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TrueShelf Inc.
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articles
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exercises
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Undergraduate
By
TrueShelf Inc.
on Sept. 29, 2013 | Updated Jan. 16, 2017
Fibonacci numbers and primes
The Fibonacci numbers are defined by \(F_1 = F_2 = 1\) and \(F_n = F_{n−1} + F_{n−2}\) for \(n \geq 3\). If \(p\) is a prime number, prove that at least one of the first \(p + 1\) Fibonacci numbers mu…
Mathematics
Combinatorics
fibonacci
pigeonhole principle
primes
0
Undergraduate
By
TrueShelf Inc.
on Sept. 28, 2013 | Updated Jan. 16, 2017
Triangles in a random graph
Let \(G = G(n, \frac{1}{2})\) be a random graph on \(n\) vertices, i.e., for each pair of verices \(i, j\), we add the edge \((i, j)\) independently with probability \(\frac{1}{2}\). Let \(T_n\) be th…
Mathematics
Graph Theory
Probability
expectation
high probability
random graphs
triangles
variance
0
Undergraduate
By
TrueShelf Inc.
on Sept. 30, 2013 | Updated Jan. 16, 2017
Stirling's Approximation
Prove the following inequality : \(e \left(\frac{n}{e}\right)^n \leq n! \leq en \left(\frac{n}{e}\right)^n\)
Mathematics
Calculus
integration
0
Undergraduate
By
TrueShelf Inc.
on Oct. 3, 2013 | Updated Jan. 16, 2017
k-partite subgraph
For each \(k\in \mathbb{N}\) and each simple graph \(G=(V,E)\), prove that \(G\) has a \(k\)-partite subgraph \(H=(V',E')\) (i.e., \(H\) has chromatic number at most \(k\)) such that …
Mathematics
Graph Theory
Probability
graph coloring
probabilistic method
0
Undergraduate
By
TrueShelf Inc.
on Oct. 3, 2013 | Updated Jan. 16, 2017
Self-complementary planar graphs
Prove that the complement of a simple planar graph with at least \(11\) vertices is nonplanar. Construct two self-complementary simple planar graphs with \(8\) vertices.
Mathematics
Graph Theory
graph complement
planar graphs
0
Undergraduate
By
TrueShelf Inc.
on Aug. 29, 2013 | Updated Jan. 16, 2017
Even Subsets
A set \(T\) is called even if it has even number of elements. Let \(n\) be a positive even integer, and let \(S_1, S_2, \dots, S_n\) be even subsets of the set \(S = \){\(1,2,\dots,n\)}. Prove that …
Mathematics
Combinatorics
counting
0
Undergraduate
By
TrueShelf Inc.
on Sept. 28, 2013 | Updated Jan. 16, 2017
Basics of Random Graphs
Let \(G = G(n, \frac{1}{2})\) be a random graph on \(n\) vertices, i.e., for each pair of verices \(i, j\), we add the edge \((i, j)\) independently with probability \(\frac{1}{2}\). Show that \(G\)…
Mathematics
Probability
high probability
random graphs
0
High School
By
TrueShelf Inc.
on Dec. 14, 2013 | Updated Jan. 16, 2017
Legendre's Theorem
Prove the following Legendre's Theorem : Legendre's Theorem : The number \(n!\) contains the prime factor \(p\) exactly \(\sum_{k \geq 1}{\lfloor \frac{n}{p^k} \rfloor}\) times.
Mathematics
Number Theory
primes
0
High School
By
TrueShelf Inc.
on Sept. 27, 2013 | Updated Jan. 16, 2017
Color of the last ball
In a large urn there are \(w\) white balls and \(b\) black balls. Beside the urn is a big pile (infinite number) of black balls. Now, we do the following. First, two balls are drawn at random from the…
Puzzles
Puzzles
invariant
0
Undergraduate
By
TrueShelf Inc.
on Oct. 12, 2013 | Updated Jan. 16, 2017
Ramsey primes
For every integer \(m \geq 1\), there exists an integer \(p_0\) such that, for all primes \(p \geq p_0\), the congruence \(x^m + y^m \equiv z^m (\mbox{mod}\ p)\) has a solution with positive \(x\), …
Mathematics
Number Theory
primes
ramsey theory
0
High School
By
TrueShelf Inc.
on May 7, 2014 | Updated Jan. 16, 2017
Sock Drawer puzzle
There are 10 socks of each of the following colors in a drawer: red blue green black white i.e., there are 50 socks. The socks are arbitrarily distributed in the drawer. You are blind-folded. …
Puzzles
Puzzles
math puzzle
pigeonhole principle
0
High School
By
TrueShelf Inc.
