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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.

WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.

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[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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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 25 minutes 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
yumeidesu
Apr 14, 2020
jasperE3
25 minutes ago
J
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Pythagorean journey on the blackboard
sarjinius   1
N 29 minutes 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
29 minutes ago
J
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Functional Equation
AnhQuang_67   2
N an hour 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
2 hours ago
jasperE3
an hour ago
J
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Assisted perpendicular chasing
sarjinius   4
N an hour 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
an hour ago
J
No more topics!
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J
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cyclic and angle bisector wanted, incircle related
parmenides51   0
Jul 21, 2020
Source: 2015 France JBMO TST 2.4
Let $ABC$ be a scalene triangle. Let $\omega$ be the inscribed circle and $I$ its center. Let $M, N, P$ be the points of contact of $\omega$ with the sides $[BC], [CA] , [AB]$ respectively. Let $\{J\} = MN \cap IC$. The line $PJ$ cuts for second time $\omega$ at $K$. Prove that:
a) $CKIP$ is cyclic
b) $(CI$ is the bisector of the angle $\angle PCK$.
0 replies
parmenides51
Jul 21, 2020
0 replies
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cyclic and angle bisector wanted, incircle related
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Reveal topic tags
geometry
angle bisector
incircle
Concyclic
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Source: 2015 France JBMO TST 2.4
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parmenides51
30629 posts
parmenides51
#1 h  Jul 21, 2020, 9:35 PM
Y by
Let $ABC$ be a scalene triangle. Let $\omega$ be the inscribed circle and $I$ its center. Let $M, N, P$ be the points of contact of $\omega$ with the sides $[BC], [CA] , [AB]$ respectively. Let $\{J\} = MN \cap IC$. The line $PJ$ cuts for second time $\omega$ at $K$. Prove that:
a) $CKIP$ is cyclic
b) $(CI$ is the bisector of the angle $\angle PCK$.
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