a_n\}_{n=1}^{m^2+1}$ is a strictly increasing sequence of $n^2+1$ positive integers, show that there exist a subsequence $\{a_{i_b}\}_b^{n+1}$ of length $n+1$ such that $a_{i_k}$ is ...
1
vote
0answers
26 views

Estimate for the co-volume of discs centered at lattice points in the plane?

Suppose I have a unimodular lattice $\Lambda = A \mathbb{Z^2}$ ($A\in SL(2,\mathbb{R})$) in the plane. I place a disc of fixed radius, $r$, around each point of $\Lambda$, so that I have a union of ...
1
vote
1answer
17 views

estimates for the largest disc not intersecting a unimodular lattice?

Are there any nice estimates for the size of the largest disc (centered anywhere) not intersecting a unimodular (i.e. covolume = 1) lattice in the plane? Maybe estimates in terms of the shortest ...
1
vote
1answer
84 views

Divide N Hot dogs among M persons

There are N hot dogs and M people and we need to divide the hot dogs equally. Now we need to calculate the minimum number of cuts required to distribute the hot dogs equally. In order to divide the ...
-1
votes
1answer
116 views

Count ways to form isosceles triangles

Their are N persons sitting on a table with N vertices.We need to count the number of isosceles triangles formed such that each vertex of the triangle is a vertex of the table and all persons seating ...
0
votes
2answers
31 views

All solution of some equation [duplicate]

Let $A=\{(m,n)\in\mathbb{N\times N}:m\neq n \text{ and } m^n=n^m\}$. It is clear that $(2,4),(4,2)\in A$. What is the solution of this equation ?
0
votes
1answer
21 views

What are the total number of ordered permutations of a list of elements (possibly repeated)?

This question is a part of a TopCoder problem. I am learning algorithms, and just got stuck at this (not homework). Suppose we have a set $A$ of integer elements, such that $n(A) = a$ (number of ...
0
votes
1answer
59 views

Conditions for equality of two binomial sums

Let $k,r,n$ be integers such that $0<k,r<n$. Let $$K=\sum^n_{i=k}k^{n-i}\binom{n-k}{i-k}^2k!(i-k)! \,\text{ and }\, R=\sum^n_{i=r}r^{n-i}\binom{n-r}{i-r}^2r!(i-r)!.$$ How to show that ...
0
votes
1answer
23 views

Count ways to take K alternative numbers from N numbers

Given N numbers from 1 to N , I need to choose K numbers (K=$0,1,2$.....$\lceil n/2 \rceil$) in such a way that if we choose number $i$ then we cannot choose numbers $i-1$ and $i+1$. In case of ...
-1
votes
0answers
66 views

Door game between alice and bob

Alice and Bob are taking a walk in the Land Of Doors which is a magical place having a series of N adjacent doors that are either open or close. After a while they get bored and decide to do ...
1
vote
1answer
71 views

Count ways to reach Nth row

Given a N*M grid I need to reach last row with following operations : ...
4
votes
2answers
80 views

partitions and their generating functions and Partitions of n

A partition of an integer, n, is one way of writing n as the sum of positive integers where the order of the addends (terms being added) does not matter. p(n, k) = number of partitions of n with k ...
0
votes
2answers
61 views

Making 24 with given number N

Initially we have a sequence of n integers: 1, 2, ..., n. In a single step, we can pick two of them, let's denote them a and b, erase them from the sequence, and append to the sequence either a + b, ...
1
vote
2answers
45 views

Generating function for number of integer solutions, no computer

How do you solve a Generating function for the number of integer solutions with no computer? Use a generating function to solve the number of integer solutions for $$x_1+x_2+x_3=17$$ Where ...
1
vote
0answers
26 views

Number of distinguishable arrangements of the word INDOOROOPILLY with three different conditions

I have the following three questions on a past final exam, I wanted to ask if I have done everything correctly. Thank you! How many distinguishable arrangements are there for the letters of the ...
0
votes
1answer
37 views

How many elements does $\mathcal{P}(A)$ have?

Let $A$ be a set of size fifteen. Let $\mathcal{P}(A)$ denote the power set of $A$, that is the set of all the subsets of $A$. How many elements does $\mathcal{P}(A)$ contain? This is the same as ...
1
vote
0answers
24 views

Ehrhart Polynomials Modulo Prime Integers

Are there any results known about computing Ehrhart Polynomials modulo prime integers?
-1
votes
1answer
252 views

Finding winner of flipping game

Alice and Bob play a game with N non-negative integers. Players take successive turns, and in each turn, they are allowed to flip active bits from any of the integers in the list. That is, they ...
0
votes
0answers
23 views

Unique sum between elements of a numerical set

I require a set of numerical elements on wich the sum of some of these elements is unique to the set, that it's to say, no other combination in the sum of elements will result the same outcome. ...
0
votes
0answers
32 views

Given a positive integer $k$, find the integer part of $n^2 /k$ for $n\ge 1$, and a related question.

For a given positive integer $k,$ I am looking for possible answers / literature about the sequence $(a_n)=([\frac{n^2}{k}])_{n=1}^\infty$, where $[x]=$the integer part of $x.$ This question is ...
1
vote
0answers
37 views

Number of divisiors of $n$ less than $m$

I'm looking for a closed- or alternative-form of the function that counts the number of divisors of an integer $n$ that are less than some integer $m$ (interested in $m < n$, obviously): $ ...
1
vote
2answers
16 views

Ordered Restricted Partition

How do I find the amount of possible ordered partition of $n$, given set of positive integer $S$? Here's an example, With $n = 4$ and $S = \{1, 3, 4\}$, we should have $4$, as $(1,1,1,1)$, $(1, 3)$, ...
4
votes
1answer
68 views

Diner Combinations, Each Pair Sits Together Exactly Once

There are $N^2$ guests at a party. How can we seat these guests at $N$ tables, in a number of rounds, so that each guest sits with every other guest exactly once? I've come up with an algorithm that ...
2
votes
0answers
62 views

Numbers which are writable as a sum of permutation pairs

We say that $N$ is writable as a sum of permutation pair $\{a,b\}$ if $a+b=N$, $a\neq b$ and $a$ and $b$ are permutations of each other (e.g. $321 = 156 + 165 = 147 + 174 = ... $). Looking at 3-digit ...
2
votes
1answer
57 views

Count ways to make total coin value [closed]

For any non-negative integer K, suppose we have exactly two coins of value 2^K (i.e., two to the power of K). Now we are given a long N. We need to find the number of different ways we can represent ...
2
votes
0answers
53 views

Minimise total cost and count ways [closed]

A country has a + b cities located in a row, which are uniformly placed. There are two large telecommunication operators in this country. The first operator will ...
2
votes
1answer
76 views

An estimate involving gaps in a subsemigroup of $(\mathbb N,+)$

I think this question can be solved by a high school student, maybe there is some trick on it or I'm forgetting something. Before my question, some background is required: Definition: A ...
0
votes
1answer
31 views

Count ways to sit men women in row of size K

Suppose we are given N men and M women.They are to sit in a row of size K such that no two women sit next to each other.What are the number of ways. Like if suppose their are 3 men and 2 women and ...
0
votes
1answer
61 views

Count numbers with prime digit

Given a number N I need to find the count of the numbers that have atleast one prime digit (2,3,5 or 7) in it. Now N can be upto 10^18.What is the best approach to solve this problem. Example : Let ...