# Tagged Questions

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### Geometrical application of generation function for permutation

It is quite well known that the generation function for permutations is represented as $$(1+x)(1+x+x^2)\dots(1+x+x^2+x^3...+x^{nā1})$$ (See, e.g., question The generating function for permutations ...
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### 2014 USAMO Problem :with Points Collinear iff Sum is Constant

Prove that there exists an infinite set of points $$\dots, \; P_{-3}, \; P_{-2},\; P_{-1},\; P_0,\; P_1,\; P_2,\; P_3,\; \dots$$ in the plane with the following property: For any three distinct ...
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### How many different right triangles are possible with the shorter side of odd length?

I was trying to solve this problem but unable to figure it out completely. I thing number of was odd integer $n$ can be the side of right triangle is number of factor of $\frac{n^2}{2}$. Can some one ...
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### number of lattice points in an n-ball

I have faced a problem in my work and I will appreciate any hint/reference as I am not much into the lattice problems. Assume an n-dimensional lattice $\Lambda_n$ with generator matrix $G$. Note that ...
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### Trying to revise a formula I was once given. How many rectangular prisms are in a $n \times n \times n$ cube?

I post it the other day. The only answer I got is that the total number of rectangular prisms in a cube is equal to ${n+1 \choose 2}^3$. But using $n=2$, I found the formula to be wrong. When counting ...
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### How many square based pyramids are in a bigger pyramids?

The biggest challenge to solve the problem is that I can't really picture a pyramid. And it is hard to make a model. The pyramids I am trying to find include those on all tiers.
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### How to find how many rectangular prisms ( including cubes) are in a n by n by n cube?

I somehow got the answer to be [(n+1)!/2!(n+1-2)!]^2 *n Each part of the equation represents the height, length, and width of the possible rectangular prism in the big cube. You can multiply the ...
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### How to find how many cubes are in a n by n by n cube?

I tried finding the answer using combinatoric by determining how many different length and width ans height are there for a cube, given the size of the bigger cube. But the formula I got turns out not ...
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### Uniqueness of projective plane of order 5

Is there a slick way to see the uniqueness of projective plane (equivalently, an affine plane) of order $5$?
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### Counting Periodic Orbits on a regular Hexagon

An orbit on a polygon is a path that a "billiards ball" (a point) would follow if it obeyed Snell's law of reflection (the angle of incidence is equal to the angle of reflection). A periodic orbit is ...
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### A probability problem with multivariate Gaussian distribution

I am a computer science guy, not a mathematician so kindly excuse me if there is any ridiculous error in my problem description. I have two clusters $C_1$ and $C_2$ in a feature space spanned by $k$ ...
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### Division of space by balls in R^n

I would like to know the generalized proof of this result: http://mathworld.wolfram.com/SpaceDivisionbySpheres.html, for $n$ dimensions. What is the maximum number of regions divided by $q$ ...
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### There is no projective plane of order $10$.

I need to determine if there is a projective plane of order $10$. The Bruck-Ryser theorem tells us that if $n \equiv2, 1 \bmod 4$, and there is a projective plane of order $n$, then $n$ is a sum of ...
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### Placing n points in a MxM square grid

I am facing an apparently well-known problem: placing $n$ points in a discrete grid so that the points are 'evenly' distributed. By evenly I mean that I would like the density of points to be nearly ...
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### Knight's metric: ellipse and parabola.

Knight's metric is a metric on $\mathbb{Z}^2$ as the minimum number of moves a chess knight would take to travel from $x$ to $y\in\mathbb{Z}^2$. What does a parabola (or an ellipse) became with this ...
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### Maximum number of pipes used to construct the polygon

Bob has been given a Summer Vacation project by his Math's Teacher. After spending almost the whole vacation sleeping,eating,playing and doing everything (except coding), Bob is now anxious about ...
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### Finding number of scalene triangles with certain perimeter?

For example, lets say you want to find the number of scalene triangles with integer side lengths such that the perimeter is less than or equal to 13. I could start listing all possible triples in ...
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### intersection of lines and planes ?

if 6 lines are drawn in a plane , what is the maximum number of parts in which plane is divided by them ? If there are 'k' lines drawn in the plane then plane is divided into how many maximum parts ? ...
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### Triangulation of surfaces and the number of edges in a triangulation.

