P/S: Belanja tah aunty Hazim dapat $400 atu ok?!

Best known for her Nazi party rally documentary "Triumph of the Will," Reifenstahl was at the top of her game when she immortalized the human physique in "Olympiad," her film about the 1936 Berlin Olympics.

I was only vaguely aware of Reifenstahl until I saw "The Wonderful, Horrible Life of Leni Riefenstahl," Ray Müller's 1993 documentary (released in German as "Macht der Bilder") made on the occasion of her 90th birthday. At the time, I was blown away by Müller's film,  but I haven't given Riefenstahl much thought since then.

During her life, which spanned more than a century, Riefenstahl was admired for her considerable talents and moxie, envied for her access to Adolph Hitler, ostracized for her involvement in National Socialism, and then accepted for her sheer staying power.

Spielberg's decision to boycott this summer's Olympic Games prompted me to do a little reading on Riefenstahl. To learn more about this chameleon who claimed that art trumps politics, I've been reading Jürgen Trimborn's gripping biography "Leni Riefenstahl: A Life."

I don't know much about Trimborn other than what is in the jacket of the book and in the reviews indicating that this 2002 book is the seminal work on Riefenstahl's life. I found myself wondering whether Trimborn was raised Catholic in light of his assessment that Riefenstahl's life was incomplete because she never confessed her sins and never apologized for profiting from her association with Hitler and his hate machine. Obviously, an apology is preferable to denial, but it still doesn't constitute atonement.

I want to read more about Riefenstahl, especially from a psychological point of view. In my estimation, she was both a textbook narcissist and a sociopath. When repeatedly confronted by courts, journalists, and filmmakers with her collaboration (at the very least) and crimes (at the worst), her strategy was: "Deny, deny, deny."

I was quite eager to run Riefenstahl's chart and was disappointed that Trimborn's book didn't have a time of birth. Biographies are great treasure troves for the astrologer looking for an accurate time of birth. Riefenstahl was born on Aug. 22, 1902 in Berlin, Germany. You can see the chart, thanks to Astrodienst, here:

http://www.astro.com/cgi/chart.cgi?rs=3&btyp=w2gw&&cid=41laaaa19347-s971800598&nhor=195

"Triumph of the Will" could be the subtitle of the Leni Riefenstahl story because her success, first as a dancer and then as the star of German "mountain films," was due entirely to her own determination and her ability to attract patrons. The artistic life that she envisioned for herself was not what her parents, particularly her father, had in mind for their daughter. In freedom-loving, modern-day America, it's hard to imagine the hurdles that Riefenstahl surmounted in a patriarchal society where women were controlled first by their fathers and then their husbands.

The self-directed, athletic life that Riefenstahl pursued epitomizes her Aries Moon, which I've seen frequently in the charts of female dancers, gymnasts, trainers, and sports therapists. Riefenstahl wasn't merely mirroring Hitler's misguided search for human physical perfection; her celebration of the martial beauty of mountain climbers, soldiers, athletes, and African tribes was part of her own aesthetic. 

If there was a time when Reifenstahl must have felt as if she were standing on top of the world, it was on Hitler's birthday (April 20) in 1938, when "Olympiad" had its premiere after two years in the making.  In looking at the transits and progressions for this day, the main thing that jumps out is the opposition of Jupiter in Aquarius to her Leo Sun. Clearly, she was at the apex of her influence.

I've been studying astrology off and on for nearly 30 years, but I still don't understand the significance of "unaspected" planets. Riefenstahl's Sun is so late in Leo that it doesn't make any connections to any other planets in her chart. Some astrologers would say the lack of aspects gives the Sun a renegade quality. Certainly Riefenstahl was an original.

What I found interesting in reading Trimborn's biography of Reifenstahl was how she responded to the transits of outer planets in her life. She was certainly dragged through the depths as Pluto made its way through Leo during World War II. But after the war she grew interested in Africa and studying the isolated Nuba tribe of Sudan as Pluto came to oppose her Sun late Leo. (I'm speaking broadly here.)

Late in her life, as Neptune in Aquarius opposed her Sun, she discovered yet another career, as an underwater photographer, which she pursued until her death.

