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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 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
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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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Three concurrent circles
jayme   1
N 26 minutes ago by jayme
Source: own?
Dear Mathlinkers,

1. ABC a triangle
2. 0 the circumcircle
3. Tb, Tc the tangents to 0 wrt. B, C
4. D the point of intersection of Tb and Tc
5. B', C' the symmetrics of B, C wrt AC, AB
6. 1b, 1c the circumcircles of the triangles BB'D, CC'D.

Prove : 1b, 1c and 0 are concurrents.

Sincerely
Jean-Louis
1 reply
2 viewing
jayme
Yesterday at 3:08 PM
jayme
26 minutes ago
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2021 SMT Guts Round 5 p17-20 - Stanford Math Tournament
parmenides51   5
N 26 minutes ago by MATHS_ENTUSIAST
p17. Let the roots of the polynomial $f(x) = 3x^3 + 2x^2 + x + 8 = 0$ be $p, q$, and $r$. What is the sum $\frac{1}{p} +\frac{1}{q} +\frac{1}{r}$ ?


p18. Two students are playing a game. They take a deck of five cards numbered $1$ through $5$, shuffle them, and then place them in a stack facedown, turning over the top card next to the stack. They then take turns either drawing the card at the top of the stack into their hand, showing the drawn card to the other player, or drawing the card that is faceup, replacing it with the card on the top of the pile. This is repeated until all cards are drawn, and the player with the largest sum for their cards wins. What is the probability that the player who goes second wins, assuming optimal play?


p19. Compute the sum of all primes $p$ such that $2^p + p^2$ is also prime.


p20. In how many ways can one color the $8$ vertices of an octagon each red, black, and white, such that no two adjacent sides are the same color?


PS. You should use hide for answers. Collected here.
5 replies
parmenides51
Feb 11, 2022
MATHS_ENTUSIAST
26 minutes ago
J
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Hard geometry
Lukariman   6
N an hour ago by Lukariman
Given circle (O) and chord AB with different diameters. The tangents of circle (O) at A and B intersect at point P. On the small arc AB, take point C so that triangle CAB is not isosceles. The lines CA and BP intersect at D, BC and AP intersect at E. Prove that the centers of the circles circumscribing triangles ACE, BCD and OPC are collinear.
6 replies
Lukariman
Yesterday at 4:28 AM
Lukariman
an hour ago
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Simple but hard
TUAN2k8   1
N 2 hours ago by Funcshun840
Source: Own
I need synthetic solution:
Given an acute triangle $ABC$ with orthocenter $H$.Let $AD,BE$ and $CF$ be the altitudes of triangle.Let $X$ and $Y$ be reflections of points $E,F$ across the line $AD$, respectively.Let $M$ and $N$ be the midpoints of $BH$ and $CH$, respectively.Let $K=YM \cap AB$ and $L=XN \cap AC$.Prove that $K,D$ and $L$ are collinear.
1 reply
1 viewing
TUAN2k8
4 hours ago
Funcshun840
2 hours ago
J
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Interesting functional equation
IvanRogers1   9
N 3 hours ago by jasperE3
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(x + y) + f(xy) + 1 = f(x) + f(y) + f(xy + 1) \forall x ,y \in \mathbb R$.
9 replies
IvanRogers1
Yesterday at 3:19 PM
jasperE3
3 hours ago
J
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Iran geometry
Dadgarnia   23
N 3 hours ago by Ilikeminecraft
Source: Iranian TST 2018, first exam, day1, problem 3
In triangle $ABC$ let $M$ be the midpoint of $BC$. Let $\omega$ be a circle inside of $ABC$ and is tangent to $AB,AC$ at $E,F$, respectively. The tangents from $M$ to $\omega$ meet $\omega$ at $P,Q$ such that $P$ and $B$ lie on the same side of $AM$. Let $X \equiv PM \cap BF $ and $Y \equiv QM \cap CE $. If $2PM=BC$ prove that $XY$ is tangent to $\omega$.

Proposed by Iman Maghsoudi
23 replies
Dadgarnia
Apr 7, 2018
Ilikeminecraft
3 hours ago
J
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Dou Fang Geometry in Taiwan TST
Li4   9
N 3 hours ago by WLOGQED1729
Source: 2025 Taiwan TST Round 3 Mock P2
Let $\omega$ and $\Omega$ be the incircle and circumcircle of the acute triangle $ABC$, respectively. Draw a square $WXYZ$ so that all of its sides are tangent to $\omega$, and $X$, $Y$ are both on $BC$. Extend $AW$ and $AZ$, intersecting $\Omega$ at $P$ and $Q$, respectively. Prove that $PX$ and $QY$ intersects on $\Omega$.