on Sept. 27, 2013 | Updated Jan. 16, 2017
Token game and Invariant
Consider the following game. Initially, there are two tokens in a row; the left one is blue and the right one is red. You can change the configuration by performing a number of moves. In each m…
Puzzles
Puzzles
invariant
0
High School
By
TrueShelf Inc.
on June 6, 2014 | Updated Jan. 16, 2017
Constructible polygons
A constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular hepta…
Mathematics
Geometry
polygon
0
Undergraduate
By
TrueShelf Inc.
on Sept. 27, 2013 | Updated Jan. 16, 2017
Minimum spanning trees
Suppose we are given a connected graph \(G\) with weights on edges, such that all edge weights are distinct. For any cycle in \(G\), prove that the maximum weight edge in the cycle cannot be in any …
Computer Science
Mathematics
Algorithms
Graph Theory
graph cut
spanning tree
trees
0
High School
By
TrueShelf Inc.
on Aug. 20, 2013 | Updated Jan. 16, 2017
Bug in a cube
A bug is sitting in one corner of a cube-shaped room. What is the shortest path to go to the extreme opposite corner (i.e., the corner that is farthest) ?
Puzzles
Puzzles
geometry puzzle
interview question
0
High School
By
TrueShelf Inc.
on June 16, 2013 | Updated Jan. 16, 2017
Pick's theorem
Let \(P\) be a polygon constructed on a grid of equal-distanced points (i.e., points with integer coordinates) such that all the polygon's vertices are grid points. Let \(I\) be the number of lattice …
Mathematics
Geometry
polygon
0
High School
By
TrueShelf Inc.
on Sept. 27, 2013 | Updated Jan. 16, 2017
Binomial coefficients
Evaluate the following sums using combinatorial methods and algebraic methods : \(\displaystyle \sum_{i=0}^{k} {m \choose i}{n \choose k-i}\) \(\displaystyle \sum_{i=0}^{n} {n \choose i}^2\) …
Mathematics
Combinatorics
binomial theorem
counting
n choose k
0
Undergraduate
By
TrueShelf Inc.
on Sept. 30, 2013 | Updated Jan. 16, 2017
Friends and Parties
Show that at a party of \(n\) people, there are two people who have the same number of friends in the party. Assume that friendship is symmetric. There are \(2n\) people at a party. Each person has a…
Mathematics
Discrete Mathematics
counting
pigeonhole principle
0
Undergraduate
By
TrueShelf Inc.
on Sept. 28, 2013 | Updated Jan. 16, 2017
Murphy's Law
Let \(A_1, A_2, \ldots ,A_n\) be independent events, and let \(T\) be the number of these events that occur. Show that the probability that none of the events occur is at most \(e^{-E[T]}\), where…
Mathematics
Probability
expectation
0
Undergraduate
By
TrueShelf Inc.
on Oct. 12, 2013 | Updated Jan. 16, 2017
Odd sets with even intersection
Let \(\mathcal{F}\) be a family of subsets of \([n]\) such that, for all \(A, B \in \mathcal{F}\) with \(A \neq B\), we have \(|A \cap B|\) is even and \(|A|\) is odd. Prove that …
Mathematics
Combinatorics
Linear Algebra
counting
0
Undergraduate
By
TrueShelf Inc.
on July 26, 2013 | Updated Jan. 16, 2017
Equal degree vertices
Prove that every simple graph with at least two vertices has two vertices of equal degree. Is the conclusion true if we allow multi-edges ?
Mathematics
Graph Theory
pigeonhole principle
0
High School
By
TrueShelf Inc.
on June 27, 2013 | Updated Jan. 16, 2017
Basics of Induction
Prove the following using induction: \(\sum_{i=1}^{n}{i} = \frac{n(n+1)}{2}\). \(\sum_{i=1}^{n}{i}^2 = \frac{n(n+1)(2n+1}{6}\). \(\sum_{i=1}^{n}{i}^3 = {(\frac{n(n+1)}{2})}^2\). …
Mathematics
Discrete Mathematics
induction
summation
0
Undergraduate
By
TrueShelf Inc.
on Sept. 28, 2013 | Updated Jan. 16, 2017
Basics of Expectation and Variance
If the random variable \(X\) takes values in non-negative integers, prove that: \(E[X] = \sum_{t=0}^\infty \Pr(X > t)\) Prove that if \(X_1\) and \(X_2\) are independent random variables then …
Mathematics
Probability
basics
expectation
variance
0
Undergraduate
By
TrueShelf Inc.
on Oct. 12, 2013 | Updated Jan. 16, 2017
Turan graphs
Prove the following : Let \(G\) be a simple graph with \(n\) vertices. If \(G\) has more than \(\lfloor \frac{n^2}{4} \rfloor\) edges, then \(G\) contains a triangle. Let \(G\) be a simple graph wit…
Mathematics
Graph Theory
extremal graph theory
0
Undergraduate
By
TrueShelf Inc.