The 1993 Uranus/Neptune conjunctions in Capricorn opposed Riefenstahl's Mars at 21 degrees of Cancer. That coincided with the international release of Müller's documentary. The film in turn precipitated a brutal re-examination of Riefenstahl's role in the rise of the Third Reich and whether the various tribunals that exonerated her after World War II were told the truth. Indeed, some of the damning documents and footage that Trimborn uses to make his case against Riefenstahl weren't available until recently.

Although Riefenstahl maintained until her dying day (Sept. 8, 2003) that her films were art, not propaganda, clearly other artists, including Steven Spielberg, understand the dangers of appearing to serve a brutal regime.

Moldova, Third Team Selection Test for IMO-BMO 2008, March 30

Problem 1

Determine a subset $latex A\subset \mathbb N^*$ having exactly $latex 5$ distinct elements, whose sum of squares equals their product.

Problem 2

Let $latex p$ be a prime number and $latex k,n$ positive integers so that $latex \gcd(p,n)=1$. Prove that $latex \displaystyle\binom{n\cdot p^k}{p^k}$ and $latex p$ are coprime.

Problem 3

In triangle $latex ABC$ the bisector of $latex \angle{ACB}$ intersects $latex AB$ at $latex D$. Consider an arbitrary circle $latex O$ passing through $latex C$ and $latex D$ and not tangent to $latex BC$ or $latex CA$. Let $latex O\cap BC=\{M\}$ and $latex O\cap CA=\{N\}$.

1.  Prove that there is a circle $latex S$ so that $latex DM$ and $latex DN$ are tangent to $latex S$ in $latex M$ and $latex N$ respectively.
2. Circle $latex S$ intersects $latex BC$ and $latex CA$ in $latex P$ and $latex Q$ respectively. Prove that the lengths of $latex MP$ and $latex NQ$ do not depend on the choice of circle $latex O$.

Problem 4

A non-empty set $latex S$ of positive integers is said to be good if there is a coloring with $latex 2008$ colors of all positive integers so that no number in $latex S$ is the sum of two different positive integers (not necessarily in $latex S$) of the same color. Find the largest value $latex t$ can take so that the set $latex S=\{a+1,a+2,\ldots,a+t\}$ is good, for any positive integer $latex a$.

Moldova, Second Team Selection Test for IMO-BMO 2008, March 29

Problem 1

Find all solutions $latex (x,y)\in\mathbb{R}\times\mathbb{R}$ of the system:

 $latex \begin{cases} x^3+3xy^2=49,\\ x^2+8xy+y^2=8y+17x.\end{cases}$

Problem 2

Let $latex a_1,a_2,\ldots,a_n$ be positive reals so that $latex a_1+a_2+\ldots+a_n\le\dfrac n2$. Find the minimal value of

   $latex \displaystyle\sqrt{a_1^2+\frac1{a_2^2}}+\sqrt{a_2^2+\frac1{a_3^2}}+\ldots+\sqrt{a_n^2+\frac1{a_1^2}}$

Problem 3

Let $latex \omega$ be the circumcircle of $latex ABC$ and let $latex D$ be a fixed point on $latex (BC)$. $latex X$ is a variable point on $latex (BC)$, $latex X\neq D$. Denote by $latex Y$ the second intersection of $latex AX$ and $latex \omega$. Prove that the circumcircle of triangle $latex XYD$ passes through a fixed point.

Problem 4

Find the number of even permutations of $latex \{1,2,\ldots,n\}$ which have no fixed points.

A UWO Gazette sketch (reposted with permission).

click image to read editorial

The spiritual leader of Tibetan Buddhists (also a Nobel peace laureate and head of the in-exile government) "was recognized at age two as the reincarnation of the 13th Dalai Lama and enthroned before he turned four. He assumed full powers at age 15, in the year that troops from Mao Zedong's newly founded communist republic entered Tibet and crushed its small army." ⁰⁷ Tibet had been effectively independent for decades before Chinese communist troops imposed Beijing's control in 1950." ⁰⁹ On 1959/03/10 a failed rebellion against Chinese rule sent the Dalai Lama into exile in India. On this year's anniversary of that rebellion, an uprising centred in the Tibetan capital, Lhasa, began. ⁰¹

"Tibetan protests spread to China's capital...with students holding a silent vigil at a Beijing university as a deadline expired in Lhasa for the surrender of those China accused of sparking the uprising... ⁰⁴

Some have accused China of cultural genocide against the Tibetan people. The Dalai Lama has referred to the new Qinghai-Tibet railway which brings thousands of Chinese migrants to Lhasa every week to further dilute the Tibetan population. And travellers are forbidden from speaking Tibetan while riding buses. ⁰¹

