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:
Turbo's en route to visit each cell of the board
by Lukaluce, Apr 14, 2025, 11:01 AM
Let 
be an integer. In a configuration of an 
board, each of the 
cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate 
counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of 
, the maximum number of good cells over all possible starting configurations.
Proposed by Melek Güngör, Turkey
be an integer. In a configuration of an 
board, each of the 
cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate 
counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of 
, the maximum number of good cells over all possible starting configurations.Proposed by Melek Güngör, Turkey
This post has been edited 1 time. Last edited by Lukaluce, Apr 14, 2025, 11:54 AM
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
Looking for the smallest ghost
by Justpassingby, Jan 17, 2022, 9:52 AM
Let 
be an odd prime number. Let 
be the sequence defined as follows: 
, and for 
, 
is the smallest integer greater than 
such that in 
there are no arithmetic progressions of length . We say that a positive integer is a ghost if it doesn’t appear in 
.
What is the smallest ghost that is not a multiple of?
Proposed by Guerrero
be an odd prime number. Let 
be the sequence defined as follows: 
, and for 
, 
is the smallest integer greater than 
such that in 
there are no arithmetic progressions of length . We say that a positive integer is a ghost if it doesn’t appear in 
.What is the smallest ghost that is not a multiple of
?Proposed by Guerrero
This post has been edited 2 times. Last edited by Justpassingby, Jan 20, 2022, 1:29 AM
Reason: Added proposer
Reason: Added proposer
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
IMO 2015 Problem 3
by liberator, Jul 14, 2015, 2:48 PM
Problem: Let 
be an acute triangle with 
. Let 
be its cirumcircle, 
its orthocenter, and 
the foot of the altitude from 
. Let 
be the midpoint of 
. Let 
be the point on 
such that 
and let 
be the point on such that 
. Assume that the points , 
, 
, and are all different and lie on in this order.
Prove that the circumcircles of triangles
and 
are tangent to each other.
Proposed by Ukraine
be an acute triangle with 
. Let 
be its cirumcircle, 
its orthocenter, and 
the foot of the altitude from 
. Let 
be the midpoint of 
. Let 
be the point on 
such that 
and let 
be the point on such that 
. Assume that the points , 
, 
, and are all different and lie on in this order.Prove that the circumcircles of triangles
and 
are tangent to each other.Proposed by Ukraine
![[asy]
/* IMO 2015 Problem 3, free script by liberator, 14 July 2015 */
import olympiad;
import cse5;
unitsize(3.2cm);
pen ddd = rgb(0.4,0.6,0.8);
pointpen=black;
defaultpen(fontsize(10pt));
pair A = dir(75);
pair B = dir(205);
pair C = dir(-25);
pair O = origin;
pair H = orthocenter(A,B,C);
pair F = foot(A,B,C);
pair D = IP(L(H,F,0,2015),unitcircle);
pair M = midpoint(B--C);
pair Ap = B+C-H;
pair Q = IP(L(M,H,0,2015), unitcircle);
pair Qp = A+Ap-Q;
pair K = IP(L(Qp,H,-0.1,2015), unitcircle);
pair P = circumcenter(D,H,K);
D(unitcircle, heavygreen);
D(circumcircle(K,Q,H), red+linewidth(1));
D(circumcircle(F,K,M), heavymagenta);
D(circumcircle(D,H,K), grey+linetype("0 2")+linewidth(1));
D(A--B--C--cycle, ddd+linewidth(1));
D(A--D, ddd);
D(A--Qp, ddd);
D(Q--Ap, ddd+dashed);
D(K--Qp, ddd+dashed);
D(H--P--K, ddd);
D(rightanglemark(C,F,A,2), ddd);
D("A",A,dir(A)*2);
D("B",B,dir(B));
D("C",C,dir(C));
D("D",D,dir(-110));
D("F",F,dir(F));
D("H",H,dir(-65)*2);
D("K",K,dir(-15)*1.6);
D("M",M,dir(-70));
D("P",P,dir(P));
D("Q",Q,dir(Q));
D("A'",Ap,dir(Ap));
D("Q'",Qp,dir(Qp));
D("\Gamma",O,dir(160)*28);
[/asy]](http://latex.artofproblemsolving.com/a/a/6/aa6ae4ba390a833b56195faffc62aaf837297443.png)
be the antipodes of 
respectively w.r.t 
be the reflection of 
are collinear. Also observe that 
are collinear, since 
be the circumcenter of 
,
:
.
to 
at 
;
from rectangle 
,
,
,
is tangent to 
.
,
is tangent to 
:
,This post has been edited 2 times. Last edited by liberator, Jul 14, 2015, 8:58 PM
with 
,
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
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