Functional equations
by hanzo.ei, Mar 29, 2025, 4:33 PM
Find all 
such that 
such that 
,
which satisfies
Student's domination
by Entei, Mar 29, 2025, 3:53 PM
Given 
students and their test results on 
different subjects, we say that student 
dominates student 
if and only if outperforms on all subjects. Assume that no two of them have the same score on the same subject, find the probability that there exists a pair of domination in class.
students and their test results on 
different subjects, we say that student 
dominates student 
if and only if outperforms on all subjects. Assume that no two of them have the same score on the same subject, find the probability that there exists a pair of domination in class.You just need to throw facts
by vicentev, Mar 29, 2025, 3:25 PM
Let 
be real numbers such that 
, and
![\[
a + \frac{1}{a} + b + \frac{1}{b} + c + \frac{1}{c} + d + \frac{1}{d} = 0.
\]](//latex.artofproblemsolving.com/a/9/0/a90c94d82e3650560635616dacaf392d300b8c91.png)
Prove that one of the numbers 
or 
is equal to 
.
be real numbers such that 
, and![\[
a + \frac{1}{a} + b + \frac{1}{b} + c + \frac{1}{c} + d + \frac{1}{d} = 0.
\]](http://latex.artofproblemsolving.com/a/9/0/a90c94d82e3650560635616dacaf392d300b8c91.png)
Prove that one of the numbers 
or 
is equal to 
.The Curious Equation for ConoSur
by vicentev, Mar 29, 2025, 3:23 PM
Find all triples 
of positive integers that satisfy the equation
![\[
x + xy + xyz = 31.
\]](//latex.artofproblemsolving.com/a/7/8/a78d1e9d66831f11e7de08c109855e7727ec8535.png)
of positive integers that satisfy the equation![\[
x + xy + xyz = 31.
\]](http://latex.artofproblemsolving.com/a/7/8/a78d1e9d66831f11e7de08c109855e7727ec8535.png)
Chile TST IMO prime geo
by vicentev, Mar 29, 2025, 2:35 AM
Let 
be a triangle with 
. Let 
be the midpoint of 
, and let 
be a point on segment such that 
. Let 
be the point of intersection, different from 
, of the circumcircle of triangle 
and line 
. Define 
and 
as the points of intersection of line 
with 
and 
, respectively. Prove that is the midpoint of 
.
be a triangle with 
. Let 
be the midpoint of 
, and let 
be a point on segment such that 
. Let 
be the point of intersection, different from 
, of the circumcircle of triangle 
and line 
. Define 
and 
as the points of intersection of line 
with 
and 
, respectively. Prove that is the midpoint of 
.Not so classic orthocenter problem
by m4thbl3nd3r, Mar 28, 2025, 4:59 PM
Let 
be circumcenter of a non-isosceles triangle 
and 
be a point in the interior of 
. Let 
be foots of perpendicular lines from to 
. Suppose that 
is cyclic and 
is the circumcenter of , 
. Prove that 
bisects 
be circumcenter of a non-isosceles triangle 
and 
be a point in the interior of 
. Let 
be foots of perpendicular lines from to 
. Suppose that 
is cyclic and 
is the circumcenter of , 
. Prove that 
bisects 
Attachments:
This post has been edited 2 times. Last edited by m4thbl3nd3r, Yesterday at 5:00 PM
A functional equation from MEMO
by square_root_of_3, Sep 1, 2022, 12:37 PM
Find all functions 
such that 
holds for all real numbers 
and 
.
such that 
holds for all real numbers 
and 
.Cute orthocenter geometry
by MarkBcc168, Jul 28, 2020, 7:49 AM
Let acute scalene triangle have orthocenter and altitude 
with 
on side 
. Let be the midpoint of side , and let 
be the reflection of over . Let 
be a point on line 
such that lines 
and are parallel, and let the circumcircles of 
and 
meet again at 
. Prove that 
.
Proposed by Daniel Hu.
have orthocenter and altitude 
with 
on side 
. Let be the midpoint of side , and let 
be the reflection of over . Let 
be a point on line 
such that lines 
and are parallel, and let the circumcircles of 
and 
meet again at 
. Prove that 
.Proposed by Daniel Hu.
Numbers not power of 5
by Kayak, Jul 17, 2019, 12:28 PM
Show that there do not exist natural numbers 
such that the numbers ![\[ (a_1)^{2018}+a_2, (a_2)^{2018}+a_3, \dots, (a_{2018})^{2018}+a_1 \]](//latex.artofproblemsolving.com/6/1/a/61ad92b49bd9b4b8839e6f6c4e81c758a11835a3.png)
are all powers of 
Proposed by Tejaswi Navilarekallu
such that the numbers ![\[ (a_1)^{2018}+a_2, (a_2)^{2018}+a_3, \dots, (a_{2018})^{2018}+a_1 \]](http://latex.artofproblemsolving.com/6/1/a/61ad92b49bd9b4b8839e6f6c4e81c758a11835a3.png)
are all powers of 
Proposed by Tejaswi Navilarekallu
This post has been edited 2 times. Last edited by v_Enhance, Feb 8, 2023, 11:43 PM
Reason: don't \cdots a list
Reason: don't \cdots a list
March 2025