nice geo

by Melid, Apr 23, 2025, 3:01 PM

Let ABCD be a cyclic quadrilateral, which is AB=7 and BC=6. Let E be a point on segment CD so that BE=9. Line BE and AD intersect at F. Suppose that A, D, and F lie in order. If AF=11 and DF:DE=7:6, find the length of segment CD.

Help me please

by sealight2107, Apr 23, 2025, 2:40 PM

Let $m,n,p,q$ be positive reals such that $m+n+p+q+\frac{1}{mnpq} = 18$. Find the minimum and maximum value of $m,n,p,q$

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?
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
Attachments:

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 \( X_0, X_1, \dots, X_{n-1} \) be \( n \geq 2 \) given points in the plane, and let \( r > 0 \) be a real number. Alice and Bob play the following game. Firstly, Alice constructs a connected graph with vertices at the points \( X_0, X_1, \dots, X_{n-1} \), 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 \( X_i \) a non-negative real number \( r_i \), for \( i = 0, 1, \dots, n-1 \), such that $\sum_{i=0}^{n-1} r_i = 1$. Bob then selects a sequence of distinct vertices \( X_{i_0} = X_0, X_{i_1}, \dots, X_{i_k} \) such that \( X_{i_j} \) and \( X_{i_{j+1}} \) are connected by an edge for every \( j = 0, 1, \dots, k-1 \). (Note that the length $k \geq 0$ is not fixed and the first selected vertex always has to be $X_0$.) Bob wins if
\[
  \frac{1}{k+1} \sum_{j=0}^{k} r_{i_j} \geq r;
  \]otherwise, Alice wins. Depending on \( n \), determine the largest possible value of \( r \) for which Bobby has a winning strategy.

Product with polynomial roots

by fungarwai, Mar 18, 2024, 11:35 AM

Vandermonde polynomial type

$\displaystyle \prod_{1\le i<j\le m} (x_j-x_i)^2
=(-1)^{\binom{m}{2}}\prod_{j=1}^m f'(x_j),
~f(x)=\prod_{i=1}^m (x-x_i)$

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 $f(x)$
by dividing $f’(x)$ with $f(x)$

$f(x)=x^2+2x-3=0$ has two real roots $x_1=-3,~x_2=1$
$f’(x)=2x+2$

$\begin{array}{c|cccccccccccc}
~ & 2 & 2 \\
-2 & ~ & -4 & 4 & -20 & 52 & -164 & 484 & -1460 & 4372 & -13124 \\
3 & ~ & ~ & 6 & -6 & 30 & -78 & 246 & -726 & 2190 & -6558 & 19686 \\
\hline
~ & 2 & -2 & 10 & -26 & 82 & -242 & 730 & -2186 & 6562 & -19682 & 19686 \end{array}$

The coefficients of the quotient show that
$\displaystyle \sum_{k=1}^2 x_k^0=2,~\sum_{k=1}^2 x_k^1=-2,~\sum_{k=1}^2 x_k^2=10,~\sum_{k=1}^2 x_k^3=-26,\dots$

This result is caused by the formula $\dfrac{f'(x)}{f(x)}=\sum_{k=0}^\infty \dfrac{1}{x^{k+1}}\left(\sum_{j=1}^n x_j^k\right)$

