Category Archives: problem

2022-12 A partition of the power set of a set

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,BP([n]) and i[k] with A,B,AB,ABQi.

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2022-11 groups with torsions

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.

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2022-10 Polynomial with root 1

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.

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2022-09 A chaotic election

Let A1,,Ak be presidential candidates in a country with n1 voters with k2. 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.

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2022-08 two sequences

For positive integers n2, let an=n/π and let bn=csc(π/n). Is an=bn for all n3?

Solutions are due May 13th (Friday), 6PM, and it will remain open if nobody solved it.

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2022-07 Coulomb potential

Prove the following identity for x,yR3:
1|xy|=1π3R31|xz|21|yz|2dz.

Solutions are due May 6th (Friday), 6PM, and it will remain open if nobody solved it.

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2022-06 A way of putting parentheses


We have an expression x0÷x1÷x2÷÷xn. A way of putting n1 left parentheses and n1 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?

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Notice on POW 2022-05 (Problem Revision)

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.

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