Compete · Serbia
Okruzno takmicenje (Regional round) 2000
1Can the plane be tiled by squares - that is, covered completely, with no two squares overlapping - in such a way that no side length is used by more than two of the squares?2Miljan and Mladen play the following game. They take turns naming divisors of \(200\), with one restriction: the number a player names must not be a divisor of any number named earlier in the game. A player …3In a triangle \(ABC\) the angle at \(A\) measures \(60^\circ\). Write \(a\), \(b\), \(c\) for the lengths of the sides \(BC\), \(CA\), \(AB\). Prove that the area of the triangle equals \[ \frac{\sqrt{3}}{4}\left(a^2 - (b-c)^2\right) . \] …4Find the sum of all seven-digit numbers whose digits are \(1, 2, 3, 3, 4, 4, 4\) in some order.5Find all real numbers \(a\), \(b\), \(c\), \(d\) for which \[ \begin{aligned} abc + ab + bc + ca + a + b + c &= 2, \\ bcd + bc + cd + db + b + c + d &= 5, \\ cda + cd + da + ac + c + d + a &= 7, \\ dab + da + ab + bd + d + a + b &= 11. \end{aligned} \] …
Work through the paper in order - each problem opens in the workspace with this paper as its trail.