Help me please
by sealight2107, Apr 23, 2025, 2:40 PM
Let
be positive reals such that
. Find the minimum and maximum value of 



Interesting combinatoric problem on rectangles
by jaydenkaka, Apr 23, 2025, 2:22 PM
Define act <Castle> as following:
For rectangle with dimensions i * j, doing <Castle> means to change its dimensions to (i+p) * (j+q) where p,q is a natural number smaller than 3.
Define 1*1 rectangle as "C0" rectangle, and define "Cn" ("n" is a natural number) as a rectangle that can be created with "n" <Castle>s.
Plus, there is a constraint for "Cn" rectangle. The constraint is that "Cn" rectangle's area must be bigger than n^2 and be same or smaller than (n+1)^2. (n^2 < Area =< (n+1)^2)
Let all "C20" rectangle's area's sum be A, and let all "C20" rectangles perimeter's sum be B.
What is A-B?
For rectangle with dimensions i * j, doing <Castle> means to change its dimensions to (i+p) * (j+q) where p,q is a natural number smaller than 3.
Define 1*1 rectangle as "C0" rectangle, and define "Cn" ("n" is a natural number) as a rectangle that can be created with "n" <Castle>s.
Plus, there is a constraint for "Cn" rectangle. The constraint is that "Cn" rectangle's area must be bigger than n^2 and be same or smaller than (n+1)^2. (n^2 < Area =< (n+1)^2)
Let all "C20" rectangle's area's sum be A, and let all "C20" rectangles perimeter's sum be B.
What is A-B?
This post has been edited 1 time. Last edited by jaydenkaka, an hour ago
(help urgent) Classic Geo Problem / Angle Chasing?
by orangesyrup, Apr 23, 2025, 1:51 PM
In the given figure, ABC is an isosceles triangle with AB = AC and ∠BAC = 78°. Point D is chosen inside the triangle such that AD=DC. Find the measure of angle X (∠BDC).
ps: see the attachment for figure
ps: see the attachment for figure
Rectangular line segments in russia
by egxa, Apr 18, 2025, 10:00 AM
Several line segments parallel to the sides of a rectangular sheet of paper were drawn on it. These segments divided the sheet into several rectangles, inside of which there are no drawn lines. Petya wants to draw one diagonal in each of the rectangles, dividing it into two triangles, and color each triangle either black or white. Is it always possible to do this in such a way that no two triangles of the same color share a segment of their boundary?
A game optimization on a graph
by Assassino9931, Apr 8, 2025, 1:59 PM
Let
be
given points in the plane, and let
be a real number. Alice and Bob play the following game. Firstly, Alice constructs a connected graph with vertices at the points
, i.e., she connects some of the points with edges so that from any point you can reach any other point by moving along the edges.Then, Alice assigns to each vertex
a non-negative real number
, for
, such that
. Bob then selects a sequence of distinct vertices
such that
and
are connected by an edge for every
. (Note that the length
is not fixed and the first selected vertex always has to be
.) Bob wins if
otherwise, Alice wins. Depending on
, determine the largest possible value of
for which Bobby has a winning strategy.














![\[
\frac{1}{k+1} \sum_{j=0}^{k} r_{i_j} \geq r;
\]](http://latex.artofproblemsolving.com/1/7/9/1795db8e4a509dd465f6ff462093ae75b04de2b4.png)


Product with polynomial roots
by fungarwai, Mar 18, 2024, 11:35 AM
Vandermonde polynomial type

Proof


Example

Proof



Example
The application of Synthetic division in Polynomial theory
by fungarwai, Feb 9, 2023, 4:40 AM
Power sum of polynomial roots
Synthetic division can be applied to find the power sums of roots of a polynomial
by dividing
with 
has two real roots 


The coefficients of the quotient show that

This result is caused by the formula
Reference: The application of Synthetic division in Polynomial theory (Chinese) Section 1.1.5
Proof
Example
Python program to generate Synthetic division in LaTeX code
pywindow code
Synthetic division can be applied to find the power sums of roots of a polynomial

by dividing






The coefficients of the quotient show that

This result is caused by the formula

Reference: The application of Synthetic division in Polynomial theory (Chinese) Section 1.1.5
Proof
Let
,
Suppose
, the Laurent series of
can be expressed as


Each term in the Laurent series is unique.
As the coefficients of the quotient are determined one by one with Synthetic division, the power sums of polynomial roots are computed.

Suppose




Each term in the Laurent series is unique.
As the coefficients of the quotient are determined one by one with Synthetic division, the power sums of polynomial roots are computed.
Example
a,b,c are roots of the equation x³-9x+9 = 0.
Find the value of -
1/a³ + 1/b³ + 1/c³
1/a⁵ + 1/b⁵ + 1/c⁵.
Reference: Sum of powers of roots

Let
Compute
with Synthetic division

i.e.
Find the value of -
1/a³ + 1/b³ + 1/c³
1/a⁵ + 1/b⁵ + 1/c⁵.
Reference: Sum of powers of roots

Let

Compute


i.e.

