# 2008-4 Limit (9/25)

Let $$a_1=\sqrt{1+2}$$,
$$a_2=\sqrt{1+2\sqrt{1+3}}$$,
$$a_3=\sqrt{1+2\sqrt{1+3\sqrt{1+4}}}$$, …,
$$a_n=\sqrt{1+2\sqrt{1+3\sqrt {\cdots \sqrt{\sqrt{\sqrt{\cdots\sqrt{1+n\sqrt{1+(n+1)}}}}}}}}$$, … .

Prove that $$\displaystyle\lim_{n\to \infty} \frac{a_{n+1}-a_{n}}{a_n-a_{n-1}}=\frac12$$.

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# Solution: 2008-3 Integer Matrices

Let A, B be $$3\times 3$$ integer matrices such that A, A+B, A+2B, A+3B, A-B, A-2B, A-3B are invertible and their inverse matrices are all integer matrices.

Prove that A+4B also has an inverse, and its inverse is again an integer matrix.

The best solution was submitted by Haewon Yoon (윤혜원), 수리과학과 2004학번. Congratulations!

Here is his Solution of Problem 2008-3.

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# 2008-3 Integer Matrices (9/18)

Let A, B be $$3\times 3$$ integer matrices such that A, A+B, A+2B, A+3B, A-B, A-2B, A-3B are invertible and their inverse matrices are all integer matrices.

Prove that A+4B also has an inverse, and its inverse is again an integer matrix.

A, B가 $$3\times 3$$ 정수 행렬이면서, A, A+B, A+2B, A+3B, A-B, A-2B, A-3B가 모두 역행렬을 가지고 그 역행렬이 모두 정수행렬이라고 하자. 이때 A+4B 역시 역행렬을 가지고 그 역행렬은 정수행렬임을 보여라.

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# Solution: 2008-2 Strange representation

Prove that if x is a real number such that $$0<x\le \frac12$$, then x can be represented as an infinite sum

$$\displaystyle x=\sum_{k=1}^\infty \frac{1}{n_k}$$,

where each $$n_k$$ is an integer such that $$\frac{n_{k+1}}{n_k}\in \{3,4,5,6,8,9\}$$.

The best solution was submitted by Byoung Chan Lee (이병찬), 수리과학과 2007학번. Congratulations!

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# 2008-2 Strange representation (9/11)

Prove that if x is a real number such that $$0<x\le \frac12$$, then x can be represented as an infinite sum

$$\displaystyle x=\sum_{k=1}^\infty \frac{1}{n_k}$$,

where each $$n_k$$ is an integer such that $$\frac{n_{k+1}}{n_k}\in \{3,4,5,6,8,9\}$$.

x가 $$0<x\le \frac12$$을 만족하는 실수일때, x는 아래와 같은 무한급수로 표현할 수 있음을 보여라.

$$\displaystyle x=\sum_{k=1}^\infty \frac{1}{n_k}$$.

여기서 각 $$n_k$$는 정수이며 $$\frac{n_{k+1}}{n_k}\in \{3,4,5,6,8,9\}$$을 만족한다.

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# The problem will be posted at Thursday 3:30PM

Following a request from a student, we will announce the problem at 3:30pm instead of noon from this week.

몇몇 학생들의 요청에 따라, 이번 주부터는 문제를 정오가 아닌, 오후 3시 30분에 공개합니다.

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# Solution: 2008-1 Distinct primes

Let $$n$$ be a positive integer. Let $$a_1,a_2,\ldots,a_k$$ be distinct integers larger than $$n^{n-1}$$ such that $$|a_i-a_j|<n$$ for all $$i,j$$.

Prove that the number of primes dividing $$a_1a_2\cdots a_k$$ is at least $$k$$.

The best solution was submitted by Chiheon Kim (김치헌), 수리과학과 2006학번. Congratulations!

This problem is equivalent to a theorem of Grimm (see his paper, A Conjecture on Consecutive Composite Numbers, The American Mathematical Monthly, Vol. 76, No. 10 (Dec., 1969), pp. 1126-1128). He conjectured that the same thing can be done without the lower bound $$n^{n-1}$$. Laishram and Shorey verified Grimm’s conjecture when $$n<19000000000$$.

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Let $$n$$ be a positive integer. Let $$a_1,a_2,\ldots,a_k$$ be distinct integers larger than $$n^{n-1}$$ such that $$|a_i-a_j|<n$$ for all $$i,j$$.
Prove that the number of primes dividing $$a_1a_2\cdots a_k$$ is at least $$k$$.
$$n$$은 양의 정수라 하자. $$n^{n-1}$$보다 큰 $$k$$개의 서로 다른 정수 $$a_1,a_2,\ldots,a_k$$가 모든 $$i,j$$에 대해서 $$|a_i-a_j|<n$$을 만족한다고 하자.
이때 $$a_1a_2\cdots a_k$$의 약수인 소수의 개수는 $$k$$개 이상임을 보여라.