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

  1. dividend=[27,-18,0];#input by user, 27x^2-18x for example
  2. divisor=[9,-9,0,1];#input by user, 9x^3-9x^2+1 for example
  3.  
  4. def roundint(x):
  5. if x == int(x):
  6. return str(int(x))
  7. else:
  8. for k in range(1,100):
  9. T=0
  10. if x*k == int(x*k):
  11. T=1;break;
  12. if T==1:
  13. return "\dfrac{"+str(int(x*k))+"}{"+str(k)+"}"
  14. else:
  15. return str(round(x,3))
  16. qlen=11;e=0;
  17. dlen=len(divisor)-1;
  18. ddlen=len(dividend);
  19. dd="~ "
  20. for k in range(0,ddlen):
  21. dividend[k]=dividend[k]/divisor[0]
  22. dd=dd+"& "+roundint(dividend[k])+" "
  23. for k in range(0,qlen-ddlen+1):
  24. dividend.append(0)
  25. dd=dd+"\\\\\n"
  26. for k in range(1,dlen+1):
  27. divisor[k]=-divisor[k]/divisor[0]
  28. q=[]
  29. q.append(dividend[0])
  30. qq="~ & "+roundint(q[0])+" "
  31. s="$\\begin{array}{c|"
  32. s2=[]
  33. for k in range(1,dlen+1):
  34. s2.append(roundint(divisor[k])+" ")
  35. for k in range(0,dlen):
  36. for i in range(0,k+1):
  37. s2[k]=s2[k]+"& ~ "
  38. for i in range(0,qlen-dlen):
  39. t=dividend[i+1]
  40. for k in range(0,dlen):
  41. tt=q[i]*divisor[k+1]
  42. s2[k]=s2[k]+"& "+roundint(tt)+" "
  43. if i-k >= 0:
  44. t=t+q[i-k]*divisor[k+1]
  45. q.append(t)
  46. if t==0:
  47. e=e+1
  48. else:
  49. e=0
  50. if e>=dlen and i>=ddlen-1:
  51. break
  52. qq=qq+"& "+roundint(t)+" "
  53. for i in range(qlen-dlen,qlen-1):
  54. if e>=dlen and i>=ddlen-1:
  55. break
  56. t=dividend[i+1]
  57. for k in range(0,dlen):
  58. if i-k < qlen-dlen:
  59. t=t+q[i-k]*divisor[k+1]
  60. q.append(t)
  61. qq=qq+"& "+roundint(t)+" "
  62. for k in range(0,qlen+1):
  63. s=s+"c"
  64. s=s+"}\n"+dd
  65. for k in range(0,dlen):
  66. s=s+s2[k]+"\\\\\n"
  67. s=s+"\\hline\n"+qq+"\\end{array}$"
  68. print(s)


pywindow code
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

$\sum_{i=1}^m \frac{1}{x-x_i}=\frac{f'(x)}{f(x)},~f(x)=\prod_{i=1}^m (x-x_i)$

Credit to the idea posted by ccmmjj in mathchina forum

Proof

$\sum_{i=1}^m \frac{-1}{(x-x_i)^2}=\frac{f''(x)f(x)-[f'(x)]^2}{[f(x)]^2}$

$\sum_{i=1}^m \frac{2}{(x-x_i)^3}=\frac{f'''(x)[f(x)]^2-3f''(x)f'(x)f(x)+2[f'(x)]^3}{[f(x)]^3}$

Identity of a conditional symmetric polynomial

Let $\displaystyle S_{n,m}=\sum_{r_1+r_2+\cdots+r_m=n} x_1^{r_1}x_2^{r_2}\cdots x_m^{r_m},~\sigma_{k,m} =\sum_{1\le j_1 < j_2 < \cdots < j_k \le m} x_{j_1} \dots x_{j_k}$
$\displaystyle \boxed{S_{j,m}=\sum_{k=1}^{m-1} (-1)^{k-1} \sigma_{k,m} S_{j-k,m}+(-1)^{m-1}\sigma_{m,m} S_{j-m,m}}$
$\displaystyle \text{or}~ \boxed{S_{j,m}=\sum_{k=1}^{j-1} (-1)^{k-1} \sigma_{k,m} S_{j-k,m}+(-1)^{j-1}\sigma_{j,m}}$

Proof

Example

$\displaystyle S_{j,K}=\sum_{k=1}^{M-1} (-1)^{k-1}
\sigma_{k,M} S_{j-k,K}+(-1)^{M-1}\sigma_{M,M}S_{j-M,K},~\forall K\le M\le m$

Proof

Example

Lagrange polynomial type

$S_{n,m}=\sum_{i=1}^m x_i^n \prod_{j=1\atop i\neq j}^m\frac{1}{x_i-x_j}
=\begin{cases}0 & 0\le n\le m-2\\1 & n=m-1\end{cases}$