Python program to generate Synthetic division in LaTeX code
dividend=[27,-18,0];#input by user, 27x^2-18x for example divisor=[9,-9,0,1];#input by user, 9x^3-9x^2+1 for example def roundint(x): if x == int(x): return str(int(x)) else: for k in range(1,100): T=0 if x*k == int(x*k): T=1;break; if T==1: return "\dfrac{"+str(int(x*k))+"}{"+str(k)+"}" else: return str(round(x,3)) qlen=11;e=0; dlen=len(divisor)-1; ddlen=len(dividend); dd="~ " for k in range(0,ddlen): dividend[k]=dividend[k]/divisor[0] dd=dd+"& "+roundint(dividend[k])+" " for k in range(0,qlen-ddlen+1): dividend.append(0) dd=dd+"\\\\\n" for k in range(1,dlen+1): divisor[k]=-divisor[k]/divisor[0] q=[] q.append(dividend[0]) qq="~ & "+roundint(q[0])+" " s="$\\begin{array}{c|" s2=[] for k in range(1,dlen+1): s2.append(roundint(divisor[k])+" ") for k in range(0,dlen): for i in range(0,k+1): s2[k]=s2[k]+"& ~ " for i in range(0,qlen-dlen): t=dividend[i+1] for k in range(0,dlen): tt=q[i]*divisor[k+1] s2[k]=s2[k]+"& "+roundint(tt)+" " if i-k >= 0: t=t+q[i-k]*divisor[k+1] q.append(t) if t==0: e=e+1 else: e=0 if e>=dlen and i>=ddlen-1: break qq=qq+"& "+roundint(t)+" " for i in range(qlen-dlen,qlen-1): if e>=dlen and i>=ddlen-1: break t=dividend[i+1] for k in range(0,dlen): if i-k < qlen-dlen: t=t+q[i-k]*divisor[k+1] q.append(t) qq=qq+"& "+roundint(t)+" " for k in range(0,qlen+1): s=s+"c" s=s+"}\n"+dd for k in range(0,dlen): s=s+s2[k]+"\\\\\n" s=s+"\\hline\n"+qq+"\\end{array}$" print(s)
pywindow code
[pywindow]
dividend=[27,-18,0];#input by user, 27x^2-18x for example
divisor=[9,-9,0,1];#input by user, 9x^3-9x^2+1 for example
def roundint(x):
if x == int(x):
return str(int(x))
else:
for k in range(1,100):
T=0
if x*k == int(x*k):
T=1;break;
if T==1:
return "\dfrac{"+str(int(x*k))+"}{"+str(k)+"}"
else:
return str(round(x,3))
qlen=11;e=0;
dlen=len(divisor)-1;
ddlen=len(dividend);
dd="~ "
for k in range(0,ddlen):
dividend[k]=dividend[k]/divisor[0]
dd=dd+"& "+roundint(dividend[k])+" "
for k in range(0,qlen-ddlen+1):
dividend.append(0)
dd=dd+"\\\\\n"
for k in range(1,dlen+1):
divisor[k]=-divisor[k]/divisor[0]
q=[]
q.append(dividend[0])
qq="~ & "+roundint(q[0])+" "
s="$\\begin{array}{c|"
s2=[]
for k in range(1,dlen+1):
s2.append(roundint(divisor[k])+" ")
for k in range(0,dlen):
for i in range(0,k+1):
s2[k]=s2[k]+"& ~ "
for i in range(0,qlen-dlen):
t=dividend[i+1]
for k in range(0,dlen):
tt=q[i]*divisor[k+1]
s2[k]=s2[k]+"& "+roundint(tt)+" "
if i-k >= 0:
t=t+q[i-k]*divisor[k+1]
q.append(t)
if t==0:
e=e+1
else:
e=0
if e>=dlen and i>=ddlen-1:
break
qq=qq+"& "+roundint(t)+" "
for i in range(qlen-dlen,qlen-1):
if e>=dlen and i>=ddlen-1:
break
t=dividend[i+1]
for k in range(0,dlen):
if i-k < qlen-dlen:
t=t+q[i-k]*divisor[k+1]
q.append(t)
qq=qq+"& "+roundint(t)+" "
for k in range(0,qlen+1):
s=s+"c"
s=s+"}\n"+dd
for k in range(0,dlen):
s=s+s2[k]+"\\\\\n"
s=s+"\\hline\n"+qq+"\\end{array}$"
print(s)
[/pywindow]
This post has been edited 1 time. Last edited by fungarwai, Feb 9, 2023, 4:57 AM
Summation with polynomial roots
by fungarwai, May 26, 2020, 12:52 PM
Reciprocal type

Credit to the idea posted by ccmmjj in mathchina forum
Proof

![$\sum_{i=1}^m \frac{-1}{(x-x_i)^2}=\frac{f''(x)f(x)-[f'(x)]^2}{[f(x)]^2}$](//latex.artofproblemsolving.com/a/b/3/ab3a6c2f444a2265b208b177fea06bfc3e7196d8.png)
![$\sum_{i=1}^m \frac{2}{(x-x_i)^3}=\frac{f'''(x)[f(x)]^2-3f''(x)f'(x)f(x)+2[f'(x)]^3}{[f(x)]^3}$](//latex.artofproblemsolving.com/9/d/a/9da9b44ce18202b3021555318f2c8ebe471836e7.png)
Identity of a conditional symmetric polynomial
Let


Proof
Example

Proof
Suppose
is true for 









Example



Lagrange polynomial type

Proof







Best Proof Credit to the idea with four variables posted by natmath at #2
Proof Credit to the idea with three variables posted by gasgous at #10
Other Proof

where




Proof
Example

Credit to the idea posted by ccmmjj in mathchina forum
Proof


![$\sum_{i=1}^m \frac{-1}{(x-x_i)^2}=\frac{f''(x)f(x)-[f'(x)]^2}{[f(x)]^2}$](http://latex.artofproblemsolving.com/a/b/3/ab3a6c2f444a2265b208b177fea06bfc3e7196d8.png)
![$\sum_{i=1}^m \frac{2}{(x-x_i)^3}=\frac{f'''(x)[f(x)]^2-3f''(x)f'(x)f(x)+2[f'(x)]^3}{[f(x)]^3}$](http://latex.artofproblemsolving.com/9/d/a/9da9b44ce18202b3021555318f2c8ebe471836e7.png)
Identity of a conditional symmetric polynomial
Let



