# Undernetmath’s Weblog

### POTD_2007

2007-12-31 last but not least of 2007 by fatherpi

Compute $\displaystyle \sum_{n=1}^\infty \frac{\lfloor\sqrt{n} \rfloor -\lfloor \sqrt{n-1}\rfloor }{n}$.

2007-12-29 by jackal

Consider a triangle $\triangle ABC$ with $|\overline{AB}|=5cm,\, |\overline{AC}|=3cm,\,|\overline{BC}|=4cm$. Compute the length of the internal bisector (cevian) of $\angle ACB$.

2007-12-28 by kmh

Consider a square ABCD with a point P inside. Prove:
$\angle BAP = \angle ABP=15^\circ \Rightarrow \triangle PCD \text{ is equilateral}$.
Solution

2007-12-21 Trigonometric Xmas by kmh

Let $\alpha,\beta,\gamma$ be the 3 angles of an arbitrary triangle. Show that $\cos(\alpha)+\cos(\beta)+\cos(\gamma)\leq\frac{3}{2}$ .

2007-12-18 by kmh

Prove or disprove that $\frac{2\cdot \binom{2^n-2}{ 2^{n-1}-1}\cdot(2^{n-1}!)^2}{2^n!}=\frac{2^{n-1}}{2^n-1}$ for all $n \in \mathbb{N}$.

2007-12-15 by yaroslav

Let A be the adjacency matrix of a graph with k triangles. Find $trace(A^3)$. Hint, first find combinatorial interpretation of $(A^k )_{ij}$

2007-12-13 by kmh

Compute the probability that in tennis tournament of 8 players the 2nd best player becomes the runner up.
(Source: F. Mosteller: 50 challenging problems in probability)
Solution

2007-12-9 by evilmasterer

Consider the following construction with $|AC|=15cm,\, |AB|=10cm,\, \angle CAB=90^\circ$.
Compute the gray area.

2007-12-6 Santa Claus’ Actuary Special by Karn

a) A random variable X has a cdf defined as follows:
$F(x)=\begin{cases} 0 & (-\infty,1) \\ \frac{x^2-2x+2}{2} & [1,2) \\ 1 & [2,\infty) \\ \end{cases}$
Determine the variance Var(X).
b) Prove that the conditional probability $P(A|B)=\frac{P(A \cap B)}{P(B)}$ is always smaller than 1.
c) Give an example of 3 events A, B, C, which are not independent, but for which $P(A \cap B \cap C)=P(A) \cdot P(B) \cdot P(C)$ holds nevertheless.

2007-12-5 by kmh

Compute the smallest area for a convex set, that intersects with both branches of the hyperbolas $xy=1$ and $xy=-1$.

2007-12-4 by R^^n

Compute ${\displaystyle \lim_{n\rightarrow \infty} \frac{1}{n} \sum_{i=1}^n exp(\frac{i}{n})}$
Solution

2007-12-3 by kmh

Show that any triangle with bisectors of equal length ($|AE|=|BD,\, \alpha=\beta,\, \gamma=\delta$) is isosceles.

2007-12-2 by kmh

Consider the following game. A coin is thrown onto a large table from a distance of 5m. The table surface is partitioned into squares with a side length of 5cm and the coin has a 3.75cm diameter. The player wins if the coin lands completely inside a square and loses otherwise. Determine the approximate probability for the player to win.

2007-11-29 by kmh

To provide an incentive for his son to improve his tennis skills dad offers him a reward for winning 2 sets in a row in a 3 set match. In each set the son faces as opponent either his dad or the tennis club champion, who’s playing a lot better than his dad. Dad offers his son 2 possible schedules for the 3 sets:
a) champion – dad – champion
Which schedule should the son pick?

2007-11-27 by Karlo

In how many ways can 2xn rectangle be tiled by dominoes (2×1,1×2)?

Consider the following isosceles trainge $\triangle ABC$ with $\beta=50^\circ$ ,$\delta=60^\circ$, $\gamma=20^\circ$.

Compute the angle $\alpha$ in M.

2007-11-23 by kmh

Find a 10 digit number, such that its nth digit equals the number of occurrences of the digit n-1 in the number itself. In other words the first digit tells you how many 0s the numbers contains, the second digit how many 1s and so forth.

2007-11-22 by karlo

At a certain deal in bridge, it was no trump, and West had to play the first card. One of her options was to play the king of hearts, and it turned out that if she (West) did so, the East-West partnership would take all thirteen tricks, no matter what. On the other hand, if West started the play with any of her other twelve cards, N-S would take all thirteen tricks, no matter what.

Given that North had the two of spades and the jack of clubs, who had the five of diamonds?

2007-11-20 by kmh

Linear Algebra Snack:
Let $k\in \mathbb{R}^n$ and $f: \mathbb{R}^n\rightarrow \mathbb{R}^n$ be defined as follows:
$f(x):=(k\cdot x) \cdot k =\cdot k \quad \forall x\in \mathbb{R}^n$
Show that f is a linear map and determine its matrix notation with regard to the canonical base.

Calculus Snack:
Compute ${\displaystyle \lim_{x\rightarrow 0} \,\cot(x)-\frac{1}{x}}$

2007-11-17 by kmh

Does there exist a triangle, such that the tangens of all 3 angles is an integer ?

Scottish Math Special by Karlo

I once knew a gal from Dundee
Whose age had the last digit 3.
The square of the first
Was her whole age reversed.
So what would the lady’s age be?
Solution

2007-11-13 by kmh

Find a 10 digit number ,such that no 2 digits are the same and the number consisting of its first n digits is divisible by n for n=1…10.

2007-11-11 by kmh

A drawer contains black and red socks. When 2 socks are drawn at random the probability for both of them being red is $\frac{1}{2}$
a) How many socks must the drawer contain at least ?
b) How many socks must the drawer contain at least, if the number of black socks is even?
(Source: F. Mosteller: 50 challenging problems in probability)
Solution

2007-11-9 by kmh

Show that for every prime p with p > 17 3 divides $p^2+2$.
Solution

2007-11-6 lhrrcc’s birthday special

Compute lhrrcc’s age by determining the only prime number p with the following property:
$f_{hector}(p!)=p,\,\text{where} f_{hector}(n):=\left \lceil \frac{ln(n)}{ln(10)} \right \rceil \quad \forall n\in\mathbb{N}$

2007-11-6 by kmh
Consider the infinite sum of the reciprocals of all natural numbers that do not contain the digit 7:
$\displaystyle{\sum_{\stackrel{n\in\mathbb{N}}{ n \text{ does not contain the digit 7}}}\frac{1}{n}}$.
Show whether the sum converges in $\mathbb{R}$.

2007-11-3 by evilmasterer

Consider 3 identical circles with radius r placed in such a way, that the center of each circle lies on the 2 other circles (see picture). Determine the red area in the center.

Solution

2007-11-2 by an unkown chatter

Show : $|x-2|\leq0.02\quad \Rightarrow \quad |\sqrt{x}-\sqrt{2}|\leq 0.01$
Solution