Let be the set of all positive integers. We say that a function is Georgian if and, for every positive integer , there exists a positive integer such that If is a Georgian function, we define, for each positive integer , as the smallest positive integer such that . Determine all positive real numbers for which there exists a Georgian function such that, for every positive integer , it holds that .
Let and be the sets of integers and rationals respectively.
a) Does there exist a partition of into three non-empty subsets such that the sets are disjoint?
b) Does there exist a partition of into three non-empty subsets such that the sets are disjoint?
Given a grid (), we color some of its cells black.A coloring is called balanced if each row and each cell contains exactly black cells.Detemine the number of balanced colorings.
Let be a quadrilateral inscribed in a circle with center and be the intersection of segments and . Let be the circumcircle of and be the circumcircle of . The tangent to at and the tangent to at meet at . The tangent to at and the tangent to at meet at . Show that .
Given triangle inscribed in with being the midpoint of . The tangents at of intersect at . Let be the projection of onto . On the perpendicular bisector of , take a point that is not on and different from M. Circle intersects at . Lines and intersect at . Prove that is an isosceles triangle.
Let be an integer. Find the smallest integer with the property that there exists a set of distinct real numbers such that each of its elements can be written as a sum of other distinct elements of the set.
Least swaps to get any labeling of a regular 99-gon
Photaesthesia9
N4 hours ago
by Blast_S1
Source: 2024 China MO, Day 2, Problem 6
Let be a regular -gon. Assign integers between and to the vertices of such that each integer appears exactly once. (If two assignments coincide under rotation, treat them as the same. ) An operation is a swap of the integers assigned to a pair of adjacent vertices of . Find the smallest integer such that one can achieve every other assignment from a given one with no more than operations.