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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.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

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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
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
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
Angles in a triangle with integer cotangents.
Stear14   0
a minute ago
In a triangle $ABC$, the point $M$ is the midpoint of $BC$ and $N$ is a point on the side $BC$ such that $BN:NC=2:1$. The cotangents of the angles $\angle BAM$, $\angle MAN$, and $\angle NAC$ are positive integers $k,m,n$.
(a) Show that the cotangent of the angle $\angle BAC$ is also an integer and equals $m-k-n$.
(b) Show that there are infinitely many possible triples $(k,m,n)$, some of which consist of Fibonacci numbers.

0 replies
+1 w
Stear14
a minute ago
0 replies
R+ FE f(f(xy)+y)=(x+1)f(y)
jasperE3   1
N 19 minutes ago by maromex
Source: p24734470
Find all functions $f:\mathbb R^+\to\mathbb R^+$ such that for all positive real numbers $x$ and $y$:
$$f(f(xy)+y)=(x+1)f(y).$$
1 reply
jasperE3
3 hours ago
maromex
19 minutes ago
An important lemma of isogonal conjugate points
buratinogigle   6
N 2 hours ago by buratinogigle
Source: Own
Let $P$ and $Q$ be two isogonal conjugate with respect to triangle $ABC$. Let $S$ and $T$ be two points lying on the circle $(PBC)$ such that $PS$ and $PT$ are perpendicular and parallel to bisector of $\angle BAC$, respectively. Prove that $QS$ and $QT$ bisect two arcs $BC$ containing $A$ and not containing $A$, respectively, of $(ABC)$.
6 replies
buratinogigle
Mar 23, 2025
buratinogigle
2 hours ago
A difficult problem [tangent circles in right triangles]
ThAzN1   48
N 2 hours ago by Autistic_Turk
Source: IMO ShortList 1998, geometry problem 8; Yugoslav TST 1999
Let $ABC$ be a triangle such that $\angle A=90^{\circ }$ and $\angle B<\angle C$. The tangent at $A$ to the circumcircle $\omega$ of triangle $ABC$ meets the line $BC$ at $D$. Let $E$ be the reflection of $A$ in the line $BC$, let $X$ be the foot of the perpendicular from $A$ to $BE$, and let $Y$ be the midpoint of the segment $AX$. Let the line $BY$ intersect the circle $\omega$ again at $Z$.

Prove that the line $BD$ is tangent to the circumcircle of triangle $ADZ$.

comment
48 replies
ThAzN1
Oct 17, 2004
Autistic_Turk
2 hours ago
IMO 2008, Question 2
delegat   63
N 2 hours ago by ezpotd
Source: IMO Shortlist 2008, A2
(a) Prove that
\[\frac {x^{2}}{\left(x - 1\right)^{2}} + \frac {y^{2}}{\left(y - 1\right)^{2}} + \frac {z^{2}}{\left(z - 1\right)^{2}} \geq 1\] for all real numbers $x$, $y$, $z$, each different from $1$, and satisfying $xyz=1$.

(b) Prove that equality holds above for infinitely many triples of rational numbers $x$, $y$, $z$, each different from $1$, and satisfying $xyz=1$.

Author: Walther Janous, Austria
63 replies
1 viewing
delegat
Jul 16, 2008
ezpotd
2 hours ago
USAMO 2003 Problem 1
MithsApprentice   69
N 2 hours ago by de-Kirschbaum
Prove that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.
69 replies
MithsApprentice
Sep 27, 2005
de-Kirschbaum
2 hours ago
d(2025^{a_i}-1) divides a_{n+1}
navi_09220114   2
N 3 hours ago by mickeymouse7133
Source: TASIMO 2025 Day 2 Problem 5
Let $a_n$ be a strictly increasing sequence of positive integers such that for all positive integers $n\ge 1$
\[d(2025^{a_n}-1)|a_{n+1}.\]Show that for any positive real number $c$ there is a positive integers $N_c$ such that $a_n>n^c$ for all $n\geq N_c$.