Proof
Distribute
balls into
boxes
Each box
has
balls
Let
be the event of box
contains at least one ball
By Inclusion–exclusion principle

satisfies for the number of any combinations of
, therefore




Each box


Let


By Inclusion–exclusion principle

satisfies for the number of any combinations of



Example
For the combination
as example





which is the same as the situation of the coefficient of

Other examples








which is the same as the situation of the coefficient of


Other examples



Proof

Suppose











Example




Lagrange polynomial type

Proof








Best Proof Credit to the idea with four variables posted by natmath at #2
Proof Credit to the idea with three variables posted by gasgous at #10
Other Proof
Let 
such that

where
The coefficient of
should be 0
i.e.![$[x^k]P(x)=0,~\forall m\le k\le n$](//latex.artofproblemsolving.com/e/e/c/eec149e97f1674238203b8de81e4b91781a0214a.png)
And then,![$[x^{m-1}]P(x)=\sum_{i=1}^m x_i^n \prod_{j=1\atop i\neq j}^m\frac{1}{x_i-x_j}$](//latex.artofproblemsolving.com/9/3/3/933e37fd232bc27983ac056623d279f3460a8f12.png)
Let
![$\begin{cases}
[x^n]P(x)=1-1=0\\
[x^{n-1}]P(x)=-\left(q_1-\sigma_{1,m}\right)=0\\
[x^{n-2}]P(x)=-\left(q_2-\sigma_{1,m} q_1+(-1)^{K_2}\sigma_{K_2,m}\right)=0\\
\vdots\\
\displaystyle [x^m]P(x)=-\left(q_{n-m}+\sum_{k=1}^{K_{n-m}-1} (-1)^k \sigma_{k,m} q_{n-m-k}+(-1)^{K_{n-m}}\sigma_{K_{n-m},m}\right)=0\\
\displaystyle [x^{m-1}]P(x)=-\left(\sum_{k=1}^{K_{n-m+1}-1} (-1)^k \sigma_{k,m} q_{n-m+1-k}+(-1)^{K_{n-m+1}}\sigma_{K_{n-m+1},m}\right)\end{cases}$](//latex.artofproblemsolving.com/e/5/f/e5f9aa5011209a8abcb849c61cae2067a13303b9.png)
As


![$[x^{m-1}]P(x)=-\left(-S_{n-m+1,m}\right)=S_{n-m+1,m}$](//latex.artofproblemsolving.com/5/4/9/549b85f0b17c81e85eb9826d6fa829e68d66ec4f.png)
Credit to the proof posted by natmath at #4

such that


where

The coefficient of

i.e.
![$[x^k]P(x)=0,~\forall m\le k\le n$](http://latex.artofproblemsolving.com/e/e/c/eec149e97f1674238203b8de81e4b91781a0214a.png)
And then,
![$[x^{m-1}]P(x)=\sum_{i=1}^m x_i^n \prod_{j=1\atop i\neq j}^m\frac{1}{x_i-x_j}$](http://latex.artofproblemsolving.com/9/3/3/933e37fd232bc27983ac056623d279f3460a8f12.png)
Let

![$\begin{cases}
[x^n]P(x)=1-1=0\\
[x^{n-1}]P(x)=-\left(q_1-\sigma_{1,m}\right)=0\\
[x^{n-2}]P(x)=-\left(q_2-\sigma_{1,m} q_1+(-1)^{K_2}\sigma_{K_2,m}\right)=0\\
\vdots\\
\displaystyle [x^m]P(x)=-\left(q_{n-m}+\sum_{k=1}^{K_{n-m}-1} (-1)^k \sigma_{k,m} q_{n-m-k}+(-1)^{K_{n-m}}\sigma_{K_{n-m},m}\right)=0\\
\displaystyle [x^{m-1}]P(x)=-\left(\sum_{k=1}^{K_{n-m+1}-1} (-1)^k \sigma_{k,m} q_{n-m+1-k}+(-1)^{K_{n-m+1}}\sigma_{K_{n-m+1},m}\right)\end{cases}$](http://latex.artofproblemsolving.com/e/5/f/e5f9aa5011209a8abcb849c61cae2067a13303b9.png)
As



![$[x^{m-1}]P(x)=-\left(-S_{n-m+1,m}\right)=S_{n-m+1,m}$](http://latex.artofproblemsolving.com/5/4/9/549b85f0b17c81e85eb9826d6fa829e68d66ec4f.png)
Credit to the proof posted by natmath at #4
https://artofproblemsolving.com/community/c4h2901132
Consider the polynomial
The desired expression is the coefficient of
of
. We know that
(and cyclically) so
But
is a quadratic so the coefficient of
and
should be
. This means that
So
The coefficient of
is
.
Consider the polynomial














where





Proof
When
,

When
,
Suppose










When

Suppose









Example
This post has been edited 23 times. Last edited by fungarwai, May 19, 2023, 12:07 PM
Chromatic polynomial of a 3×n grid
by fungarwai, Apr 26, 2020, 1:26 AM
Denote Chromatic polynomial of a m×n grid as 
Chromatic polynomial of a 2×n grid has been found and named as ladder graph

