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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...