2025年台湾数学奥林匹克（TMO）竞赛试题（PDF版，无答案）

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,