Path: cnn.Princeton.EDU!cbgw1.lucent.com!cbgw2.lucent.com!worldnet.att.net!hunter.premier.net!hammer.uoregon.edu!newsfeed.kreonet.re.kr!not-for-mailFrom: Oum Sangil <sangil@hugsvr.kaist.ac.kr>Newsgroups: sci.mathSubject: 15th Korean Undergraduate Math Competition (1996)Date: 10 Feb 1997 14:39:46 +0900Organization: HUG in KAISTLines: 70Message-ID: <x5ohdtmd7h.fsf@hugsvr.kaist.ac.kr>NNTP-Posting-Host: hugsvr.kaist.ac.krX-Newsreader: Gnus v5.3/Emacs 19.34To keep my promise, I post this set of problems. 1996.11.24.I have no solution of problem 5 in the morning. Can anybody solve that?<Morning, 6 problems, 2hours>1. Prove or disprove: There is a sequence of positive real numbers {a_n}  such that        both \sum_{n=1}^\infty a_n/(n^2) and \sum_{n=1}^\infty 1/a_n  converges.2. A matrix N is a nilpotent if there is an integer k>1 such that N^k=0.   For any matrix A and nilpotent N, if AN=NA then                 det(A+N)=det(A).3. \lim_{n\to\infty} \sum_k=1^{2n} (1/{2n\choose k} )^n = ?4. Define f(x)=\int_0^{\pi/2}  t^x \cos t dt where x>=0. Show that there  exist x such that f(x)=4/5.5. There is a twice-differentiable function f:(0,1]-> R^+  (R^+ is the set  of postive real numbers.) satisfying the following:  i) f(1)<=1  ii) f(x)/x is a increasing function.  iii)  f''(x)>=0  Let 0<a<=b<1. Prove the following:   1) 1+f(a)-f(b) >= ln(b)/ln(a)   2) a^f(a) + b^f(b) >= a^f(b) + b^f(a)6. Prove that there is no integer n such that f(n)=n^3+7 is a square number.<Afternoon, 6 problems, 2hours>1. A positive integer k is given. Let n be a postive integer such that  2^{k+1} divides n. Let S be the set {1^k, 2^k, 3^k, ... , n^k}. Prove that  there exist subsets A, B such that        A \union B = S,         A \cap   B = \emptyset, and         \sum_{x\in A} x = \sum_{x\in B} x.2. Let f(x)=x^5 + x + 1.             | 1 0  -2 |        A = | 2 -1 -2 |            | 0  0  3 |   Give an example of X, 3\times 3 matrix which satisfies f(X)=A.3. A=(a_ij) is a n\times n matrix. If \sum_{j=1}^n j a_ij = i, then    A has an eigenvalue 1.4. For all postivie integer n, prove this inequality:     | 1- 1/2 + 1/3 - ... + (-1)^n 1/(n+1)  - ln(2) | <= 1/{2(n+1)}5. Let n>1 be an odd integer.   1) Show that if n is a prime number, then n cannot be represented as a sum    of 3 continuous positive integers.  2) Show that if n cannot be represented as a sum of 3 continuous positive    integers, then n is a prime number.6. Let S(x)=4[x]-2[2x]+1. Let f(x) be a uniformly continuous funtion.  Show that if n is a positive integer,        \lim_{n\to\infty} \int_0^1 f(x)S(nx) dx = 0.  Note that [x] is the greatest integer which is not larger than x.   -- Oum Sang-ilKAIST Mathematical Problem Solving Group, HUGhttp://hugsvr.kaist.ac.kr/~sangil/sangil@math.kaist.ac.kr