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k a March Highlights and 2025 AoPS Online Class Information
jlacosta   0
Mar 2, 2025
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

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Do you have plans this summer? There are so many options to fit your schedule and goals whether attending a summer camp or taking online classes, it can be a great break from the routine of the school year. Check out our summer courses at AoPS Online, or if you want a math or language arts class that doesn’t have homework, but is an enriching summer experience, our AoPS Virtual Campus summer camps may be just the ticket! We are expanding our locations for our AoPS Academies across the country with 15 locations so far and new campuses opening in Saratoga CA, Johns Creek GA, and the Upper West Side NY. Check out this page for summer camp information.

Be sure to mark your calendars for the following events:
[list][*]March 5th (Wednesday), 4:30pm PT/7:30pm ET, HCSSiM Math Jam 2025. Amber Verser, Assistant Director of the Hampshire College Summer Studies in Mathematics, will host an information session about HCSSiM, a summer program for high school students.
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0 replies
jlacosta
Mar 2, 2025
0 replies
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a! + b! = 2^{c!}
parmenides51   5
N 2 minutes ago by TigerOnion
Source: 2023 Austrian Mathematical Olympiad, Junior Regional Competition , Problem 4
Determine all triples $(a, b, c)$ of positive integers such that
$$a! + b! = 2^{c!}.$$
(Walther Janous)
5 replies
parmenides51
Mar 26, 2024
TigerOnion
2 minutes ago
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Geometry
srnjbr   1
N 10 minutes ago by Curious_Droid
In triangle ABC, D is the leg of the altitude from A. l is a variable line passing through D. E and F are points on l such that AEB=AFC=90. Find the locus of the midpoint of the line segment EF.
1 reply
srnjbr
2 hours ago
Curious_Droid
10 minutes ago
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Area problem
MTA_2024   1
N 17 minutes ago by Curious_Droid
Let $\omega$ be a circle inscribed inside a rhombus $ABCD$. Let $P$ and $Q$ be variable points on $AB$ and $AD$ respectively, such as $PQ$ is always the tangent line to $\omega$.
Prove that for any position of $P$ and $Q$ the area of triangle $\triangle CPQ$ is the same.
1 reply
MTA_2024
2 hours ago
Curious_Droid
17 minutes ago
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Graph Theory in China TST
steven_zhang123   1
N 20 minutes ago by Photaesthesia
Source: China TST Quiz 4 P3
For a positive integer \( n \geq 6 \), find the smallest integer \( S(n) \) such that any graph with \( n \) vertices and at least \( S(n) \) edges must contain at least two disjoint cycles (cycles with no common vertices).
1 reply
steven_zhang123
4 hours ago
Photaesthesia
20 minutes ago
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PIE Help
Rice_Farmer   2
N Today at 3:33 AM by EaZ_Shadow
How many ways are there to color the $8$ regions of a three-set Venn Diagram with $3$ colors such that each color is used at least once? Two colorings are considered the same if one can be reached from the other by rotation and reflection.

Non ai way btw pls
2 replies
Rice_Farmer
Yesterday at 11:47 PM
EaZ_Shadow
Today at 3:33 AM
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Inequality striker
giangtruong13   3
N Today at 2:46 AM by anduran
Let $a,b,c >0$ satisfy that $a+b+c \leq 3 $. Prove that $$\frac{1}{a^2+b^2+c^2}+\frac{362}{ab+bc+ca} \geq 121$$
3 replies
giangtruong13
Yesterday at 4:33 PM
anduran
Today at 2:46 AM
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Inequalities
sqing   1
N Today at 2:37 AM by sqing
Let $ a,b\geq 0 $ and $ ab+\frac{1}{4}(a-1)^2(b-1)^2(a-b)^2\geq 1.$ Prove that
$$a+b\geq 2$$Let $ a,b\geq 0 $ and $ ab+\frac{1}{9}(a-\frac{1}{2})^2(b-1)^2(a-b)^2\geq 1.$ Prove that
$$a+b\geq 2$$
1 reply
sqing
Today at 2:11 AM
sqing
Today at 2:37 AM
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reshuffle cupcakes
SYBARUPEMULA   1
N Today at 2:33 AM by mathprodigy2011
I have six cupcakes, each of which was served on a plate. Only $N$ of them have the same color. I want to reshuffle the cupcakes such that each plate is replaced by a different color cupcake. Determine how many ways to reshuffle when:

(a) $N = 2$.
(b) $N = 3$.
1 reply
SYBARUPEMULA
Today at 2:17 AM
mathprodigy2011
Today at 2:33 AM
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Help on base numbers
Rice_Farmer   3
N Today at 2:06 AM by mathprodigy2011
The number $b$ is called beautiful if the are at least $17$ integers $0\le N<b$ such that in base $b,$ we have that $N$ and $N^2$ end in the same digit. Find the sum of the two smallest beautiful numbers.

