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Explore the Back-StoryIn a convex quadrilateral PQRS, the areas of triangles and are in the ratio . A line through cuts at and at such that . Prove that is the midpoint of and is the midpoint of .

Given positive real numbers such that . Prove that there exist at least one positive integer such that .

Find the number of permutations of the integers that satisfy the chain of inequalities.

In the figure, is a diameter of the circle, where , and . Find the length of and hence find the length of the altitude from to .

A math contest consists of objective type questions and fill in the blanks questions. From a school some number of students took the test and it was noticed that all students had attempted exactly out of the questions. Let , be the six fill in the blanks questions. Let be the number of students who attempted both questions and . If the sum of all the and is , then find the number of students who took the test in the school.

Find all positive integer triples that satisfy the equation

The perimeter of is 2 and its sides are . Prove that .

A circular disc is divided into equal sectors and one of different colours is used to colour each sector. No two adjacent sectors can have the same colour. Find the number of such distinct colourings possible.

In a convex quadrilateral PQRS, the areas of triangles and are in the ratio . A line through cuts at and at such that . Prove that is the midpoint of and is the midpoint of .

Given positive real numbers such that . Prove that there exist at least one positive integer such that .

Find the number of permutations of the integers that satisfy the chain of inequalities.

In the figure, is a diameter of the circle, where , and . Find the length of and hence find the length of the altitude from to .

A math contest consists of objective type questions and fill in the blanks questions. From a school some number of students took the test and it was noticed that all students had attempted exactly out of the questions. Let , be the six fill in the blanks questions. Let be the number of students who attempted both questions and . If the sum of all the and is , then find the number of students who took the test in the school.

Find all positive integer triples that satisfy the equation

The perimeter of is 2 and its sides are . Prove that .

A circular disc is divided into equal sectors and one of different colours is used to colour each sector. No two adjacent sectors can have the same colour. Find the number of such distinct colourings possible.

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