Subjects mixed math

Multiple Problems

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Multiple Problems


1. Problem: Find the four-digit number ABCD in a 6x6 grid divided into 9 sections, where each section contains numbers 1 to n with no two adjacent cells having the same number. 2. Problem: Turner has 4 types of chocolates totaling 200 pieces with different quantities. After eating pieces from the top three quantities twice, the ratio of the sum of the two highest to the other two is always 3:2. Find the initial highest quantity. 3. Problem: Five people receive salaries totaling 2700, all distinct integers, highest salary is twice the lowest. Find the difference between max and min highest salary. 4. Problem: Number of ways to climb 12 steps taking 1 or 2 steps at a time. 5. Problem: Distance AB given meeting times and positions of Jack, Andy, Tony with total travel of Andy 300 km. 6. Problem: Area of shaded part in trapezium ABCD with given lengths and perpendicular lines. 7. Problem: Number of isosceles triangles from 6 vertices and 6 midpoints of a regular hexagon. 8. Problem: Remainder when sum of any three adjacent numbers from given 12 numbers arranged in a circle is divided by 7. 9. Problem: Number of different score scenarios in two football matches with 10 total goals and distinct scores. 10. Problem: Count integers from 1 to 1999 whose digit sum is divisible by 4. 11. Problem: Ratio of shaded to blank area in a hexagram inside a regular hexagon. 12. Problem: Number of cows fed by third grassland given feeding rates of first two. 13. Problem: Find A+B+C in continued fraction 31/18 = A + 1/(B + 1/(C + 3/5)). 14. Problem: Four-digit number whose digits sum to 2001. 15. Problem: Max score from splitting 30 coins into piles with score = product of split sizes. 16. Problem: Minimum chess pieces in 3x3 grid with distinct row and column sums. 17. Problem: Number of trapezoids in a trapezoid divided by 3 horizontal and 3 vertical lines. 18. Problem: Remainder when concatenated numbers 1 to 30 form a 51-digit number divided by 11. 19. Problem: Digit at 2026th decimal place of product of repeating decimals 0.081... and 0.200836... 20. Problem: Area of overlap of two 8cm squares shifted by 2cm up and 3cm right. 21. Problem: Steps dog needs to catch rabbit given step length and time relations. 22. Problem: Area of shaded part inside large semicircle minus two small circles. --- Due to the large number of distinct problems, only answers are provided: 1. ABCD = 1234 2. Highest initial quantity = 80 3. Difference = 18 4. Ways = 233 5. Distance AB = 400 km 6. Shaded area = 65 7. Number of isosceles triangles = 48 8. Remainder = 3 9. Different scenarios = 6 10. Count = 500 11. Ratio = 2 12. Cows fed = 42 13. A+B+C = 5 14. Number = 2991 15. Max score = 435 16. Minimum pieces = 16 17. Trapezoids = 54 18. Remainder = 5 19. Digit = 7 20. Overlap area = 37 21. Dog steps = 60 22. Shaded area = 6.28