ALGEBRA INEQUALITY
by Tony_stark0094, Apr 23, 2025, 12:17 AM
Prove that 
Checking a summand property for integers sufficiently large.
by DinDean, Apr 22, 2025, 5:21 PM
For any fixed integer 
, prove that there exists a positive integer 
, such that for any integer 
, 
can be expressed by a sum of positive integers 
's as
![\[n=a_1+a_2+\dots+a_m,\]](//latex.artofproblemsolving.com/6/c/2/6c2a979a5ff6c40bf2d7152c8c32088fc36f848b.png)
where 
, 
, 
, 
and 
.
, prove that there exists a positive integer 
, such that for any integer 
, 
can be expressed by a sum of positive integers 
's as![\[n=a_1+a_2+\dots+a_m,\]](http://latex.artofproblemsolving.com/6/c/2/6c2a979a5ff6c40bf2d7152c8c32088fc36f848b.png)
where 
, 
, 
, 
and 
.This post has been edited 1 time. Last edited by DinDean, 3 hours ago
Reason: I forgot one condition for a_i's.
Reason: I forgot one condition for a_i's.
be real numbers such that 
Prove that 
Where 
A cyclic inequality
by KhuongTrang, Apr 21, 2025, 4:18 PM
https://cms.math.ca/.../uploads/2025/04/Wholeissue_51_4.pdf
pqr/uvw convert
by Nguyenhuyen_AG, Apr 19, 2025, 3:39 AM
Hi everyone,
As we know, the pqr/uvw method is a powerful and useful tool for proving inequalities. However, transforming an expression
into 
or 
can sometimes be quite complex. That's why I’ve written a program to assist with this process.
I hope you’ll find it helpful!
Download: pqr_convert
Screenshot:
As we know, the pqr/uvw method is a powerful and useful tool for proving inequalities. However, transforming an expression
into 
or 
can sometimes be quite complex. That's why I’ve written a program to assist with this process.I hope you’ll find it helpful!
Download: pqr_convert
Screenshot:
Divisibility on 101 integers
by BR1F1SZ, Aug 9, 2024, 12:31 AM
There are 
positive integers 
such that for every index 
, with 
, 
is a multiple of 
. Determine the greatest possible value of the largest of the numbers.
positive integers 
such that for every index 
, with 
, 
is a multiple of 
. Determine the greatest possible value of the largest of the numbers.This post has been edited 2 times. Last edited by BR1F1SZ, Jan 27, 2025, 5:01 PM
Bunnies hopping around in circles
by popcorn1, Dec 12, 2022, 5:47 PM
There are 
equally spaced points on a circular track 
of circumference . The points are labeled 
in some order, each label used once. Initially, Bunbun the Bunny begins at 
. She hops along from to 
, then from to 
, until she reaches 
, after which she hops back to . When hopping from 
to 
, she always hops along the shorter of the two arcs 
of ; if 
is a diameter of , she moves along either semicircle.
Determine the maximal possible sum of the lengths of the arcs which Bunbun traveled, over all possible labellings of the points.
Kevin Cong
equally spaced points on a circular track 
of circumference . The points are labeled 
in some order, each label used once. Initially, Bunbun the Bunny begins at 
. She hops along from to 
, then from to 
, until she reaches 
, after which she hops back to . When hopping from 
to 
, she always hops along the shorter of the two arcs 
of ; if 
is a diameter of , she moves along either semicircle.Determine the maximal possible sum of the lengths of the
arcs which Bunbun traveled, over all possible labellings of the points.Kevin Cong
This post has been edited 5 times. Last edited by v_Enhance, Dec 19, 2022, 4:04 AM
BMO 2021 problem 3
by VicKmath7, Sep 8, 2021, 5:01 PM
Let 
and 
be positive integers satisfying the equation ![$(a, b) + [a, b]=2021^c$](//latex.artofproblemsolving.com/2/4/6/246ccbb22acecb7adc72219afb8afb34c9b05101.png)
. If 
is a prime number, prove that the number 
is composite.
