# Solution: 2009-3 Intersecting family

Let $$\mathcal F$$ be a collection of subsets (of size r) of a finite set E such that $$X\cap Y\neq\emptyset$$ for all $$X, Y\in \mathcal F$$. Prove that there exists a subset S of E such that $$|S|\le (2r-1)\binom{2r-3}{r-1}$$ and $$X\cap Y\cap S\neq\emptyset$$ for all $$X,Y\in\mathcal F$$.

The best solution was submitted by Hyung Ryul Baik (백형렬), 수리과학과 2003학번. Congratulations!

Click here for his Solution of Problem 2009-3.

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# 2009-4 Initial values

Let $$a_0=a$$ and $$a_{n+1}=a_n (a_n^2-3)$$. Find all real values $$a$$ such that the sequence $$\{a_n\}$$ converges.

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# Status:2009-3 Intersecting family

So far the problem remains open.

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# 2009-3 Intersecting family

Let $$\mathcal F$$ be a collection of subsets (of size r) of a finite set E such that $$X\cap Y\neq\emptyset$$ for all $$X, Y\in \mathcal F$$. Prove that there exists a subset S of E such that $$|S|\le (2r-1)\binom{2r-3}{r-1}$$ and $$X\cap Y\cap S\neq\emptyset$$ for all $$X,Y\in\mathcal F$$.

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# Solution: 2009-2 Sequence of Log

Let $$a_1<\cdots$$ be a sequence of positive integers such that $$\log a_1, \log a_2,\log a_3,\cdots$$ are linearly independent over the rational field $$\mathbb Q$$. Prove that $$\lim_{k\to \infty} a_k/k=\infty$$.

The best solution was submitted by SangHoon Kwon (권상훈), 수리과학과 2006학번. Congratulations!

Click here for his Solution of Problem 2009-2.

There were 3 other submitted solutions which will earn points: 김치헌+3, 김린기+3,  조강진+2.

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# 2009-2 Sequence of Log

Let $$a_1<\cdots$$ be a sequence of positive integers such that $$\log a_1, \log a_2,\log a_3,\cdots$$ are linearly independent over the rational field $$\mathbb Q$$. Prove that $$\lim_{k\to \infty} a_k/k=\infty$$.

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# Solution: 2009-1 Integer or not

Chiheon Kim (김치헌)

Let $$a_1\le a_2\le \cdots \le a_n$$ be integers. Prove that

$$\displaystyle\prod_{1\le j<i\le n} \frac{a_i-a_j}{i-j}$$

is an integer.

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

Click here for his Solution of Problem 2009-1.

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# Status: 2009-1 Integer or not

So far 5 solutions were submitted but I am not sure whether any of them is absolutely correct. They (이병찬, 류연식, 권상훈, 김린기, 조강진) will all receive 2 points each.
Here is the origin of typical mistakes: if x|z and y|z, then xy|z.
The problem remains open.

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# 2009-1 Integer or not

Let $$a_1\le a_2\le \cdots \le a_n$$ be integers. Prove that

$$\displaystyle\prod_{1\le j<i\le n} \frac{a_i-a_j}{i-j}$$

is an integer.

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# We will resume on February 6th, Friday!

Welcome back! We will continue our POW. Starting from this semester, we will have a new set of rules. First of all, the problems will be posted on every Friday 3PM. (Note that we now have a tea time every day at 3PM.) The problem will be still posted on the blackboard in the room 1401, E6-1 bldg as well as on this website.

We will also revise our way of selecting winners of the semester. At each week, students will earn points.

• +2 if the submitted solution seems reasonable. (meaning: good try, it may be correct but I don’t have enough time to check it carefully)
• +3 if the submitted solution is correct,
• +4 if it was selected as the best solution. (meaning: correct and well written)

At the end of each semester, we will select the 1st, 2nd, and 3rd students as we did in the previous semester.

How to submit your solution: read the “rules” section.

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