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k a June Highlights and 2025 AoPS Online Class Information
jlacosta   0
Jun 2, 2025
Congratulations to all the mathletes who competed at National MATHCOUNTS! If you missed the exciting Countdown Round, you can watch the video at this link. Are you interested in training for MATHCOUNTS or AMC 10 contests? How would you like to train for these math competitions in half the time? We have accelerated sections which meet twice per week instead of once starting on July 8th (7:30pm ET). These sections fill quickly so enroll today!

[list][*]MATHCOUNTS/AMC 8 Basics
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[*]AMC 10 Problem Series[/list]
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Summer camps are starting this month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have a transformative summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following upcoming events:
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0 replies
jlacosta
Jun 2, 2025
0 replies
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k i Adding contests to the Contest Collections
dcouchman   1
N Apr 5, 2023 by v_Enhance
Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add
1 reply
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
J
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k i Zero tolerance
ZetaX   49
N May 4, 2019 by NoDealsHere
Source: Use your common sense! (enough is enough)
Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:


To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.


More specifically:

For new threads:


a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.

Examples:
Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿)
Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"


b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.

Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".


c) Good problem statement:
Some recent really bad post was:
[quote]$lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$[/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.


For answers to already existing threads:


d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve $x^{3}+y^{3}=z^{3}$, do not answer with "$x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like "$x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.

e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.



To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!


Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).

The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.
49 replies
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
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Easy Number Theory
math_comb01   42
N 11 minutes ago by heheman
Source: INMO 2024/3
Let $p$ be an odd prime and $a,b,c$ be integers so that the integers $$a^{2023}+b^{2023},\quad b^{2024}+c^{2024},\quad a^{2025}+c^{2025}$$are divisible by $p$.
Prove that $p$ divides each of $a,b,c$.
$\quad$
Proposed by Navilarekallu Tejaswi
42 replies
1 viewing
math_comb01
Jan 21, 2024
heheman
11 minutes ago
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A function on a 2D grid
Rijul saini   3
N an hour ago by alexheinis
Source: India IMOTC 2025 Day 4 Problem 2
Does there exist a function $f:\{1,2,...,2025\}^2 \rightarrow \{1,2,...,2025\}$ such that:

$\bullet$ for any positive integer $i \leqslant 2025$, the numbers $f(i,1),f(i,2),...,f(i,2025)$ are all distinct, and
$\bullet$ for any positive integers $i \leqslant 2025$ and $j \leqslant 2024$, $f(f(i,j),f(i,j+1))=i$?

Proposed by Shantanu Nene
3 replies
Rijul saini
Jun 4, 2025
alexheinis
an hour ago
J
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3 lines concurrent with 1 circle
parmenides51   4
N an hour ago by LeYohan
Source: 2021 Irish Mathematical Olympiad P8
A point $C$ lies on a line segment $AB$ between $A$ and $B$ and circles are drawn having $AC$ and $CB$ as diameters. A common tangent to both circles touches the circle with $AC$ as diameter at $P \ne C$ and the circle with $CB$ as diameter at $Q \ne C$.
Prove that $AP, BQ$ and the common tangent to both circles at $C$ all meet at a single point which lies on the circumference of the circle with $AB$ as diameter.
4 replies
parmenides51
May 30, 2021
LeYohan
an hour ago
J
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Integer Polynomial with P(a)=b, P(b)=c, and P(c)=a
Brut3Forc3   30
N 2 hours ago by megahertz13
Source: 1974 USAMO Problem 1
Let $ a,b,$ and $ c$ denote three distinct integers, and let $ P$ denote a polynomial having integer coefficients. Show that it is impossible that $ P(a) = b, P(b) = c,$ and $ P(c) = a$.
30 replies
Brut3Forc3
Mar 13, 2010
megahertz13
2 hours ago
J
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Divisors Formed by Sums of Divisors
tobiSALT   2
N 2 hours ago by lksb
Source: Cono Sur 2025 #2
We say that a pair of positive integers $(n, m)$ is a minuan pair if it satisfies the following two conditions:

1. The number of positive divisors of $n$ is even.
2. If $d_1, d_2, \dots, d_{2k}$ are all the positive divisors of $n$, ordered such that $1 = d_1 < d_2 < \dots < d_{2k} = n$, then the set of all positive divisors of $m$ is precisely
$$ \{1, d_1 + d_2, d_3 + d_4, d_5 + d_6, \dots, d_{2k-1} + d_{2k}\} $$
Find all minuan pairs $(n, m)$.
2 replies
tobiSALT
Yesterday at 4:24 PM
lksb
2 hours ago
J
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Random NT property
MTA_2024   1
N 2 hours ago by MTA_2024
Prove that any number equivalent to $1$ mod 3 can be written as the sum of one square and 2 cubes.
Note:This is obviously all in integers
1 reply
MTA_2024
Thursday at 11:00 PM
MTA_2024
2 hours ago
J
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Cyclic Quadrilateral in a Square
tobiSALT   4
N 2 hours ago by lksb
Source: Cono Sur 2025 #1
Given a square $ABCD$, let $P$ be a point on the segment $BC$ and let $G$ be the intersection point of $AP$ with the diagonal $DB$. The line perpendicular to the segment $AP$ through $G$ intersects the side $CD$ at point $E$. Let $K$ be a point on the segment $GE$ such that $AK = PE$. Let $Q$ be the intersection point of the diagonal $AC$ and the segment $KP$.
Prove that the points $E, K, Q,$ and $C$ are concyclic.
4 replies
1 viewing
tobiSALT
Yesterday at 4:20 PM
lksb
2 hours ago
J
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power sum system of equations in 3 variables
Stear14   1
N 2 hours ago by Stear14
Given that
$x^2+y^2+z^2=8\ ,$
$x^3+y^3+z^3=15\ ,$
$x^5+y^5+z^5=100\ .$

Find the value of $\ x+y+z\ .$
1 reply
Stear14
May 25, 2025
Stear14
2 hours ago
J
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Euler Line Madness
raxu   76
N 3 hours ago by zuat.e
Source: TSTST 2015 Problem 2
Let ABC be a scalene triangle. Let $K_a$, $L_a$ and $M_a$ be the respective intersections with BC of the internal angle bisector, external angle bisector, and the median from A. The circumcircle of $AK_aL_a$ intersects $AM_a$ a second time at point $X_a$ different from A. Define $X_b$ and $X_c$ analogously. Prove that the circumcenter of $X_aX_bX_c$ lies on the Euler line of ABC.
(The Euler line of ABC is the line passing through the circumcenter, centroid, and orthocenter of ABC.)

Proposed by Ivan Borsenco
76 replies
raxu
Jun 26, 2015
zuat.e
3 hours ago
J
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IMO 2006 Slovenia - PROBLEM 5
Valentin Vornicu   70
N 3 hours ago by bjump
Let $P(x)$ be a polynomial of degree $n > 1$ with integer coefficients and let $k$ be a positive integer. Consider the polynomial $Q(x) = P(P(\ldots P(P(x)) \ldots ))$, where $P$ occurs $k$ times. Prove that there are at most $n$ integers $t$ such that $Q(t) = t$.
70 replies
Valentin Vornicu
Jul 13, 2006
bjump
3 hours ago
J
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The angle bisectors are perpendicular/parallel
Entei   1
N 3 hours ago by RANDOM__USER
Source: Own
In $\triangle{ABC}$, let two altitudes $BE$ and $CF$ meet at the orthocenter $H$. Let the tangents of circle $(ABC)$ from $B$ and $C$ meet at a point $T$. $AT$ meets $EF$ at $X$. $M$ is the midpoint of $BC$. Prove that the angle bisector of $\angle{XHM}$ is perpendicular to the angle bisector of $\angle{BAC}.$

