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Math Forum :: View topic – Functional equation

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Arne

Joined: 21 Mar 2004Posts: 1

Location: Belgium

Posted: Sun May 09, 2004 10:54 pm    Post subject: Functional equation

Determine all real values of a for which there exists a UNIQUE function f such that
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Math Forum :: View topic – Problem in Trigonometry and Complex numbers

Problem in Trigonometry and Complex numbers
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Michael

Joined: 19 Jan 2004Posts: 17

Posted: Tue Oct 19, 2004 8:45 pm    Post subject: Problem in Trigonometry and Complex numbers

How to show that
[cos(2pi/13)+cos(10pi/13)][cos(4pi/13)+cos(6pi/13)][cos(8pi/13)+cos(12pi/13)] = 1/8 ?
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Peter

Frequent VisitorJoined: 18 Jan 2004Posts: 110

Location: Hong Kong

Posted: Fri Oct 22, 2004 1:20 am    Post subject: Trigonometry

= , by

, by using successively
._________________

Therefore do not worry about tomorrow, for tomorrow will worry about itself. Each day has enough trouble of its own.
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Math Forum :: View topic – prime ideal

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ckcdog

Joined: 23 Apr 2004Posts: 4

Posted: Sun Oct 17, 2004 1:25 am    Post subject: prime ideal

May I ask is there any relationship between the prime ideal of A and the prime ideal of Aq, where Aq denotes the localization at q (q is another prime ideal of A)?

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Math Forum :: View topic – past CMO

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yptsoi

Frequent VisitorJoined: 24 Jan 2004Posts: 34

Location: Hong Kong

Posted: Mon Oct 04, 2004 6:51 pm    Post subject: past CMO

Can anyone who joined CMO before tell me the cutting scores of gold, silver and bronze awards? Thanks.

Carto

Frequent Visitor
Joined: 29 Oct 2003Posts: 321

Location: Hong Kong

Posted: Wed Oct 13, 2004 11:38 am    Post subject:

年年唔同嫁喎。我果年（99 年，北京）30 分都有三等奬。_________________

世上沒有完美的人完美的事，而我們的責任就是要令自己的表演達至最精彩最完美。所以魔術一直都沒有停頓下來，與時代一起進步去追求無止境的完美。

kelvinkklee

Frequent VisitorJoined: 20 Jan 2004Posts: 41

Location: Diocesan Boys’ School

Posted: Wed Oct 13, 2004 4:23 pm    Post subject:

what is the full mark of one problem?

Andy

Frequent VisitorJoined: 28 Oct 2003Posts: 265

Location: Hong Kong

Posted: Wed Oct 13, 2004 10:41 pm    Post subject:

每題 21 分，6 題共 126 分。 可是根據我的經驗，每題的分數都必定是 3 的倍數。簡單來說，這和 IMO 的每題 7 分的制度一樣，只是分數都乘了 3 而已。_________________

Patience and tolerance are necessarily demanded Year-round.

yptsoi

Frequent VisitorJoined: 24 Jan 2004Posts: 34

Location: Hong Kong

Posted: Wed Oct 13, 2004 11:11 pm    Post subject:

I see… how about 二等獎 or 一等獎 in general? For example what’re they in last year?

Tc

Frequent VisitorJoined: 25 Oct 2003Posts: 132

Location: Hong Kong

Posted: Fri Oct 15, 2004 12:03 am    Post subject:

If you say about last year… bronze: 5*3 = 15 silver: 14*3 = 42

gold (i don’t know if it is correct): 28*3 = 84

yptsoi

Frequent VisitorJoined: 24 Jan 2004Posts: 34

Location: Hong Kong

Posted: Fri Oct 15, 2004 5:30 pm    Post subject:

I see… thanks Tc. Last question: How many people got gold, silver and bronze respectively in last year?

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Math Forum :: View topic – past CMO

Author Message

yptsoi

Frequent VisitorJoined: 24 Jan 2004Posts: 34

Location: Hong Kong

Posted: Mon Oct 04, 2004 6:51 pm    Post subject: past CMO

Can anyone who joined CMO before tell me the cutting scores of gold, silver and bronze awards? Thanks.

Carto

Frequent Visitor
Joined: 29 Oct 2003Posts: 321

Location: Hong Kong

Posted: Wed Oct 13, 2004 11:38 am    Post subject:

年年唔同嫁喎。我果年（99 年，北京）30 分都有三等奬。_________________

世上沒有完美的人完美的事，而我們的責任就是要令自己的表演達至最精彩最完美。所以魔術一直都沒有停頓下來，與時代一起進步去追求無止境的完美。

kelvinkklee

Frequent VisitorJoined: 20 Jan 2004Posts: 41

Location: Diocesan Boys’ School

Posted: Wed Oct 13, 2004 4:23 pm    Post subject:

what is the full mark of one problem?

Andy

Frequent VisitorJoined: 28 Oct 2003Posts: 265

Location: Hong Kong

Posted: Wed Oct 13, 2004 10:41 pm    Post subject:

每題 21 分，6 題共 126 分。 可是根據我的經驗，每題的分數都必定是 3 的倍數。簡單來說，這和 IMO 的每題 7 分的制度一樣，只是分數都乘了 3 而已。_________________

Patience and tolerance are necessarily demanded Year-round.

yptsoi

Frequent VisitorJoined: 24 Jan 2004Posts: 34

Location: Hong Kong

Posted: Wed Oct 13, 2004 11:11 pm    Post subject:

I see… how about 二等獎 or 一等獎 in general? For example what’re they in last year?