Note. Here $d(m)$ denotes the number of positive divisors of the positive integer $m$.
2 replies
navi_09220114
Monday at 11:51 AM
mickeymouse7133
3 hours ago
Funky function
TheUltimate123   22
N 3 hours ago by jasperE3
Source: CJMO 2022/5 (https://aops.com/community/c594864h2791269p24548889)
Find all functions \(f:\mathbb R\to\mathbb R\) such that for all real numbers \(x\) and \(y\), \[f(f(xy)+y)=(x+1)f(y).\]
Proposed by novus677
22 replies
TheUltimate123
Mar 20, 2022
jasperE3
3 hours ago
Inequality with x+y+z=1.
FrancoGiosefAG   1
N 3 hours ago by Blackbeam999
Let $x,y,z$ be positive real numbers such that $x+y+z=1$. Show that
\[ \frac{x^2-yz}{x^2+x}+\frac{y^2-zx}{y^2+y}+\frac{z^2-xy}{z^2+z}\leq 0. \]
1 reply
1 viewing
FrancoGiosefAG
Yesterday at 8:36 PM
Blackbeam999
3 hours ago
Find all numbers
Rushil   10
N 3 hours ago by SomeonecoolLovesMaths
Source: Indian RMO 1994 Problem 3
Find all 6-digit numbers $a_1a_2a_3a_4a_5a_6$ formed by using the digits $1,2,3,4,5,6$ once each such that the number $a_1a_2a_2\ldots a_k$ is divisible by $k$ for $1 \leq k \leq 6$.
10 replies
Rushil
Oct 25, 2005
SomeonecoolLovesMaths
3 hours ago
Some number theory
EeEeRUT   3
N 4 hours ago by MathLuis
Source: Thailand MO 2025 P9
Let $p$ be an odd prime and $S = \{1,2,3,\dots, p\}$
Assume that $U: S \rightarrow S$ is a bijection and $B$ is an integer such that $$B\cdot U(U(a)) - a \: \text{ is a multiple of} \: p \: \text{for all} \: a \in S$$Show that $B^{\frac{p-1}{2}} -1$ is a multiple of $p$.
3 replies
EeEeRUT
May 14, 2025
MathLuis
4 hours ago
Gcd
Rushil   5
N 4 hours ago by SomeonecoolLovesMaths
Source: Indian RMO 1994 problem 5
Let $A$ be a set of $16$ positive integers with the property that the product of any two distinct members of $A$ will not exceed 1994. Show that there are numbers $a$ and $b$ in the set $A$ such that the gcd of $a$ and $b$ is greater than 1.
5 replies
Rushil
Oct 25, 2005
SomeonecoolLovesMaths
4 hours ago
Solve the system
Rushil   20
N 4 hours ago by SomeonecoolLovesMaths
Source: 0
Solve the system of equations for real $x$ and $y$: \begin{eqnarray*} 5x \left( 1 + \frac{1}{x^2 + y^2}\right) &=& 12 \\ 5y \left( 1 - \frac{1}{x^2+y^2} \right) &=& 4 . \end{eqnarray*}
20 replies
Rushil
Oct 25, 2005
SomeonecoolLovesMaths
4 hours ago
Angles made with the median
BBNoDollar   1
N 4 hours ago by Ianis
Determine the measures of the angles of triangle \(ABC\), knowing that the median \(BM\) makes an angle of \(30^\circ\) with side \(AB\) and an angle of \(15^\circ\) with side \(BC\).
1 reply
BBNoDollar
6 hours ago
Ianis
4 hours ago
Do not try to bash on beautiful geometry
ItzsleepyXD   9
N May 2, 2025 by Captainscrubz
Source: Own , Mock Thailand Mathematic Olympiad P9
Let $ABC$be triangle with point $D,E$ and $F$ on $BC,AB,CA$
such that $BE=CF$ and $E,F$ are on the same side of $BC$
Let $M$ be midpoint of segment $BC$ and $N$ be midpoint of segment $EF$
Let $G$ be intersection of $BF$ with $CE$ and $\dfrac{BD}{DC}=\dfrac{AC}{AB}$
Prove that $MN\parallel DG$
9 replies
ItzsleepyXD
Apr 30, 2025
Captainscrubz
May 2, 2025
Do not try to bash on beautiful geometry
G H J
Source: Own , Mock Thailand Mathematic Olympiad P9
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ItzsleepyXD
147 posts
#1 • 1 Y
Y by Dasfailure
Let $ABC$be triangle with point $D,E$ and $F$ on $BC,AB,CA$
such that $BE=CF$ and $E,F$ are on the same side of $BC$
Let $M$ be midpoint of segment $BC$ and $N$ be midpoint of segment $EF$
Let $G$ be intersection of $BF$ with $CE$ and $\dfrac{BD}{DC}=\dfrac{AC}{AB}$
Prove that $MN\parallel DG$
Z K Y
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moony_
22 posts
#2
Y by
ddit for AEGF and point D: pairs (DE, DF), (DA, DG), (DB, DC) are in involution. Now move this lines to point A in parallel: (AK, AI), (AL, AD), (Ainf, Ainf) and we wanna proof that AL is bissector of angle BAC. Now we move this involution on line BC: (K, I), (L, D), (inf, inf).
CI/CD = CA/CF => CI/BK = BE/CF * CA/BA * CD/BD = 1 => CI = BK => this involution is symmetry by point M => D and L are symmetrical by M => BL/CL = BA/CA => BL - bissector of angle BAC => DG || AL || MN => DG || MN

