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9. In a 2×3 table, there are 10 rectangles which consist of an even number of unit squares.
How many rectangles are there in a 6×9 table which consist of an even number of unit squares?
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10. Find the smallest positive common multiple of 4 and 6 such that each digit is either 4 or 6, there is at least one 4 and there is at least one 6.
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11. We have two kinds of isosceles triangles each with two sides of length 1. The acute triangle has a 30^{\circ} angle between the two equal sides, and the right triangle has a right angle between the two equal sides. We place a sequence of isosceles triangles around a point according to the following rules. The n-th isosceles triangle is a right isosceles triangle if n is a multiple of 3, and an acute isosceles triangle if it is not. Moreover, the n-th and (n+1)-st isosceles triangles share a common side, as shown in the diagram below. What is the smallest value of n>1 such that the n-th isosceles triangle coincides with the 1-st one?
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12. When the digits of a two-digit number are reversed, the new number is at least 3 times as large as the original number. How many such two-digit numbers are there?
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13. In the quadrilateral ABCD, AB=CD, $\angle BCD=57^{\circ}$, and $\angle ADB + \angle CBD = 180^{\circ}$. Find the value of $\angle BAD$.
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14. Squares on an infinite chessboard are being painted. As shown in the diagram below, three squares (lightly shaded) are initially painted. In the first step, we paint all squares (darkly shaded) which share at least one edge with squares already painted. The same rule applies in all subsequent steps. Find the number of painted squares after one hundred steps.
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15. The rows of a 2011×4024 chessboard are numbered from 1 to 2011 from bottom to top, and the columns from 1 to 4024 from left to right. A snail starts crawling from the cell on row 1 and column 1 along row 1. Whenever it is about to crawl off the chessboard or onto a cell which it has already visited, it will make a left turn and then crawl forwards in a straight line. Thus it follows a spiraling path until it has visited every cell. Find the sum of the row number and the column number of the cell where the path ends. (The answer is 3+2=5 for a 4×5 table.)
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