"Journalists have been barred from entering any of the Tibetan regions, despite Chinese promises to allow free movement of journalists in this Olympic year." ¹⁰

According to the Chinese government, the violent protests led to the deaths of 16 people. But spokespersons for the Dalai Lama's camp say the figure for deaths is as high as 80. ⁰⁸

In a subsequent "military crackdown...carried out under a fog of secrecy with foreign media expelled from the region...Chinese authorities stripped Tibetans of basic rights like free speech and freedom of religion. ⁰²

"India hosts the Dalai Lama in the India city of Dharamsala, seat of the self-proclaimed Tibetan government-in-exile... More than 2,000 Tibetans gathered...from all over northeastern India for their biggest rally in the area in years, demanding the United Nations investigate reports of killings of protesters in China." ⁰⁷

Tibetan-Canadians upset with China's crackdown in their homeland staged a loud but peaceful 2008/03/17 protest outside the Chinese consulate in Toronto.

"Canada's Tamil community includes about 250,000 people – the largest Tamil community outside Sri Lanka." ⁰⁵

Complicating Canadian reaction to the China/Tibet situation, a Toronto resident who was soliciting cash donations in British Columbia for the LTTE, has been charged under this country's Antiterrorism Act which was passed in 2001. It's reported to be "the first time police in Canada have pressed charges under federal antiterrorism financing legislation." ⁰⁵ The LTTE (commonly known as Tamil Tigers) separatist group "was blacklisted in Canada as a terrorist outfit shortly after the Conservatives formed the government in 2006." ⁰⁵

Next month, Toronto Mayor David Miller plans to visit the western city of Chongqing, which has a sister-city relationship with the Canadian city. The trip is being co-ordinated with a similar mission for businesses organized by the Canada-China Business Council. A spokesperson for the council is reported to have referred to the recent events in Tibet as being "distractions." ⁰¹

The Canadian government had been remaining largely silent. But Foreign Minister Maxime Bernier is reported to have delivered a 'one-China policy' speech to Asian ambassadors behind closed doors in Ottawa on 2008/03/12 and a text of that speech indicates that he said "we recognize the Government of the People's Republic of China as the sole legitimate government of China." ⁰⁶

According to Charles Burton (former Canadian diplomat in China and currently a Brock University professor), the "subtle but meaningful change in Conservative government rhetoric on China...would suggest that Canada supports Chinese government claims to legitimate sovereignty over Taiwan and over Tibet." ⁰⁶

"A Canadian team of about 330 athletes, plus 140 support staff, will participate in the 2008 Summer Olympiad Aug. 8-24 in Beijing. ⁰³

France's Foreign Minister Bernard Kouchner says that although "France had no plans to boycott the entire Olympic Games...the European Union should consider boycotting the opening ceremony of the Beijing Olympics if violence continues in Tibet." ⁰⁷

"French President Nicolas Sarkozy said...that he cannot rule out the possibility he might boycott the opening ceremony of the Beijing Olympics if China continues its crackdown in Tibet [and] an official from France's state television company said the broadcaster would likely boycott the games if coverage was censored." ¹²

The COC suggests that the Tibetan situation "isn't even being raised by a single one of its 90-plus members who would need to vote on a boycott." ⁰²

"The Canadian Olympic Committee agreed with European Union sport ministers and various countries' Olympic committees Monday in ruling out the possibility of a boycott of the Games... The IOC's position is that it is a sports organization and can't lecture China on political matters. The COC echoes that sentiment." ⁰³

But "the rhetoric was turned up a notch...when 600 people protested in Lausanne, Switzerland, demanding the IOC call off a section of the Olympic torch relay in Tibet." ⁰⁸

Beijing plans for the torch to be carried to the top of Mount Everest. Organizing committee Executive Vice-President Jiang Xiaoyu has said that the ascent to the top of the world's tallest mountain would be the "highlight" of the relay and "a great feat in Olympic history." ¹¹

The 1976 Montreal Games were the target of an anti-apartheid boycott because some of the Olympic countries maintained sports relations with the former apartheid regime of South Africa. In 1980, the U.S. led a boycott of the Moscow Olympics in order to protest the Soviet Union's invasion of Afghanistan. Eastern Bloc countries retaliated with a subsequent boycott of the 1984 Los Angeles games. Both boycotts were viewed by many as being ineffective, and a similar large number of people felt that the athletes were victimized by the politics.

