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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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0 replies
jlacosta
Apr 2, 2025
0 replies
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$f(xy)=xf(y)+yf(x)$
yumeidesu   2
N an hour ago by jasperE3
Find $f: \mathbb{R} \to \mathbb{R}$ such that $f(x+y)=f(x)+f(y), \forall x, y \in \mathbb{R}$ and $f(xy)=xf(y)+yf(x), \forall x, y \in \mathbb{R}.$
2 replies
+1 w
yumeidesu
Apr 14, 2020
jasperE3
an hour ago
J
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Pythagorean journey on the blackboard
sarjinius   1
N an hour ago by alfonsoramires
Source: Philippine Mathematical Olympiad 2025 P2
A positive integer is written on a blackboard. Carmela can perform the following operation as many times as she wants: replace the current integer $x$ with another positive integer $y$, as long as $|x^2 - y^2|$ is a perfect square. For example, if the number on the blackboard is $17$, Carmela can replace it with $15$, because $|17^2 - 15^2| = 8^2$, then replace it with $9$, because $|15^2 - 9^2| = 12^2$. If the number on the blackboard is initially $3$, determine all integers that Carmela can write on the blackboard after finitely many operations.
1 reply
sarjinius
Mar 9, 2025
alfonsoramires
an hour ago
J
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Functional Equation
AnhQuang_67   2
N 2 hours ago by jasperE3
Find all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $$2\cdot f\Big(\dfrac{-xy}{2}+f(x+y)\Big)=xf(y)+y(x), \forall x, y \in \mathbb{R} $$











2 replies
AnhQuang_67
3 hours ago
jasperE3
2 hours ago
J
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Assisted perpendicular chasing
sarjinius   4
N 2 hours ago by X.Allaberdiyev
Source: Philippine Mathematical Olympiad 2025 P7
In acute triangle $ABC$ with circumcenter $O$ and orthocenter $H$, let $D$ be an arbitrary point on the circumcircle of triangle $ABC$ such that $D$ does not lie on line $OB$ and that line $OD$ is not parallel to line $BC$. Let $E$ be the point on the circumcircle of triangle $ABC$ such that $DE$ is perpendicular to $BC$, and let $F$ be the point on line $AC$ such that $FA = FE$. Let $P$ and $R$ be the points on the circumcircle of triangle $ABC$ such that $PE$ is a diameter, and $BH$ and $DR$ are parallel. Let $M$ be the midpoint of $DH$.
(a) Show that $AP$ and $BR$ are perpendicular.
(b) Show that $FM$ and $BM$ are perpendicular.
4 replies
sarjinius
Mar 9, 2025
X.Allaberdiyev
2 hours ago
J
No more topics!
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concurrent wanted, symmetric of projections of vertices on sides wrt centroid
parmenides51   2
N Apr 24, 2023 by Ashe
Source: 2015 Saudi Arabia GMO TST II p3
Let $ABC$ be a triangle and $G$ its centroid. Let $G_a, G_b$ and $G_c$ be the orthogonal projections of $G$ on sides $BC, CA$, respectively $AB$. If $S_a, S_b$ and $S_c$ are the symmetrical points of $G_a, G_b$, respectively $G_c$ with respect to $G$, prove that $AS_a, BS_b$ and $CS_c$ are concurrent.

Liana Topan
2 replies
parmenides51
Aug 1, 2020
Ashe
Apr 24, 2023
J
concurrent wanted, symmetric of projections of vertices on sides wrt centroid
G H J
Reveal topic tags
geometry
concurrency
concurrent
symmetry
Centroid
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G H BBookmark kLocked kLocked NReply
Source: 2015 Saudi Arabia GMO TST II p3
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parmenides51
30629 posts
parmenides51
#1 h  Aug 1, 2020, 7:37 AM
Y by
Let $ABC$ be a triangle and $G$ its centroid. Let $G_a, G_b$ and $G_c$ be the orthogonal projections of $G$ on sides $BC, CA$, respectively $AB$. If $S_a, S_b$ and $S_c$ are the symmetrical points of $G_a, G_b$, respectively $G_c$ with respect to $G$, prove that $AS_a, BS_b$ and $CS_c$ are concurrent.

Liana Topan
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Prod55
127 posts
Prod55
#2 h  Aug 1, 2020, 8:36 AM • 1 Y
Y by llplp
Let $\rm AS_a\cap BC\equiv K,AG\cap BC \equiv M,Q\equiv KG\cap AD$ and $\rm AD\perp BC,D\in BC$
Since $\rm GG_a=GS_a $ we have that $\rm QD=QA$.Also $\rm \dfrac{KG}{KQ}=\dfrac{GG_a}{DQ}=2\dfrac{GG_a}{AD}=2\dfrac{MG}{MA}=\dfrac{2}{3}$ so $\rm G$ is the centoid of $\rm ADK$.Therefore $\rm DM=MK$.
Hence ,the lines $\rm AS_a,BS_b,CS_c$ are concurrent at the isotomic conjugate point of orthocenter of triangle $\rm ABC$.
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Ashe
27 posts
Ashe
#3 h  Apr 24, 2023, 3:41 PM
Y by
Can this problem be solved using the fact MSb = MG = MSc?
So, I added the point La which is symmetrical point of G respect to Ga. Using this new point, I proved MSb = MG = MSc but I don't know how to finish the task.
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