Algebraic Manipulation
by Darealzolt, May 30, 2025, 4:01 AM
It is known that 
that satisfies
![\[
a^3+b^3=1957
\]](//latex.artofproblemsolving.com/5/d/8/5d8d2a40734ec6b0a9149f9a4ae3a5828d5055f3.png)
![\[
(a+b)(a+1)(b+1)=2014
\]](//latex.artofproblemsolving.com/3/0/0/300f2d16e669a1d8d9c54b357a6eaf7a8eb5d826.png)
Hence, find the value of 
that satisfies![\[
a^3+b^3=1957
\]](http://latex.artofproblemsolving.com/5/d/8/5d8d2a40734ec6b0a9149f9a4ae3a5828d5055f3.png)
![\[
(a+b)(a+1)(b+1)=2014
\]](http://latex.artofproblemsolving.com/3/0/0/300f2d16e669a1d8d9c54b357a6eaf7a8eb5d826.png)
Hence, find the value of 
Vieta Substitution
by Darealzolt, May 30, 2025, 3:42 AM
Let 
be the roots of the equation 
. Hence, find the value of 
if
![\[
Z=\frac{\alpha+\beta+\gamma+\delta}{\alpha(\beta+\gamma+\delta)+\beta(\gamma+\delta)}+\alpha\beta\gamma\delta[\alpha\beta(\gamma+\delta)+\gamma\delta(\alpha+\beta)]
\]](//latex.artofproblemsolving.com/4/2/5/425d9100db9e90264b50fc7c4b062ddb53c64068.png)
be the roots of the equation 
. Hence, find the value of 
if![\[
Z=\frac{\alpha+\beta+\gamma+\delta}{\alpha(\beta+\gamma+\delta)+\beta(\gamma+\delta)}+\alpha\beta\gamma\delta[\alpha\beta(\gamma+\delta)+\gamma\delta(\alpha+\beta)]
\]](http://latex.artofproblemsolving.com/4/2/5/425d9100db9e90264b50fc7c4b062ddb53c64068.png)
Floor Function Series
by Darealzolt, May 30, 2025, 3:20 AM
Let 
denote the greatest integer less than or equal to 
. Hence find the value of 
, if
![\[
M = \left\lfloor \frac{1^2}{3} \right\rfloor + \left\lfloor \frac{2^2}{3} \right\rfloor + \left\lfloor \frac{3^2}{3} \right\rfloor + \dots + \left\lfloor \frac{39^2}{3} \right\rfloor + \left\lfloor \frac{40^2}{3} \right\rfloor
\]](//latex.artofproblemsolving.com/1/d/4/1d4576c5f14b3f43a99f273eb2e8133388945135.png)
denote the greatest integer less than or equal to 
. Hence find the value of 
, if![\[
M = \left\lfloor \frac{1^2}{3} \right\rfloor + \left\lfloor \frac{2^2}{3} \right\rfloor + \left\lfloor \frac{3^2}{3} \right\rfloor + \dots + \left\lfloor \frac{39^2}{3} \right\rfloor + \left\lfloor \frac{40^2}{3} \right\rfloor
\]](http://latex.artofproblemsolving.com/1/d/4/1d4576c5f14b3f43a99f273eb2e8133388945135.png)
Vincentian Numbers
by Darealzolt, May 30, 2025, 2:41 AM
A number is called 
if within that number exists a digit 
that appears exactly 
times in that number, hence find the number of that consist of 4 digits (Numbers may contain a 0)
if within that number exists a digit 
that appears exactly 
times in that number, hence find the number of that consist of 4 digits (Numbers may contain a 0)
,
.
.
Prove atleast one from a,b,c is 2
by Darealzolt, May 29, 2025, 11:31 AM
Let 
be real numbers, such that
![\[
a^2+b^2+c^2+abc=5
\]](//latex.artofproblemsolving.com/7/c/8/7c8044347b7e25cb21dbef42c65ec351cefd4699.png)
![\[
a+b+c=3
\]](//latex.artofproblemsolving.com/6/4/a/64a3034ae02dd28aa6220ee5c37a22d608074173.png)
Prove that atleast one of the numbers is equal to 
.
be real numbers, such that![\[
a^2+b^2+c^2+abc=5
\]](http://latex.artofproblemsolving.com/7/c/8/7c8044347b7e25cb21dbef42c65ec351cefd4699.png)
![\[
a+b+c=3
\]](http://latex.artofproblemsolving.com/6/4/a/64a3034ae02dd28aa6220ee5c37a22d608074173.png)
Prove that atleast one of the numbers is equal to 
.Find the sum of all the products a_i a_j
by Darealzolt, May 29, 2025, 11:24 AM
Among the 100 constants 
,there are 
equal to 
, and 
equal to 
. Find the sum of all the products 
, where 
.
,there are 
equal to 
, and 
equal to 
. Find the sum of all the products 
, where 
.Isogonal and isotomic conjugates
by V0305, May 26, 2025, 9:16 PM
Loading poll details...
1. Do you think isogonal conjugates should be renamed to angular conjugates?
2. Do you think isotomic conjugates should be renamed to cevian conjugates?
Please answer truthfully
Credit to Stead for this renaming idea
2. Do you think isotomic conjugates should be renamed to cevian conjugates?
Please answer truthfully
Credit to Stead for this renaming idea
This post has been edited 2 times. Last edited by V0305, May 26, 2025, 9:17 PM
Interesting Geometry
by captainmath99, May 25, 2025, 5:11 PM
Let ABC be a right triangle such that 
. A circle O with center C has a radius of 2. Let P be a point on the circle O.
a)What is the minimum value of
?
b) What is the minimum value of
?
. A circle O with center C has a radius of 2. Let P be a point on the circle O.a)What is the minimum value of
?
b) What is the minimum value of
?
This post has been edited 1 time. Last edited by captainmath99, May 26, 2025, 1:21 PM
"Where wisdom and valor fail, all that remains is faith. . . And it can overcome all." -Toa Mata Tahu
September 2024
June 2024
May 2024
June 2023
February 2023
July 2020
January 2020
December 2019
November 2018
October 2018
August 2018
January 2018
October 2017
September 2017
August 2017
July 2017
June 2017
May 2017
April 2017
March 2017
February 2017
December 2016
November 2016
October 2016
September 2016
July 2016
June 2016
May 2016
March 2016
February 2016
December 2015
October 2015
September 2015
August 2015
July 2015
June 2015
February 2015
January 2015
November 2014
October 2014
September 2014
August 2014
July 2014
June 2014
May 2014
April 2014
March 2014
February 2014
January 2014
December 2013
November 2013
October 2013
September 2013
August 2013
July 2013
May 2013
April 2013
March 2013
February 2013
January 2013
December 2012
November 2012
October 2012
September 2012
August 2012
July 2012
June 2012
May 2012
April 2012