Tc

Frequent VisitorJoined: 25 Oct 2003Posts: 132

Location: Hong Kong

Posted: Fri Oct 15, 2004 12:03 am    Post subject:

If you say about last year… bronze: 5*3 = 15 silver: 14*3 = 42

gold (i don’t know if it is correct): 28*3 = 84

yptsoi

Frequent VisitorJoined: 24 Jan 2004Posts: 34

Location: Hong Kong

Posted: Fri Oct 15, 2004 5:30 pm    Post subject:

I see… thanks Tc. Last question: How many people got gold, silver and bronze respectively in last year?

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Math Forum :: View topic – algebra problem

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sheep

Joined: 14 Oct 2004Posts: 1

Posted: Thu Oct 14, 2004 12:49 pm    Post subject: algebra problem

Can somebody help me ? ^.^ 1. Suppose G is a cyclic group with order n,

a. Prove that if r,s are relatively prime, then Crxs is isomorpsic to CrXCs.

b. Prove that if r is prime, then Crxr is not isomorphic to CrXCr.

Thanks a lot !

Milton

Frequent VisitorJoined: 27 Oct 2003Posts: 176

Location: HKUST Math

Posted: Thu Oct 14, 2004 11:53 pm    Post subject:

Let and be the generator of and . (a) follows from noting that is the generator of , which is of order .

To prove (b), note that is not cyclic as its elements have order not greater than . It seems we do not need to be prime here.

_________________Sometimes Truth is meanlingless;What means is how you believe in.

偶爾，真相並沒有意義；意義在於你怎樣相信。

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Math Forum :: View topic – past CMO

Author Message

yptsoi

Frequent VisitorJoined: 24 Jan 2004Posts: 34

Location: Hong Kong

Posted: Mon Oct 04, 2004 6:51 pm    Post subject: past CMO

Can anyone who joined CMO before tell me the cutting scores of gold, silver and bronze awards? Thanks.

Carto

Frequent Visitor
Joined: 29 Oct 2003Posts: 321

Location: Hong Kong

Posted: Wed Oct 13, 2004 11:38 am    Post subject:

年年唔同嫁喎。我果年（99 年，北京）30 分都有三等奬。_________________

世上沒有完美的人完美的事，而我們的責任就是要令自己的表演達至最精彩最完美。所以魔術一直都沒有停頓下來，與時代一起進步去追求無止境的完美。

kelvinkklee

Frequent VisitorJoined: 20 Jan 2004Posts: 41

Location: Diocesan Boys’ School

Posted: Wed Oct 13, 2004 4:23 pm    Post subject:

what is the full mark of one problem?

Andy

Frequent VisitorJoined: 28 Oct 2003Posts: 265

Location: Hong Kong

Posted: Wed Oct 13, 2004 10:41 pm    Post subject:

每題 21 分，6 題共 126 分。 可是根據我的經驗，每題的分數都必定是 3 的倍數。簡單來說，這和 IMO 的每題 7 分的制度一樣，只是分數都乘了 3 而已。_________________

Patience and tolerance are necessarily demanded Year-round.

yptsoi

Frequent VisitorJoined: 24 Jan 2004Posts: 34

Location: Hong Kong

Posted: Wed Oct 13, 2004 11:11 pm    Post subject:

I see… how about 二等獎 or 一等獎 in general? For example what’re they in last year?

Tc

Frequent VisitorJoined: 25 Oct 2003Posts: 132

Location: Hong Kong

Posted: Fri Oct 15, 2004 12:03 am    Post subject:

If you say about last year… bronze: 5*3 = 15 silver: 14*3 = 42

gold (i don’t know if it is correct): 28*3 = 84

yptsoi

Frequent VisitorJoined: 24 Jan 2004Posts: 34

Location: Hong Kong

Posted: Fri Oct 15, 2004 5:30 pm    Post subject:

I see… thanks Tc. Last question: How many people got gold, silver and bronze respectively in last year?

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Math Forum :: View topic – Number of solutions (in N) of an equation

In relation with a probabilistic problem for concrete numbers I came upon the generalization of an idea and there’s a problem:

Let’s suppose we are throwing by special cubes with sides numbered serially by 1,2,…,m and the probability of toss of each number is the same according to the classical definition of probability). Let denote the sum of all numbers on the sides of all cubes shortly after the last toss. Which sum has the highest probability.

Let be random variables that represents the value of the toss on cubes in this order and . Then should be the value of the most probable sum on the sides of cubes (or in case that isn’t integer let’s take the closest two integers which E X is situated between). At last it’s clear that , then:

But I would like to solve it more classically and exactly, so first I need to solve this:

Let be some natural numbers such that , then the esencial question is how many (orderly) somes does there exist such that each x_i is a natural number from 1 to maximally and:

.
Could anybody give me any analytic formula (depending on ) for number of all convenient some (or any hint).

Obviously, the number of all convenient somes is also the number of all adic combinations with reprise of elements such that each element can appear minimally once and maximally times.

I’ve tried to use the principle of inclusion and exclusion, but it doesn’t give me the right result.

Thank you….

Math Forum :: View topic – 2004 China Western Mathematical Olympiad (25/9/04-30/9/04)

Q5:

Lemma:

Prove of lemma:

(Induction) Cases n = 2, 3: Obvious by calculation. Suppose cases n = k – 2, k – 1 are true. When n = k,

Thus the lemma is true. Thus

Thus .

(It doesn’t need to be written in “expanded form”, does it? )

Q8:

Suppose x = a + bi, y = b + ci, z = c + ai. Hence: a = Re(x) b = Re(y) c = Re(z) And the original statement becomes:

Assume . Then:

.

Notice that since a,b,c > 0, we have .

Moreover, the cosine values do not change if we scale the whole system, so let’s assume |x|, |y|, |z|