# Solution: 2016-21 Bound on the number of divisors

For a positive integer $$n$$, let $$d(n)$$ be the number of positive divisors of $$n$$. Prove that, for any positive integer $$M$$, there exists a constant $$C>0$$ such that $$d(n) \geq C ( \log n )^M$$ for infinitely many $$n$$.

The best solution was submitted by Kim, Taegyun (김태균, 2016학번). Congratulations!

Here is his solution of problem 2016-21.

Alternative solutions were submitted by 신준형 (수리과학과 2015학번, +3), 장기정 (수리과학과 2014학번, +3, alternative solution), 국윤범 (수리과학과 2015학번, +3), 이상민 (수리과학과 2014학번, +3), 윤준기 (전기및전자공학부 2014학번, +3), 이정환 (수리과학과 2015학번, +3), Ivan Adrian Koswara (전산학부 2013학번, +3), 조준영 (수리과학과 2012학번, +3), 이시우 (포항공대 수학과 2013학번, +3).

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# 2016-22 Computing the Determinant

Let $$M_n=(a_{ij})_{ij}$$ be an $$n\times n$$ matrix such that $a_{ij}=\binom{2(i+j-1)}{i+j-1}.$ What is $$\det M_n$$?

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# Solution: 2016-20 Finding a subspace

Let $$V_1,V_2,\ldots$$ be countably many $$k$$-dimensional subspaces of $$\mathbb{R}^n$$. Prove that there exists an $$(n-k)$$-dimensional subspace $$W$$ of $$\mathbb{R}^n$$ such that $$\dim V_i\cap W=0$$ for all $$i$$.

The best solution was submitted by Shin, Joonhyung (신준형, 수리과학과 2015학번). Congratulations!

Here is his solution of problem 2016-20.

Alternative solutions were submitted by 김태균 (2016학번, +3), 국윤범 (수리과학과 2015학번, +3), 장기정 (수리과학과 2014학번, +3, alternative solution). One incorrect solution was submitted.

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# 2016-21 Bound on the number of divisors

For a positive integer $$n$$, let $$d(n)$$ be the number of positive divisors of $$n$$. Prove that, for any positive integer $$M$$, there exists a constant $$C>0$$ such that $$d(n) \geq C ( \log n )^M$$ for infinitely many $$n$$.

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# Solution: 2016-19 Zeta function

Let
$P(k) = \sum_{i_1=1}^{\infty} \dots \sum_{i_k=1}^{\infty} \frac{1}{i_1 \dots i_k (i_1 + \dots + i_k)}$
for a positive integer $$k$$. Find $$\zeta(k+1) / P(k)$$, where $$\zeta$$ is the Riemann-zeta function.

The best solution was submitted by Lee, Sangmin (이상민, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2016-19.

Alternative solutions were submitted by 신준형 (수리과학과 2015학번, +3), 장기정 (수리과학과 2014학번, +3), 이종원 (수리과학과 2014학번, +3), 김태균 (2016학번, +3).

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# 2016-20 Finding a subspace

Let $$V_1,V_2,\ldots$$ be countably many $$k$$-dimensional subspaces of $$\mathbb{R}^n$$. Prove that there exists an $$(n-k)$$-dimensional subspace $$W$$ of $$\mathbb{R}^n$$ such that $$\dim V_i\cap W=0$$ for all $$i$$.

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# Solution: 2016-18 Partitions with equal sums

Suppose that we have a list of $$2n+1$$ integers such that whenever we remove any one of them, the remaining can be partitioned into two lists of $$n$$ integers with the same sum. Prove that all $$2n+1$$ integers are equal.

The best solution was submitted by Joonhyung Shin (신준형, 수리과학과 2015학번). Congratulations!

Here is his solution of problem 2016-18.

Alternative solutions were submitted by 국윤범 (수리과학과 2015학번, +3), 장기정 (수리과학과 2014학번, +3), 이종원 (수리과학과 2014학번, +3, solution), 김태균 (2016학번, +3), 윤준기 (전기및전자공학부 2014학번, +3), 김재현 (2016학번, +3), 채지석 (2016학번, +3), 강한필 (2016학번, +3), Ivan Adrian Koswara (전산학부 2013학번, +3), 김기현 (수리과학과 대학원생, +3). One incorrect solution was received.

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$P(k) = \sum_{i_1=1}^{\infty} \dots \sum_{i_k=1}^{\infty} \frac{1}{i_1 \dots i_k (i_1 + \dots + i_k)}$
for a positive integer $$k$$. Find $$\zeta(k+1) / P(k)$$, where $$\zeta$$ is the Riemann-zeta function.