Prove for any x≥1 that
(∞∑n=0(n+x)−2)2≥2∞∑n=0(n+x)−3.
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Prove for any x≥1 that
(∞∑n=0(n+x)−2)2≥2∞∑n=0(n+x)−3.
Consider the power set P([n]) consisting of 2n subsets of [n]={1,…,n}.
Find the smallest k such that the following holds: there exists a partition Q1,…,Qk of P([n]) so that there do not exist two distinct sets A,B∈P([n]) and i∈[k] with A,B,A∪B,A∩B∈Qi.
Does there exists a finitely generated group which contains torsion elements of order p for all prime numbers p?
Solutions for POW 2022-11 are due July 4th (Saturday), 12PM, and it will remain open if nobody solved it.
Prove or disprove the following:
For any positive integer n, there exists a polynomial Pn of degree n2 such that
(1) all coefficients of Pn are integers with absolute value at most n2, and
(2) 1 is a root of Pn=0 with multiplicity at least n.
Let A1,…,Ak be presidential candidates in a country with n≥1 voters with k≥2. Candidates themselves are not voters. Each voter has her/his own preference on those k candidates.
Find maximum m such that the following scenario is possible where Ak+1 indicates the candidate A1: for each i∈[k], there are at least m voters who prefers Ai to Ai+1.
For positive integers n≥2, let an=⌈n/π⌉ and let bn=⌈csc(π/n)⌉. Is an=bn for all n≠3?
Solutions are due May 13th (Friday), 6PM, and it will remain open if nobody solved it.
Prove the following identity for x,y∈R3:
1|x−y|=1π3∫R31|x−z|21|y−z|2dz.
Solutions are due May 6th (Friday), 6PM, and it will remain open if nobody solved it.
POW will resume on Apr. 29.
We have an expression x0÷x1÷x2÷⋯÷xn. A way of putting n−1 left parentheses and n−1 right parenthese is called ‘parenthesization’ if it is valid and completely clarify the order of operations. For example, when n=3, we have the following five parenthesizations.
((x0÷x1)÷x2)÷x3,(x0÷(x1÷x2))÷x3,(x0÷x1)÷(x2÷x3),
x0÷((x1÷x2)÷x3),x0÷(x1÷(x2÷x3)).
(a) For an integer n, how many parenthesizations are there?
(b) For each parenthesization, we can compute the expression to obtain a fraction with some variables in the numerator and some variable in the denominator. For an integer n, determine which fraction occur most often. How many times does it occur?
I hope you noticed the day this problem appeared was April fool’s day. However, we sincerely apologize to the students got confused about the problem description, and we found that many students already submitted the solution corresponding to the original problem.
Hence We revise the problem as the following:
Show that there do not exist perfect squares a, b, c such that a2+b2=c2, provided that a, b, c are nonzero integers.
You should actually provide the full valid proof i.e. the solution like ‘It is the special case of some famous theorem hence it is trivial’ will not be graded. Please resubmit your solution if you already submitted the solution for the previous version.
We accept the solution until April 11 Monday, 6PM.