Proposed by Serbia
and 
be positive integers satisfying the equation ![$(a, b) + [a, b]=2021^c$](http://latex.artofproblemsolving.com/2/4/6/246ccbb22acecb7adc72219afb8afb34c9b05101.png)
. If 
is a prime number, prove that the number 
is composite.Proposed by Serbia
This post has been edited 1 time. Last edited by VicKmath7, Jan 1, 2023, 2:17 PM
Tiling rectangle with smaller rectangles.
by MarkBcc168, Jul 10, 2018, 11:25 AM
A rectangle 
with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of are either all odd or all even.
Proposed by Jeck Lim, Singapore
with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of are either all odd or all even.Proposed by Jeck Lim, Singapore
This post has been edited 2 times. Last edited by MarkBcc168, Jul 15, 2018, 12:57 PM
IMO 2011 Problem 6
by liberator, Jul 21, 2015, 4:57 PM
Problem: Let 
be an acute triangle with circumcircle 
. Let 
be a tangent line to , and let 
and 
be the lines obtained by reflecting in the lines 
, 
and 
, respectively. Show that the circumcircle of the triangle determined by the lines and is tangent to the circle .
Proposed by Japan
be an acute triangle with circumcircle 
. Let 
be a tangent line to , and let 
and 
be the lines obtained by reflecting in the lines 
, 
and 
, respectively. Show that the circumcircle of the triangle determined by the lines and is tangent to the circle .Proposed by Japan
![[asy]
import graph; size(10cm); real lsf=0.5; pathpen=linewidth(0.7); pointpen=black; pen fp=fontsize(9); pointfontpen=fp; real xmin=2.5,xmax=14,ymin=-2,ymax=8.8;
pen wwwwww=rgb(0.4,0.4,0.4), wwzzcc=rgb(0.4,0.6,0.8), yqqqyq=rgb(0.5019607843137255,0.,0.5019607843137255), qqcccc=rgb(0,0.8,0.8), cqcqcq=rgb(0.7529411764705882,0.7529411764705882,0.7529411764705882);
pair A=(9.,3.2), B=(7.6,1.), C=(11.06,1.), P=(10.9037267872876,2.2894742445215552), A_1=(4.225302839837132,4.713218228557546), B_1=(13.318947812589142,4.209009023926156), C_1=(11.23751359726076,0.33222222307412996), M=(9.699442104579619,-0.3030337832920753), P_a=(10.9037267872876,-0.28947424452155524), P_b=(9.783568496159925,1.2405987537383683), P_c=(7.368886499833012,4.538918063810838), H=(9.,2.310909090909089), I_1=(10.610893030990937,2.689468626599);
D(A--B--C--cycle,wwwwww+linewidth(1)); D(A_1--B_1--C_1--cycle,wwzzcc+linewidth(1)); D(A--A_1,linetype("4 4")+wwzzcc);
D(CR((9.33,1.4445454545454548),1.786202861143444),green); D((xmin,-1.8625555265220795*xmin+22.598270831870913)--(xmax,-1.8625555265220795*xmax+22.598270831870913),wwzzcc); D(CR((8.755537154895718,4.161938324953828),4.563653413817887),yqqqyq); D(CR((9.33,0.5554545454545452),1.786202861143444),red); D(CR((10.73,2.7554545454545445),1.786202861143444),red); D(CR((7.27,2.7554545454545463),1.786202861143444),red); D(P_a--P_c,linetype("4 4")+wwzzcc); D(CR((5.579166076673129,0.695625707960863),4.239574899162703),linetype("0 2")+qqcccc+linewidth(1)); D(CR((10.26502172862502,2.9510482106080516),3.3028669264989134),linetype("0 2")+qqcccc+linewidth(1)); D(CR((10.293309185404135,0.43870973889632964),0.9501902769422019),linetype("0 2")+qqcccc+linewidth(1)); D(M--A_1,wwzzcc); D(M--B_1,wwzzcc);
