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Wednesday, October 11, 2017

A tale of two cities

In an island with 257 cities, there are 2018 railroad segments such that each segment connects exactly 2 cities, and each pair of cities are connected by at most 1 segment. A group of 10 cities is called "clustered" if each pair of those 10 is connected by a segment.

Prove that there exist 2 unconnected cities that we can connect without increasing the number of clustered groups.

Solution

Proof by contradiction. Suppose we cannot connect two cities without increasing the number of clustered groups.

Let $n=257$ be the number of cities. Take all possible groups of 10 cities, ${n \choose 10 }$ of them, and count how many segments there are among them. Of course, if the group is clustered, it should have ${10 \choose 2} = 45$ segments. Let $S$ be the sum of unconnected pairs, over all possible groups of 10 cities.

Now let us count $S$ in a different way. Take 2 connected cities $A$ and $B$. The segment $AB$ contributes to $S$ only when the chosen group of 10 cities include $AB$, so this segment is counted ${n-2 \choose 8}$ times. Because there are 2018 segments, then $S$ must be:

$$S = {n-2 \choose 8}.(2018)$$

Note that $2018 < 2020 = 8n-36$, so $S < {n-2 \choose 8}.(8n-26)$. That means, on average, there are $S / {n \choose 10 }$ segments.

Non-adjacent subsets

Find the number of subsets of $n$ such that if $i$ does not belong in the subset then either $i+1$ or $i-1$ belongs to the subset. (Hint: use double recursion, use fibonacci series)

Tuesday, October 10, 2017

symmetric 2 variable function

Suppose $S = \{1,2,\dots,n\}$ and $f: S \times S \to R$ satisfies the following: $$f(i,j) + f(j,i) = 0 \forall i,j \in S$$ Now for any two number $i,j$, we say that $i$ is superior to $j$ if there exists a $k$ (not necessarily distinct from $i,j$) such that $f(i,k) + f(k,j) \geq 0$. Show that there exists a number $x \in S$ such that $x$ is superior to all elements of $S$.

Sunday, October 8, 2017

Combinatorial Proof

Give a combinatorial proof of this identity: $$n(n+1)2^{n-2} = \sum_{k=1}^n k^2 {n \choose k}$$ Solution

Let's say in a class there are $n$ students and you are asked to select one or more students, with the caveat that Joe must be selected. For each student that is selected, the whole class gets 1 dollar. So what is the expected value of the class' income?

Or if you want a non-probabilistic way to think about it, then over all the possible ways to select students, what is the sum of class income in all of those scenarios?

First way to count: Say there are $k \geq 1$ students who are selected (including Joe). There are $n-1 \choose k-1$ ways to choose the students (because Joe must be selected). So this particular way of selecting the students has $k {n-1 \choose k-1} = \frac{k^2}{n} {n \choose k}$ income. So the sum of all income over all scenarios is $\sum_{k=1}^n \frac{k^2}{n} {n \choose k}$

Second way to count: Consider a student, not Joe. How many dollars does he contribute to the total class income? He contributes 1 dollar everytime he is selected, which is $2^{n-2}$ times (because both he and Joe must be selected for him to contribute, so the other $n-2$ students can be chosen freely, selected or not selected). There are $(n-1)2^{n-2}$ contributions from these, because there are $n-1$ students that are not Joe.

But how many dollars did Joe contribute? He ALWAYS contributes 1 dollar for every selection, and there are $2^{n-1}$ selections total. So there are $2^{n-1}$ contributions from Joe.

So the total contribution is $(n-1)2^{n-2} + 2^{n-1} = (n+1)2^{n-2}$

Wednesday, September 20, 2017

Coloring polygon vertices

The vertices of a regular $n$-gon ($n \geq 3$) are to be colored with one of the $k$ colors such that no two adjacent vertices may have the same color. Let $f(n)$ be the number of legal coloring. Find a closed formula for $f(n)$.

Solution:

Suppose we sever one of the edges temporarily, so that we're left with a "string" of length $n$ where the first and last vertices are not connected. In order to color this string, we may color the first vertex $k$ ways, but each of the subsequent vertices may only be colored $k-1$ ways, avoiding the color of the previous vertex. Suppose $g(n)$ is the number of ways to color a string of length $n$, then $g(n) = k(k-1)^{n-1}$

Now, take one of these colored strings of length $n$ and join the first and last vertices. The resulting polygon may or may not be a valid coloring. It is not a valid coloring if and only if the first and last vertices have the same color. The number of such invalid coloring is the same as $f(n-1)$. There is a bijection between the invalid coloring where the first and last vertices have the same color and a valid coloring of an $(n-1)$-gon.

