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}


Solution


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.

evilmasterer_geometry.png
Solution


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.
sl.png


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
b) dad – champion – dad
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)?


2007-11-26 by YourShadow
Consider the following isosceles triangle \triangle ABC with \beta=50^\circ ,\delta=60^\circ, \gamma=20^\circ.

triangle_problem_1_potd.jpg

Compute the angle \alpha in M.
Solution


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 =<k,x>\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.
3 circles
Solution


2007-11-2 by an unkown chatter
Show : |x-2|\leq0.02\quad \Rightarrow \quad |\sqrt{x}-\sqrt{2}|\leq 0.01
Solution

1 Comment »

  1. here is the solution for POTD on 2007-11-22 by karlo

    As we know that west has the king of hearts, we can deduce that east must have the ace of hearts (to win the first trick. we know west cannot have the ace because he could lead that and win a trick, and we know that north or south cannot have it because they could win the first trick after the king is played). This also tells us that west cannot have any more hearts in his hand (playing one of these would also result in east winning the first trick). Also based on the fact that not leading with the king of hearts results in a 13-trick win for N/S we know that all the other cards in west’s hand are losers. We can also deduce that East cannot have any more hearts in his deck, if this were so then its possible for him to play a heart other than ace hearts in the first trick and west would win the trick, and we know his hand is full of losers, hence north/south win a trick.
    Assuming the king-heart lead, after the first trick the play begins with east. For him to win the next 12 hands no matter what, then the order of the cards played is irrelevant, they must all be winners. The first easter egg in the question is that its no trumps, so its quite obvious that all the other cards in East’s hand are of the same suit (although we already know it isnt hearts). Based on the fact we know that any other of the cards led by west results in a loss, this means that any of the remaining 12 cards in easts hand cannot be shared with west, otherwise west could lead another card which could be won by east.
    Seeing as they all have to be the same suit, we can rule out clubs, as the jack of clubs held by north would allow north to win a club-lead hand assuming the first card played is lower than the jack. East has to possess spades or diamonds.
    If, in the non-Kh led game, north wins a trick, at some point he has to lead 2 spades which is always going to be a loser. By rights he cannot hold more than 12 cards of one suit (11 for diamonds/hearts) which means he could play 2 spades first off and lose the trick. Therefore like west, for the game result to be no matter what, he must have a hand full of losers. Seeing as in the non-Kh led game, the king heart and ace of hearts are still in play in west/east hands, no tricks can be started with a heart, so every heart in his hand has to be a loser. Now we know that north’s hand constists of 2 spades, jack clubs, 2-queen of hearts.
    Now we know that south has to have all the winners, like east he needs a hand filled mostly of one suit and at least one “entry card”. There has to be one card to cover the jack of clubs loser (the ace of clubs). We know that if west leads a non-kh card, south has to win every trick, which by elimination means west must hold most of the remaining clubs. Any club lead would result in south winning the hand with ace of clubs.
    Because north contains 2 of spades, and each of south and east contain at least one other winner, west must contain 2 diamonds. As already stated, east and west cannot share any other suit in common other than the single heart cards. For this reason east’s hand is made up of ace hearts, 3-ace spades, and the answer to the question is: South holds 5 diamonds.
    the cards 3d, 4d, kc and qc do not need to be specified as any placement of these cards in west’s or south’s hand will result in the same outcome as long as all of south’s diamonds or remaining clubs are higher than west’s

    W KH TC 9C 8C 7C 6C 5C 4C 3C 2C 2D(3D 4D)(KC QC)
    N QH JH TH 9H 8H 7H 6H 5H 4H 3H 2H JC 2S
    E AH AS KS QS JS TS 9S 8S 7S 6S 5S 4S 3S
    S AC AD KD QD JD TD 9D 8D 7D 6D 5D(3D 4D)(KC QC)

    Comment by ScReeeeaM — September 22, 2008 @ 7:05 pm

    Reply

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