My recurrence equation found for Chromatic polynomial of a 3×n grid is as follow





![[asy]
unitsize(5mm);
int i, j, k;
k=0;
for(i=0; i<k+7; i=i+1){
if(i!=2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
draw((0,j)--(k+6,j));
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$=$", (7,1));
k=8;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+6,2));
for(i=k; i<k+7; i=i+1){
if(i!=10){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (15,1));
k=16;
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
draw((k,j)--(k+5,j));
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
dot((k+6,0.5));dot((k+6,1.5));
draw((k+5,0)--(k+6,0.5));
draw((k+5,1)--(k+6,0.5));
draw((k+5,1)--(k+6,1.5));
draw((k+5,2)--(k+6,1.5));
[/asy]](//latex.artofproblemsolving.com/c/c/a/ccad5fb03d2ced72204eb38d6229c6a87af277d3.png)
![[asy]
unitsize(5mm);
int i, j, k;
label("$~$", (0,1));label("$=$", (15,1));
k=16;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+6,2));
draw((k+6,1)--(k+6,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+6,0));dot((k+6,1));dot((k+6,2));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (23,1));
k=24;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+6,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
draw((k+6,0)--(k+6,2));dot((k+6,0));dot((k+6,2));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (31,1));
k=32;
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
draw((k,j)--(k+5,j));
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
dot((k+6,0.5));dot((k+6,1.5));
draw((k+5,0)--(k+6,0.5));
draw((k+5,1)--(k+6,0.5));
draw((k+5,1)--(k+6,1.5));
draw((k+5,2)--(k+6,1.5));
[/asy]](//latex.artofproblemsolving.com/0/0/8/008cd57e022024e061740eba373c9f5fdc60a831.png)
![[asy]
unitsize(5mm);
int i, j, k;
label("$~$", (0,1));label("$=[(t-1)^3-(t-2)^2]$", (17,1));
k=22;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (28,1));
k=29;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+6,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
draw((k+6,0)--(k+6,2));dot((k+6,0));dot((k+6,2));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]](//latex.artofproblemsolving.com/5/8/d/58dff26c2b4e23d5dde2a97708f9204f946a1c00.png)
![[asy]
unitsize(5mm);
int i, j, k;
label("$~$", (0,1));label("$=(t^3-4t^2+7t-5)$", (17,1));
k=22;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (28,1));
k=29;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+6,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
draw((k+6,0)--(k+6,2));dot((k+6,0));dot((k+6,2));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]](//latex.artofproblemsolving.com/a/3/a/a3a8c515b7c76985dca0801af3ee0ced2fe0d097.png)
![[asy]
unitsize(5mm);
int i, j, k;
k=0;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+6,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
draw((k+6,0)--(k+6,2));dot((k+6,0));dot((k+6,2));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$=$", (7,1));
k=8;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+6,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+6,0));dot((k+6,2));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (15,1));
k=16;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
draw((k+5,2)--(k+6,0));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+6,0));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]](//latex.artofproblemsolving.com/e/4/1/e4130b105f551a3b95bbc3e78b9e3c0523bb3555.png)
![[asy]
unitsize(5mm);
int i, j, k;
label("$~$", (-7,1));label("$=$", (7,1));
k=8;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+6,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+6,0));dot((k+6,2));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (15,1));
k=16;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+6,0));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$+$", (23,1));
k=24;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
draw((k+5,0)..(k+6,1)..(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]](//latex.artofproblemsolving.com/8/4/1/841a9f991afce666e147b88b6bbc1fd04f2d5dce.png)
![[asy]
unitsize(5mm);
int i, j, k;
label("$~$", (-14,1));label("$=[(t-1)^2-(t-1)]$", (3,1));
k=8;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$+$", (15,1));
k=16;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
draw((k+5,0)..(k+6,1)..(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]](//latex.artofproblemsolving.com/7/1/4/714494d58ee3a803a519415729713ffdc37b7f69.png)
![[asy]
unitsize(5mm);
int i, j, k;
label("$~$", (-11,1));label("$=(t-1)(t-2)$", (3,1));
k=7;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$+$", (14,1));
k=15;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
draw((k+5,0)..(k+6,1)..(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]](//latex.artofproblemsolving.com/f/3/f/f3f83503229bcc11d071aa08134eef47bd57746b.png)
![[asy]
unitsize(5mm);
int i, j, k;
k=0;
for(i=0; i<k+7; i=i+1){
if(i!=2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
draw((0,j)--(k+6,j));
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$=[(t-1)^3-(t-2)^2]$", (11,1));
k=16;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (22,1));
k=23;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+6,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
draw((k+6,0)--(k+6,2));dot((k+6,0));dot((k+6,2));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]](//latex.artofproblemsolving.com/5/5/0/5506ed426381e3eb388ecc348f6b53f3c8da3490.png)
![[asy]
unitsize(5mm);
int i, j, k;