on Oct. 1, 2013 | Updated Jan. 16, 2017
Graphs, Matrices and Walks
Let \(G\) be a directed graph (possibly with self-loops) with vertices \(v_1, \dots , v_n\). Let \(M\) be the adjacency matrix of \(G\). Prove that \(M_{ij}^k\) (i.e., that \([i][j]^{th}\) entry of \…
Mathematics
Graph Theory
Linear Algebra
adjacency matrix
matrices
matrix multiplication
0
Undergraduate
By
TrueShelf Inc.
on Sept. 28, 2013 | Updated Jan. 16, 2017
Minimum of Random Subsets
Let \(S = \){\(1,2,\dots,n\)}. Let \(A,B\) be two random subsets of \(S\). Let \(\min(A)\) denote the minimum number in the set \(A\). What is the probability that \(\min(A)= \min(B)\) ? Evaluate th…
Mathematics
Probability
probability
0
Undergraduate
By
TrueShelf Inc.
on Oct. 12, 2013 | Updated Jan. 16, 2017
Two-distance sets
Let \(d_1, d_2 \in \mathbb{R}\), and let \(S \subset \mathbb{R}^n\) be a set of vectors such that \(|| x - y || \in \){\(d_1, d_2\)} for all \(x, y \in S\). Prove that there exists such a set \(S\) …
Mathematics
Combinatorics
distance
vectors
0
High School
By
TrueShelf Inc.
on May 7, 2014 | Updated Jan. 16, 2017
Overlapping hour and minute hands
At what times are the hour and minute hands of a clock are exactly on top of each other ? At what times are the minute and second hands of a clock are exactly on top of each other ?
Puzzles
Puzzles
math puzzle
0
Undergraduate
By
TrueShelf Inc.
on Sept. 28, 2013 | Updated Jan. 16, 2017
Hamiltonicity of Line Graphs
Let \(G=(V,E)\) be a simple undirected graph. The line graph \(L(G)\) of \(G\) is defined as follows: Each vertex in \(L(G)\) corresponds to an edge in \(G\), and two vertices are connected by an edg…
Mathematics
Graph Theory
hamiltonian cycle
line graph
0
Undergraduate
By
TrueShelf Inc.
on Oct. 4, 2013 | Updated Jan. 16, 2017
Trees and non-planarity
Let \(V\) be a set of vertices, and let \(T_1=(V,E_1)\), \(T_2=(V,E_2)\), and \(T_3=(V,E_3)\) be three trees on the vertices of \(V\) with disjoint sets of edges: …
Mathematics
Graph Theory
planar graphs
trees
0
Undergraduate
By
TrueShelf Inc.
on Sept. 28, 2013 | Updated Jan. 16, 2017
Extremal graphs
Let \(G\) be a simple undirected graph with \(2n\) nodes and no triangles (i.e., cycles of length \(3\)). Prove that \(\mathcal G\) has at most \(n^2\) edges.
Mathematics
Graph Theory
extremal graph theory
triangles
0
High School
By
TrueShelf Inc.
on Sept. 30, 2013 | Updated Jan. 16, 2017
Using Binomial Theorem
Prove the following using binomial theorem and/or mathematical induction Let \(a\), \(b\) and \(n\) be natural numbers, prove that \(\frac{(a+\sqrt{b})^n + (a-\sqrt{b})^n}{2}\) is also a natural num…
Mathematics
Discrete Mathematics
binomial theorem
induction
0
Undergraduate
By
TrueShelf Inc.
on Sept. 27, 2013 | Updated Jan. 16, 2017
Fibonacci numbers and Induction
The Fibonacci numbers, \(F_0, F_1, F_2, \dots\) , are defined recursively by the equations \(F_0 = 0\), \(F_1 = 1\), and \(F_n = F_{n-1} + F_{n-2},\) for \(n > 1\). Prove that …
Mathematics
Discrete Mathematics
fibonacci
induction
tiling
0
Undergraduate
By
TrueShelf Inc.
on Dec. 14, 2013 | Updated Jan. 16, 2017
Product of Primes
Prove that \(\displaystyle \prod_{p \leq x} p \leq 4^{x-1}\) for all real \(x \geq 2\). Here the product is taken over all prime numbers \(p \leq x\).
