All Questions
28
questions
5
votes
0
answers
76
views
Number of loops of a ball bouncing in a room with obstacles
Introduction
With a friend of mine we were studying the following problem: given a $m\times n$ grid draw this pattern (I don't know how to describe it in words)
The first image has $3$ loops and the ...
5
votes
2
answers
262
views
Shading the entire $i$-th row and $j$-th column of an $m\times n$ grid when $\gcd(i,m)>1$ and $\gcd(j,n)>1$, how many grids leave $x$ cells unshaded?
Is there a way of cleverly counting the following scenario?
Given an $m\times n$ grid, with $m$ and $n$ relatively prime, imagine shading a subset of the squares of an $m\times n$ grid using this ...
- 1,755
0
votes
2
answers
59
views
Difference between geometric approach and counting of dots in a right-triangular arrangement
If I have some dots and arrange them geometrically in a right triangle such that the width (or bottom) of the triangle has $20$ dots and the height of the triangle also has $20$ dots my intuitive ...
- 1,609
1
vote
0
answers
33
views
Different slopes defined by nesting $m$ polygons
I know that the vertices of a regular $n$-gon determines the total of $n$ different slopes. We nest the total of $m \in \mathbb{N}$ polygons by drawing a $(1/2)n$-gon inscribed inside the original ...
user830143
2
votes
1
answer
183
views
Minimum possible distance between $n$ grid points
We are given a grid and a set $S$ of $n$ points on it (i.e points in the plane with integer coordinates). We define the diamatar $diam(S)$ of $S$ to be the maximum possible distance between two ...
- 2,510
0
votes
3
answers
513
views
What is the maximum number of T-shaped polyominos (shown below) that can be put into a 6x6 grid without any overlaps? The blocks can be rotated.
I just drew the figure and manually tried the question but I am wondering is there a way to do this problem via permutations and combinations.
PS: I got answer as 7.
- 77
1
vote
0
answers
122
views
PDFs for Olympiad preparation
Could someone please recommend me some pdf files containing theory for topics that come up often in maths olympiads? I'm currently working through one about inequalities, and I'm really enjoying it. I ...
- 41
10
votes
2
answers
739
views
Tiling a rectangle with both rational and irrational sided squares
We define a 'tiling of rectangle with squares' as the process of dividing the rectangle into finitely many squares so that they do not overlap and cover up the whole rectangle.
Here is my question:
...
- 156
2
votes
2
answers
97
views
Single marble stacking operation that fills out a 3-dimensional space?
My question is:
Is there a way to stack marbles by using only a single one-marble stacking operation such that an infinite 3-dimensional stack is constructed?
For example:
In 1-dimension one can ...
- 2,259
1
vote
2
answers
126
views
Square Chessboard Problem [duplicate]
Show that there is a $6$ x $4$ board whose squares are all black or white, where no rectangle has the four vertex squares of the same color. Also show that on each $7$
x $4$ board whose squares are ...
- 1,591
4
votes
0
answers
1k
views
Counting the Number of Lattice Points in an $n$-Dimensional Sphere
Let $S_n(R)$ denote the number of lattice points in an $n$-dimensional "sphere" with radius $R$. For clarification, I am interested in lattice points found both strictly inside the sphere, and on its ...
- 1,022
8
votes
2
answers
237
views
Is it true that the number is divisible by $p$?
Question: Let $a, b, c$ be positive integers and $p>3$ be a prime ($ a$ isn't divisible by $p$).
Consider a quadratic polynomial $P(x) = ax^2+bx+c$, and assume that there exists $2 p-1$ ...
- 1,705
3
votes
1
answer
80
views
how can I prove that $p=7,n=2$ is the only solution (sum of divisors)?
Question: Find every pair of $(n,p)$ in which $n$ is a positive integer and $p$ is an odd prime number so that the sum of every positive divisor of $p^{2^n-1}$ is a square number.
It can be seen that ...
- 1,705
5
votes
1
answer
136
views
Is it possible to prove that $M$ is an integer with $p+M \over x$ is always an integer?
Given a prime number $p$ and a set $S$ of $n$ rational numbers. Multiply all $n$ rational numbers we get a number $M$. For each number $x$ in the set $S$, we have $\frac{p+M}{x}$ is an integer. Is it ...
- 1,705
5
votes
0
answers
2k
views
Good books to learn olympiad geometry,number theory, combinatorics and more
I want to start learning olympiad mathematics more seriously, and I would like to have advice on some good books or pdfs to learn with.
I have background but not a big background. For example I know ...
- 2,510