label("$~$", (-10,1));label("$=[(t-1)^3-(t-2)^2-(t-1)^2+(t-1)]$", (11,1));
k=20;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (26,1));
k=27;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
draw((k+5,0)..(k+6,1)..(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]](//latex.artofproblemsolving.com/7/3/3/733697e2a3a467e510d7879f3055f05c795c1a85.png)
![[asy]
unitsize(5mm);
int i, j, k;
label("$~$", (0,1));label("$=(t^3-5t^2+10t-7)$", (11,1));
k=16;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (22,1));
k=23;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
draw((k+5,0)..(k+6,1)..(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]](//latex.artofproblemsolving.com/3/4/9/349dd217f185ae317a81907e981089971471f2a0.png)
![[asy]
unitsize(5mm);
int i, j, k;
k=0;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
draw((k+5,0)..(k+6,1)..(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$=$", (7,1));
k=8;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (15,1));
k=16;
draw((k,0)--(k+4,0));
draw((k,1)--(k+6,1));
draw((k,2)--(k+4,2));
draw((k+4,2)--(k+6,1));
draw((k+4,0)--(k+6,1));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+5,1));dot((k+6,1));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]](//latex.artofproblemsolving.com/a/8/f/a8f5fc3cfe6ba188709f0e8b2630944bd06b866d.png)
![[asy]
unitsize(5mm);
int i, j, k;
k=0;
draw((k,0)--(k+4,0));draw((k+4,0)--(k+6,1));
draw((k,1)--(k+6,1));
draw((k,2)--(k+4,2));draw((k+4,2)--(k+6,1));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+5,1));dot((k+6,1));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$=$", (7,1));
k=8;
draw((k,0)--(k+4,0));draw((k+4,0)--(k+6,1));
draw((k,1)--(k+4,1));draw((k+5,1)--(k+6,1));
draw((k,2)--(k+4,2));draw((k+4,2)--(k+6,1));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+5,1));dot((k+6,1));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (15,1));
k=16;
draw((k,0)--(k+4,0));draw((k+4,0)--(k+6,1));
draw((k,1)--(k+6,1));
draw((k,2)--(k+4,2));draw((k+4,2)--(k+6,1));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+6,1));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]](//latex.artofproblemsolving.com/5/1/7/51704fbb6319701529de22e8f81ce492a41c9cd3.png)
![[asy]
unitsize(5mm);int i, j, k;
label("$=$", (7,1));
k=8;draw((k,0)--(k+4,0));draw((k+4,0)--(k+6,1));
draw((k,1)--(k+4,1));draw((k+5,1)--(k+6,1));
draw((k,2)--(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+5,1));dot((k+6,1));
label("...", (k+2,0.5));label("...", (k+2,1.5));
label("$-$", (15,1));
k=16;draw((k,0)--(k+4,0));
draw((k,1)--(k+4,1));draw((k+4,2)--(k+5,1));
draw((k,2)--(k+4,2));
draw((k+4,0)..(k+6,1)..(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+5,1));
label("...", (k+2,0.5));label("...", (k+2,1.5));
label("$-$", (23,1));
k=24;draw((k,0)--(k+4,0));
draw((k,1)--(k+4,1));draw((k,2)--(k+4,2));
draw((k+4,2)--(k+6,1));draw((k+4,0)--(k+6,1));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+6,1));
label("...", (k+2,0.5));label("...", (k+2,1.5));
label("$+$", (31,1));
k=32;draw((k,0)--(k+4,0));draw((k,1)--(k+4,1));draw((k,2)--(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));label("...", (k+2,1.5));
[/asy]](//latex.artofproblemsolving.com/4/9/3/493ee1edd4e4c9d449607a55c576b9d0b14833d3.png)
![[asy]
unitsize(5mm);int i, j, k;
label("$=$", (7,1));
k=8;draw((k,0)--(k+4,0));draw((k+4,0)--(k+6,1));
draw((k,1)--(k+4,1));draw((k+5,1)--(k+6,1));
draw((k,2)--(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+5,1));dot((k+6,1));
label("...", (k+2,0.5));label("...", (k+2,1.5));
label("$-$", (15,1));
k=16;draw((k,0)--(k+4,0));
draw((k,1)--(k+4,1));draw((k+4,2)--(k+5,1));
draw((k,2)--(k+4,2));
draw((k+4,0)..(k+6,1)..(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}dot((k+5,1));
label("...", (k+2,0.5));label("...", (k+2,1.5));label("$-$", (23,1));
k=24;draw((k,0)--(k+4,0));
draw((k,1)--(k+4,1));draw((k,2)--(k+4,2));
draw((k+4,0)--(k+6,1));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){if(i!=k+2){dot((i,j));}}}
dot((k+6,1));label("...", (k+2,0.5));label("...", (k+2,1.5));label("$+$", (31,1));
k=32;draw((k,0)--(k+4,0));draw((k,1)--(k+4,1));draw((k,2)--(k+4,2));
draw((k+4,0)..(k+5,1)..(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));label("...", (k+2,1.5));
label("$+$", (39,1));
k=40;draw((k,0)--(k+4,0));draw((k,1)--(k+4,1));draw((k,2)--(k+4,2));
for(i=k; i<k+5; i=i+1){if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));label("...", (k+2,1.5));
[/asy]](//latex.artofproblemsolving.com/b/d/4/bd4ce072118f581f5d3d6dc4885be0787e07f5e4.png)
![[asy]
unitsize(5mm);
int i, j, k;
label("$~$", (-14,1));label("$=[(t-1)^2-(t-1)+1]$", (3,1));
k=8;
draw((k,0)--(k+4,0));
draw((k,1)--(k+4,1));
draw((k,2)--(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-(t-2)$", (14,1));
k=16;
draw((k,0)--(k+4,0));
draw((k,1)--(k+4,1));
draw((k,2)--(k+4,2));
draw((k+4,0)..(k+5,1)..(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]](//latex.artofproblemsolving.com/b/a/5/ba5910812cd1e515e16525ab2fcc921a17a0bd8a.png)
![[asy]
unitsize(5mm);
int i, j, k;
label("$~$", (-10,1));label("$=(t^2-3t+3)$", (3,1));
k=6;
draw((k,0)--(k+4,0));
draw((k,1)--(k+4,1));
draw((k,2)--(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-(t-2)$", (12,1));
k=14;
draw((k,0)--(k+4,0));
draw((k,1)--(k+4,1));
draw((k,2)--(k+4,2));
draw((k+4,0)..(k+5,1)..(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]](//latex.artofproblemsolving.com/0/0/b/00b8214c224cb9ec778201a5d00b3566d09184f9.png)




