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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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[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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Strange angle condition and concyclic points
lminsl   126
N 22 minutes ago by cj13609517288
Source: IMO 2019 Problem 2
In triangle $ABC$, point $A_1$ lies on side $BC$ and point $B_1$ lies on side $AC$. Let $P$ and $Q$ be points on segments $AA_1$ and $BB_1$, respectively, such that $PQ$ is parallel to $AB$. Let $P_1$ be a point on line $PB_1$, such that $B_1$ lies strictly between $P$ and $P_1$, and $\angle PP_1C=\angle BAC$. Similarly, let $Q_1$ be the point on line $QA_1$, such that $A_1$ lies strictly between $Q$ and $Q_1$, and $\angle CQ_1Q=\angle CBA$.

Prove that points $P,Q,P_1$, and $Q_1$ are concyclic.

Proposed by Anton Trygub, Ukraine
126 replies
lminsl
Jul 16, 2019
cj13609517288
22 minutes ago
J
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Geo with unnecessary condition
egxa   7
N 23 minutes ago by ehuseyinyigit
Source: Turkey Olympic Revenge 2024 P4
Let the circumcircle of a triangle $ABC$ be $\Gamma$. The tangents to $\Gamma$ at $B,C$ meet at point $E$. For a point $F$ on line $BC$ which is not on the segment $BC$, let the midpoint of $EF$ be $G$. Lines $GB,GC$ meet $\Gamma$ again at points $I,H$ respectively. Let $M$ be the midpoint of $BC$. Prove that the points $F,I,H,M$ lie on a circle.

Proposed by Mehmet Can Baştemir
7 replies
egxa
Aug 6, 2024
ehuseyinyigit
23 minutes ago
J
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Functional equations
hanzo.ei   19
N 24 minutes ago by GreekIdiot
Source: Greekldiot
Find all $f: \mathbb R_+ \rightarrow \mathbb R_+$ such that $f(xf(y)+f(x))=yf(x+yf(x)) \: \forall \: x,y \in \mathbb R_+$
19 replies
hanzo.ei
Mar 29, 2025
GreekIdiot
24 minutes ago
J
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Functional Equation
AnhQuang_67   3
N 28 minutes ago by GreekIdiot
Find all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $$2\cdot f\Big(\dfrac{-xy}{2}+f(x+y)\Big)=xf(y)+y(x), \forall x, y \in \mathbb{R} $$











3 replies
AnhQuang_67
4 hours ago
GreekIdiot
28 minutes ago
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AX = BX wanted, common external tangent of disjoint circles, 3rd circle
parmenides51   0
Feb 18, 2021
Source: 2019 AESC MSU Internet Olympiad 10.5 - Kolmogorov Boarding School - Интернет-олимпиада СУНЦ МГУ
A common external tangent $\ell$ is drawn to disjoint circles $\omega_1$ and $\omega_2$. Through the points of tangency $\ell$ with the circles, a circle $\omega_3$ is drawn, intersecting $\omega_1$ and $\omega_2$ for the second time at points $A$ and $B$, respectively. Through points $A$ and $B$ tangents to $\omega_1$ and $\omega_2$, respectively, are drawn, which intersect at point $X$, and points $A, B, X$ lie in the same half-plane wrt $\ell$, and point $X$ lies outside $\omega_3$, . Prove that $AX = BX$.
0 replies
parmenides51
Feb 18, 2021
0 replies
J
AX = BX wanted, common external tangent of disjoint circles, 3rd circle
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Source: 2019 AESC MSU Internet Olympiad 10.5 - Kolmogorov Boarding School - Интернет-олимпиада СУНЦ МГУ
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parmenides51
30629 posts
parmenides51
#1 h  Feb 18, 2021, 11:21 AM
Y by
A common external tangent $\ell$ is drawn to disjoint circles $\omega_1$ and $\omega_2$. Through the points of tangency $\ell$ with the circles, a circle $\omega_3$ is drawn, intersecting $\omega_1$ and $\omega_2$ for the second time at points $A$ and $B$, respectively. Through points $A$ and $B$ tangents to $\omega_1$ and $\omega_2$, respectively, are drawn, which intersect at point $X$, and points $A, B, X$ lie in the same half-plane wrt $\ell$, and point $X$ lies outside $\omega_3$, . Prove that $AX = BX$.
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