มาแล้วครับ. IMO 2005 วันแรก ข้อแรกเรขาคณิต , ข้อสองทฤษฎีจำนวน , ข้อสาม อสมการ ใครจะลองคิดข้อไหนดู ก็ลองได้เลยครับ.
\[ \bf{IMO \,\,2005 \,\, Day 1} \]
1. Six points are chosen on the sides of an equilateral triangle \( ABC, A_1, A_2\,\) on\( \,BC, B_1, B_2 \,\)on\( \,CA\,\) and\( \,C_1, C_2\,\) on \(\,AB\,\) , such that they are the vertices of a convex hexagon \(\,A_1A_2B_1B_2C_1C_2\,\) with equal side lengths.
Prove that the lines \(\,A_1B_2, B_1C_2\,\) and \(\,C_1A_2\,\) are concurrent
2.Let \(a_1,a_2,\ldots \)be a sequence of integers with infinitely many positive and negative terms. Suppose that for every positive integer \(n\) the numbers \(a_1,a_2,\ldots,a_n \)leave \(n\) different remainders upon division by \(n\).
Prove that every integer occurs exactly once in the sequence \(a_1,a_2,\ldots.\)
3.Let \(x,y,z\) be three positive reals such that \(xyz\geq 1.\) Prove that \[ \displaystyle \frac { x^5-x^2 }{x^5+y^2+z^2} + \frac {y^5-y^2}{x^2+y^5+z^2} + \frac {z^5-z^2}{x^2+y^2+z^5} \geq 0\]