Sources:
01. 2008/03/18 - Globe&Mail: Tibetan-Canadians protest Chinese crackdown
02. 2008/03/20 - LFP: Protesters to call for Olympic boycott
03. 2008/03/17 - Globe&Mail: Canada won't boycott Beijing
04. 2008/03/18 - Globe&Mail: Tibetans stage protest in Beijing
05. 2008/03/17 - Globe&Mail: Alleged solicitation for banned terror group...
06. 2008/03/18 - Globe&Mail: Canada subtly signalling change of policy...
07. 2008/03/18 - Globe&Mail: France hints at Olympic boycott over Tibet
08. 2008/03/18 - Globe&Mail: Canada will not boycott Olympics, COC says
09. 2008/03/18 - Globe&Mail: China says Dalai Lama trying to ruin Olympics
10. 2008/03/18 - Globe&Mail: Photo evidence of Tibet horror comes to light
11. 2008/03/18 - Globe&Mail: China ready to hold talks with Dalai Lama
12. 2008/03/25 - Globe&Mail: Sarkozy won't rule out opening ceremony boycott

Other Links:
China e-Lobby
2008/03/16 - Toronto Star: China quells Tibet protest
2008/03/16 - Toronto Star: Tibetan riots spread, Lhasa in lockdown
2008/03/16 - Toronto Star: China blocks YouTube.com over Tibet videos
2008/03/17 - Toronto Star: China pledges harsh measures
2008/03/18 - Toronto Star: China's Olympic hurdle in Tibet
2008/03/19 - Globe&Mail: French minister backtracks on Olympic boycott
2008/03/19 - Toronto Star: Beijing admits riots spread to provinces...
2008/03/20 - Globe&Mail: China admits shooting Tibetan protesters
2008/03/20 - Toronto Star: China sends troops to Tibetan areas
2008/03/20 - Canada.com: Demonstrators chant for Tibet on Parliament Hill
2008/03/20 - Globe&Mail: Harper urges China to use 'restraint' in Tibet
2008/03/21 - Globe&Mail: Tibet's move; 50 years in the making
2008/03/21 - Globe&Mail: Why Tibet is boiling over
2008/03/21 - Toronto Star: Controversy won't delay China mission
2008/03/21 - Globe&Mail: Thousands of troops push into western China
2008/03/21 - Globe&Mail: China shuts down 25 websites
2008/03/21 - Toronto Star: China tightens control
2008/03/21 - CNEWS: Several hundred people take to Vancouver streets
2008/03/22 - Globe&Mail: China launches media campaign on Tibet unrest
2008/03/22 - Toronto Star: China steps up riot manhunt
2008/03/23 - Toronto Star: The other thorn in China's side
2008/03/23 - Globe&Mail: China lashes out at critics
2008/03/23 - Globe&Mail: China protest draws hundreds
2008/03/23 - CNEWS: Hundreds of Tibetan supporters march in Toronto
2008/03/24 - Toronto Star: Hundreds rally in Toronto for free Tibet
2008/03/24 - Toronto Star: Protests mar Olympic torch lighting
2008/03/24 - Globe&Mail: Tibet supporters take to streets of Toronto
2008/03/24 - Globe&Mail: China jails man who urged rights over Olympics
2008/03/24 - Globe&Mail: Protest disrupts Olympic flame-lighting ceremony
2008/03/24 - Globe&Mail: IOC president in 'silent diplomacy' with China
2008/03/24 - canada.com: [poll] Most Canadians opposed to Olympic boycott
2008/03/25 - Globe&Mail: The great call of China...
2008/03/25 - Toronto Star: Olympic torch run inflames Tibet protesters
2008/03/25 - Toronto Star: China slams jail door on Olympic dissent

Jag har alltid varit positiv till att Olympiska Spelen i Peking borde bojkottas. Den kinesiska regimen undertrycker yttrandefriheten, kränker de mänskliga rättigheterna - och religionsfrihet är inte att prata om. Men när scrollade ner och läste kommentarerna kom jag på andra tankar...

