ΗΜΕΡΑ 1η
1. Find the greatest positive integer 
for which is divisible by all positive integers whose cube is not greater than 
for which is divisible by all positive integers whose cube is not greater than 2. Let 
We want to partition 
into two disjoint sets such that both sets do not contain two different numbers whose sum is a power of 
Find the number of such partitions.
We want to partition into two disjoint sets such that both sets do not contain two different numbers whose sum is a power of Find the number of such partitions. 3. Let 
be a chord of the circle 
not passing through its center and let 
be the midpoint of 
Let 
be a variable point on different from 
and 
and 
be the point of intersection of the tangent lines at of circumcircle of 
and at of circumcircle of 
Show that all 
lines pass through a fixed point.
be a chord of the circle not passing through its center and let be the midpoint of Let be a variable point on different from and and be the point of intersection of the tangent lines at of circumcircle of and at of circumcircle of Show that all lines pass through a fixed point.4. Find the greatest real number for which
for which 
for all non-negative real numbers 
satisfying 
satisfying 1. Let 
be the side-lengths of a triangle, 
be the inradius and 
be the corresponding exradius. Show that
be the side-lengths of a triangle, be the inradius and be the corresponding exradius. Show that 
.
4. Let 
be a connected simple graph. When we add an edge to (between two unconnected vertices) using at most 
edges we can reach to any vertex from any vertex. Find the maximum number of edges to be used to reach any vertex from any vertex in the original graph, i.e. in the graph before we add an edge.
be a connected simple graph. When we add an edge to (between two unconnected vertices) using at most edges we can reach to any vertex from any vertex. Find the maximum number of edges to be used to reach any vertex from any vertex in the original graph, i.e. in the graph before we add an edge.
Δεν υπάρχουν σχόλια:
Δημοσίευση σχολίου