![$(t^3-5t^2+10t-7)a_n-a_{n+1}=a_n-(t^2-3t+3)a_{n-1}+(t-2)[(t^3-5t^2+10t-7)a_{n-1}-a_n]$](//latex.artofproblemsolving.com/1/3/6/136a5617117c972b269369c99a2286c2caeac6b1.png)

![[asy]
unitsize(5mm);int i, j, k;
label("$a_1:$", (0,1));
k=2;
for(i=k; i<k+1; i=i+1){
draw((i,0)--(i,2));
for(j=0; j<3; j=j+1){dot((i,j));}}
label("$a_2:$", (5,1));
k=7;
for(i=k; i<k+2; i=i+1){
draw((i,0)--(i,2));
for(j=0; j<3; j=j+1){
draw((k,j)--(k+1,j));
dot((i,j));}}
label("$a_3:$", (11,1));
k=13;
for(i=k; i<k+3; i=i+1){
draw((i,0)--(i,2));
for(j=0; j<3; j=j+1){
draw((k,j)--(k+2,j));
dot((i,j));}}
[/asy]](//latex.artofproblemsolving.com/f/3/6/f369465809f70343e7a3bc7e9f094de0c6a5da6a.png)




Chromatic polynomial of a 2×n grid has been found and named as ladder graph

My recurrence equation found for Chromatic polynomial of a 3×n grid is as follow





![[asy]
unitsize(5mm);
int i, j, k;
k=0;
for(i=0; i<k+7; i=i+1){
if(i!=2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
draw((0,j)--(k+6,j));
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$=$", (7,1));
k=8;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+6,2));
for(i=k; i<k+7; i=i+1){
if(i!=10){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (15,1));
k=16;
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
draw((k,j)--(k+5,j));
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
dot((k+6,0.5));dot((k+6,1.5));
draw((k+5,0)--(k+6,0.5));
draw((k+5,1)--(k+6,0.5));
draw((k+5,1)--(k+6,1.5));
draw((k+5,2)--(k+6,1.5));
[/asy]](http://latex.artofproblemsolving.com/c/c/a/ccad5fb03d2ced72204eb38d6229c6a87af277d3.png)
![[asy]
unitsize(5mm);
int i, j, k;
label("$~$", (0,1));label("$=$", (15,1));
k=16;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+6,2));
draw((k+6,1)--(k+6,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+6,0));dot((k+6,1));dot((k+6,2));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (23,1));
k=24;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+6,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
draw((k+6,0)--(k+6,2));dot((k+6,0));dot((k+6,2));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (31,1));
k=32;
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
draw((k,j)--(k+5,j));
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
dot((k+6,0.5));dot((k+6,1.5));
draw((k+5,0)--(k+6,0.5));
draw((k+5,1)--(k+6,0.5));
draw((k+5,1)--(k+6,1.5));
draw((k+5,2)--(k+6,1.5));
[/asy]](http://latex.artofproblemsolving.com/0/0/8/008cd57e022024e061740eba373c9f5fdc60a831.png)
![[asy]
unitsize(5mm);
int i, j, k;
label("$~$", (0,1));label("$=[(t-1)^3-(t-2)^2]$", (17,1));
k=22;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (28,1));
k=29;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+6,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
draw((k+6,0)--(k+6,2));dot((k+6,0));dot((k+6,2));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]](http://latex.artofproblemsolving.com/5/8/d/58dff26c2b4e23d5dde2a97708f9204f946a1c00.png)
![[asy]
unitsize(5mm);
int i, j, k;
label("$~$", (0,1));label("$=(t^3-4t^2+7t-5)$", (17,1));
k=22;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (28,1));
k=29;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+6,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
draw((k+6,0)--(k+6,2));dot((k+6,0));dot((k+6,2));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]](http://latex.artofproblemsolving.com/a/3/a/a3a8c515b7c76985dca0801af3ee0ced2fe0d097.png)
![[asy]
unitsize(5mm);
int i, j, k;
k=0;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+6,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
draw((k+6,0)--(k+6,2));dot((k+6,0));dot((k+6,2));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$=$", (7,1));
k=8;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+6,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+6,0));dot((k+6,2));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (15,1));
k=16;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
draw((k+5,2)--(k+6,0));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+6,0));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]](http://latex.artofproblemsolving.com/e/4/1/e4130b105f551a3b95bbc3e78b9e3c0523bb3555.png)
![[asy]
unitsize(5mm);
int i, j, k;
label("$~$", (-7,1));label("$=$", (7,1));
k=8;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+6,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+6,0));dot((k+6,2));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (15,1));
k=16;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+6,0));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$+$", (23,1));
k=24;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
draw((k+5,0)..(k+6,1)..(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]](http://latex.artofproblemsolving.com/8/4/1/841a9f991afce666e147b88b6bbc1fd04f2d5dce.png)
![[asy]
unitsize(5mm);
int i, j, k;
label("$~$", (-14,1));label("$=[(t-1)^2-(t-1)]$", (3,1));
k=8;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$+$", (15,1));
k=16;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
draw((k+5,0)..(k+6,1)..(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]](http://latex.artofproblemsolving.com/7/1/4/714494d58ee3a803a519415729713ffdc37b7f69.png)
![[asy]
unitsize(5mm);
int i, j, k;
label("$~$", (-11,1));label("$=(t-1)(t-2)$", (3,1));
k=7;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$+$", (14,1));
k=15;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
draw((k+5,0)..(k+6,1)..(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]](http://latex.artofproblemsolving.com/f/3/f/f3f83503229bcc11d071aa08134eef47bd57746b.png)
![[asy]
unitsize(5mm);
int i, j, k;
k=0;
for(i=0; i<k+7; i=i+1){
if(i!=2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
draw((0,j)--(k+6,j));
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$=[(t-1)^3-(t-2)^2]$", (11,1));
k=16;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (22,1));