Mathematics
Number Theory
primes
0
Undergraduate
By
TrueShelf Inc.
on Sept. 27, 2013 | Updated Jan. 16, 2017
Tournament Graphs
A tournament is a directed graph obtained by assigning a direction for each edge in an undirected complete graph. That is, it is an orientation of a complete graph, or equivalently a directed graph i…
Mathematics
Graph Theory
hamiltonian path
tournament
0
High School
By
TrueShelf Inc.
on Sept. 28, 2013 | Updated Jan. 16, 2017
Bijective counting
Let \(S = {1,2,...,n}\). How many ordered pairs \((A,B)\) of subsets of \(S\) are there that satisfy \(A \subseteq B\) ? Let \(S = {1,2,...,n}\). How many ordered pairs \((A,B)\) of subsets of \(S\) …
Mathematics
Combinatorics
bijection
counting
0
Undergraduate
By
TrueShelf Inc.
on Sept. 28, 2013 | Updated Jan. 16, 2017
Chromatic number of interval graphs
An undirected simple graph \(G = (V,E)\) is an interval graph if and only if there exists a family of intervals \({I_u}_{u\in V}\) such that \(I_u\) intersects \(I_v\) if and only if \((u,v)\in E\). …
Mathematics
Graph Theory
clique number
cliques
graph coloring
interval graph
0
Undergraduate
By
TrueShelf Inc.
on Sept. 28, 2013 | Updated Jan. 16, 2017
Primes and divisibility
Prove the following without using Fermat's little theorem or Euler's totient theorem. Here \(a\ |\ b\) means \(a\) divides \(b\). Prove that for every prime number \(p\) and every pair of integers \…
Mathematics
Number Theory
divisibility
primes
0
Undergraduate
By
TrueShelf Inc.
on Sept. 28, 2013 | Updated Jan. 16, 2017
Runs in coin tosses
You have a biased coin i.e., each coin toss is a head with probability \(p\) and is a tail with probability \(1-p\). Let \(X\) be the number of runs in \(n\) independent tosses of this coin. Here, run…
Mathematics
Probability
expectation
variance
0
Undergraduate
By
TrueShelf Inc.
on Sept. 28, 2013 | Updated Jan. 16, 2017
Task assignment using Hall's Theorem
You are given a collection of tasks each of which must be assigned a time slot in the range {\(1,\dots,n\)}. Each task \(j\) has an associated interval \(I_j = [s_j,t_j]\) and the time slot assigned t…
Mathematics
Graph Theory
assignment
halls theorem
0
High School
By
TrueShelf Inc.
on Oct. 21, 2014 | Updated Jan. 16, 2017
Farmers and Chickens
Three farmers were selling chickens at the local market. One farmer had 10 chickens to sell, another had 16 chickens to sell, and the last had 26 chickens to sell. In order not to compete with each …
Mathematics
linear equations
0
Undergraduate
By
TrueShelf Inc.
on Sept. 28, 2013 | Updated Jan. 16, 2017
Crossing number
The crossing number \(cr(G)\) of a graph \(G\) is the lowest number of edge crossings of a plane drawing of the graph \(G\). Prove that the crossing number of \(K_n\) (the complete graph on \(n\) ve…
Mathematics
Graph Theory
complete graph
crossing number
planar graphs
0
Undergraduate
By
TrueShelf Inc.
on April 23, 2014 | Updated Jan. 16, 2017
Putnam A5 2005
Evaluate \(\int_0^1 \frac{\ln{(x+1)}}{x^2 + 1} dx\).
Mathematics
Calculus
integration
0
Undergraduate
By
TrueShelf Inc.
on Sept. 28, 2013 | Updated Jan. 16, 2017
Partitioning planar graphs
Let \(G=(V,E)\) simple planar graph. Prove that \(V\) can be partitioned into three disjoint sets \(V = V_1 \cup V_2 \cup V_3\), such that the induced subgraphs on \(V_1\), \(V_2\) and \(V_3\) are acy…
Mathematics
Graph Theory
acyclic
graph partition
planar graphs
0
High School
By
TrueShelf Inc.
on May 7, 2014 | Updated Jan. 16, 2017
Adding the Numbers 1 to 100
What is the sum of the integers from 1 to 100 ? Origin : See this anecdote.
Puzzles
Puzzles
math puzzle
0
Undergraduate
By
TrueShelf Inc.
on Oct. 3, 2013 | Updated Jan. 16, 2017
Leaves in a tree
Prove that every tree has at least two leaves (i.e., vertices of degree \(1\)). Prove that every tree with maximum degree \(\Delta>1\) has at least \(\Delta\) leaves.
Mathematics
Graph Theory
counting
trees
0
High School
By
TrueShelf Inc.
on May 7, 2014 | Updated Jan. 16, 2017
Quarter past three
How many degrees are there in the angle between the hour and minute of a clock when the time is a quarter past three ?
