Solve the following mathematical problems:

Prove that a parallelogram inscribed in a circle must be a rectangle.

60 men could complete a piece of work in 250 days. They worked together for 200 days. After that the work had to be stopped for 10 day to bad weather. How many more men should be engaged to complete the work in time?

The ratio of the numbers of boys and girls in a school was 5 : 3. Some new boys and girls were admitted to the school, in the ratio 5 : 7. At this, the total number of students in the school became 1,200 and the ratio of boys to girls changed to 7: 5. What was the number of students in the school before new admissions?

An article manufactured by a company consists of two parts A and B. In the process of manufacture of part A, 9 out of 100 are likely to be defective. Similarly, 5 out of 100 are likely to be defective in the process of manufacture of part B. What will be the probability that the assembled part will not be defective?

Three runners A, B and C run a race, with runner A finishing 12m ahead of runner B and 18m ahead of runner C, in another race of same type runner B finished 8m ahead of runner C. Each runner travels the entire distance at a constant speed. Find the length of the race.

‘A’ began a small business with a certain amount of money. After four months from the start of the business. 'B' joined the business with an amount which was Tk. 6,000 less than 'A's initial investment. 'C' joined the business after seven months from the start of business with an amount which was Tk. 2,000 less than A's initial investment. At the end of the year total investment repented was Tk. 1,42,000. What will be A's share in the profit, if 'B' received Tk. 8,000 as profit share ?