Två av dessa stack ut lite extra - Gurka i Burkas och Uppsala Students. Gurka i Burka (här förkortat GiB) menar på att en bojkott av själva olympiaden och sponsorerna skulle innebära en social, ekonomisk och demokratisk katastrof för "kinesen på gatan". Det han vill säga är:

Det är helt otroligt hur ett idrottsevenemang kan engagera en hel värld på det här viset. Att få anordna ett OS är ju otroligt ärorikt, och skulle ett land bli bojkottat när det är deras tur är det ju självklart det måste kännas nedvärderande.

GiB nämner inte så mycket om att Kina inte följer de mänskliga rättigheterna, vilket gör att jag inte kan köpa det helt. Det är ju trots allt en stor grej, men det är här Uppsala Student kommer in.

Vad han/hon gör är att lägga upp en artikel skriven av Lars Ellström som kommentar (se länk ovan). Lars berättar om när britterna startade de så kallade Opiumkrigen. För Kinas del innebar de en total katastrof.

Sedan dess har västvärlden försökt lägga Kina under deras fötter på alla möjliga sätt. Han menar på att kampen för alla kinesiska politiker och samhällsorganisationer har sedan dess varit att överleva och återupprätta Kinas storhet och styrka, och därigenom vinna världens respekt.

Inte förrän på 1970-talet börjar ekonomin växa igen - riktigt ordentligt. Numera är Kina den tredje största handelsstaten i världen, enligt Ellström.

"Allt är därmed inte bra, men det har blivit mycket bättre, och det finns all anledning att stödja denna utveckling, både för Kina och för världen. [...]

I det här perspektivet kändes det både följdriktigt och passande att Beijing 2001 fick förtroendet att arrangera de olympiska spelen 2008.
För kineser i allmänhet, oavsett politisk eller ideologisk hemvist, skulle detta bli det ögonblick man så länge längtat efter: att Kina skulle stå i centrum för världens positiva intresse. Samtidigt skulle olympiaden kunna bli en välkommen slutpunkt för mer än ett århundrade av splittring och konfrontation och inledningen till en ny era av samarbete."

Det här fick mig att byta ben.

Okej, allt är inte perfekt i Kina. Men betyder det att vi bara ska stå vid sidan som en elak storebror och säga: "Skyll dig själv"? Jag tycker inte det låter rättvist.

Missuppfatta mig inte, jag gillar fortfarande inte Kinesiska regimen som behandlar människor som skit - förföljer kristna exempelvis - men nu har jag mer förståelse för dem. De har länge haft det tufft och jag tycker vi borde hjälpa till för att få dem på fötter igen. OS i Peking kan vara en språngbräda.

Solusi
Saya akan buktikan dengan induksi, tapi ruas kiri selalu tetap, sehingga harus diubah dahulu. Pertidaksamaan dapat diperkuat menjadi

$latex \displaystyle\left(1+\dfrac{1}{1^3}\right)\left(1+\dfrac{1}{2^3}\right)\left(1+\dfrac{1}{3^3}\right)\cdots\left(1+\dfrac{1}{n^3}\right)\le3-\dfrac1n$.

Jelas bahwa ini benar untuk $latex n=1$. Asumsikan ini benar untuk $latex n=k$, sehingga

$latex \displaystyle\left(1+\dfrac{1}{1^3}\right)\left(1+\dfrac{1}{2^3}\right)\left(1+\dfrac{1}{3^3}\right)\cdots\left(1+\dfrac{1}{k^3}\right)\le3-\dfrac1k$.

Kalikan kedua ruas dengan

$latex \displaystyle\left(1+\dfrac{1}{(k+1)^3}\right)$.

Ruas kanan menjadi

$latex \displaystyle\left(3-\dfrac1k\right)\left(1+\dfrac{1}{(k+1)^3}\right)$.

Dapat dibuktikan dengan mudah bahwa

$latex \displaystyle\left(3-\dfrac1k\right)\left(1+\dfrac{1}{(k+1)^3}\right)<\left(3-\dfrac{1}{k+1}\right)$.

Maka terbukti.

Moldova, First Team Selection Test for IMO-BMO 2008, March 3

Problem 1

Let $latex p$ be a prime number. Solve in $latex \mathbb{N}_0\times\mathbb{N}_0$ the equation $latex x^3+y^3-3xy=p-1$.

Problem 2

We say the set $latex \{1,2,\ldots,3k\}$ has property $latex D$ if it can be split into disjoint triples, so that in each such triple, one number is the sum of the other two. Prove that

1. The set $latex \{1,2,\ldots,3324\}$ has property $latex D$.
2. The set $latex \{1,2,\ldots,3309\}$ hasn't property $latex D$.

