លំហាត់ប្រលងគណិតពិភពលោកឆ្នាំ២០០៨ សំរាប់ថ្ងៃទី១ ( IMO 2008 )

Madrid-Spain

Version: English

First Day
Wednesday , July 16 , 2008

Probleam 1:

Let H be the orthocenter of an acute-angled triangle ABC. The circle Γ_A centered at the midpoint of BC and passing through H intersects the sideline BC at points A₁ and A₂. Similarly, define the points B₁, B₂, C₁ and C₂.

Prove that six points A₁ , A₂, B₁, B₂, C₁ and C₂ are concyclic.

Probleam 2:

(i) If x, y and z are real numbers, different from 1, such that xyz = 1 prove that

(ii) Prove that equality case is achieved for infinitely many triples of rational numbers x, y and z.

Probleam 3:

Prove that there are infinitely many positive integers n such that n²+1 has a prime divisor greater than

~ by khmerempire on July 16, 2008.