作者boggart0803 (幻形怪)
看板IMO_Taiwan
标题[问题] IMO 2008 day1
时间Wed Jul 16 23:15:03 2008
Problem 1
Let H be the orthocenter of an acute-angled triangle ABC.
The circle G_A centered at the midpoint of BC and passing
through H intersects the sideline BC at points A_1 and A_2.
Similarly, define the points B_1, B_2, C_1, C_2.
Prove that six points A_1, A_2, B_1, B_2, C_1, C_2 are concyclic.
Problem 2
(i) If x, y and z are three real numbers, all different
from 1, such that xyz=1, then prove that Σ(x^2/(x-1)^2)>=1
(ii) Prove that equality is achieved for infinitely many
triples of rational numbers x, y and z.
Problem 3
Prove that there are infinitely many positive integers n
such that n^2+1 has a prime divisor greater than 2n+sqrt(2n)
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※ 编辑: boggart0803 来自: 59.117.196.96 (07/17 00:26)
1F：推 myflame:疑~哪边有修改@_@ 07/17 00:30
2F：推 Dawsen:学弟加油 07/17 02:52
3F：→ boggart0803:小小错字XD 07/17 08:12
4F：推 Dawsen:第二题蛮难的，等号成立部分除了硬凑还有别的解法吗？ 07/20 11:51
5F：→ boggart0803:找a小题算几的成立条件?? 07/20 16:59
6F：推 LimSinE:2.(b)其实是常规题 07/26 16:55