គណិតវិទ្យា​អូឡាំពិច​អាស៊ីប៉ាស៊ីភិច​២០០៥


Problem 1. Prove that for every irrational real number a, there are irrational real numbers
b and b_0 so that a + b and ab_0 are both rational while ab and a + b_0 are both irrational.

Problem 2. Let a; b and c be positive real numbers such that abc = 8. Prove that
\displaystyle \frac{a^2}{\sqrt{(1+a^3)(1+b^3)}}+\frac{b^2}{\sqrt{(1+b^3)(1+c^3)}} +\frac{c^2}{\sqrt{(1+c^3)(1+a^3)}} \geq \frac{4}{3}

Problem 3. Prove that there exists a triangle which can be cut into 2005 congruent
triangles.

Problem 4. In a small town, there are n \times n houses indexed by (i, j) for 1 \leq i, j \leq n with (1, 1) being the house at the top left corner, where i and j are the row and column indices, respectively. At time 0, a fire breaks out at the house indexed by (1, c), where c \leq \frac{n}{2}.
During each subsequent time interval \left[t;t+1\right], the fire fighters defend a house which is not
yet on fire while the fire spreads to all undefended neighbors of each house which was on
fire at time t. Once a house is defended, it remains so all the time. The process ends when
the fire can no longer spread. At most how many houses can be saved by the fire fighters?
A house indexed by (i; j) is a neighbor of a house indexed by (k; l) if |i-k| + |j-l| = 1.

Problem 5. In a triangle ABC, points M and N are on sides AB and AC, respectively, such that MB = BC = CN. Let R and r denote the circumradius and the inradius of the triangle ABC, respectively. Express the ratio MN=BC in terms of R and r.

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