Matematické Fórum

Skrytý text:

Step 1. Let denotes the real number for which



We know that the auxilliary function



is a continuos increasing function with and as . Hence there exists exactly one real number for which the above equality holds true. Moreover, one can easily deduce that .

Step 2. Next, we transform the equality as follows



where is a sufficiently large positive integer s.t. both terms on the left hand side are positive. Now, computing the tangent of the left and right hand side gives



For the numerator on the left hand side we conclude that



a finite value, whereas the limit of the right hand side is



This indicates that the denominator on the left hand side satisfies



which implies the equality



Step 3. In the next step we use known formulas



and



We apply this formulas to transform the identity above



After simplifications we obtain the biquadratic equation



Solving this equation under the restrictive condition we get the final answer, i.e.,





Remark. Note that the problem can be stated equivalently in a more difficult modification as



because of