IMAGE
1 reply
Entei
4 hours ago
RANDOM__USER
3 hours ago
J
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Product of injective/surjective functions
WakeUp   7
N 3 hours ago by lpieleanu
Source: Romanian TST 2001
a) Let $f,g:\mathbb{Z}\rightarrow\mathbb{Z}$ be one to one maps. Show that the function $h:\mathbb{Z}\rightarrow\mathbb{Z}$ defined by $h(x)=f(x)g(x)$, for all $x\in\mathbb{Z}$, cannot be a surjective function.

b) Let $f:\mathbb{Z}\rightarrow\mathbb{Z}$ be a surjective function. Show that there exist surjective functions $g,h:\mathbb{Z}\rightarrow\mathbb{Z}$ such that $f(x)=g(x)h(x)$, for all $x\in\mathbb{Z}$.
7 replies
WakeUp
Jan 16, 2011
lpieleanu
3 hours ago
J
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Multiple of power of two.
dgrozev   4
N 4 hours ago by Assassino9931
Source: Bulgarian TST, 2020, p2
Given two odd natural numbers $ a,b$ prove that for each $ n\in\mathbb{N}$ there exists $ m\in\mathbb{N}$ such that either $ a^mb^2-1$ or $ b^ma^2-1$ is multiple of $ 2^n.$
4 replies
dgrozev
Aug 4, 2020
Assassino9931
4 hours ago
J
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Parallelity and equal angles given, wanted an angle equality
BarisKoyuncu   6
N 4 hours ago by expsaggaf
Source: 2022 Turkey JBMO TST P4
Given a convex quadrilateral $ABCD$ such that $m(\widehat{ABC})=m(\widehat{BCD})$. The lines $AD$ and $BC$ intersect at a point $P$ and the line passing through $P$ which is parallel to $AB$, intersects $BD$ at $T$. Prove that
$$m(\widehat{ACB})=m(\widehat{PCT})$$
6 replies
BarisKoyuncu
Mar 15, 2022
expsaggaf
4 hours ago
J
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J
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BC = BD + DA if AB=AC, <A = 100^o, BD angle bisector
parmenides51   3
N Aug 25, 2021 by zerononnatural
Source: Indonesia INAMO Shortlist 2008 G10
Given a triangle $ABC$ with $AB = AC$, angle $\angle A = 100^o$ and $BD$ bisector of angle $\angle B$. Prove that $$BC = BD + DA.$$
3 replies
parmenides51
Aug 25, 2021
zerononnatural
Aug 25, 2021
J
BC = BD + DA if AB=AC, <A = 100^o, BD angle bisector
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Reveal topic tags
geometry
equal segments
isosceles
L
G H BBookmark kLocked kLocked NReply
Source: Indonesia INAMO Shortlist 2008 G10
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parmenides51
30653 posts
parmenides51
#1 h  Aug 25, 2021, 9:09 PM
Y by
Given a triangle $ABC$ with $AB = AC$, angle $\angle A = 100^o$ and $BD$ bisector of angle $\angle B$. Prove that $$BC = BD + DA.$$
This post has been edited 2 times. Last edited by parmenides51, Aug 25, 2021, 9:10 PM
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OlympusHero
17020 posts
OlympusHero
#2 h  Aug 25, 2021, 10:07 PM
Y by
There's probably a super slick solution but we'll just take the obvious route.

Solution
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i3435
1350 posts
i3435
#3 h  Aug 25, 2021, 10:24 PM
Y by
Sol
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zerononnatural
5 posts
zerononnatural
#4 h  Aug 25, 2021, 11:19 PM
Y by
Let $T$ be on $BC$ such that $BD = BT$. With a simple angle chasing we found that $TD = TC$, so we just need to prove that $TC = TD = DA$, but this is simple since $BADT$ is cyclic, then $\angle TAD = \angle ATD = 20º$. And we are done.
This post has been edited 1 time. Last edited by zerononnatural, Aug 25, 2021, 11:19 PM
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