So: $$f(n) = g(n) - f(n-1)$$ Now, $f(3) = k(k-1)(k-2)$. So: $$f(4) = k(k-1)^3 - f(3) = k(k-1)(k^2 -3k+3) =(k-1)((k-1)^3+1)$$ $$f(5) = k(k-1)^4 - f(4) = k(k-1)(k^3-4k^2+6k-4) = (k-1)((k-1)^4-1)$$ $$f(6) = k(k-1)^5 - f(5) = (k-1)((k-1)^5+1)$$ So we can see a pattern, which we can prove via induction easily.

$$f(n) = (k-1)((k-1)^{n-1}+(-1)^n)$$

ODE discrete version

If $f$ is a function that is defined over natural numbers ($f:N \to R$), let $f^*$ denote the function $f^*(n) = f(n) - f(n-1)$. Find all functions such that $$n(f^*)^*(n) + (n-1)f^*(n) = f(n)$$ for all $n > 2$

Solution We can check that $f(n) = A(n-1) + B/2^n$ for any constant $A,B$ satisfies the given equation. Now we show that there's nothing else.

Define $g = f^*+f$. The equation is identical to: $$n(f^*+f)^*(n) =(f^*+f)(n)$$ $$ n(g(n) - g(n-1)) = ng^*(n) = g(n)$$ $$\frac{g(n)}{n} = \frac{g(n-1)}{n-1}$$ for all $n>2$, therefore $g(n)/n$ must be a constant. Remember that $g(1)$ is not defined because $f$ is only defined for $n \geq 1$ thus $g$ is only defined for $n \geq 2$. Furthermore, the value $g(2)$ depends on $f(1)$ and $f(2)$.

Case 1: $g(2) = 0$ In this case then $g(n) = 0 \forall n$. $$(f^*+f)(n) = 0$$ $$f(n) = f(n-1)/2$$ so we easily get $f(n) = k/2^n$ for some constant $k$.

Case 1: $g(2) \neq 0$ Note that if $f$ is a solution to the problem then $kf$ is also a solution to the problem, for all $k$ real numbers. If we replace $f$ by $kf$, then $g$ is also replaced by $kg$. So WLOG, we may assume that $g(2) = 2$

Because $g(n) / n$ is a constant then $$g(n) / n = g(2) / 2 = 1$$ so $g(n) = n \forall n >2$ $$f(n) = \frac{f(n-1) + n}{2}$$ for all $n > 2$.

Because $g(2) = 2 = 2f(2) - f(1)$ then $f(2) = 1+ f(1)/2$ Now notice this pattern: $$f(3) = \frac{f(2) + 3}{2} = 2 + f(1)/4$$ $$f(4) = \frac{f(3) + 4}{2} = 3 + f(1)/8$$ Easy to prove by induction that $f(n) = n-1 + f(1)2^{1-n}$.

From the two cases above, it's clear that any solution of the equation must be in the form of $f(n) = A(n-1) + B/2^n$ for any constant $A,B$

Thursday, August 11, 2016

Polynomial and divisibility

Find an integer polynomial $P(x)$ with the lowest degree that satisfies the following:

$P(x)$ is divisible by $x^2+x+1$

$P(x)+3$ is divisible by $x^{2017} -1$

Solution

Let $Q(x) = x^2+x+1$ and $R(x) = x^{2016} + \dots + 1$. It's easy to show that $Q$ and $R$ are relatively prime.

The LCM for both divisors is thus $(x-1)Q(x)R(x)$ which has degree 2019.

Now if $P$ and $P'$ both satisfy the problem statement, then $P-P'$ must be divisible by $(x-1)Q(x)R(x)$.

On the other hand, if $P$ satisfies the problem statement, then $P + A(x).(x-1)Q(x)R(x)$ also satisfies the problem statement for any integer polynomial $A(x)$.

Therefore, we can find at most one $P(x)$ with degree less than 2019, because if there were two such polynomials, their difference must be divisible by $(x-1)Q(x)R(x)$ but that difference has degree less than 2019, so the difference must be zero. That one polynomial is our answer.

To find it: let $t$ be the root of $Q(x)$. We want to find $P(x)$ such that:

$$P(x)+3 = S(x).(x-1).R(x)$$ for some $S(x)$, with $S(x) = ax+b$. We know that it's possible for $S$ to be a linear function because we've established that there exists a $P$ with degree less than 2019, and $R$ has degree 2016.

$$P(t) + 3 = (at+b).(t-1).R(t)$$

$P(t) = 0$ because $P(x)$ is divisible by $Q(x)$ and $Q(t) = 0$.

$R(t) = t^{2016} + \dots + t + 1 = 1$ because every 3 consecutive terms in $t^{2016} + \dots + t$ sum to zero (because $Q(t) = 0$).

So: $$ 3 = (at+b)(t-1)$$ $$at^2 + (b-a)t - b-3 = 0$$ $$a(-t-1) + (b-a)t - b-3 = 0$$ $$ (b-2a)t=a+b+3$$

Solving for the system:$b-2a = 0, a+b+3 = 0$ gives us: $a = -1, b= -2$ so $S(x) = -(x+2)$

Plugging in to the equations, we find:

$$P(x) = -(x+2)(x^{2017}-1) - 3 = -x^{2018} - 2x^{2017} + x -1 = -x^2(x^{2016}-1) -2x(x^{2016}-1) -( x^2+x+1) $$

Each term in the right-most expression is divisible by $Q(x)$ because $(x^{2016}-1)$ is divisible by $x^3-1$ so $P(x)$ is also divisible by $Q(x)$