# 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!

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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!

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

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!

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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.