Τετάρτη, 30 Μαΐου 2012

▪ Turkey JBMO Team Selection Test 2012

ΗΜΕΡΑ 1η
1. Find the greatest positive integer 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. 
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.
4. Find the greatest real number for which 
for all non-negative real numbers satisfying
ΗΜΕΡΑ 2η
1. Let be the side-lengths of a triangle, be the inradius and be the corresponding exradius. Show that 
 .
2. Find all positive integers and prime numbers for which is a perfect square. 
3. Show that for all real numbers satisfying
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.
Πηγή: artofproblemsolving
Η λύση του 3ου θέματος της πρώτης ημέρας: Φραγκάκης Νίκος 

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