》╋|||《数学オリンピック 16》╋|||《
【1】
Let n be a positive integer and let a[1], a[2], a[3], …, a[k] (k≧2) be distinct integers in the set 1, 2, …, n such that n divides a[i](a[i+1] - 1) for i = 1, 2, …, k-1.
Prove that n does not divide a[k](a[1] - 1).
(Proposed by Ross Atkins, Australia)
Let n be a positive integer and let a[1], a[2], a[3], …, a[k] (k≧2) be distinct integers in the set 1, 2, …, n such that n divides a[i](a[i+1] - 1) for i = 1, 2, …, k-1.
Prove that n does not divide a[k](a[1] - 1).
(Proposed by Ross Atkins, Australia)
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405 ：IMO 2009 Bremen, Germany Day 1 - 15 July 2009：2009/07/16(木) 10:34:04
【2】
Let ABC be a triangle with circumcentre 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.
(Proposed by Sergei Berlov, Russia)
Let ABC be a triangle with circumcentre 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.
(Proposed by Sergei Berlov, Russia)
406 ：IMO 2009 Bremen, Germany Day 1 - 15 July 2009：2009/07/16(木) 10:34:58
【3】
Suppose that s[1], s[2], s[3], … is a strictly increasing sequence of positive integers such that the sub-sequences s[s[1]], s[s[2]], s[s[3]], … and s[s[1]+1], s[s[2]+1], s[s[3]+1], … are both arithmetic progressions.
Prove that the sequence s[1], s[2], s[3], … is itself an arithmetic progression.
(Proposed by Gabriel Carroll, USA)
Suppose that s[1], s[2], s[3], … is a strictly increasing sequence of positive integers such that the sub-sequences s[s[1]], s[s[2]], s[s[3]], … and s[s[1]+1], s[s[2]+1], s[s[3]+1], … are both arithmetic progressions.
Prove that the sequence s[1], s[2], s[3], … is itself an arithmetic progression.
(Proposed by Gabriel Carroll, USA)
407 ：１３２人目の素数さん：2009/07/16(木) 17:25:32
第１問は楽勝
第２、３問はまだ解けていません
第２、３問はまだ解けていません
408 ：１３２人目の素数さん：2009/07/16(木) 18:22:25
3問目はやはり難しい
5時間考えてるがサッパリ糸口がつかめん
5時間考えてるがサッパリ糸口がつかめん
409 ：IMO 2009 Bremen, Germany Day 2 - 16 July 2009：2009/07/17(金) 00:51:17
【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 ∠BEK = 45°.
Find all possible values of ∠CAB.
(Jan Vonk, Belgium, Peter Vandendriessche, Belgium and Hojoo Lee, Korea)
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 ∠BEK = 45°.
Find all possible values of ∠CAB.
(Jan Vonk, Belgium, Peter Vandendriessche, Belgium and Hojoo Lee, Korea)
410 ：IMO 2009 Bremen, Germany Day 2 - 16 July 2009：2009/07/17(金) 00:52:05
【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.)
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.)
411 ：IMO 2009 Bremen, Germany Day 2 - 16 July 2009：2009/07/17(金) 00:52:53
【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 point in M.
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 point in M.