Proof

$S_{n,m}=\sum_{i=1}^m x_i^n \prod_{j=1\atop i\neq j}^m\frac{1}{x_i-x_j}
=\begin{cases}0 & 0\le n\le m-2\\1 & n=m-1\\
\displaystyle
\boxed{\sum_{r_1+r_2+\cdots+r_m=n-m+1} x_1^{r_1}x_2^{r_2}\cdots x_m^{r_m}}
 & n\ge m
\end{cases}$

$S_{m,m}=\sum a$
$S_{m+1,m}=\sum a^2+\sum ab$
$S_{m+2,m}=\sum a^3+\sum a^2 b+\sum abc$
$S_{m+3,m}=\sum a^4+\sum a^3 b+\sum a^2 b^2+\sum a^2 bc+\sum abcd$

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

$S_{n,m}=\sum_{i=1}^m x_i^n \prod_{j=1\atop i\neq j}^m\frac{1}{x_i-x_j}=\begin{cases}0 & 0\le n\le m-2\\1 & n=m-1\\
\displaystyle
\boxed{\sum_{r_1 + 2r_2 + \cdots + mr_m = s \atop r_1\ge 0, \ldots, r_m\ge 0} (-1)^n \frac{(r_1 + r_2 + \cdots + r_m)!}{r_1!r_2! \cdots r_m!} \prod_{i=1}^m (-\sigma_{i,m})^{r_i}} & n=s+m-1\ge m
\end{cases}$

where $\sigma_{k,m} =\sigma_k (x_1 , \ldots , x_m )=\sum_{1\le  j_1 < j_2 < \cdots < j_k \le m} x_{j_1} \dots x_{j_k}$

$S_{m,m}=\sigma_{1,m}$
$S_{m+1,m}=\sigma_{1,m}^2-\sigma_{2,m}$
$S_{m+2,m}=\sigma_{1,m}^3-2\sigma_{1,m}\sigma_{2,m}+\sigma_{3,m}$
$S_{m+3,m}=\sigma_{1,m}^4-3\sigma_{1,m}^2\sigma_{2,m}+\sigma_{2,m}^2-2\sigma_{1,m}\sigma_{3,m}+\sigma_{4,m}$

Proof

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 $P(G_{m,n},t)$

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

$P(G_{2,n},t)=t(t-1)(t^2-3t+3)^{n-1}$

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

$P(G_{3,n},t)=(t^3-5t^2+11t-10)P(G_{3,n-1},t)-(t^4-7t^3+19t^2-24t+11)P(G_{3,n-2},t)$
$~=(t-2)(t^2-3t+5)P(G_{3,n-1},t)-(t-1)(t^3-6t^2+13t-11)P(G_{3,n-2},t)$

$P(G_{3,1},t)=t(t-1)^2,~P(G_{3,2},t)=t(t-1)(t^2-3t+3)^2$
$P(G_{3,3},t)=(t^3-5t^2+11t-10)t(t-1)(t^2-3t+3)^2
-(t^4-7t^3+19t^2-24t+11)t(t-1)^2$
$~=t(t-1)(t^7-11t^6+55t^5-161t^4+298t^3-350t^2+244t-79)$

[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]
[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]
[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]
[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][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][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]
[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]
[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]
[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]
[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]
[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]
[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]
[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]
[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]
[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]


[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]
[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]
$a_n=(t^3-5t^2+10t-7)a_{n-1}-b_{n-1}$
$b_n=a_n-(t^2-3t+3)a_{n-1}+(t-2)b_{n-1}$

$b_{n-1}=(t^3-5t^2+10t-7)a_{n-1}-a_n$
$b_n=(t^3-5t^2+10t-7)a_n-a_{n+1}$
$(t^3-5t^2+10t-7)a_n-a_{n+1}=a_n-(t^2-3t+3)a_{n-1}+(t-2)[(t^3-5t^2+10t-7)a_{n-1}-a_n]$
$a_{n+1}=(t^3-5t^2+11t-10)a_n-(t^4-7t^3+19t^2-24t+11)a_{n-1}$

[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]

$a_1=t(t-1)^2,~a_2=t(t-1)(t^2-3t+3)^2$
$a_3=(t^3-5t^2+11t-10)t(t-1)(t^2-3t+3)^2-(t^4-7t^3+19t^2-24t+11)t(t-1)^2$
$~=t(t-1)(t^7+11t^6+55t^5-161t^4+298t^3-350t^2+244t-79)$
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