Mathematics
Puzzles
Geometry
Puzzles
geometry puzzle
0
Undergraduate
By
TrueShelf Inc.
on Oct. 3, 2013 | Updated Jan. 16, 2017
Saturated vertices and Maximum matchings
Let \(S\) be a set of vertices saturated by a matching \(M\) in a graph \(G\). Prove that some maximum matching also saturates all of \(S\). Must the statement be true for every maximum matching ? …
Mathematics
Graph Theory
matching
3
High School
By
TrueShelf Inc.
on Nov. 14, 2016 | Updated Feb. 24, 2017
IMO 2016 Problem 5
The equation \((x-1)(x-2)(x-3)...(x-2016) = (x-1)(x-2)(x-3)...(x-2016)\) is written on a board, with 2016 linear factors on each side. What is the least possible value of \(k\) for which it is possi…
Mathematics
Combinatorics
imo
imo 2016
polynomials
0
High School
By
TrueShelf Inc.
on Aug. 28, 2013 | Updated Jan. 16, 2017
7 points inside a hexagon
Consider a hexagon \(H\) with side length 1. Given any 7 points inside \(H\), show that at least two points are separated by a distance of at most 1.
Puzzles
Puzzles
geometry puzzle
pigeonhole principle
0
Undergraduate
By
TrueShelf Inc.
on Oct. 3, 2013 | Updated Jan. 16, 2017
Eulerian facts
State whether each of the following statements are TRUE or FALSE. Your answers should be accompanied by a proof. Every Eulerian bipartite graph has an even number of edges. Every Eulerian simple gr…
Mathematics
Graph Theory
eulerian graph
true or false
1
High School
By
TrueShelf Inc.
on July 21, 2016 | Updated Jan. 16, 2017
IMO 2016 Problem 3
Let \(P = A_1, A_2 \dots A_k\) be a convex polygon on the plane. The vertices \(P = A_1, A_2 \dots A_k\) have integral coordinates and lie on a circle. Let \(S\) be the area of \(P\). An odd positive …
Mathematics
Geometry
polygon
0
Undergraduate
By
TrueShelf Inc.
on Dec. 13, 2013 | Updated Jan. 16, 2017
Every edge in a triangle
For \(n \geq 3\), determine the minimum number of edges in a connected \(n\)-vertex graph in which every edge belongs to a triangle.
Mathematics
Graph Theory
triangles
0
Undergraduate
By
TrueShelf Inc.
on Sept. 29, 2013 | Updated Jan. 16, 2017
Dirichlet approximation
(Trivial approximation) For \(x \in \mathbb{R}\) and \(n \in \mathbb{Z}^+\), there is a rational number \(\frac{p}{q}\), with \(1 \leq q \leq n\), such that …
Mathematics
Combinatorics
pigeonhole principle
0
Undergraduate
By
TrueShelf Inc.
on Aug. 22, 2013 | Updated Jan. 16, 2017
Graphs and Fermat's Little Theorem
Given a prime number \(n\), let \(\mathbb{Z}_n\), denote the set of congruence classes of integers modulo \(n\). Let \(a\) be a natural number having no common prime factors with \(n\); multiplication…
Mathematics
Graph Theory
Number Theory
digraphs
fermats little theorem
0
Undergraduate
By
TrueShelf Inc.
on Sept. 27, 2013 | Updated Jan. 16, 2017
Graph complement
Let \(G\) be an undirected graph without self-loops and multi-edges. The complement of graph \(G\) is a graph \(\overline{G}\) on the same vertices such that two vertices of \(\overline{G}\) are adjac…
Mathematics
Graph Theory
connectivity
graph complement
0
Undergraduate
By
TrueShelf Inc.
on Oct. 3, 2013 | Updated Jan. 16, 2017
Regular planar graphs
Prove or disprove : For each \(n\in \mathbb{N}\), there is a simple connected \(4\)-regular planar graph with more than \(n\) vertices. Prove that a planar, simple, connected, \(6\)-regular graph d…
Mathematics
Graph Theory
connectivity
planar graphs
regular graphs
0
Undergraduate
By
TrueShelf Inc.
on June 18, 2013 | Updated Jan. 16, 2017
Putnam B6 1998
Prove that, for any integers \(a, b, c\), there exists a positive integer \(n\) such that \(\sqrt{n^3 + an^2 + bn + c}\) is not an integer.