Problem 3

Let $latex \Gamma(I,r)$ and $latex \Gamma(O,R)$ be the incircle and circumcircle, respectively, of triangle $latex ABC$. Consider all triangles $latex A_iB_iC_i$ which are simultaneously inscribed in $latex \Gamma(O,R)$ and circumscribed to $latex \Gamma(I,r)$. Prove that the centroids of the triangles $latex A_iB_iC_i$ lie on a circle.

Problem 4

A non-zero polynomial $latex S\in\mathbb{R}[X,Y]$ is called homogeneous of degre $latex d$ if there is a positive integer $latex d$ so that $latex S(\lambda x,\lambda y)=\lambda^dS(x,y)$ for any $latex \lambda\in\mathbb{R}$. Let $latex P,Q\in\mathbb{R}[X,Y]$ so that $latex Q$ is homogeneous and $latex P$ divides $latex Q$ (that is $latex P|Q$). Prove that $latex P$ is homogeneous too.

Solusi
Misalkan banyaknya kacang adalah $latex a$, orang pertama mengambil $latex b$ kacang, orang kedua mengambil $latex c$ kacang, dan seterusnya sampai orang kelima mengambil $latex f$ kacang. Paginya, $latex g$ kacang diterima masing-masing orang.

Pertama-tama, $latex a$ dikurangi 1 (diberi ke monyet) kemudian diambil $latex b$ oleh orang pertama. Maka sisanya $latex 4b$, dikurangi 1 (diberi ke monyet) lalu diambil $latex c$ oleh orang kedua. Dan seterusnya. Maka didapat persamaan-persamaan:

$latex a=5b+1$ menjadi $latex a+4=5(b+1)$

$latex 4b=5c+1$ menjadi $latex (b+1)=\dfrac54(c+1)$

$latex 4c=5d+1$ menjadi $latex (c+1)=\dfrac54(d+1)$

$latex 4d=5e+1$ menjadi $latex (d+1)=\dfrac54(e+1)$

$latex 4e=5f+1$ menjadi $latex (e+1)=\dfrac54(f+1)$

$latex 4f=5g+1$ menjadi $latex (f+1)=\dfrac54(g+1)$

Maka akan didapat

$latex a+4=\dfrac{5^6}{4^5}(g+1)$.

Nilai minimum $latex g+1$ adalah $latex 4^5$. Maka nilai minimum $latex a$ adalah

$latex \dfrac{5^6}{4^5}(4^5)-4=15261$.

Solusi
Mulai dari pertidaksamaan AM-GM,

$latex \dfrac{1+2+3+...+n}{n}\ge\sqrt[n]{1\cdot2\cdot3\ldots\cdot n}$.

Gunakan rumus barisan untuk ruas kiri,

$latex \dfrac{\frac{n(n+1)}{2}}{n}\ge\sqrt[n]{1\cdot2\cdot3\ldots\cdot n}$.

Sederhanakan menjadi

$latex \dfrac{n+1}{2}\ge\sqrt[n]{n!}$,

atau

$latex \displaystyle\left(\dfrac{n+1}{2}\right)^n\ge n!$.

Hmm,…… true, just another year added to my life, but this one will of course be very special. 2007 has been quite off-beat and I did enjoy it.

SPECIAL BECAUSE …

1. Only thrice in my life I have been genuinely very happy and content, and 2007 gave me two of them. Once in March, when I got the news of my selection for the Bio-olympiad camp, and another in June ( 29th to be precise ), when learnt of my selection in ISI. Both are special because they were long-cherished dreams come true. I was fed up with the gimmicky lab-works in school, and the bio-camp will be memorable for the vital hardcore lab-exposures I was badly yearning for, AND the camaraderie there. The rest of the year has been pretty drab in terms of labs, in fact I guess this is the only sacrifice I had to make by opting out of medical even after allotment in counseling. But the next yr for sure is going to make up for all this, yes. In fact, at the end of the year I may end up better off than the 2nd yr college counterparts!Well, what can I tell about the enormity of selection in ISI? It is literally future-defining. My first love has always been maths, and I badly wanted to be here. Yet, only due to lack of practice, the orientation was slipping away from me all the time, from end-11th. True I finished the maths syllabus by June last yr, but after that I left doing maths altogether, barring a few special ISI-types, …… and, the grades suffered. I couldn’t even qualify the regional math Olympiad in 12th, which I did in 11th! Though I recovered completely, my future with maths was very much depending on this very result, because not getting in ISI means going on to study medical, where it is an unknown territory. But after getting in ISI I can still can go on with bio, my second love, maybe not here, but elsewhere. And above all, this institute is a totally different ball-game. After all, some try and try and try to be the best, some are born to be ISI-ans.

