For Lower Secondary Schools
1. The first positive integer numbers are written consecutively in a certain order. Call the resulting number . Is a multiple of ?
2. Let be a nonisosceles triangle, where is the shortest side. Choose a point in the opposite ray of such that . Prove that .
3. Let be positive reals such that . Prove that
4. Solve the equation in .
5. Let denote the altitude of a right triangle , right angle at and suppose that , where are the feet of the altitude from to and , respectively. Find the measures of the angles of triangle .
For Upper Secondary Schools
1. Find all such that .
2. Let be a sequence given by
3. Let and denote the three sides of a triangle . Its altitudes are and the radius of its three escribed circles are . Prove that
Toward Mathematical Olympiad
1. In a quadrilateral , where meets at , and the angle bisector of the angles meets at . Prove that the midpoints of are colinear.
2. Prove that
3. Find the number of binary strings of length in which the substring occurs exactly twice.
4. Let be a function such that and
Find the limit .
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