Consider a ball that moves inside an acute-angled triangle along a straight line, unit it hits the boundary, which is when it changes direction according to the mirror law, just like a ray of light (angle of incidence = angle of reflection). Prove that there exists a triangular periodic path for the ball, as pictured below.
A bit tricky invariant with 98 numbers on the board.
Nuran20100
an hour ago
Source: Azerbaijan Al-Khwarizmi IJMO TST 2025
The numbers are written on the board.In each step,two random numbers and are chosen and deleted.Then,the number is written instead.What will be the number remained on the board after the last step.
Taking antipode on isosceles triangle's circumcenter
Nuran20100
an hour ago
Source: Azerbaijan Al-Khwarizmi IJMO TST 2025
In isosceles triangle, the condition is satisfied. Point is taken on the circumcircle of such that .A line parallel to which passes from intersects and respectively at and .Show that circumcircle of passes from circumcenter of .
Let be the unit circle in the complex plane. Let be the map given by . We define and for . The smallest positive integer such that is called the period of . Determine the total number of points in of period .
(Hint : )
Let denote the set of natural numbers, and let ,, be nine distinct tuples in . Show that there are three distinct elements in the set whose product is a perfect cube.
Proof: After inversion around , and is the intersection of line with line . Thus the map becomes projective after inversion. Since inversion preserves cross ratios, the map itself is projective, as desired.
Now we induct on , the number of points among that are equal to . The base case is true and this follows from the fact that the radical axes of concur.
Suppose the statement holds for and we prove it for . WLOG . The maps are both projective, hence it suffices to verify the statement for three positions of . We may take (and have the result from induction hypothesis), (in which case the lines concur at ) and the point on such that is cyclic.