soooo nice geo ^_^
Attachments:
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moony_
22 posts
#3
Y by
ItzsleepyXD wrote:
Let $ABC$be triangle with point $D,E$ and $F$ on $BC,AB,CA$
such that $BE=CF$ and $E,F$ are on the same side of $BC$
Let $M$ be midpoint of segment $BC$ and $N$ be midpoint of segment $EF$
Let $G$ be intersection of $BF$ with $CE$ and $\dfrac{BD}{DC}=\dfrac{AC}{AB}$
Prove that $MN\parallel DG$


another solution:
Let ABA'C - parallelogram
BD/DC = CA/BA = BA'/CA' => A'D - bissector of angle BA'C.
G, D, A' collinear cuz parallelogram bissector lemma
GD || bissector of angle BAC || MN => GD || MN

you can proove parallelogram bissector lemma by using pappus theorem for lines AB and AC and points E, B, inf and F, C, inf on them

it may be easier than first one... hahaha

upd: point names on pic are wrong, sry
Attachments:
This post has been edited 2 times. Last edited by moony_, Apr 30, 2025, 12:38 PM
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FarrukhBurzu
5 posts
#4
Y by
Is $BE=CF$ true for D E F symmetrically ?
This post has been edited 1 time. Last edited by FarrukhBurzu, Apr 30, 2025, 7:59 PM
Z K Y
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Tkn
33 posts
#6
Y by
[asy]
size(9cm);
defaultpen(fontsize(11pt));
    
pair A = (-0.25,2);
pair B = (-1,0);
pair C = (2,0);

path C1 = circle(B,1);
path C2 = circle(C,1);

pair E1 = intersectionpoint(C1, A--B);
pair F = intersectionpoint(C2, A--C);

pair M = (E1+F)/2;
pair N1 = (B+C)/2;

pair P = 2N1-A;
pair Q = 2M-A;

pair R = extension(Q,F,B,P);
pair S1 = extension(E1,Q,C,P);
pair T = extension(B,F,C,E1);
pair D = extension(T,P,B,C);
pair U = extension(F,Q,C,E1);

draw(A--B--C--cycle, black);
draw(E1--Q--F, black);
draw(B--P--C, black);
draw(R--Q--S1, black+dashed);
draw(E1--F, black);
draw(B--F,blue);
draw(C--E1,blue);
draw(T--P, blue+dashed);
draw(M--N1, red+dashed);

        
dot(A);
dot(B);
dot(C);
dot(E1);
dot(F);
dot(P);
dot(Q);
dot(R);
dot(S1);
dot(M,red);
dot(N1,red);
dot(T,blue);
dot(D);

label("$A$", A, N, black);
label("$B$", B, SW, black);
label("$C$", C, E, black);
label("$E$", E1, NW, black);
label("$F$", F, NE, black);
label("$Q$", Q, 1.3W+0.5S, black);
label("$P$", P, S, black);
label("$R$", R, SW, black);
label("$S$", S1, SE, black);
label("$N$", N1, SW, black);
label("$M$", M, N, black);
label("$G$", T, S+0.5W, black);
label("$D$", D, NE, black);
[/asy]

