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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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Hard functional equation
Hopeooooo   33
N 2 hours 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
2 hours ago
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series and factorials?
jenishmalla   9
N 2 hours 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
1 viewing
jenishmalla
Mar 15, 2025
mpcnotnpc
2 hours ago
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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
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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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Inequalities
sqing   4
N Yesterday at 3:28 PM by sqing
Let $ a, b,c\geq 0 $ and $ 2a+3b+ 4c=11.$ Prove that
$$a+ab+abc\leq\frac{49}{6}$$Let $ a, b,c\geq 0 $ and $ 2a+3b+ 4c=10.$ Prove that
$$a+ab+abc\leq\frac{169}{24}$$Let $ a, b,c\geq 0 $ and $ 2a+3b+ 4c=14.$ Prove that
$$a+ab+abc\leq\frac{63+5\sqrt 5}{6}$$Let $ a, b,c\geq 0 $ and $ 2a+3b+ 4c=32.$ Prove that
$$a+ab+abc\leq48+\frac{64\sqrt{2}}{3}$$
4 replies
sqing
Apr 1, 2025
sqing
Yesterday at 3:28 PM
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Geo Mock #9
Bluesoul   1
N Yesterday at 3:19 PM by vanstraelen
Consider $\triangle{ABC}$ with $AB=12, AC=22$. The points $D,E$ lie on $AB,AC$ respectively, such that $\frac{AD}{BD}=\frac{AE}{CE}=3$. Extend $CD,BE$ to meet the circumcircle of $\triangle{ABC}$ at $P,Q$ respectively. Let the circumcircles of $\triangle{ADP}, \triangle{AEQ}$ meet at points $A,T$. Extend $AT$ to $BC$ at $R$, given $AR=16$, find $[ABC]$.
1 reply
Bluesoul
Apr 1, 2025
vanstraelen
Yesterday at 3:19 PM
J
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Regarding IMO prepartion
omega2007   0
Yesterday at 3:14 PM
<Hey Everyone'>
I'm 10 grader student and Im starting prepration for maths olympiad..>>> From scratch (not 2+2=4 )

Do you haves compilled resources of Handouts,
PDF,
Links,
List of books topic wise
which are shared on AOPS (and from your prespective) for maths olympiad and any useful thing, which will help me in boosting Maths olympiad prepration.
0 replies
omega2007
Yesterday at 3:14 PM
0 replies
J
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Geo Mock #6
Bluesoul   1
N Yesterday at 1:59 PM by vanstraelen
Consider triangle $ABC$ with $AB=5, BC=8, AC=7$, denote the incenter of the triangle as $I$. Extend $BI$ to meet the circumcircle of $\triangle{AIC}$ at $Q\neq I$, find the length of $QC$.
1 reply
Bluesoul
Apr 1, 2025
vanstraelen
Yesterday at 1:59 PM
J
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Congruence
Ecrin_eren   1
N Yesterday at 1:39 PM by Ecrin_eren
Find the number of integer pairs (x, y) satisfying the congruence equation:

3y² + 3x²y + y³ ≡ 3x² (mod 41)

for 0 ≤ x, y < 41.

1 reply
Ecrin_eren
Thursday at 10:34 AM
Ecrin_eren
Yesterday at 1:39 PM
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Probability
Ecrin_eren   1
N Yesterday at 1:38 PM by Ecrin_eren
In a board, James randomly writes A , B or C in each cell. What is the probability that, for every row and every column, the number of A 's modulo 3 is equal to the number of B's modulo 3?