Determine all integers $1 \le m, 1 \le n \le 2009$, for which
\begin{align*} \prod_{i=1}^n \left( i^3 +1 \right) = m^2 \end{align*}

Problem 1

by SpectralS, Jul 10, 2012, 5:24 PM

Given triangle $ABC$ the point $J$ is the centre of the excircle opposite the vertex $A.$ This excircle is tangent to the side $BC$ at $M$, and to the lines $AB$ and $AC$ at $K$ and $L$, respectively. The lines $LM$ and $BJ$ meet at $F$, and the lines $KM$ and $CJ$ meet at $G.$ Let $S$ be the point of intersection of the lines $AF$ and $BC$, and let $T$ be the point of intersection of the lines $AG$ and $BC.$ Prove that $M$ is the midpoint of $ST.$

(The excircle of $ABC$ opposite the vertex $A$ is the circle that is tangent to the line segment $BC$, to the ray $AB$ beyond $B$, and to the ray $AC$ beyond $C$.)

Proposed by Evangelos Psychas, Greece

Composite sum

by rohitsingh0812, Jun 3, 2006, 5:39 AM

Let $ a$, $ b$, $ c$, $ d$, $ e$, $ f$ be positive integers and let $ S = a+b+c+d+e+f$.
Suppose that the number $ S$ divides $ abc+def$ and $ ab+bc+ca-de-ef-df$. Prove that $ S$ is composite.

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 $x$ in the array can be changed into either $\lceil x\rceil $ or $\lfloor x\rfloor $ so that the row-sums and column-sums remain unchanged. (Note that $\lceil x\rceil $ is the least integer greater than or equal to $x$, while $\lfloor x\rfloor $ is the greatest integer less than or equal to $x$.)
Attachments:
This post has been edited 1 time. Last edited by orl, Oct 23, 2004, 12:58 PM

Notable algebra methods with proofs and examples

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fungarwai
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  • Nice blog!

    by Inconsistent, Mar 18, 2024, 2:41 PM

  • hey, nice blog! really enjoyed the content here and thank you for this contribution to aops. Sure to subscribe! :)

    by thedodecagon, Jan 22, 2022, 1:33 AM

  • thanks for this

    by jasperE3, Dec 3, 2021, 10:01 PM

  • I am working as accountant and studying as ACCA student now.
    I graduated applied mathematics at bachelor degree in Jinan University but I still have no idea to find a specific job with this..

    by fungarwai, Aug 28, 2021, 4:54 AM

  • Awesome algebra blog :)

    by Euler1728, Mar 22, 2021, 5:37 AM

  • I wonder if accountants need that kind of math tho

    by Hamroldt, Jan 14, 2021, 10:55 AM

  • Nice!!!!

    by Delta0001, Dec 12, 2020, 10:20 AM

  • this is very interesting i really appericate it :)

    by vsamc, Oct 29, 2020, 4:42 PM

  • this is god level

    by Hamroldt, Sep 4, 2020, 7:48 AM

  • Super Blog! You are Pr0! :)

    by Functional_equation, Aug 23, 2020, 7:43 AM

  • Great blog!

    by freeman66, May 31, 2020, 5:40 AM

  • cool thx! :D

    by erincutin, May 18, 2020, 4:55 PM

  • How so op???

    by Williamgolly, Apr 30, 2020, 2:42 PM

  • Beautiful

    by Al3jandro0000, Apr 25, 2020, 3:11 AM

  • Nice method :)

    by Feridimo, Jan 23, 2020, 5:05 PM

  • This is nice!

    by mufree, May 26, 2019, 6:40 AM

  • Wow! So much Algebra.

    by AnArtist, Mar 15, 2019, 1:19 PM

  • :omighty: :omighty:

    by AlastorMoody, Feb 9, 2019, 5:17 PM

  • 31415926535897932384626433832795

    by lkarhat, Dec 25, 2018, 11:53 PM

  • rip 0 shouts and 0 comments until now

    by harry1234, Nov 17, 2018, 8:56 PM

20 shouts
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