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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
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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
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Graduate
By
Shiva Kintali
on June 11, 2012 | Updated Dec. 6, 2017
Embedding complete bipartite graphs
Let \(S\) be an orientable surface of genus \(g \geq 0\). Prove that for every \(g \geq 0\) there exists an integer \(t\) such that \(K_{3,t}\) cannot be drawn on \(S\) without any crossings. What is…
Mathematics
Graph Theory
graph embedding
0
Undergraduate
By
Shiva Kintali
on May 19, 2013 | Updated Dec. 6, 2017
Large min-degree implies perfect matching
Let \(G\) be a bipartite graph with partitions \(X\) and \(Y\) such that \(|X|=|Y|=n\). The degree of each vertex in \(G\) is at least \(n/2\). Prove that \(G\) has a perfect matching.
Mathematics
Graph Theory
matching
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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
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
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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