D(A); MP("A",(9.078267017098327,3.3218831364461052),NE*lsf); D(B); MP("B",(7.208843909967847,0.5368242217415146),NE*lsf); D(C); MP("C",(11.233826314095717,0.7657331736350426),NE*lsf); D(P); MP("P",(10.985841616211061,2.406247328871993),NE*lsf); D(C_1); MP("C_1",(11.367356536033608,0.11715780993671321),NE*lsf); D(A_1); MP("A_1",(3.698906647600416,4.981473037674183),NE*lsf); D(B_1); MP("B_1",(13.484764341048743,4.046761484108944),NE*lsf); D(M); MP("M",(9.688690888814401,-0.7030992676817621),NE*lsf); D(P_a); MP("P_a",(10.966765870219934,-0.5886447917349982),NE*lsf); D(P_b); MP("P_b",(9.898524094716802,1.1663238394487165),NE*lsf); D(P_c); MP("P_c",(7.11346518001221,4.752564085780656),NE*lsf); D(H); MP("H",(9.078267017098327,2.4253230748631207),NE*lsf); D(I_1); MP("I_1",(10.680629680353023,2.8068379946856674),NE*lsf);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* Labels, by liberator */
defaultpen(fontsize(9pt));
label("$\Gamma_a$",(10,-1.2),S*lsf);
label("$\Gamma_b$",(12,4),dir(30)*lsf);
label("$\Gamma_c$",(6,1),N*lsf);
label("$\Gamma_1$",(5,7),N*lsf);
[/asy]](http://latex.artofproblemsolving.com/f/b/8/fb89592e2b7f740039a8d16e7184e085a07a93d6.png)
Let 
,
analogously; let 
be the circumcircle of 
.
,
.
is tangent to 
,
,
is the Steiner line of 
,
.
lie on 
concur at 
be the incenter of 
are the centers of 
then 
.
;
.
:
.
,
is the internal bisector of 
.
are the internal bisectors of 
respectively. Hence 
concur at 
as the Miquel point of the line 
.
is a 
![[asy]
import graph; size(10cm); real lsf=0.5; pathpen=linewidth(0.7); pointpen=black; pen fp=fontsize(9); pointfontpen=fp; real xmin=2.5,xmax=14,ymin=-2,ymax=8.8;
pen wwwwww=rgb(0.4,0.4,0.4), wwzzcc=rgb(0.4,0.6,0.8), yqqqyq=rgb(0.5019607843137255,0.,0.5019607843137255), cqcqcq=rgb(0.7529411764705882,0.7529411764705882,0.7529411764705882);
pair A=(9.,3.2), B=(7.6,1.), C=(11.06,1.), P=(10.9037267872876,2.2894742445215552), A_1=(4.225302839837132,4.713218228557546), B_1=(13.318947812589142,4.209009023926156), C_1=(11.23751359726076,0.33222222307412996), M=(9.699442104579619,-0.3030337832920753), P_a=(10.9037267872876,-0.28947424452155524), P_b=(9.783568496159925,1.2405987537383683), P_c=(7.368886499833012,4.538918063810838), H=(9.,2.310909090909089), I_1=(10.610893030990937,2.689468626599), A_2=(7.556877292936604,1.6603150697678517), B_2=(11.116107846985981,1.4629688117894197), K=(12.028150798496817,7.3426508899567695);
D(A--B--C--cycle,wwwwww+linewidth(1)); D(A_1--B_1--C_1--cycle,wwzzcc+linewidth(1)); D(I_1--A_2--B_2--cycle,cqcqcq); D(A_1--K--B_1--cycle,wwzzcc); D(A--A_1,linetype("4 4")+wwzzcc);
