Let ,, be real numbers different from 0 for which + + = 0 holds
a) Prove that (+)(+)(+)≠ 0
b) Let = + + and = + + . Prove that numbers and are both positive or both negative.
Let be a positive integer. A board with a format is divided in equal squares.Determine all integers ≥3 such that the board can be covered in (or ) pieces so that there is exactly one empty square in each row and each column.
Let ABC be an acute triangle with an incenter .The Incircle touches sides and in and ,respectively. Lines CI and EF intersect at . The point ≠ is on the line AI so that =.If is the foot of the altitude from in triangle ,prove that points , and are colinear.
Let be an acute scalene triangle and let be a point in its interior. Let ,, be projections of onto triangle sides ,,, respectively. Find the locus of points such that ,, are concurrent and .
Source: Serbian selection contest for the IMO 2025
Determine the smallest positive real number such that there exists a sequence of positive real numbers ,, with the property that for every it holds that: Proposed by Pavle Martinović