Intereting Posts

Closed-form expression for $\sum_{k=0}^n\binom{n}kk^p$ for integers $n,\,p$
What exactly is a number?
Last Digits of a Tetration
$\mathbb{F}_p/(X^2+X+1)$ is a field iff $p \equiv 2 \bmod 3$
weak convergence of product of weakly and strongly convergent $L^{2}$ sequences in $L^{2}$
Should I put number combinations like 1111111 onto my lottery ticket?
Modular arithmetic division
If $a\in \mathrm{clo}(S)$, does it follow that there exists a sequence of points in $S$ that converges to $a$?
Is whether a set is closed or not a local property?
Why is polynomial convolution equivalent to multiplication in $F/(x^n-1)$?
Interpretation of a combinatorial identity
Simplying linear recurrence sum with binomials
Maximizing and minimizing dot products
Arrangement of integers in a row such that the sum of every two adjacent numbers is a perfect square.
Infinite Product $\prod\limits_{k=1}^\infty\left({1-\frac{x^2}{k^2\pi^2}}\right)$

There are several circles inside a square of side length $1$. The sum of the circumferences of the circles is $10$. Prove that there exists a line that intersects at least $4$ of the circles.

I know I have to use expected value to prove this, but how? Casework wouldn’t possibly work and I can’t really think of other stratigies. Thanks in advanced for providing a solution!

- Average distance between two randomly chosen points in unit square (without calculus)
- Improper Random Variables
- If a coin toss is observed to come up as heads many times, does that affect the probability of the next toss?
- Expected time to roll all 1 through 6 on a die
- Proving Pascal's Rule : ${{n} \choose {r}}={{n-1} \choose {r-1}}+{{n-1} \choose r}$ when $1\leq r\leq n$
- positive martingale process

- Problems on expected value
- Roulette betting system probability
- Given enough time, what are the chances I can come out ahead in a coin toss contest?
- $K$ consecutive heads with a biased coin?
- Distribution of Difference of Chi-squared Variables
- A bridge hand void in one suit
- Probability that $xy = yx$ for random elements in a finite group
- What is the correct way to think about this yet another balls/boxes problem?
- How do you prove that if $ X_t \sim^{iid} (0,1) $, then $ E(X_t^{2}X_{t-j}^{2}) = E(X_t^{2})E(X_{t-j}^{2})$?
- how to derive the mean and variance of a Gaussian Random variable?

**Basic approach.** Let the unit square be the square bounded by the four lines $x = 0$, $x = 1$, $y = 0$, and $y = 1$. Further let $f(u)$ be the number of circles that intersect the vertical line $x = u$. What is the expected value of $f(u)$ over the interval $[0, 1]$? In other words, what is $\int_{u=0}^1 f(u) \, du$? What can you say about the *maximum* of $f(u)$ over that same interval?

This is more or less a variation on the pigeonhole principle.

- Finding a closed form expression for $\sum_{i=1}^{n-1}\csc{\frac{i\pi}{n}}$
- How to find the direction vector of a ball falling off an ellipsoid?
- Find the Fourier transform of $\frac1{1+t^2}$
- Minimal Polynomial of $\sqrt{2}+\sqrt{3}+\sqrt{5}$
- Can $A, B$ fail to commute if $e^A=e^B=e^{A+B}=id$?
- Let $D$ be a UFD. If an element of $D$ is not a square in $D$ then is it true that it is not a square in the fraction field of $D$?
- Recommended (free) software to plot points in 3d
- Examples of sets whose cardinalities are $\aleph_{n}$, or any large cardinal. (not assuming GCH)
- What is the precise definition of 'uniformly differentiable'?
- Number of homomorphisms between two cyclic groups.
- Expression for $\int_0^1 x^n(1-x)^{n}/(1+x^2) \ dx$
- CW construction of Lens spaces Hatcher
- Counting Set Partitions with Constraints
- Prove that $\prod\limits_{i=1}^n \frac{2i-1}{2i} \leq \frac{1}{\sqrt{3n+1}}$ for all $n \in \Bbb Z_+$
- How to find number of prime numbers up to to N?