Compete · Serbia
Opštinsko takmičenje (Municipal round) 1995
1Determine the smallest six-digit number whose digits are all different and which is divisible by \(11\).2Determine the smallest natural number which, when divided by \(4\), \(6\), \(8\), \(10\) and \(12\), leaves the remainders \(2\), \(4\), \(6\), \(8\) and \(10\) respectively.3If \(a^{2} + a + 1 = 0\), what is the value of \[ a^{1995} + \frac{1}{a^{1995}} \, ? \]4Every cell of a \(3 \times 3\) board is to be painted in one of \(9\) colors so that all \(9\) colors are used. How many differently colored boards can be made? (Two boards are colored differently if they …5Two people are talking. Person \(A\) says: "If we win at football, we will also win at basketball." Person \(B\) says: "If we do not win at basketball, we will win at football." Person \(C\) remarks: "At …
Work through the paper in order - each problem opens in the workspace with this paper as its trail.