k=23;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+6,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
draw((k+6,0)--(k+6,2));dot((k+6,0));dot((k+6,2));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]](http://latex.artofproblemsolving.com/5/5/0/5506ed426381e3eb388ecc348f6b53f3c8da3490.png)
![[asy]
unitsize(5mm);
int i, j, k;
label("$~$", (-10,1));label("$=[(t-1)^3-(t-2)^2-(t-1)^2+(t-1)]$", (11,1));
k=20;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (26,1));
k=27;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
draw((k+5,0)..(k+6,1)..(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]](http://latex.artofproblemsolving.com/7/3/3/733697e2a3a467e510d7879f3055f05c795c1a85.png)
![[asy]
unitsize(5mm);
int i, j, k;
label("$~$", (0,1));label("$=(t^3-5t^2+10t-7)$", (11,1));
k=16;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (22,1));
k=23;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
draw((k+5,0)..(k+6,1)..(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]](http://latex.artofproblemsolving.com/3/4/9/349dd217f185ae317a81907e981089971471f2a0.png)
![[asy]
unitsize(5mm);
int i, j, k;
k=0;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
draw((k+5,0)..(k+6,1)..(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$=$", (7,1));
k=8;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (15,1));
k=16;
draw((k,0)--(k+4,0));
draw((k,1)--(k+6,1));
draw((k,2)--(k+4,2));
draw((k+4,2)--(k+6,1));
draw((k+4,0)--(k+6,1));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+5,1));dot((k+6,1));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]](http://latex.artofproblemsolving.com/a/8/f/a8f5fc3cfe6ba188709f0e8b2630944bd06b866d.png)
![[asy]
unitsize(5mm);
int i, j, k;
k=0;
draw((k,0)--(k+4,0));draw((k+4,0)--(k+6,1));
draw((k,1)--(k+6,1));
draw((k,2)--(k+4,2));draw((k+4,2)--(k+6,1));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+5,1));dot((k+6,1));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$=$", (7,1));
k=8;
draw((k,0)--(k+4,0));draw((k+4,0)--(k+6,1));
draw((k,1)--(k+4,1));draw((k+5,1)--(k+6,1));
draw((k,2)--(k+4,2));draw((k+4,2)--(k+6,1));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+5,1));dot((k+6,1));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (15,1));
k=16;
draw((k,0)--(k+4,0));draw((k+4,0)--(k+6,1));
draw((k,1)--(k+6,1));
draw((k,2)--(k+4,2));draw((k+4,2)--(k+6,1));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+6,1));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]](http://latex.artofproblemsolving.com/5/1/7/51704fbb6319701529de22e8f81ce492a41c9cd3.png)
![[asy]
unitsize(5mm);int i, j, k;
label("$=$", (7,1));
k=8;draw((k,0)--(k+4,0));draw((k+4,0)--(k+6,1));
draw((k,1)--(k+4,1));draw((k+5,1)--(k+6,1));
draw((k,2)--(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+5,1));dot((k+6,1));
label("...", (k+2,0.5));label("...", (k+2,1.5));
label("$-$", (15,1));
k=16;draw((k,0)--(k+4,0));
draw((k,1)--(k+4,1));draw((k+4,2)--(k+5,1));
draw((k,2)--(k+4,2));
draw((k+4,0)..(k+6,1)..(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+5,1));
label("...", (k+2,0.5));label("...", (k+2,1.5));
label("$-$", (23,1));
k=24;draw((k,0)--(k+4,0));
draw((k,1)--(k+4,1));draw((k,2)--(k+4,2));
draw((k+4,2)--(k+6,1));draw((k+4,0)--(k+6,1));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+6,1));
label("...", (k+2,0.5));label("...", (k+2,1.5));
label("$+$", (31,1));
k=32;draw((k,0)--(k+4,0));draw((k,1)--(k+4,1));draw((k,2)--(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));label("...", (k+2,1.5));
[/asy]](http://latex.artofproblemsolving.com/4/9/3/493ee1edd4e4c9d449607a55c576b9d0b14833d3.png)
![[asy]
unitsize(5mm);int i, j, k;
label("$=$", (7,1));
k=8;draw((k,0)--(k+4,0));draw((k+4,0)--(k+6,1));
draw((k,1)--(k+4,1));draw((k+5,1)--(k+6,1));
draw((k,2)--(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+5,1));dot((k+6,1));
label("...", (k+2,0.5));label("...", (k+2,1.5));
label("$-$", (15,1));
k=16;draw((k,0)--(k+4,0));
draw((k,1)--(k+4,1));draw((k+4,2)--(k+5,1));
draw((k,2)--(k+4,2));
draw((k+4,0)..(k+6,1)..(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}dot((k+5,1));
label("...", (k+2,0.5));label("...", (k+2,1.5));label("$-$", (23,1));
k=24;draw((k,0)--(k+4,0));
draw((k,1)--(k+4,1));draw((k,2)--(k+4,2));
draw((k+4,0)--(k+6,1));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){if(i!=k+2){dot((i,j));}}}
dot((k+6,1));label("...", (k+2,0.5));label("...", (k+2,1.5));label("$+$", (31,1));
k=32;draw((k,0)--(k+4,0));draw((k,1)--(k+4,1));draw((k,2)--(k+4,2));
draw((k+4,0)..(k+5,1)..(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));label("...", (k+2,1.5));
label("$+$", (39,1));
k=40;draw((k,0)--(k+4,0));draw((k,1)--(k+4,1));draw((k,2)--(k+4,2));
for(i=k; i<k+5; i=i+1){if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));label("...", (k+2,1.5));
[/asy]](http://latex.artofproblemsolving.com/b/d/4/bd4ce072118f581f5d3d6dc4885be0787e07f5e4.png)
![[asy]
unitsize(5mm);
int i, j, k;
label("$~$", (-14,1));label("$=[(t-1)^2-(t-1)+1]$", (3,1));
k=8;
draw((k,0)--(k+4,0));
draw((k,1)--(k+4,1));
draw((k,2)--(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-(t-2)$", (14,1));
k=16;
draw((k,0)--(k+4,0));
draw((k,1)--(k+4,1));
draw((k,2)--(k+4,2));
draw((k+4,0)..(k+5,1)..(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]](http://latex.artofproblemsolving.com/b/a/5/ba5910812cd1e515e16525ab2fcc921a17a0bd8a.png)
![[asy]
unitsize(5mm);
int i, j, k;
label("$~$", (-10,1));label("$=(t^2-3t+3)$", (3,1));
k=6;
draw((k,0)--(k+4,0));
draw((k,1)--(k+4,1));
draw((k,2)--(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-(t-2)$", (12,1));
k=14;
draw((k,0)--(k+4,0));
draw((k,1)--(k+4,1));
draw((k,2)--(k+4,2));
draw((k+4,0)..(k+5,1)..(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]](http://latex.artofproblemsolving.com/0/0/b/00b8214c224cb9ec778201a5d00b3566d09184f9.png)