I am reading a chapter on surfaces and triangulations, but I think I am losing the plot. I am reading page 650 of this book ...
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### combinatorics and geometry

how many different triangles with a perimeter of 15cm can be constructed that have integral multiples of sides?? MY APPROACH : (a+b+c)=15 and three sides(...
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### Number of ways to make grid

I need to construct a L x 3 grid as shown below But i can use only two shapes to make it which are : Here L is the number of small square boxes in each row. I can rotate the shapes as I want. I ...
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### Issue concerning enumerating vertices in a prism (number of two adjacent vertices can only differ by a certain amount)

There are 100 vertices in a prism with a 50-gon as its base. Those vertices are assigned integers 1 to 100 (inclusive) in a random order. Each number can only be assigned once. The objective is to ...
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### Intersection of lines on a plane

Suppose we have n lines on a plane such that there are $k_2$ points where two lines intersect, $k_3$ points where three lines intersect, ... , $k_n$ points where n lines intersect. How many segments ...
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### Count number of triangles

If we are given N lines and out of N lines M is the set of parallel lines and we are also provided M integers m1,m2,m3.. number of parallel lines in each set. Here 2<=m1,m2,m3ā¦<=N and ...
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### Existence and unicity of a complete bounded cell in a generic hyperplane arrangement.

Let $n>d$ be integers and $H_1,\ldots,H_n$ be hyperplanes in $\mathbb{R}^d$ in generic position. By generic position I mean that if we change slightly their position, then the configuration does ...
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### Maximum of the minimal distance of a set of points in an equilateral triangle

In this question, a closed triangle on a plane is a set of all points in its area and on its boundary, while an open triangle excludes its boundary. Now, the problems: Let $T$ be an equilateral ...
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### How many triangle can be drawn with those points? [duplicate]

There are 7 points on the circumference of a circle.How many acute triangle can be drawn with those points. please help me to solve this problem.
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### $7$ points inside a circle at equal distances

BdMO 2014 There are $7$ points on a circle.Any 2 consecutive points are at equal distance from one another.How many acute angled triangles can you form taking any 3 of these points? I believe ...
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### Separating points on a plane

BdMO 2011 There are $25$ points on a plane, no three of which lie on a line. Find the minimum number of lines needed to separate them from one another. Can we assume that the points lie on a ...
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### Rectilinear polygons winding around a torus

A simple rectilinear polygon on the plane the difference between the number of interior convex angles ($90^{\circ}$) and that of interior concave angles ($270^{\circ}$) is always $4$. Consider a ...
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### How to solve this cube puzzle question? [closed]

How to solve questions which is based on one main question? (This question is asked in big IT MNC's aptitude test) A cube is colored orange on one face , pink on the opposite face , brown on one face ...
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### Number of classes of K-sets

I am having a plane in N dimension. Th distance between 2 points (a1,a2,...,aN) and (b1,b2,...,bN) is max{|a1-b1|, |a2-b2|, ..., |aN-bN|}. I need to to know how many K-sets exist(here K-set refers to ...
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### Existence Of Congruent Triangles

Two triangles $ABC$, $XYZ$ are "good" when $AB=XY$, $AC=XZ$, $\angle ABC=\angle XYZ$. That is, when two segments are equal and a not-included angle is equal, they are "good". There are $n$ ...
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### The largest regular m-gon that fits inside a regular n-gon

This question just popped into my head while doing some "for fun" math. More precisely: Let $m,n\in\Bbb{Z};m,n>2$. Let $P$ be a regular $n$-gon (let's say $P$ is the convex hull of the $n$ $n$th ...
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### Cover the n-sphere with sub-hemispherical caps

Original Question (answered): Define a cap (x,Phi) to be the set of all points of the sphere that are within an angle Phi of the point x. $0 \le \phi < \frac{\pi}{2}$. (define the angle ...
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### Symmetry group of the geometric realization of a simplicial complex

Question: Given $\Delta$ an abstract simplicial complex, can one find a geometric realization of $\Delta$ whose symmetry group is isomorphic to $Aut(\Delta)$? Relevant definitions: Let $\Delta$ an ...
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### Possible combinations for a cube

When drawing one of two possible diagonals on each side of a cube, how many unique patterns are possible with regard to all sides of the cube and all possible diagonal orientations. I am stuck on ...