Compete · Serbia
Okruzno takmicenje (Regional round) 1999
1A convex pentagon \(A_1A_2A_3A_4A_5\) is given. Let \(B_1\), \(B_2\), \(B_3\), \(B_4\) be the midpoints of the sides \(A_1A_2\), \(A_2A_3\), \(A_3A_4\), \(A_4A_5\), in that order, and let \(M\) be the …2The polynomial \(P\) is given by \[ P(x) = x^{2000} - 2000x^{1999} + 2000x^{1998} - \cdots + 2000x^{2} - 2000x + 2000 . \] Compute \(P(1999)\).3How many pairs \((x, y)\) of rational numbers satisfy \(2x^{2} + 5y^{2} = 1\)?4A set \(A\) is given. Among its subsets a relation \(\sim\) is defined by \[ X \subseteq A, \quad Y \subseteq A, \qquad X \sim Y \iff X \cap Y \neq \varnothing . \] Determine whether \(\sim\) is reflexive, …5Let \(M\) be an interior point of a parallelogram \(ABCD\). Prove that \[ MA + MB + MC + MD < \text{the perimeter of } ABCD . \]
Work through the paper in order - each problem opens in the workspace with this paper as its trail.