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High School
By
True IMO
on Nov. 14, 2016 | Updated Jan. 4, 2018
International Mathematical Olympiad 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 p…
Mathematics
Combinatorics
imo
imo 2016
polynomials
1
High School
By
True IMO
on July 1, 2017 | Updated Jan. 4, 2018
International Mathematical Olympiad 2015 Problem 2
Determine all triples of positive integers such that each of the numbers is a power of 2.
Mathematics
Algebra
imo
imo 2015
2
High School
By
True IMO
on July 21, 2016 | Updated Jan. 4, 2018
International Mathematical Olympiad 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
imo
imo 2016
polygon
0
Undergraduate
By
123forman
on May 8, 2014 | Updated Dec. 6, 2017
Democratic distribution of wealth
A crew of 100 pirates have captured 100 laptops. The pirates are ranked 1 through 100, with pirate 100 being the pirate king. Captured booty is distributed as follows. The pirate king proposes a dis…
Mathematics
Discrete Mathematics
Game Theory
induction
0
High School
By
Shiva Kintali
on June 27, 2013 | Updated Jan. 4, 2018
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
1
Undergraduate
By
Shiva Kintali
on Sept. 27, 2013 | Updated Jan. 4, 2018
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
High School
By
Shiva Kintali
on Sept. 30, 2013 | Updated Jan. 4, 2018
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 …
Mathematics
Discrete Mathematics
binomial theorem
induction
0
Undergraduate
By
Shiva Kintali
on June 16, 2012 | Updated Dec. 6, 2017
L shaped tiling
Consider a \(2^n × 2^n\) board with one (arbitrarily chosen) square removed, as in the following figure for \(n = 3\). Prove that any such board can be tiled (without gaps or overlaps) by \(L\)-shaped…
Puzzles
Puzzles
induction
tiling
0
High School
By
True IMO
on Oct. 9, 2012 | Updated Jan. 4, 2018
International Mathematical Olympiad 1998 Problem 3
For any positive integer \(n\), let \(d(n)\) denote the number of positive divisors of \(n\) (including \(1\) and \(n\) itself). Determine all positive integers \(k\) such that \(d(n^2)/d(n) = k\) for…
Mathematics
Number Theory
divisibility
imo
imo 1998
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