1 reply
Ecrin_eren
Thursday at 11:21 AM
Ecrin_eren
Yesterday at 1:38 PM
J
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Excalibur Identity
jjsunpu   9
N Yesterday at 12:21 PM by fruitmonster97
proof is below
9 replies
jjsunpu
Thursday at 3:27 PM
fruitmonster97
Yesterday at 12:21 PM
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.problem.
Cobedangiu   2
N Yesterday at 12:06 PM by Lankou
Find the integer coefficients after expanding Newton's binomial:
$$(\frac{3}{2}-\frac{2}{3}x^2)^n (n \in Z)$$
2 replies
Cobedangiu
Yesterday at 6:20 AM
Lankou
Yesterday at 12:06 PM
J
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Inequalities
sqing   23
N Yesterday at 11:43 AM by sqing
Let $ a,b,c\geq 0 $ and $a+b+c=1$. Prove that
$$a^2+b^2+ ab +21abc\leq\frac{512}{441}$$Equality holds when $a=b=\frac{38}{21},c=\frac{5}{214}.$
$$a^2+b^2+ ab +19abc\leq\frac{10648}{9747}$$Equality holds when $a=b=\frac{22}{57},c=\frac{13}{57}.$
$$a^2+b^2+ ab +22abc\leq\frac{15625}{13068}$$Equality holds when $a=b=\frac{25}{66},c=\frac{8}{33}.$
23 replies
sqing
Mar 26, 2025
sqing
Yesterday at 11:43 AM
J
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Puzzling p&c question
Hunter87   0
Yesterday at 8:51 AM
There are 9 numbered tickets (distinct) to be distributed (1-9), and 7 contestants. Each contestant must get atleast one ticket. Two PARTICULAR contestants (say, A and B) can never get tickets with adjacent numbers and any contestant can get more than 1 ticket. All tickets are to be distributed.
In how many ways can this be done?
0 replies
Hunter87
Yesterday at 8:51 AM
0 replies
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2008 JBMO Shortlist G8
parmenides51   4
N May 6, 2023 by StarLex1
Source: 2008 JBMO Shortlist G8
The side lengths of a parallelogram are $a, b$ and diagonals have lengths $x$ and $y$. Knowing that $ab = \frac{xy}{2}$, show that $\left( a,b \right)=\left( \frac{x}{\sqrt{2}},\frac{y}{\sqrt{2}} \right)$ or $\left( a,b \right)=\left( \frac{y}{\sqrt{2}},\frac{x}{\sqrt{2}} \right)$.
4 replies
parmenides51
Oct 10, 2017
StarLex1
May 6, 2023
J
2008 JBMO Shortlist G8
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Reveal topic tags
geometry
JBMO
L
G H BBookmark kLocked kLocked NReply
Source: 2008 JBMO Shortlist G8
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parmenides51
30629 posts
parmenides51
#1 h  Oct 10, 2017, 4:33 PM • 2 Y
Y by Adventure10, Mango247
The side lengths of a parallelogram are $a, b$ and diagonals have lengths $x$ and $y$. Knowing that $ab = \frac{xy}{2}$, show that $\left( a,b \right)=\left( \frac{x}{\sqrt{2}},\frac{y}{\sqrt{2}} \right)$ or $\left( a,b \right)=\left( \frac{y}{\sqrt{2}},\frac{x}{\sqrt{2}} \right)$.
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TemetNosce
10 posts
TemetNosce
#2 h  Jun 13, 2022, 5:47 PM • 1 Y
Y by AlperenINAN
Official solution can be found here: https://pregatirematematicaolimpiadejuniori.wordpress.com/2016/07/27/shl-2008/
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Lemmas
11 posts
Lemmas
#3 h  May 5, 2023, 6:27 PM
Y by
You can use 2(a²+b²)=x²+y²
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Almora
41 posts
Almora
#4 h  May 5, 2023, 8:38 PM
Y by
Complex plane suffices. $\blacksquare$
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StarLex1
812 posts
StarLex1
#5 h  May 6, 2023, 11:16 AM
Y by
Stewart
$(\frac{x^2+y^2}{4}) = \frac{a^2+b^2}{2}$
since $ab=\frac{xy}{2}$
$a+b=\frac{x+y}{\sqrt{2}}$
sub $b = \frac{xy}{2a}$
$2a^2+a(-\sqrt{2}x-\sqrt{2}y)+xy=0$
note that (a,b) is symmetric so (a,b) sols to the following quadratic eqn
leading that $(a) = (\frac{x}{\sqrt{2}} ,\frac{y}{\sqrt{2}})$ same wise with b
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