$(A) 2520 (B) 3560 (C) 4240 (D) 5040 (E) 6060$
3 replies
Rice_Farmer
Yesterday at 8:35 PM
mathprodigy2011
Today at 2:06 AM
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PHP + Fibonacci
SomeonecoolLovesMaths   5
N Today at 2:03 AM by OronSH
The Question

I have tried to search this problem both inside/outside AoPS but was unable to find a solution, specifically with the use of PHP in it. It would be really helpful if someone posted a solution to it.
5 replies
SomeonecoolLovesMaths
Yesterday at 10:39 PM
OronSH
Today at 2:03 AM
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Inequality
jokehim   2
N Today at 1:10 AM by jokehim
Problem. Given non-negative real numbers $a,b,c$ satisfying $a+b+c=3.$ Prove that $$\sqrt{a^2+2abc+b^2}+\sqrt{b^2+2abc+c^2}+\sqrt{c^2+2abc+a^2}\le 6.$$Proposed by Phan Ngoc Chau
2 replies
jokehim
Yesterday at 7:34 AM
jokehim
Today at 1:10 AM
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Cool one
MTA_2024   6
N Today at 12:19 AM by MTA_2024
Prove that for all real numbers $a$ and $b$ verifying $a>b>0$ . $$(n+1) \cdot b^n \leq \frac{a^{n+1}-b^{n+1}}{a-b} \leq (n+1) \cdot a^n $$
6 replies
1 viewing
MTA_2024
Yesterday at 9:09 PM
MTA_2024
Today at 12:19 AM
J
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Prove three lines are concurrent
Gimbrint   2
N Yesterday at 11:19 PM by greenturtle3141
*Note: Whenever I have an equation, AoPS gives the error "New users are not allowed to post images in the Community." To circumvent this I've removed all dollar signs from the equations. Sorry for making this unreadable! If anyone has a better solution, please notify me!

Let ABC be a triangle with |AB|\neq|BC| and the circle \omega, passing through the points A and C, intersect sides AB and BC again at points D and E respectively. The tangents to \omega at the points A and E intersect at X. Prove that AC, DE and BX are concurrent.

I stumbled into this result whilst playing around with the nine-point circle. I wasn't able to find this on the internet. If anybody finds it, please reply and give me advice on how to search for such geometry problems! I don't know how to prove this. I can't figure out how to use the tangency.
2 replies
Gimbrint
Yesterday at 10:01 PM
greenturtle3141
Yesterday at 11:19 PM
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Excursion Solver #15
SomeonecoolLovesMaths   4
N Yesterday at 9:09 PM by Levieee
If $2=p_1<p_2<p_3<....<p_n$ where $p_i$ are primes, show that $p_1p_2....p_n + 1$ can never be a perfect square.
4 replies
SomeonecoolLovesMaths
Apr 24, 2024
Levieee
Yesterday at 9:09 PM
J
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J
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NT Functional divisibility
VicKmath7   3
N Oct 26, 2021 by VicKmath7
Source: Serbia TST 2021, P6
Let $S=\{1,2, \ldots ,10^{10}\}$. Find all functions $f:S \rightarrow S$, such that $$f(x+1)=f(f(x))+1 \pmod {10^{10}}$$for each $x \in S$ (assume $f(10^{10}+1)=f(1)$).
3 replies
VicKmath7
Oct 25, 2021
VicKmath7
Oct 26, 2021
J
NT Functional divisibility
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Reveal topic tags
number theory
algebra
L
G H BBookmark kLocked kLocked NReply
Source: Serbia TST 2021, P6
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VicKmath7
1385 posts
VicKmath7
#1 h  Oct 25, 2021, 2:37 PM
Y by
Let $S=\{1,2, \ldots ,10^{10}\}$. Find all functions $f:S \rightarrow S$, such that $$f(x+1)=f(f(x))+1 \pmod {10^{10}}$$for each $x \in S$ (assume $f(10^{10}+1)=f(1)$).
This post has been edited 4 times. Last edited by VicKmath7, Oct 12, 2023, 6:20 PM
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StarLex1
812 posts
StarLex1
#2 h  Oct 26, 2021, 2:46 AM
Y by
$f(2) = f(f(1))+1 $
$f(3)= f(f(f(1))+1 )+1$
$f(4) = f(f(f(f(1))+1 )+1)+1$
$f(5) = f( f(f(f(f(1))+1 )+1)+1)+1$
.
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$f(10^{10}) = f(10^{10}-1)+1 $

$f(10^{10}) = f(............f(f(1))+1)+1$
$f(10^{10}) = f(...........f(f(1))+1)+1+1+.......)$
$10^{10}-2$ times
$f(f(f(..........f(f(1))+1.....+1+1+1)$

$f(x) = (10^{10})k+x$
for (k=integer)
This post has been edited 4 times. Last edited by StarLex1, Oct 26, 2021, 2:53 AM
Reason: edit
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RevolveWithMe101
482 posts
RevolveWithMe101
#3 h  Oct 26, 2021, 12:44 PM
Y by
VicKmath7 wrote:
Let $S=${$1,2,...,10^{10}$}. Find all functions from $S$ to itself, such that $f(x+1)=f(f(x))+1(mod 10^{10})$ for each $x$ from $S$.

$x = 10^{10}$, you get $f(10^{10} + 1)$ in the LHS, which is not defined?
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VicKmath7
1385 posts
VicKmath7
#4 h  Oct 26, 2021, 3:07 PM
Y by
Now it is, check again.
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