Math Excellence Exam – Thai Nguyen Province (Grade 10, 2012–2013)

1. a) Solve the inequality \[ (\sqrt{x+5}-\sqrt{x-3})\left(1+\sqrt{x^{2}+2x-15}\right) \ge 8. \]

   b) Solve the system of equations \[ \begin{cases} 81x^{3}y^{2} - 81x^{2}y^{2} + 33xy^{2} - 29y^{2} = 4,\\[4pt] 25y^{3} + 9x^{2}y^{3} - 6xy^{3} - 4y^{2} = 24. \end{cases} \]

2. Find all real numbers \(x\) satisfying \[ x + \left[\frac{x}{6}\right] = \left[\frac{x}{2}\right] + \left[\frac{2x}{3}\right], \] where \([t]\) denotes the greatest integer not exceeding \(t\).

3. In a convex quadrilateral \(ABCD\), suppose that \(CD = DA\) and \(\angle DAB = \angle ABC < 90^\circ\). The line through \(D\) and the midpoint of \(BC\) meets line \(AB\) at \(E\). Prove that \(\angle BEC = \angle DAC\).

4. Consider \(100\) pairwise distinct sets satisfying:

   • For any \(10\) sets among them, there exist two sets \(A, B\) such that \(A \subset B\).
   • For any three sets \(A, B, C\) among them with \(A \supset B\) and \(A \supset C\), we have \(B \subset C\) or \(C \subset B\).

   Prove that one can choose \(12\) sets \(A_1, A_2, \ldots, A_{12}\) among the given \(100\) sets such that \[ A_1 \subset A_2 \subset \cdots \subset A_{12}. \]

5. Let \(a, b, c\) be positive real numbers satisfying \[ a + 2b + 3c \ge 20. \] Find the minimum value of \[ L = a + b + c + \frac{3}{a} + \frac{9}{2b} + \frac{4}{c}. \]

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