Mathematics
Number Theory
perfect square
0
Undergraduate
By
TrueShelf Inc.
on Oct. 3, 2013 | Updated Jan. 16, 2017
Minimum-weight spanning path
Let \(G\) be an undirected weighted complete graph. Iteratively select the edge of least weight such that the edges selected so far form a disjoint union of paths. After \(n-1\) steps, the result is a…
Mathematics
Graph Theory
counter example
hamiltonian path
path
0
Undergraduate
By
TrueShelf Inc.
on Sept. 28, 2013 | Updated Jan. 16, 2017
Basics of Ramsey's theory
A special case of Ramsey's theorem states that for any pair of positive integers \((r,s)\), there exists a least positive integer \(R(r,s)\) such that for any complete graph on \(R(r,s)\) vertices, wh…
Mathematics
Graph Theory
extremal graph theory
ramsey theory
0
High School
By
TrueShelf Inc.
on June 18, 2013 | Updated Jan. 16, 2017
Calling out integers
Two players \(A\) and \(B\) are playing a game. They take turns and call out an integer during their turn. The first player to call out \(50\) wins. They must follow these rules : The first player \…
Mathematics
Puzzles
Game Theory
Puzzles
math puzzle
0
High School
By
TrueShelf Inc.
on May 5, 2014 | Updated Jan. 16, 2017
Wine and water mixing problem
In the wine and water mixing problem, one starts with two barrels, one holding wine and the other an equal volume of water. A cup of wine is taken from the wine barrel and added to the water. A cup of…
Puzzles
Puzzles
math puzzle
0
Undergraduate
By
TrueShelf Inc.
on Aug. 28, 2013 | Updated Jan. 16, 2017
Putnam A6 2005
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
High School
By
TrueShelf Inc.
on Sept. 28, 2013 | Updated Jan. 16, 2017
Limits sine cosine tangent
Evaluate the following limits : \(\lim_{x \to 0} {\frac{\sqrt{1+{\sin}^2x^2}-{\cos}^3x^2}{x^3{\tan}x}}\)
Mathematics
Calculus
cosine
limits
sine
tangent
0
Undergraduate
By
TrueShelf Inc.
on June 24, 2013 | Updated Jan. 16, 2017
Putnam A4 2005
Let \(H\) be an \(n \times n\) matrix all of whose entries are \(\pm 1\) and whose rows are mutually orthogonal. Suppose \(H\) has an \(a \times b\) sub matrix whose entries are all \(1\). Show that …
Mathematics
Matrix Theory
matrices
0
Undergraduate
By
TrueShelf Inc.
on Sept. 28, 2013 | Updated Jan. 16, 2017
Summations and Combinations
Prove the following : \(\sum_{k=0}^{\lfloor n/2\rfloor} (-1)^k {n-k \choose k} \cdot 2^{n-2k} = n + 1\) \(\sum_{k=0}^n {2k \choose k}{2n-2k \choose n-k} = 4^n\) …
Mathematics
Combinatorics
n choose k
summation
0
Undergraduate
By
TrueShelf Inc.
on Sept. 28, 2013 | Updated Jan. 16, 2017
Asymptotic Summations
Prove the following : \(\sum_{i=1}^{n} \frac{\log i}{i} = \Theta ((\log n)^2)\) \( \sum_{i=0}^{n} |\sin(i)| = \Theta(n)\) where \(i\) is in radians.
Mathematics
Discrete Mathematics
asymptotic analysis
0
Undergraduate
By
TrueShelf Inc.
on May 21, 2014 | Updated Jan. 16, 2017
99 fair coins
Person \(A\) flips 99 fair coins and obtains \(a\) heads. Person \(B\) flips 100 fair coins and obtains \(b\) heads. What is the probability that \(a < b\) ?
Mathematics
Probability
conditional probability
interview question
0
Undergraduate
By
TrueShelf Inc.
on Oct. 3, 2013 | Updated Jan. 16, 2017
Planar graphs and girth
Let \(G\) be an \(n\)-vertex simple connected planar graph with girth \(k\). Prove that \(G\) has at most \((n-2)\cdot \frac{k}{k-2}\) edges. Use this to prove that the Petersen graph is nonplanar. …
Mathematics
Graph Theory
eulers formula
girth
petersen graph
planar graphs
0
Undergraduate
By
TrueShelf Inc.
on April 16, 2014 | Updated Jan. 16, 2017
Not square-free numbers
Suppose that \(m\) is a positive integer that is not square-free. Show that there exist integers \(a_1\) and \(a_2\) such that \(a_1 \not\equiv a_2 (\mbox{mod}\ m)\), but …
Mathematics
Number Theory
modular arithmetic
0
Undergraduate
By
TrueShelf Inc.
on Sept. 3, 2013 | Updated Jan. 16, 2017
Parliament Pacification
In a parliament, each member has at most three enemies. (We assume that enmity is always mutual). Is the following statement TRUE (or) FALSE ? One can always divide the parliament into two chambers …
Mathematics
Puzzles
Graph Theory
Puzzles
math puzzle
0
Undergraduate
By
TrueShelf Inc.
on Sept. 28, 2013 | Updated Jan. 16, 2017
Ramsey points
Let \(P\) be a set of \(n\) points in the plane, such that each 4-tuple forms a convex 4-gon. Then \(P\) forms a convex n-gon. Let \(P\) be a set of five points in the plane, with no three points col…
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on May 29, 2014 | Updated Jan. 16, 2017
JEE-Main 2013 Mathematics 35
All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of 10 to each of the students. Which of the following statistical measures will not change even aft…
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on May 29, 2014 | Updated Jan. 16, 2017
JEE-Advanced 2013 Mathematics 55
A rectangular sheet of fixed perimeter with sides having their lengths in the ratio 8 : 15 is converted into an open rectangular box by folding after removing squares of equal area from all four corne…
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on June 13, 2013 | Updated Jan. 16, 2017
Among the following sorting algorithms, which one has the least worst-case running time asymptotically ?
