[問題] IMO 2009 day2

[boggart0803 (幻形怪)]

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
     incenter of triangle ADC. Suppose that ∠BEK=45. Find allpossible values
     of ∠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),
     and f(b+f(a)-1). (A triangle is non-degenerate if its vertices are not
     collinear.)

Problem 6. Let a_1, a_2, ..., a_n be distinct positive integers and let M be
     a set of n-1 positive integers not containing s=a_1+a_2+...+a_n. A
     grasshopper is to jump along the real axis, starting at the point 0 and
     making n jumps to the right with lengths a_1, a_2, ..., a_n in some order.
     Prove that the order can be chosen in such a way that the grasshopper
     never lands on any points in M.

--
※ 發信站: 批踢踢實業坊(ptt.cc) 
◆ From: 60.244.116.125
→ boggart0803:根據最新消息 第六題大陸隊只有一個全作出來.....
推 LimSinE:我覺得這次的3,6看起來都很有趣
推 phear:喜歡嗎?XD
推 myflame:是我誤會第五題的意思嗎...怎麼不像第五題的難度orz
→ boggart0803:他的確不太有第五題的難度XD