Proposed by kyou46, Li4, Revolilol.
9 replies
Li4
Apr 26, 2025
WLOGQED1729
3 hours ago
J
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Equal segments in a cyclic quadrilateral
a_507_bc   4
N 4 hours ago by AylyGayypow009
Source: Greece JBMO TST 2023 P2
Consider a cyclic quadrilateral $ABCD$ in which $BC = CD$ and $AB < AD$. Let $E$ be a point on the side $AD$ and $F$ a point on the line $BC$ such that $AE = AB = AF$. Prove that $EF \parallel BD$.
4 replies
a_507_bc
Jul 29, 2023
AylyGayypow009
4 hours ago
J
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Geometry
AlexCenteno2007   0
4 hours ago
Source: NCA
Let ABC be an acute triangle. The altitudes from B and C intersect the sides AC and AB at E and F, respectively. The internal bisector of ∠A intersects BE and CF at T and S, respectively. The circles with diameters AT and AS intersect the circumcircle of ABC at X and Y, respectively. Prove that XY, EF, and BC meet at the exsimilicenter of BTX and CSY
0 replies
AlexCenteno2007
4 hours ago
0 replies
J
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Assam Mathematics Olympiad 2023 Category III Q16
SomeonecoolLovesMaths   3
N 4 hours ago by nyacide
$n$ is a positive integer such that the product of all its positive divisors is $n^3$. Find all such $n$ less than $100$.
3 replies
SomeonecoolLovesMaths
Sep 11, 2024
nyacide
4 hours ago
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Tangents to a cyclic quadrilateral
v_Enhance   24
N 5 hours ago by hectorleo123
Source: ELMO Shortlist 2013: Problem G9, by Allen Liu
Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\omega$ whose diagonals meet at $F$. Lines $AB$ and $CD$ meet at $E$. Segment $EF$ intersects $\omega$ at $X$. Lines $BX$ and $CD$ meet at $M$, and lines $CX$ and $AB$ meet at $N$. Prove that $MN$ and $BC$ concur with the tangent to $\omega$ at $X$.

Proposed by Allen Liu
24 replies
v_Enhance
Jul 23, 2013
hectorleo123
5 hours ago
J
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subsets of {1,2,...,mn}
N.T.TUAN   11
N 5 hours ago by MathLuis
Source: USA TST 2005, Problem 1
Let $n$ be an integer greater than $1$. For a positive integer $m$, let $S_{m}= \{ 1,2,\ldots, mn\}$. Suppose that there exists a $2n$-element set $T$ such that
(a) each element of $T$ is an $m$-element subset of $S_{m}$;
(b) each pair of elements of $T$ shares at most one common element;
and
(c) each element of $S_{m}$ is contained in exactly two elements of $T$.

Determine the maximum possible value of $m$ in terms of $n$.
11 replies
N.T.TUAN
May 14, 2007
MathLuis
5 hours ago
J
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PIE practice
Serengeti22   4
N Today at 1:34 AM by Andyluo
Does anybody know any good problems to practice PIE that range from mid-AMC10/12 level - early AIME level for pracitce.
4 replies
Serengeti22
May 12, 2025
Andyluo
Today at 1:34 AM
J
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Strange angle condition and concyclic points
lminsl   128
N Today at 1:29 AM by Giant_PT
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
128 replies
lminsl
Jul 16, 2019
Giant_PT
Today at 1:29 AM
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no of integer soultions of ||x| - 2020| < 5 - IOQM 2020-21 p5
parmenides51   9
N Apr 19, 2025 by AshAuktober
Find the number of integer solutions to $||x| - 2020| < 5$.
9 replies
parmenides51
Jan 18, 2021
AshAuktober
Apr 19, 2025
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no of integer soultions of ||x| - 2020| < 5 - IOQM 2020-21 p5
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Reveal topic tags
algebra
inequalities
absolute value
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parmenides51
30652 posts
parmenides51
#1 h  Jan 18, 2021, 8:27 AM
Y by
Find the number of integer solutions to $||x| - 2020| < 5$.
This post has been edited 1 time. Last edited by parmenides51, Jan 18, 2021, 8:28 AM
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MrOreoJuice
594 posts
MrOreoJuice
#2 h  Jan 18, 2021, 8:36 AM • 1 Y
Y by Ash_Blanc
sol
This post has been edited 2 times. Last edited by MrOreoJuice, Jan 18, 2021, 8:39 AM
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cubingsoda
19220 posts
cubingsoda
#3 h  Jan 18, 2021, 8:40 AM • 3 Y
Y by centslordm, a.shyam.25, suvamkonar
enumeration
This post has been edited 2 times. Last edited by cubingsoda, Jan 18, 2021, 9:04 AM
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TuZo
19351 posts
TuZo
#4 h  Jan 18, 2021, 8:47 AM • 1 Y
Y by centslordm
forester2015 wrote:
sol

Solution posted befor!
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cubingsoda
19220 posts
cubingsoda
#5 h  Jan 18, 2021, 8:51 AM • 3 Y
Y by centslordm, a.shyam.25, suvamkonar
TuZo wrote:
forester2015 wrote:
sol

Solution posted befor!

yes but it is a different soution
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TuZo
19351 posts
TuZo
#6 h  Jan 18, 2021, 9:02 AM • 3 Y
Y by centslordm, Mango247, Mango247
Your "solution" it is not a solution, just an enumeration! ;)
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cubingsoda
19220 posts
cubingsoda
#7 h  Jan 18, 2021, 9:03 AM • 3 Y
Y by centslordm, a.shyam.25, suvamkonar
TuZo wrote:
Your "solution" it is not a solution, just an enumeration! ;)

oh ok i edited
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RamAgarwal2009
5 posts
RamAgarwal2009
#8 h  Aug 30, 2023, 1:01 PM
Y by
The answer is 18
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S_14159
48 posts
S_14159
#9 h  Mar 10, 2025, 5:57 PM • 1 Y
Y by Exponent11
We have $-5<|x|-2020<5$, now $|x|>2015$ or $|x|<2025$ which give $\boxed{18}$ number of solutions.
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AshAuktober
1007 posts
AshAuktober
#10 h  Apr 19, 2025, 9:11 AM
Y by
We have $|x| \in \{2016, 2017, \dots, 2024\$, which gives us $\boxed{18}$ solutions. $\square$
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