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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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Hard functional equation
Hopeooooo   33
N an hour ago by jasperE3
Source: IMO shortlist A8 2020
Let $R^+$ be the set of positive real numbers. Determine all functions $f:R^+$ $\rightarrow$ $R^+$ such that for all positive real numbers $x$ and $y:$
\[f(x+f(xy))+y=f(x)f(y)+1\]
Ukraine
33 replies
Hopeooooo
Jul 20, 2021
jasperE3
an hour ago
J
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series and factorials?
jenishmalla   9
N an hour ago by mpcnotnpc
Source: 2025 Nepal ptst p4 of 4
Find all pairs of positive integers \( n \) and \( x \) such that
\[ 1^n + 2^n + 3^n + \cdots + n^n = x! \]
(Petko Lazarov, Bulgaria)
9 replies
jenishmalla
Mar 15, 2025
mpcnotnpc
an hour ago
J
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Geo with unnecessary condition
egxa   8
N 2 hours ago by ErTeeEs06
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
8 replies
egxa
Aug 6, 2024
ErTeeEs06
2 hours ago
J
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USAMO 2000 Problem 3
MithsApprentice   9
N 4 hours ago by Anto0110
A game of solitaire is played with $R$ red cards, $W$ white cards, and $B$ blue cards. A player plays all the cards one at a time. With each play he accumulates a penalty. If he plays a blue card, then he is charged a penalty which is the number of white cards still in his hand. If he plays a white card, then he is charged a penalty which is twice the number of red cards still in his hand. If he plays a red card, then he is charged a penalty which is three times the number of blue cards still in his hand. Find, as a function of $R, W,$ and $B,$ the minimal total penalty a player can amass and all the ways in which this minimum can be achieved.
9 replies
MithsApprentice
Oct 1, 2005
Anto0110
4 hours ago
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Areas and areas
jred   1
N Sep 7, 2014 by Jettywang828
Source: 1983 all soviet union grade 8
As shown in the figure below, the four triangles colored in yellow have equal areas. Show that the three uncolored quadrilaterals also have equal areas.
1 reply
jred
Sep 7, 2014
Jettywang828
Sep 7, 2014
J
Areas and areas
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geometry unsolved
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Source: 1983 all soviet union grade 8
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jred
290 posts
jred
#1 h  Sep 7, 2014, 3:32 PM • 2 Y
Y by Adventure10, Mango247
As shown in the figure below, the four triangles colored in yellow have equal areas. Show that the three uncolored quadrilaterals also have equal areas.
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Jettywang828
79 posts
Jettywang828
#2 h  Sep 7, 2014, 8:14 PM • 2 Y
Y by Adventure10, Mango247
I will denote the area of a polygon $ABC...X$ by $|ABC...X|$.

If the vertices of the triangle are $A$ at the top, $B$ at left, and $C$ at right, the foot of the cevian from $A$ is $D$, the one from $B$ is $E$, and the one from $C$ is $F$, and the vertices of the small triangle in the middle are $X$, $Y$, $Z$ as top, left, right (Note: you should probably draw you own diagram, because that's a bunch of labels to remember):

If the desired were true, then we would have that $|AXZE|+|XYZ|=|BYXF|+|AFX|$, or $|AYE|=|AYB|$. This implies that cevian $AD$ bisects $BE$; likewise, $BE$ bisects $CF$ and $CF$ bisects $AD$.

So that's where I'm at right now: that it suffices to prove the above using the given information. Someone help please? :D
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