answer:1. **Coloring the Chessboard:** - We color the 19 times 19 chessboard as follows: - Squares with both even x and y coordinates are colored red. - Squares with both odd x and y coordinates are colored green. - All other squares are colored white. 2. **Counting the Squares:** - The total number of squares on the chessboard is 19 times 19 = 361. - Red squares: These are the squares where both coordinates are even. The even coordinates range from 0 to 18, giving us 10 even numbers. Thus, the number of red squares is 10 times 10 = 100. - Green squares: These are the squares where both coordinates are odd. The odd coordinates range from 1 to 19, giving us 10 odd numbers. Thus, the number of green squares is 10 times 10 = 100. - White squares: The remaining squares are white. Therefore, the number of white squares is 361 - 100 - 100 = 161. 3. **Movement Constraints:** - A pawn on a green square will move to a white square in the next move and then to a red square in the subsequent move. - A pawn on a red square will move to a white square in the next move and then to a green square in the subsequent move. - A pawn on a white square will move to either a red or green square in the next move. 4. **Maximum Number of Pawns:** - Since no two pawns can occupy the same square, we need to ensure that the number of pawns on red and green squares does not exceed the number of available squares. - The maximum number of pawns on red squares is 100. - The maximum number of pawns on green squares is 100. - The maximum number of pawns on white squares is 161. 5. **Total Number of Pawns:** - The total number of pawns that can be placed on the board is the sum of the maximum number of pawns on red, green, and white squares: [ 100 + 100 + 161 = 361 ] 6. **Feasibility of the Arrangement:** - To ensure that the game can continue indefinitely, we can arrange the pawns in an 18 times 18 square in the bottom left corner of the board. This arrangement ensures that the pawns can be moved right, up, left, and down in a cyclic manner without violating the movement constraints. The final answer is boxed{361}.

answer:1. **Position and Nature of Point P**: - Point P is the centroid of the tetrahedron, being equidistant from all vertices. 2. **Distances from P to Each Face**: - The distance from centroid to any face of a regular tetrahedron is consistent due to symmetry and equals frac{h}{3} where h is the height from a vertex to the opposite face. - Height h can be calculated using the Pythagorean theorem; height of a face (an equilateral triangle) with side 2 units is sqrt{3} by h_f = frac{sqrt{3}}{2} times 2. Then, height of tetrahedron h = sqrt{4-frac{4}{3}} = sqrt{frac{8}{3}}. - For centroid P, distance from each face d_p = frac{sqrt{frac{8}{3}}}{3} = frac{sqrt{24}}{9}. - Sum of distances to four faces s = 4 times frac{sqrt{24}}{9} = frac{4sqrt{24}}{9}. 3. **Distances from P to Each Edge**: - Sum of perpendicular distances from the centroid of a tetrahedron to its edges adds up to an expression related to the tetrahedron’s circumradius, but here we use a property: sum of distances from centroid to edges in symmetric geometry like regular tetrahedron is proportional to its height through a constant factor (frac{1}{2} for centroid). - Therefore, S = 6 times frac{2}{2} = 6 units. 4. **Ratio Calculation**: - frac{s}{S} = frac{frac{4 sqrt{24}}{9}}{6} = frac{4 sqrt{6}}{18} = frac{sqrt{6}}{3}. The final ratio frac{s}{S} equals frac{sqrt{6}}{3}. frac{sqrt{6}{3}} The final answer is C) boxed{frac{sqrt{6}}{3}}.

answer:Given: acosB + bsinA = c, Using the sine law, we have: sinC = sinAcosB + sinBsinA ① Also, since A + B + C = π, sinC = sin(A + B) = sinAcosB + cosAsinB ② From ① and ②, we get sinA = cosA, which implies tanA = 1. Since A ∈ (0, π), we have A = frac {π}{4} Given a = sqrt {2}, using the cosine law, 2 = b² + c² - 2bccosA = b² + c² - sqrt {2}bc ≥ 2bc - sqrt {2}bc = (2 - sqrt {2})bc This leads to bc ≤ frac {2}{2- sqrt {2}}. The equality holds if and only if b = c. Thus, the area of triangle ABC is given by S = frac {1}{2}bcsinA = frac { sqrt {2}}{4}bc ≤ frac { sqrt {2}}{4} × frac {2}{2- sqrt {2}} = boxed{frac { sqrt {2}+1}{2}}. The equality holds if and only if b = c, which means the maximum area of triangle ABC is boxed{frac { sqrt {2}+1}{2}}. This problem involves applying the sine and cosine laws and the area formula for a triangle. Additionally, it requires understanding trigonometric identity transformations, making it a moderate level problem.

answer:Let's assume the cost price of one article is C and the selling price of one article is S. According to the given information, the selling price of 100 articles is equal to the cost price of 40 articles. So we can write the equation: 100S = 40C Now, we want to find the ratio of the selling price (S) to the cost price (C) to determine the loss or gain percent. We can rearrange the equation to solve for S/C: S/C = 40/100 S/C = 2/5 This means that the selling price is 2/5 of the cost price. To find the loss percent, we need to see how much less the selling price is compared to the cost price. Loss = C - S Loss Percent = (Loss/C) * 100% We know that S = 2C/5, so we can substitute this into the Loss equation: Loss = C - (2C/5) Loss = (5C/5) - (2C/5) Loss = (3C/5) Now, we can calculate the Loss Percent: Loss Percent = [(3C/5)/C] * 100% Loss Percent = (3/5) * 100% Loss Percent = 60% Therefore, there is a boxed{60%} loss on the articles.