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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
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
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
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
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 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
High School
By
Shiva Kintali
on June 1, 2013 | Updated Dec. 6, 2017
The Sixth Sense
In the following, you are allowed to put any mathematical symbols on the left-hand side of the \("="\) sign to make the left-hand side evaluate to \(6\). For example, \(2+2+2=6\). Do this for all the …
Puzzles
Puzzles
math puzzle
0
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
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 June 30, 2012 | Updated Dec. 6, 2017
Coloring graphs with odd cycles
Prove that a graph with at most two odd cycles has chromatic number of at most 3. Let \(G\) be a graph where every two odd cycles have at least a vertex in common. We call such graphs nicely-odd grap…
Mathematics
Graph Theory
graph coloring
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