Intereting Posts

Showing that a group of order $pq$ is cyclic if it has normal subgroups of order $p$ and $q$
Is there a tangent of $x\sin(1/x)$ at $x = 0$?
Proof of equivalence of algebraic and geometric dot product?
Primary/Elementary Pedagogy: What is the rationale for the absent '+' in mixed fractions?
Show that $n$ does not divide $2^n – 1$ where $n$ is an integer greater than $1$?
Is there a way to solve for $x$ in $\cos^{-1}(ax) / \cos^{-1}(bx) = c$?
Cross product of two vectors, given magnitudes and angle
On the Factor group $\Bbb Q/\Bbb Z$
Simplifying $\sum_{i=0}^n i^k\binom{n}{2i+1}$
About the branch-cut in the complex logarithm
Number of positive integral solutions to $x+y+z+w=20$ with $x<y<z<w$ and $x,y,z,w\geq1\;?$
integer as sum of three binomials
$n$th derivative of $e^x \sin x$
Matrix multiplication: interpreting and understanding the process
Find all $x$ such that $x^6 = (x+1)^6$.

Consider a equilateral triangle of total area 1. Suppose 7 points are chosen inside. Show that some 3 points form a triangle of area $\leq\frac 14$.

- Fascinating induction problem with numerous interpretations
- Average distance between two randomly chosen points in unit square (without calculus)
- Check if a point is within an ellipse
- Orthonormal vectors in Polar coordinates, show $\hat{e}_R=\frac{(x,y,z)}{r}$
- Volume of Region in 5D Space
- Irrational numbers in reality
- Let $A_1,A_2,..,A_n$ be the vertices of n sides of a regular polygon such that $1/A_1.1/A_2=1/A_1.1/A_3+1/A_1.1/A_4$ then value of $n$ must be?
- How to partition area of an ellipse into odd number of regions?
- What hexahedra have faces with areas of exactly 1, 2, 3, 4, 5, and 6 units?
- Nested sequences of balls in a Banach space

Choose one point $p$ and draw a line from $p$ to each of the other six points.

We can order the six points $a,b,c,d,e,f$ going clockwise around point $p$ and draw lines from $a$ to $b$, from $b$ to $c$, from $c$ to $d$, from $d$ to $e$ and from $e$ to $f$.

This gives five disjoint triangles which fit inside the triangle of area $1$.

Therefore the total area of the five triangles is less than or equal to one and at least one triangle has area less than or equal to $\frac 15$.

- Proof strategy – Stirling numbers formula
- Which step in this process allows me to erroneously conclude that $i = 1$
- If $f$ is a non-constant analytic function on $B$ such that $|f|$ is a constant on $\partial B$, then $f$ must have a zero in $B$
- Two equivalent definitions of a.s. convergence of random variables.
- Hypergeometric formulas for the Rogers-Ramanujan identities?
- Can $ \int_0^{\pi/2} \ln ( \sin(x)) \; dx$ be evaluated with “complex method”?
- Finding the number of elements of order two in the symmetric group $S_4$
- Permutations and Derangements
- Convergence in quadratic mean and in mean
- Roots of polynomials over finite fields
- Importance of determining whether a number is squarefree, using geometry
- $Z(I:J)$ is the Zariski closure of $Z(I)-Z(J)$
- Does there exist a function satisfying $\sup \int\vert u\vert=0$?
- Once and for all – “Rational numbers” – because of ratio, or because they make sense?
- Fast $L^{1}$ Convergence implies almost uniform convergence