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Undergraduate
By
aa1062
on July 22, 2012 | Updated Dec. 6, 2017
Prisoners finding numbers
A prison contains \(n\) prisoners, labeled \(1, 2, 3, \dots, n\). One day the warden announces that he is going to set up a room with \(n\) drawers in it, labeled \(1, 2, 3, \dots, n\). He will then …
Puzzles
Puzzles
strategy
0
Undergraduate
By
Shiva Kintali
on May 19, 2013 | Updated Dec. 6, 2017
Self-complementary graphs
Let \(G\) be a self-complementary graph (i.e., \(G\) is isomorphic to its complement) on \(n\) vertices. Prove that \(n \equiv 0\ (mod\ 4)\) or \(n \equiv -1\ (mod\ 4)\). Prove that \(G\) has a cut…
Mathematics
Graph Theory
graph complement
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Undergraduate
By
Shiva Kintali
on June 7, 2012 | Updated Dec. 6, 2017
Number of edges in a quasi-planar graph
A graph \(G(V,E)\) is called quasi-planar if it can be drawn in the plane with no three pairwise crossing edges. Prove that a quasi-planar graph with \(|V| = n\) vertices has at most \(O(n^{3/2})\) ed…
Mathematics
Graph Theory
planar graphs
0
Undergraduate
By
True Putnam
on Aug. 28, 2013 | Updated Jan. 4, 2018
Putnam 2005 A6
Let \(n\) be given, \(n \geq 4\), and suppose that \(P_1,P_2, \dots,P_n\) are \(n\) randomly, independently and uniformly, chosen points on a circle. Consider the convex \(n\)-gon whose vertices are \…
Mathematics
Probability
uniform distribution
0
Undergraduate
By
Shiva Kintali
on July 17, 2012 | Updated Dec. 6, 2017
Number of triangles in a graph
Prove that a simple graph with \(n\) vertices and \(m\) edges has at least \(\frac{m}{3n}(4m − n^2)\) triangles.
Mathematics
Graph Theory
counting
0
Undergraduate
By
aa1062
on July 22, 2012 | Updated Dec. 6, 2017
Pecking order
A researcher is studying the social dynamics of chicken coops. In each coop, for each pair of chickens \(A\) and \(B\), there is a pecking relationship: either \(A\) pecks \(B\) or \(B\) pecks \(A\) (…
Mathematics
Graph Theory
tournament
0
High School
By
Shiva Kintali
on Jan. 5, 2018
Evaluate Riemann Zeta function at all positive even integers
Reimann zeta function is the following \(\zeta(s) = \displaystyle\sum_{n=1}^{\infty}{\frac{1}{n^s}}\) Euler proved the following …
Mathematics
Mathematical Analysis
infinite series
riemann zeta function
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