![$(t^3-5t^2+10t-7)a_n-a_{n+1}=a_n-(t^2-3t+3)a_{n-1}+(t-2)[(t^3-5t^2+10t-7)a_{n-1}-a_n]$](http://latex.artofproblemsolving.com/1/3/6/136a5617117c972b269369c99a2286c2caeac6b1.png)

![[asy]
unitsize(5mm);int i, j, k;
label("$a_1:$", (0,1));
k=2;
for(i=k; i<k+1; i=i+1){
draw((i,0)--(i,2));
for(j=0; j<3; j=j+1){dot((i,j));}}
label("$a_2:$", (5,1));
k=7;
for(i=k; i<k+2; i=i+1){
draw((i,0)--(i,2));
for(j=0; j<3; j=j+1){
draw((k,j)--(k+1,j));
dot((i,j));}}
label("$a_3:$", (11,1));
k=13;
for(i=k; i<k+3; i=i+1){
draw((i,0)--(i,2));
for(j=0; j<3; j=j+1){
draw((k,j)--(k+2,j));
dot((i,j));}}
[/asy]](http://latex.artofproblemsolving.com/f/3/6/f369465809f70343e7a3bc7e9f094de0c6a5da6a.png)



This post has been edited 17 times. Last edited by fungarwai, Aug 27, 2021, 12:50 PM
product of all integers of form i^3+1 is a perfect square
by AlastorMoody, Apr 6, 2020, 12:09 PM
Problem 1
by SpectralS, Jul 10, 2012, 5:24 PM
Given triangle
the point
is the centre of the excircle opposite the vertex
This excircle is tangent to the side
at
, and to the lines
and
at
and
, respectively. The lines
and
meet at
, and the lines
and
meet at
Let
be the point of intersection of the lines
and
, and let
be the point of intersection of the lines
and
Prove that
is the midpoint of 
(The excircle of
opposite the vertex
is the circle that is tangent to the line segment
, to the ray
beyond
, and to the ray
beyond
.)
Proposed by Evangelos Psychas, Greece























(The excircle of







Proposed by Evangelos Psychas, Greece
Composite sum
by rohitsingh0812, Jun 3, 2006, 5:39 AM
Let
,
,
,
,
,
be positive integers and let
.
Suppose that the number
divides
and
. Prove that
is composite.







Suppose that the number




IMO ShortList 1998, combinatorics theory problem 1
by orl, Oct 22, 2004, 3:22 PM
A rectangular array of numbers is given. In each row and each column, the sum of all numbers is an integer. Prove that each nonintegral number
in the array can be changed into either
or
so that the row-sums and column-sums remain unchanged. (Note that
is the least integer greater than or equal to
, while
is the greatest integer less than or equal to
.)







This post has been edited 1 time. Last edited by orl, Oct 23, 2004, 12:58 PM
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