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
IMO 2012 Problem 4
by liberator, Oct 5, 2018, 8:45 PM
This was a pretty tricky functional equation, especially for a P4...
Problem: Find all functions
such that, for all integers 
that satisfy 
, the following equality holds:
![\[f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).\]](//latex.artofproblemsolving.com/a/9/e/a9edf1a3207d29facc970b6ee288d4fabdd43a4b.png)
(Here 
denotes the set of integers.)
Proposed by Liam Baker, South Africa
Problem: Find all functions
such that, for all integers 
that satisfy 
, the following equality holds:![\[f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).\]](http://latex.artofproblemsolving.com/a/9/e/a9edf1a3207d29facc970b6ee288d4fabdd43a4b.png)
(Here 
denotes the set of integers.)Proposed by Liam Baker, South Africa
,
and two pathological ones: 
for any 
.
implies that 
.
implies 
,![$$\big[f(a)+f(b)-f(a+b)\big]^2=4f(a)f(b).$$](http://latex.artofproblemsolving.com/9/6/2/9628037cf5e447b568f9c5eee86d2de4f04c1147.png)
Let 
denote the above equation. Then ![$P(a,a)\implies f(2a)\big[f(2a)-4f(a)\big]=0$](http://latex.artofproblemsolving.com/e/6/0/e600d7841028e324f93ed64811d153132dfb8ba2.png)
,
for each 
.
for some 
;
,
is periodic with period 
.
,
.
.
,
.
.![[asy]
unitsize(0.6cm);
real e = 1;
MP("k",origin,E);
D((e,0)--(2,0));
MP("4k",(2,0),E);
D((4,-1)--(2+e,0)--(4,1));
MP("k",(4,1),E);
MP("9k",(4,-1),E);
D((6,0)--(4+e,1)--(6,2));
D((6,0)--(4+e,-1)--(6,-2));
DPA((6,0.3)--(7,-0.3)^^(6,-0.3)--(7,0.3),red);
MP("0",(6,2),E);
MP("\color{red}4k",(6,0),E);
MP("16k",(6,-2),E);
MP("f(1)",(0,3),E);
MP("f(2)",(2,3),E);
MP("f(3)",(4,3),E);
MP("f(4)",(6,3),E);
[/asy]](http://latex.artofproblemsolving.com/4/c/b/4cbcba452f921e2ef6cad5f6377172915491d4fd.png)
and 
.
,
is the only solution in this case. Suppose that 
for 
.
and 
implies 
.
and 
implies 
.
and 
implies 
.
and 
implies 
.
October 2018
April 2018
April 2016
December 2014
October 2014
September 2014
August 2014