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on Jan. 16, 2017 | Updated Jan. 27, 2017
JEE Advanced 2016 Paper 1 Mathematics Question 45
In a triangle XYZ, let x, y, z be the lengths of sides opposite to the angles X, Y, Z respectively, and \(2s = x + y + z\). If \(\displaystyle\frac{s-x}{4} = \frac{s-y}{3} = \frac{s-z}{2}\) and area o…
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JEE Advanced 2016 Paper 2 Mathematics Question 39
The value \(\displaystyle\sum_{k=1}^{13}\frac{1}{\text{sin}{\left(\frac{\pi}{4} + \frac{(k-1)\pi}{6}\right)}\text{sin}{\left(\frac{\pi}{4} + \frac{k\pi}{6}\right)}}\) is equal to
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on Jan. 16, 2017 | Updated Jan. 27, 2017
JEE Advanced 2016 Paper 1 Mathematics Question 49
Let RS be the diameter of the circle \(x^2 + y^2 = 1\), where S is the point \((1,0)\). Let P be a variable point (other than R and S) on the circle and tangents to the circle at S and P meet at the p…
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on Jan. 9, 2017 | Updated Jan. 27, 2017
JEE Advanced 2016 Paper 1 Mathematics Question 40
A computer producing factory has only two plants \(T_1\) and \(T_2\). Plant \(T_1\) produces 20% and plant \(T_2\) produces 80% of the total computers produced. 7% of computers produced in the factory…
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on Jan. 16, 2017 | Updated Jan. 27, 2017
JEE Advanced 2016 Paper 1 Mathematics Question 46
A solution curve of the differential equation \(\displaystyle(x^2 + xy + 4x + 2y + 4) \frac{dy}{dx} - y^2 = 0, \ \ \ x > 0\), passes through the point \((1,3)\). Then the solution curve
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on Jan. 17, 2017
JEE Advanced 2016 Paper 1 Mathematics Question 52
The total number of distinct \(x \in [0,1]\) for which \(\displaystyle\int_0^x \frac{t^2}{1+t^4} dt = 2x - 1\) is
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JEE Advanced 2016 Paper 1 Mathematics Question 50
The total number of distinct \(x \in \mathbb{R}\) for which \(\left| \begin{array}{ccc} x & x^2 & 1+x^3 \\ 2x & 4x^2 & 1+8x^3 \\ 3x & 9x^2 & 1+27x^3 \end{array} \right| = 10 \), is
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on Jan. 16, 2017 | Updated Jan. 27, 2017
JEE Advanced 2016 Paper 1 Mathematics Question 44
Let \(P = \left[ \begin{array}{ccc} 3 & -1 & -2 \\ 2 & 0 & \alpha \\ 3 & -5 & 0 \end{array} \right] \), where \(\alpha \in \mathbb{R}.\) Suppose \(Q = [q_{ij}]\) is a matrix such that \(PQ = kI\), whe…
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JEE Advanced 2016 Paper 2 Mathematics Question 41
Area of the region \(\displaystyle\{(x,y) \in \mathbb{R}^2 : y \geq \sqrt{|x+3|}, 5y \leq x + 9 \leq 15\}\) is equal to
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on Jan. 9, 2017 | Updated Jan. 27, 2017
JEE Advanced 2016 Paper 1 Mathematics Question 37
Let \(-\frac{\pi}{6} < \theta < -\frac{\pi}{12}\). Suppose \(\alpha_1\) and \(\beta_1\) are the roots of the equation \(x^2 - 2 x \text{sec}\theta + 1 = 0\) and \(\alpha_2\) and \(\beta_2\) are the r…
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JEE Advanced 2016 Paper 2 Mathematics Question 37
Let \(P = \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1 \end{array} \right] \) and \(I\) be the identity matrix of order 3. If \(Q = [q_{ij}]\) is a matrix such that …
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Matrix Theory
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on Jan. 29, 2017
JEE Advanced 2016 Paper 2 Mathematics Question 42
Let \(P\) be the image of the point \((3,1,7)\) with respect to the plane \(x - y + z = 3\). Then the equation of the plane passing through \(P\) and containing the straight line …
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on Jan. 9, 2017 | Updated Jan. 27, 2017
JEE Advanced 2016 Paper 1 Mathematics Question 39
Let \(S = \displaystyle\{ x \in (-\pi, \pi) : x \neq 0, \pm \frac{\pi}{2}\}\). The sum of all distinct solutions of the equation …
