※ 引述《hiei81 (最爱)》之铭言： ※ 引述《chaogold (GREECEimoimcomin￾N￾N5)》之铭言： : Here are the 6 problems of IMO 2004 (july 12-13 in Athen) : Problem 1 : ABC is acute angle triangle with AB<>AC. The circle with diameter BC : intersects the lines AB and AC respectively at M and N. O is the : midpoint of BC. The bisectors of <BAC and <MON intersect at R. : Prove that the circumcircles of thev triangles BMR and CNR have a : common point lying on the line BC. First note RM=RN but ⊿AMR不全等於⊿ANR => A、M、R、N共圆 => ∠AMR+∠ANR=180' => BMR, CNR外接圆共点於BC上某一点 : Problem 2 : Find all polynomials f with real coefficients such that, for all : reals a,b,c such that ab+bc+ca = 0, we have the relation : f(a-b) + f(b-c) + f(c-a) = 2 f(a+b+c) Let a=b=c=0 => 3f(0) = 2f(0) => f(0)=0 Let b=c=0 => f(a)+f(0)+f(-a)=2f(a) => f is an even function => f(x)= Sigma a x^2k 且a = 0 2k 0 Let s=a-b, t=b-c => c-a= -s-t且a+b+c= √(s^2+st+t^2) => f(s)+f(t)+f(s+t) = 2f(√(s^2+st+t^2)) 因偶函数，可假设s>0, t>0 Let g(x)= Sigma a x^k => f(x)=g(x^2) 2k => g(s^2)+g(t^2)+g(s^2+2st+t^2)=2g(s^2+st+t^2) ... (1) Let s=t => 2g(s^2)+g(4s^2) = 2g(3s^2) if deg(g)>2 for sufficient large s，左式>右式 => deg(g)=2 or 1 => g(x)= ax^2+bx代回(1)均成立 Therefore, f(x) = ax^4+bx^2, a,b属於R : Problem 3 : Define a "hook" to be a figure made up of six unit squares as shown : in the figure below, or any of the figures obtained by rotations and : reflections to this figure. : [ ][ ][ ] : [ ] [ ] : [ ] : Determine all mxn rectangles that can be covered without gaps and : without overlaps with hooks such that no point of a hook covers area : outside the rectangle Guess: 3N*4M , N,M为自然数 : Problem 4 : Let n>=3 be an integer. Let t[1],...,t[n] be positive real numbers : such that : n^2+1>(t[1]+...+t[n])(1/t[1]+...+1/t[n]) : Show that, for all distinct i,j,k, t[i],t[j],t[k] are the side : lengths of a triangle Without loss of generality, assume that t[1]<=t[2]<=...<=t[n] Let S=(t[1]+...+t[n])(1/t[1]+...+1/t[n]) k+1 1 When n=3, let t[2]=kt[1] => S=((k+1)t[1]+t[3])(----- + -----) k+1 t[3] t[1] kt[1] t[3] =(k+1)^2/k+1 + --------- + (k+1) ---- k t[1] t[3] Let t[3]=ut[1]=> u>=k>=1，且由微分知(k+1)u/k + (k+1)/u 在u=√k时有最小值 又k>=1 => k>=√k => u>=√k =>u越大，S越大 当u=1+k时 S=(2+2k)(1+1/k+1/(k+1))=6+2k+2/k >= 6+4 = 3^2+1 => u>=1+k时，S>=n^2+1，但已知S<n^2+1，故u<1+k => t[3]<t[1]+t[2]三者可为三角形之三边 考虑原题，仅需证明t[n]<t[1]+t[2]即可 用数学归纳法，n=3时，已知原式成立，设n=k时，原式成立，则n=k+1时 设S'表示1至k项，S表示1至k+1项，若S'>=n^2+1 => S=S'+1+ t[k+1](1/t[1]+1/t[2]+...+1/t[k])+(t[1]+t[2]+...+t[k])/t[k+1] t[k+1] t[1] t[k+1] t[k] =S'+1+ (------ + ------) + ... + (------ + ------) t[1] t[k+1] t[k] t[k+1] >=(k^2+1)+1+2+2+2+...+2 = k^2+2k+1 +1 = (k+1)^2+1 --><-- 故S'<n^2+1 => 由归纳假设知t[k]<t[1]+t[2] 若t[k+1]>=t[1]+t[2]，注意由柯西不等式，S'>=k^2 => S=S'+1+ t[k+1](1/t[1]+1/t[2]+...+1/t[k])+(t[1]+t[2]+...+t[k])/t[k+1] >= k^2+1+t[k+1]/t[1]+t[k+1]/t[2]+(t[1]+t[2])/t[k+1] t[k+1] t[3] t[k+1] t[k] + (------ + ------) + ... + (------ + ------) t[3] t[k+1] t[k] t[k+1] >= k^2+1+(t[1]+t[2])/t[1]+ (t[1]+t[2])/t[2]+1 + 2 + 2 + ... +2 =k^2+1+1+1+1+t[2]/t[1]+t[1]/t[2]+2(k-2) >= k^2+2k+2 = (k+1)^2 + 1 得证 : Problem 5 : In a convex quadrilateral ABCD, the diagonal BD bisects neither <ABC : nor <CDA. A point P lies inside ABCD and satisfies <PBC = <DBA and : <PDC = <BDA. : Prove that ABCD are concyclic if and only if AP = CP. 由已知，B-P-C不共线 若<A=<C=90' 且<QBC=<DBA, <RDC=<BDA => BQ平行RD => P点不存在 所以<A≠<C≠90' 不妨设<A > 90' => <A > <C => <ABD+<ADB < <CDB+<DBC => <ABD < <CBD或<ADB < <CDB 不妨设<ABD < <CBD，若<ADB > <CDB => P点不存在，故<ADB < <CBD (=>) 由共圆性质证得<BPD = 2<C，令圆心O，圆上两点E、F且B-P-E、D-P-F => 弧BAD = 弧ECF(圆内角) => 弦BE=弦DF => BE、DF等弦心距 <--> => O在<DPE之分角线上 => OP 过弧BF的中点M，又<BDA=<PDC => M亦为弧AC之中点 => OP直线过弧AC之中点 => OP直线垂直平分AC => AP=CP (<=) 先睡觉，有空再想... : Problem 6 : A positive integer is alternating if every two consecutive digits in : its decimal representation are of different parity. Find all : positive integers n such that n has a multiple which is alternating. -- 永不平息的海浪一波一波打来人生的 挑战与试炼 曾经 我是岩石，尽一切力量承受着 於是 海浪在我身上留下一道道的伤痕 曾经 我是沙子 於是我 随。波。逐。流 现在 我是海洋 看着一道道海浪打去 沈淀下最美丽的结晶 --

※ 发信站: 批踢踢实业坊(ptt.cc) ◆ From: 140.112.18.71 -- 永不平息的海浪一波一波打来人生的 挑战与试炼 曾经 我是岩石，尽一切力量承受着 於是 海浪在我身上留下一道道的伤痕 曾经 我是沙子 於是我 随。波。逐。流 现在 我是海洋 看着一道道海浪打去 沈淀下最美丽的结晶 --

※ 发信站: 批踢踢实业坊(ptt.cc) ◆ From: 140.112.18.71