?

Suppose φ (x) is the distance from x to the nearest integer. Define a function B(x) = Σ (0->∞) φ (2^n x)/2^n.

Find C∫(0->1) B(x) dx

C means Cauchy Integral

試求∫[0~∞] cos (t^3) dt 之值

我12:00到達計程車站﹐剛好有9個人在我前面﹐這時有一架計程車來到將第一個客人載走。接著我埋首看書﹐到12:05分時剛好又有一架計程車來到將第五個客人載走。

假定兩架計程車來到之間的時間差服從指數分配﹐平均數是180秒。則我的 95% C.I. 上車時間是甚麼?

假定有一條在x-y面上的曲線 y=f(x)﹐其中一點固定在原點。考慮一小球(質量 m) 從這條曲線位於(x,y)的地點釋放﹐則其到達原點的時間T(y)不難計算出來。現在考慮相反的問題﹐給定T(y)﹐是否可以找出一條對應的y=f(x) (或者參數式 x=f(s),y=g(s) ) 曲線出來?

具體地說﹐請以最簡單的情況T(y)=C 導出相關方程。

提示：與 Laplace Transform﹐Integral equation 有關

Let a1, a2, . . . , an be distinct positive integers and let M be a set of n − 1 positive integers not containing s = a1 +a2 + +an. A grasshopper is to jump along the real axis, starting at the point 0 and making n jumps to the right with lengths a1, a2, . . . , an in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in M.

But now at last the sacred influence

Of light appears, and from the walls of Heaven

Shoots far into the bosom of dim Night

A glimmering dawn. Here Nature first begins

Her fardest verge, and Chaos to retire,

As from her utmost works, a broken foe,

With tumult less and with less hostile din

John Milton, [1667 and 1671] PARADISE LOST, Second Book

Here Hills and Vales, the Woodland and the Plain,

Here Earth and Water seem to strive again,

Not Chaos-like together crush’d and bruis’d,

But as the world, harmoniously confus’d:

Where Order in Variety we see,

And where, tho' all things differ, all agree.

Pope, Windsor Forest

Prove that if the graph G has 2n vertices and more than n^2 edges, then G contains at least n triangles

Explain why S3=SU(2) is not the same group as S1 x S1 x S1

找出兩條曲線y=1+(x^2/4)和y=x/2之間的最短距離。

Let f:N->N and f(n+1)>f(f(n)),find the expression of f(n)

Suppose that f is a continuous real-valued function on R and suppose that for any x and y in R

f(x+y)+f(x-y)=2[f(x)+f(y)]

Provw that there is a constant a such that f(x)=ax^2 for all real x

Find the exact value of

(1+1/2^2)(1+1/3^2)...(1+1/n^2)...

假定有4男{a,b,c,d}4女{A,B,C,D}現在婚姻介紹所想撮合他們﹐他們之間的喜歡對方的程度如下。

a {1 1 2 0}

b {3 1 2 2}

c {1 2 3 4}

d {0 1 2 0}

A {0 1 1 2}

B {2 3 1 2}

C {1 0 2 3}

D {4 1 2 3}

1代表最喜歡﹐4代表最不喜歡。0代表一定不會和該人結婚。試問婚姻介紹所應如何安排這4對男女。(註：1婚姻介紹所可以不幫某人撮合。如男士d。2安排的結果應該是穩定的﹐例如a最最喜歡A而A也最最喜歡a﹐則aA應配成一對。)

Let B be a set of more than 2^(n+1)/n distinct points with coordinates of the form (1,1, . . . ,1) in n-dimensional space, with n >= 3. Show that there are three distinct points in B which are the vertices of an equilateral triangle.

Let f(x) be a polynomial with integer coefficients. Define a sequence a0, a1, . . . of integers such that a0 = 0 and an+1 = f(an) for all n >=0. Prove that if there exists a positive integer m for which am = 0 then either a1 = 0 or a2 = 0.

Show that the number of incongruent triangles with integer sides and perimeter n is given by

p(n-3 | parts in {2,3,4})

我擲一顆骰子﹐直到1-6六個數字都出現。例如

1,2,3,4,5,6 擲了6次

1,2,2,5,3,4,1,6 擲了8次

我應該至少要擲多少次才能肯定有8/9的概率會六個面都出現呢?

考慮一元二次方程x^2+px+q=0,其中p和q是實數且在[-1,1]上服從均勻分佈。求根x是實數的概率

甲國的公路系統是這樣的: 在每個交叉路口有三條公路交會。(該國的公路系統是封閉的)證明該公路系統有下列性質: 假定一司機從任一交叉路口A1 出發延著三條路中任一條到下一路口A2 向右轉開到下一路口A3。在A3 向左轉, 這樣繼續下去, 一次右轉一次左轉。那麼他最後會回到出發點A1。

請問gamma函數的最小值是如何求出來的