2. I am someone who very much likes to sit for exams. But 2007 has managed to exhaust even me. Erm, …… appearing in 40 exams and 4 interviews doesn’t seem less, eh? ( Not to mention the ‘countably infinite’ mock tests! ) April holds the ‘most in one month’ record, 11 exams! It looked like this …
April 1st – CBSE PMT prelims,
April 2nd – Biology practical board exam,
April 3rd – Physics practical board exam,
April 4th – Chemistry practical board exam,
April 11th – Maths board exam,
April 16th – Environmental Studies board exam,
April 18th – Bengali board exam,
April 22nd – WB Joint ent. ( 3 papers of 2 hrs each ),
April 29th – AIEEE ( All-India Engg. Ent. ).

3. I started blogging. Now I am a competent programmer. My FM radio adapter was repeatedly ‘getting ill’ last yr, so had to spend a very boring new-music-less 6 months or so, but now, thanks to the net facilities in our institute, I am growing my collection.

A FEW GOOD MEN …( Of course not in any order )

Those I would be leaving behind …...

1. My maths teacher, Tapas Sir ( Or rather, Tapasbabu, the Bengali way! ). An enormous source of inspiration and positivity. He kept faith in me, when even I was losing faith on myself.

2. My chemistry teacher, Samirbabu, the ‘headsir’ of our school. An encyclopedia, a genius of good-old type chemistry, though maybe tad weak on the MCQ front ( for muggles, MCQ means multiple choice questions, favorite in India for competitive exams ).

3. My bio teacher, Abhjitbabu. I’d call him the human face of biology. Not flawless, but more importantly, a person to quickly concede any error if pointed, he taught the subject with a certain passion, and curious nuggets of information.

4. P.G.Kale, a professor at the bio camp. An endearing fellow to all of us there. His heartfelt cordiality impressed me.

Those I would be going with ……

5. B.V.Rao – Our probability teacher in ISI. Well, he’d top this list. You can’t dislike anything about this short, elderly person. A world-famous probabilist and measure-theorist, he’s literally a man with no enemies, a perfect gentleman, a great teacher, and above all, very close to the students. He’d be in the ‘a few good men’ list of each and every ISI-an, I can bet.

6. Saurabh Ghosh – Our young, short and chubby Statistics teacher, dear ‘sau-daa’ of us all. Again someone close to students, yet very professional and straightforward in academic matters.

ON PERSONAL FRONT …

I have grown up.

Let $latex E=MS\cap AB,F=MT\cap AB$, two lines passing through E and F respectively and perpendicular to AB intersect OS and OT at X and Y respectively. A line through C cuts the circle O at P and Q ,let $latex R=MP\cap AB$ , Z be the center of the circumcircle of $latex \Delta PQR$ .Show that X,Y,Z are collinear.
2. A national number $latex x$ is called “good” if it satisfies: $latex x=p/q>1$ with $latex p,q$ being positive integers, $latex \gcd (p,q)=1$ and there exists constant numbers $latex \alpha,N$ such that for any integer $latex n\geq N$,

$latex |\{x^n\}-\alpha|\leq\dfrac{1}{2(p+q)}$

Find all the “good” numbers.

3.There are $latex 63$ points arbitrarily on the circle $latex C$ with its diameter being $latex 20$. Let $latex S$ denote the number of triangles whose vertices are three of the $latex 63$ points and the length of its sides is no less than $latex 9$. Fine the maximum of $latex S$ .

Day 2

4. Fine all functions $latex f:\mathbb{Q}^+\to\mathbb{Q}^+$ such that

$latex f(x)+f(y)+2xyf(xy)=\dfrac{f(xy)}{f(x+y)}\, \forall x,y\in\mathbb{Q}^+$.

.Here we denote by $latex \mathbb{Q}^+$ be the positive rational number set.

