Practice library
Problems
1How many integers \(n\) with \(10 \le n < 100000\) are divisible by \(4\), contain no digit \(0\) in their decimal representation, and have no two adjacent digits equal?Open2In the expression \[ *\,1 * 3 * 3^{2} * 3^{3} * \cdots * 3^{1997} * 3^{1998} \] Arkadije and Branislav take turns replacing one of the stars by \(+\) or by \(-\), one star per move, until no star is left. …3In a group of \(20\) people, every person chooses ten of the other nineteen and sends one letter to each of them. Prove that there are two people who sent a letter to each other.4How many isosceles trapezoids with integer side lengths have perimeter \(2005\)? (A trapezoid here means a quadrilateral with exactly two parallel sides, so a parallelogram is not one. Trapezoids with …5The number \(1\) is written on a board \(2005\) times. A move consists of erasing two of the numbers written on the board and writing, in their place, one quarter of their sum. The move is repeated until …6A \(2004 \times 2004\) board is completely tiled by pieces of size \(1 \times 4\); each piece covers four cells of a single row (call it horizontal) or four cells of a single column (vertical). Can the …7A set \(\mathcal{A}\) of \(2000\) points in the plane contains no three collinear points. Prove that these points can be joined by \(1000\) blue, \(1000\) red and \(1000\) yellow segments in such a way …
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