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Last Poster
Do you need to attend mop
averageguy 5
N
by babyzombievillager
So I got accepted into a summer program and already paid the fee of around $5000 dollars. It's for 8 weeks (my entire summer) and it's in person. I have a few questions
1. If I was to make MOP this year am I forced to attend?
2.If I don't attend the program but still qualify can I still put on my college application that I qualified for MOP or can you only put MOP qualifier if you actually attend the program.
1. If I was to make MOP this year am I forced to attend?
2.If I don't attend the program but still qualify can I still put on my college application that I qualified for MOP or can you only put MOP qualifier if you actually attend the program.
5 replies
Is this F.E.?
Jackson0423 1
N
by jasperE3
Let the set
be finite.Find all functions
satisfying the following condition for all real numbers 
:![\[
f(x - 1 - f(x)) = f(x) - x - 1.
\]](http://latex.artofproblemsolving.com/3/a/5/3a51479b61a5c025ed81f539e964ce96b5857f0f.png)
1 reply
IMO Shortlist 2014 N2
hajimbrak 31
N
by Sakura-junlin
Determine all pairs 
of positive integers such that ![\[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\]](//latex.artofproblemsolving.com/e/c/7/ec778edf97100f3b19350d8a817317ebf1f391bd.png)
Proposed by Titu Andreescu, USA
of positive integers such that ![\[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\]](http://latex.artofproblemsolving.com/e/c/7/ec778edf97100f3b19350d8a817317ebf1f391bd.png)
Proposed by Titu Andreescu, USA
31 replies
Permutations of Integers from 1 to n
Twoisntawholenumber 76
N
by maromex
Source: 2020 ISL C1
Let 
be a positive integer. Find the number of permutations 
, 
, 
of the
sequence
, 
, 
, satisfying
.
Proposed by United Kingdom
be a positive integer. Find the number of permutations 
, 
, 
of thesequence
, 
, 
, satisfying
.Proposed by United Kingdom
76 replies
Another right angled triangle
ariopro1387 5
N
by aaravdodhia
Source: Iran Team selection test 2025 - P7
Let 
be a right angled triangle with 
.Let 
be the midpoint of 
, and 
be an arbitrary point on 
. The reflection of 
over 
intersects lines 
and at 
and 
, respectively. The circumcircles of 
and intersect again at 
. Prove that the center of the circumcircle of 
lies on 
.
be a right angled triangle with 
.Let 
be the midpoint of 
, and 
be an arbitrary point on 
. The reflection of 
over 
intersects lines 
and at 
and 
, respectively. The circumcircles of 
and intersect again at 
. Prove that the center of the circumcircle of 
lies on 
.
5 replies
Inequality
srnjbr 5
N
by mihaig
For real numbers a, b, c and d that a+d=b+c prove the following:
(a-b)(c-d)+(a-c)(b-d)+(d-a)(b-c)>=0
(a-b)(c-d)+(a-c)(b-d)+(d-a)(b-c)>=0
5 replies
IMO 2014 Problem 4
ipaper 170
N
by lpieleanu
Let and be on segment of an acute triangle such that 
and 
. Let and 
be the points on 
and 
, respectively, such that is the midpoint of and is the midpoint of 
. Prove that the intersection of 
and 
is on the circumference of triangle .
Proposed by Giorgi Arabidze, Georgia.
and be on segment of an acute triangle such that 
and 
. Let and 
be the points on 
and 
, respectively, such that is the midpoint of and is the midpoint of 
. Prove that the intersection of 
and 
is on the circumference of triangle .Proposed by Giorgi Arabidze, Georgia.
170 replies
Tilted Students Thoroughly Splash Tiger part 2
DottedCaculator 20
N
by cj13609517288
Source: ELMO 2024/5
In triangle with 
and 
, let be the midpoint of 
. Choose point on the extension of 
past 
and point on segment 
such that lies on 
. Let 
be on the opposite side of 
from 
such that 
and 
. Let 
intersect the circumcircle of 
again at 
, and let 
intersect the circumcircle of 
again at 
. Prove that , 
, and 
are collinear.
Tiger Zhang
with 
and 
, let be the midpoint of 
. Choose point on the extension of 
past 
and point on segment 
such that lies on 
. Let 
be on the opposite side of 
from 
such that 
and 
. Let 
intersect the circumcircle of 
again at 
, and let 
intersect the circumcircle of 
again at 
. Prove that , 
, and 
are collinear.Tiger Zhang
20 replies
Prefix sums of divisors are perfect squares
CyclicISLscelesTrapezoid 37
N
by SimplisticFormulas
Source: ISL 2021 N3
Find all positive integers with the following property: the 
positive divisors of have a permutation 
such that for 
, the number 
is a perfect square.
with the following property: the 
positive divisors of have a permutation 
such that for 
, the number 
is a perfect square.
37 replies
Serbian selection contest for the IMO 2025 - P5
OgnjenTesic 3
N
by atdaotlohbh
Source: Serbian selection contest for the IMO 2025
Determine the smallest positive real number 
such that there exists a sequence of positive real numbers 
, 
, with the property that for every it holds that:
![\[
a_1 + \cdots + a_{n+1} < \alpha \cdot a_n.
\]](//latex.artofproblemsolving.com/9/b/7/9b73deb46032844b324288b02470ff0623ac0ff1.png)
Proposed by Pavle Martinović
such that there exists a sequence of positive real numbers 
, 
, with the property that for every it holds that:![\[
a_1 + \cdots + a_{n+1} < \alpha \cdot a_n.
\]](http://latex.artofproblemsolving.com/9/b/7/9b73deb46032844b324288b02470ff0623ac0ff1.png)
Proposed by Pavle Martinović
3 replies