5. Let $latex x_1,\cdots x_n(n>1)$ be $latex n$ real numbers satisfing:

$latex A=|x_1+\cdots x_n|\not =0$ and $latex B=\max_{1\leq i<j\leq n}|x_j-x_i|\not =0$. Prove that for any $latex n$ vectors $latex \vec{\alpha_i}$ in the plane, there exists a permutations $latex (k_1,\cdots,k_n)$ of the numbers $latex (1,\cdots,n)$ such that $latex |\sum_{i=1}^nx_{k_i}\vec{\alpha_i}|\geq\dfrac{AB}{2A+B}\max_{1\leq i\leq n}|\vec{\alpha_i}|.$

6.Let $latex n$ be a positive integer, let $latex A$ be a subset of $latex \{1,2,\cdots,n\}$ ,satisfing for any two numbers $latex x,y\in A$ ,the least common multiple of $latex x,y$ not more than $latex n$.

Show that $latex |A|\leq 19\sqrt{n}+5$.

$latex p(0) = 0, p(1) = \dfrac{1}{2}$,

$latex p(2) = \dfrac{2}{3},\cdots, p(n) =\dfrac{n}{n+1}$ . Find $latex p(n+1)$.

Solution 1 .

If $latex q(x)=(x+1)p(x)-x$ then $latex \deg q=n+1$ and $latex q(k)=0\,\forall k=\overline{0,n}$ , therefore $latex q(x)\equiv a\cdot x\cdot (x-1)\cdots (x-n)\, (1)$, from this, setting $latex x=-1$ we have $latex a=\dfrac{(-1)^{n+1}}{(n+1)!}\, (2)$.

By $latex (1)$ and $latex (2)$ we have $latex q(n+1)=(-1)^{n+1}$, therefore $latex p(n+1)=\dfrac{(-1)^{n+1}+n+1}{n+2} $.

Solution 2.

By Lagrange’s Interpolation Formula, we have $latex p(x)\equiv\sum_{i=0}^np(i)\cdot \prod_{j\not =i}\dfrac{x-j}{i-j}$.

From above, if setting $latex x=n+1$ then

$latex p(n+1)=\dfrac{1}{n+2}\cdot\sum_{i=0}^ni\cdot\binom{n+2}{i+1}\cdot (-1)^{n-i}$

$latex =\dfrac{1}{n+2}\cdot\sum_{k=1}^{n+1}(k-1)\cdot\binom{n+2}{k}\cdot (-1)^{n-k+1}$

$latex =-\dfrac{1}{n+2}\cdot (\sum_{k=0}^{n+2}k\cdot\binom{n+2}{k}\cdot (-1)^{n-k+2}-\\ (n+2)-\sum_{k=0}^{n+2}(-1)^{n-k+2}\cdot\binom{n+2}{k}+ (-1)^{n+2}+1)$

$latex =-\dfrac{1}{n+2}\cdot((\sum_{k=0}^{n+2}\binom{n+2}{k}\cdot x^k\cdot (-1)^{n-k+2})'_{x=1}-\\ (n+2)-(1-1)^{n+2}+(-1)^{n+2}+1) $

$latex =-\dfrac{1}{n+2} \cdot\left(((x-1)^{n+2})'_{x=1}-n-1+(-1)^{n+2}\right)$

$latex =\dfrac{(-1)^{n+1}+n+1}{n+2} \,\Box$.

Proof. WLOG, assume that $latex a = 0$ and $latex n\geq 3$, then we need to prove $latex \sum_{k = 0}^n( - 1)^{n + k}\binom{n}{k}( k)^n = n!\, (*)$. By Lagrange's Interpolation Formula, we have $latex x^n - (x - 1)(x - 2)\cdots (x - n) \equiv \sum_{k = 1}^nk^n\cdot\prod_{i\not = k}\dfrac{x - i}{k - i}\, (**)$. Now, in $latex (**)$, setting $latex x = 0$ we have $latex ( - 1)^{n + 1}\cdot n! = \sum_{k = 1}^nk^n\cdot \dfrac{1}{k}\cdot \dfrac{( - 1)^{n - 1}\cdot n!}{(k - 1)!\cdot (n - k)!\cdot ( - 1)^{n - k}}$, from this and $latex \binom{n}{k} = \dfrac{n!}{k!\cdot (n - k)!}$ we have $latex (*)$ is true. $latex \Box$