2025年台湾数学奥林匹克(TMO) (2025年2月5日) 时长：4小时 分值：每题7分，满分35分. Problem 1.Given triangle ABC,let 1,J,I be its incenter,the excenter with respect to A,and circumcircle,respectively.Let A'be the antipodal point of A on I,and M be the midpoint of the are BAC on I.Let X be the reflection of A across A',and Y be the reflection of A across M.Prove that IJXY are concyclic. 题1.设三角形ABC的内心为L,角A所对的旁心为J,外接圆为「.令A'为A在P上 的对径点，M为T上弧BAC的中点.设点A关于点A和点M的对称点分别为X和Y. 证明：I,,X,Y四点共圆. Problem 2.Let a,b.c,d be four positive reals such that abc+abd+acd+bcd 1.Determine all possible values for (ab cd)(ac+bd)(ad +bc). 题2.设正实数a,b,c,d满足abc+abd+acd+bcd=1,试确定 (ab +cd)(ac+bd)(ad +bc) 的所有可能值。 Problem 3.For any pair of coprime positive integers a and b,define f(a,b)to be the smallest nonnegative integer k such that b divides ak+1.Prove that if a and b are coprime positive integers satisfying f(a,b)-f(b,a)=2, then there exists a prime number p such that p2 divides a+b. 题3.对任意一对互质的正整数a和b,定义f(a,b)为使得b整除ak+1的最小非负整数 k.证明：若a与b是互质的正整数，且满足 f(a,b)-f(b,a)=2,
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