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on Jan. 16, 2017 | Updated Jan. 18, 2017
JEE Advanced 2016 Paper 1 Mathematics Question 47
Let \(f : \mathbb{R} \rightarrow \mathbb{R}\), \(g : \mathbb{R} \rightarrow \mathbb{R}\) and \(h : \mathbb{R} \rightarrow \mathbb{R}\) be differentiable functions such that \(f(x) = x^3 + 3x + 2\), …
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on Jan. 17, 2017 | Updated Jan. 27, 2017
JEE Advanced 2016 Paper 1 Mathematics Question 54
Let \(z = \displaystyle\frac{-1 + \sqrt{3}i}{2}\) where \(i = \sqrt{-1}\), and \(r, s \in \{1,2,3\}\). Let \(P = \left[ \begin{array}{cc} (-z)^r & z^{2s} \\ z^{2s} & z^r \end{array} \right]\) and …
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on Jan. 17, 2017 | Updated Jan. 27, 2017
JEE Advanced 2016 Paper 1 Mathematics Question 51
Let \(m\) be the smallest positive integer such that the coefficient of \(x^2\) in the expansion of \((1+x)^2 + (1+x)^3 + \dots + (1+x)^{49} + (1+mx)^{50}\) is \((3n+1)\ \ {}^{51}C_{3}\) for some posi…
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on Jan. 16, 2017
JEE Advanced 2016 Paper 1 Mathematics Question 42
Consider a pyramid OPQRS located in the first octant \(( x \geq 0, y \geq 0, z \geq 0 )\) with O as origin, and OP and OR along the x-axis and the y-axis, respectively. The base OPQR of the pyramid is…
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on Jan. 9, 2017 | Updated Jan. 27, 2017
JEE Advanced 2016 Paper 1 Mathematics Question 38
A debate club consists of 6 girls and 4 boys. A team of 4 members is to be selected from this club including the selection of a captain (from among these 4 members) for the team. If the team has to in…
Mathematics
Combinatorics
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on Jan. 16, 2017 | Updated Jan. 27, 2017
JEE Advanced 2016 Paper 1 Mathematics Question 43
Let \(f : (0, \infty) \rightarrow \mathbb{R}\) be a differentiable function such that \(f' (x) = 2 - \frac{f(x)}{x}\) for all \(x \in (0 , \infty)\) and \(f(1) \neq 1\). Then
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on Jan. 17, 2017
JEE Advanced 2016 Paper 1 Mathematics Question 53
Let \(\alpha, \beta \in \mathbb{R}\) be such that \(\displaystyle\lim_{x \rightarrow 0} \frac{x^2 \text{sin} (\beta{x})}{\alpha{x} - \text{sin}x} = 1\). Then \(6(\alpha + \beta)\) equals
Mathematics
Calculus
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limits
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on Jan. 16, 2017
JEE Advanced 2016 Paper 1 Mathematics Question 48
The circle \(C_1 : x^2 + y^2 = 3\), with centre at O, intersects the parabola \(x^2 = 2y\) at the pont P in the first quadrant. Let the tangent to the circle \(C_1\) at P touches other two circles …
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Geometry
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on May 27, 2014 | Updated Jan. 16, 2017
JEE-Main 2013 Mathematics 31
The circle passing through \((1, −2)\) and touching the axis of \(x\) at \((3, 0)\) also passes through the point
Mathematics
Geometry
circle
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on Jan. 29, 2017
JEE Advanced 2016 Paper 2 Mathematics Question 40
The value of \(\displaystyle\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}}\frac{x^2\text{cos}x}{1+e^x}dx\) is equal to
Mathematics
Calculus
integration
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on Jan. 7, 2017 | Updated Jan. 27, 2017
JEE Advanced 2016 Paper 1 Mathematics Question 41
The least value of \(\alpha \in \mathbb{R}\) for which \(4\alpha {x}^2 + \frac{1}{x} \geq 1\), for all \(x > 0\), is
Mathematics
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on Jan. 29, 2017
JEE Advanced 2016 Paper 2 Mathematics Question 38
Let \(b_i > 1\) for \(i = 1, 2, \dots, 101\). Suppose \(\text{log}_e{b_1}, \text{log}_e{b_2}, \dots, \text{log}_e{b_{101}}\) are in Arithmetic Progression (A.P.) with the common difference …
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