IMO 2009

Problem 1. Let n be a positive integer and let a1, a2, a3, ..., ak ( k≥ 2) be distinct integer in the set { 1, 2, ..., n} such that n divides ai(ai+1-1) for i = 1, 2, ..., k-1. Prove that n does not divide ak(a1-1).

Problem 2. Let ABC be a triangle with circumcenter O. The points P and Q are interior points of the sides CA and AB respectively. Let K, L and M be the midpoints of the segments BP, CQ and PQ. respectively, and let Γ be the circle passing through K, L and M. Suppose that the line PQ is tangent to the circle Γ. Prove that OP = OQ.

Problem 3. Suppose that s1, s2, s3, ... is a strictly increasing sequence of positive integers such that the sub-sequences ss_{1}, ss_{2}, ss_{3}, ... and ss_{1+1}, ss_{2+1}, ss_{3+1}, ... are both arithmetic progressions. Prove that the sequence s1, s2, s3, ... is itself an arithmetic progression.

Problem 4. Let ABC be a triangle with AB = AC. The angle bisectors of CAB and ABC meet the sides BC and CA at D and E, respectively. Let K be the incentre of triangle ADC. Suppose that angle BEK = 45. Find all possible values of angle CAB.

Problem 5. Determine all functions f from the set of positive integers to the set of positive integers such that, for all positive integers a and b, there exists a non-degenerate triangle with sides of lengths a, f(b), abd f(b+f(a)-1).

Problem 6. Let a1, a2, ..., an be distinct positive integers and let M be a set of 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.

所有跟帖： 

• 回複：IMO 2009 -康MM- ♀ (563 bytes) () 07/24/2009 postreply 04:00:28

• China's rank -botong- ♂ (243 bytes) () 07/24/2009 postreply 08:29:10

• 回複：China's rank -於德利- ♂ (77 bytes) () 07/24/2009 postreply 19:54:05

• Problem 6 -botong- ♂ (422 bytes) () 07/24/2009 postreply 10:32:52

• 通常第六題不是最難的 -累死算了- ♂ (148 bytes) () 07/26/2009 postreply 01:05:48

• 通常第六題是最難的 -botong- ♂ (408 bytes) () 07/26/2009 postreply 09:25:21

• 有點問題吧 -dynamic- ♂ (226 bytes) () 07/26/2009 postreply 22:14:34

• 回複：有點問題吧 -botong- ♂ (209 bytes) () 07/27/2009 postreply 11:01:10

• 回複：回複：有點問題吧 -dynamic- ♂ (79 bytes) () 07/27/2009 postreply 18:12:36

• 回複：回複：回複：有點問題吧 -botong- ♂ (91 bytes) () 07/28/2009 postreply 10:22:54

• Don't understand Problem 6 -dayuwang- ♂ (166 bytes) () 07/24/2009 postreply 12:29:19

• 回複：Don't understand Problem 6 -botong- ♂ (79 bytes) () 07/24/2009 postreply 12:59:16

• That makes sense. Thanks. -dayuwang- ♂ (0 bytes) () 07/24/2009 postreply 16:25:42

• Where do you see that M is a given set with n-1 points? -Noneandnone- ♂ (0 bytes) () 07/24/2009 postreply 18:32:58

• 康MM把這條件去掉了：） -botong- ♂ (78 bytes) () 07/24/2009 postreply 19:16:06

• 好家夥，前５名：中國，日本，南北朝鮮，俄羅斯 -於德利- ♂ (180 bytes) () 07/24/2009 postreply 19:52:41

• 還有一個：＃２３ 澳大利亞，５／６是中國人 -於德利- ♂ (0 bytes) () 07/25/2009 postreply 09:38:27

• 第2題 -說了就走- ♂ (29 bytes) () 07/25/2009 postreply 11:14:19

• Problem 4 -HavingFun2- ♂ (29 bytes) () 07/26/2009 postreply 11:22:39

• 回複：Problem 4 -HavingFun2- ♂ (63 bytes) () 08/18/2009 postreply 12:32:56

• problem 2 is too easy -zlinux- ♂ (230 bytes) () 07/26/2009 postreply 12:00:43

• problem 1 -屋漏痕- ♂ (1080 bytes) () 07/27/2009 postreply 10:29:08

• Problem 6 -屋漏痕- ♂ (521 bytes) () 07/27/2009 postreply 12:16:59

• 回複：Problem 6 -dynamic- ♂ (217 bytes) () 07/27/2009 postreply 18:18:33

• 回複：回複：Problem 6 -屋漏痕- ♂ (224 bytes) () 07/28/2009 postreply 05:53:51

• 回複：回複：回複：Problem 6 -dynamic- ♂ (195 bytes) () 07/28/2009 postreply 09:14:51

• 回複：回複：回複：回複：Problem 6 -屋漏痕- ♂ (79 bytes) () 07/28/2009 postreply 10:07:10

• still off a littlt bit. -屋漏痕- ♂ (0 bytes) () 07/28/2009 postreply 10:58:07

• This one might be right, but I might be wrong. -屋漏痕- ♂ (1042 bytes) () 07/28/2009 postreply 12:39:19

• The solution to problem 6 and thanks to dynamic -botong- ♂ (2810 bytes) () 07/28/2009 postreply 17:28:00

• excellent proof, needs correction in some places. -dynamic- ♂ (673 bytes) () 07/28/2009 postreply 23:44:13

• Thanks -botong- ♂ (58 bytes) () 07/29/2009 postreply 08:15:04

• Here is a new version -botong- ♂ (2693 bytes) () 07/29/2009 postreply 08:34:38

• Great! -亂彈- ♂ (0 bytes) () 07/30/2009 postreply 17:49:50