欧几里得-真题-答案.docx

2020 Euclid Contest Tuesday, April 7, 2020 (in North America and South America) Wednesday, April 8, 2020 (outside of North America and South America) Solutions 2020 Euclid Contest Solutions Page 2 1. (a) Solution 1 If x = ?2, then 3x + 6 x + 2 = 3(x + 2) x + 2 = 3. In other words, for every x = ?2, the expression is equal to 3. 3x + 6 Therefore, when x = 11, we get = 3. x + 2 Solution 2 When x = 11, we obtain 3x + 6 x + 2 = 3(11) + 6 11 + 2 = 39 13 = 3. (b) Solution 1 The point at which a line crosses the y-axis has x-coordinate 0. Because A has x-coordinate ?1 and B has x-coordinate 1, then the midpoint of AB is on the y-axis and is on the line through A and B, so is the point at which this line crosses the x-axis. The midpoint of A(?1, 5) and B(1, 7) is 2(5 + 7) or (0, 6). 1 1 Therefore, the line that passes through A(?1, 5) and B(1, 7) has y-intercept 6. Solution 2 7 ? 5 2 The line through A(?1, 5) and B(1, 7) has slope 1 ? (?1) 2 = = 1. Since the line passes through B(1, 7), its equation can be written as y ? 7 = 1(x ? 1) or y = x + 6. The line with equation y = x + 6 has y-intercept 6. (c) First, we ?nd the coordinates of the point at which the lines with equations y = 3x + 7 and y = x + 9 intersect. Equating values of y, we obtain 3x + 7 = x + 9 and so 2x = 2 or x = 1. When x = 1, we get y = x + 9 = 10. Thus, these two lines intersect at (1, 10). Since all three lines pass through the same point, the line with equation y = mx + 17 passes through (1, 10). Therefore, 10 = m · 1 + 17 which gives m = 10 ? 17 = ?7. 2. (a) Suppose that m has hundreds digit a, tens digit b, and ones (units) digit c. From the given information, a, b and c are distinct, each of a, b and c is less than 10, a = bc, and c is odd (since m is odd). The integer m = 623 satis?es all of these conditions. Since we are told there is only one such number, then 623 must be the only answer. Why is this the only possible value of m? We note that we cannot have b = 1 or c = 1, otherwise a