## Solve for missing angles of a right triangle using inverse trigonometry

Inverse Trigonometric Functions We used a special function, one of the trig functions, to take an angle of a triangle and find the side length. Here we will do the opposite, take the side lengths and find the angle....

## Solve for a missing side of a right triangle using trigonometric ratios

There are six special functions that describe the relationship between the sides of a right triangle and the angles of the triangle. We will discuss three of the functions here. The three functions are called the sine, cosine, and...

## prove that there exist two positive integers from the infinite sequence

solution IMO 2015 (combinatoric shortlist problem) Problem 5: Consider an infinite sequence a1, a2, …. of positive integers with for all . Suppose that for any two distinct indices i and j we have . Prove that there exist...

## Combinatoric : problem about determine the outcome of the game

Solution IMO 2015 (shortlist combinatoric problem) Problem 4: Let n be a positive integer. Two players A and B play a game in which they take turns choosing positive integers . The rules of the game are: A player...

## Determine the minimum value for good partitions

Solution IMO 2015 ( combinatoric shortlist problem 3) Problem 3: For a finite set A of positive integers, we call a partition of A into two disjoint nonempty subsets A1 and A2 good if the least common multiple of...

## Showing that there exist a balance set consisting of n point

Solution Combinatoric IMO 2015 ( Shortlist problem) Problem 2 : Let V be a ﬁnite set of points in the plane. We say that V is balanced if for any two distinct points , there exists a point such...

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