D(CR((9.33,1.4445454545454548),1.786202861143444),green); D((xmin,-1.8625555265220795*xmin+22.598270831870913)--(xmax,-1.8625555265220795*xmax+22.598270831870913),wwzzcc); D(CR((8.755537154895718,4.161938324953828),4.563653413817887),yqqqyq); D(P_a--P_c,linetype("4 4")+wwzzcc); D(CR((5.579166076673129,0.695625707960863),4.239574899162703),linetype("0 2")+red+linewidth(1)); D(CR((10.26502172862502,2.9510482106080516),3.3028669264989134),linetype("0 2")+red+linewidth(1)); D(CR((10.293309185404135,0.43870973889632964),0.9501902769422019),linetype("0 2")+red+linewidth(1)); D(M--A_1,wwzzcc); D(M--B_1,wwzzcc); D((xmin,0.33698371098991503*xmin+3.289359997533001)--(xmax,0.33698371098991503*xmax+3.289359997533001),wwzzcc); D((xmin,-2.4276798224809775*xmin+36.54314988522595)--(xmax,-2.4276798224809775*xmax+36.54314988522595),wwzzcc); D(I_1--A_2--B_2--cycle,wwzzcc);
D(A); MP("A",(9.078267017098327,3.3218831364461052),NE*lsf); D(B); MP("B",(7.208843909967847,0.5368242217415146),NE*lsf); D(C); MP("C",(11.233826314095717,0.7657331736350426),NE*lsf); D(P); MP("P",(10.985841616211061,2.406247328871993),NE*lsf); D(C_1); MP("C_1",(11.367356536033608,0.11715780993671321),NE*lsf); D(A_1); MP("A_1",(3.698906647600416,4.981473037674183),NE*lsf); D(B_1); MP("B_1",(13.484764341048743,4.046761484108944),NE*lsf); D(M); MP("M",(9.688690888814401,-0.7030992676817621),NE*lsf); D(P_a); MP("P_a",(10.966765870219934,-0.5886447917349982),NE*lsf); D(P_b); MP("P_b",(9.898524094716802,1.1663238394487165),NE*lsf); D(P_c); MP("P_c",(7.11346518001221,4.752564085780656),NE*lsf); D(H); MP("H",(9.078267017098327,2.4253230748631207),NE*lsf); D(I_1); MP("I_1",(10.680629680353023,2.8068379946856674),NE*lsf); D(A_2); MP("A_2",(7.266071147941229,1.9484294250849374),NE*lsf); D(B_2); MP("B_2",(11.386432282024735,1.5096872672890087),NE*lsf); D(K); MP("K",(12.07315913770532,7.5376230004852465),NE*lsf);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* Labels, by liberator */
defaultpen(fontsize(9pt));
label("$\Gamma_1$",(5,7),N*lsf);
[/asy]](http://latex.artofproblemsolving.com/3/0/c/30cdce277f8b9fc56322d3d0a0baa20ca56c49e3.png)
Suppose the tangents to 
at 
respectively intersect at 
.
;
.
,
.
.
,
are collinear.
,
and 
are homothetic.
By a converse of Reim's theorem, 
are tangent at This post has been edited 2 times. Last edited by liberator, Jul 22, 2015, 1:35 PM
USAMO 2002 Problem 4
by MithsApprentice, Sep 30, 2005, 7:55 PM
Let 
be the set of real numbers. Determine all functions 
such that ![\[ f(x^2 - y^2) = x f(x) - y f(y) \]](//latex.artofproblemsolving.com/6/8/f/68f78e0df627f6ac04052433c01627bf16f0af8e.png)
for all pairs of real numbers 
and 
.
be the set of real numbers. Determine all functions 
such that ![\[ f(x^2 - y^2) = x f(x) - y f(y) \]](http://latex.artofproblemsolving.com/6/8/f/68f78e0df627f6ac04052433c01627bf16f0af8e.png)
for all pairs of real numbers 
and 
.This post has been edited 1 time. Last edited by MithsApprentice, Sep 30, 2005, 7:56 PM
October 2018
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