Let $P$ and $Q$ denote the reflection of $A$ across $N$ and $M$, respectively.
Let $R=\overleftrightarrow{FQ}\cap\overleftrightarrow{BP}$, and $S=\overleftrightarrow{EQ}\cap \overleftrightarrow{CP}$
Note that $AM=MQ$, and $AN=NP$. It implies that $AEQF$, and $ABPC$ are parallelograms.
Since $BE=CF$, we have $QS=RP=RQ=PS$. This indicates that $PRQS$ is a rhombus. Furthermore, the line $\overleftrightarrow{PQ}$ must bisect $\angle{BPC}$.
By the given ratio condition of the point $D$:
$$\frac{BD}{CD}=\frac{AC}{AB}=\frac{BP}{CP},$$so, $D$ lies on the angle bisector of $\angle{BPC}$; that is $D\in \overleftrightarrow{PQ}$.
Next, it suffices to show that $G$ also lies on $\overleftrightarrow{PQ}$, and it could be proceed using Menelaus' theorem on $\triangle{ABF}$ and the line $\overleftrightarrow{EGC}$:
\begin{align*}
    1&=\frac{BE}{AE}\cdot \frac{AC}{CF}\cdot \frac{FG}{GB}\\
    &=\frac{AE}{BE}\cdot \frac{CF}{AC}\cdot \frac{GB}{FG}\\
    &=\frac{FQ}{QR}\cdot \frac{RP}{BP}\cdot \frac{GB}{GF}.
\end{align*}The converse of Menelaus' theorem also holds for $\triangle{FBR}$, so $G\in \overleftrightarrow{PQ}$.
The last step is to observe that $A$ is the homothetic center sending $\overline{MN}\mapsto \overline{QP}$. Therefore, $\overleftrightarrow{PQ}\parallel\overleftrightarrow{MN}$.
In conclusion $\overleftrightarrow{MN}\parallel\overleftrightarrow{GD}$ as desired.
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cj13609517288
1922 posts
#7
Y by
Swap the names of $E$ and $F$.

Move $E$ linearly along $AC$, then $F$ moves linearly along $AB$ and $N$ moves linearly. Assume WLOG $AB<AC$, then let the value of $E$ when $F=A$ be $P$. Then when $F=B$ and $F=A$ we get that $N$ lies on the line through $M$ and the midpoint of $AP$. A homothety of scale factor $2$ from $A$ takes this line through $D$ parallel to the $A$-angle bisector. A simple bary (or MMP) argument shows that $G$ moves on a line (since it's degree $1+1-1=1$). When $E=P$ we get $G=P$ which does lie on the desired line. When $E=A$ we get $G=DP\cap AB$ which does lie on the desired line. $\blacksquare$
This post has been edited 1 time. Last edited by cj13609517288, May 1, 2025, 3:55 PM
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DottedCaculator
7357 posts
#8
Y by
Beautiful problem with a beautiful solution!

Let $D=(0:c:b)$, $E=(t:c-t:0)$, and $F=(t:0:b-t)$. Then, $M=(0:1:1)$ and $N=((b+c)t:b(c-t):c(b-t))$, so $MN$ has direction $(b+c:-b:-c)$. In addition, $G=(t:c-t:b-t)$, so $DG$ also has direction $(t(b+c):(c-t)(b+c)-c(b+c-t):(b-t)(b+c)-b(b+c-t))=(t(b+c):-bt:-ct)=(b+c:-b:-c)$. Therefore, $MN$ and $DG$ are parallel.
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Assassino9931
1361 posts
#9
Y by
In this config it is also well known (follows through the midpoint of $BF$ or $CE$) that $MN$ is parallel to the angle bisector of $\angle BAC$.
In fact, the foot of this bisector is isotomic with $D$ by the angle bisector property.
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Ianis
415 posts
#10
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Assassino9931 wrote:
In this config it is also well known (follows through the midpoint of $BF$ or $CE$) that $MN$ is parallel to the angle bisector of $\angle BAC$.
In fact, the foot of this bisector is isotomic with $D$ by the angle bisector property.

Indeed! And hence it suffices to prove that if $G'$ is the reflection of $G$ wart to $M$ then $G'$ lies on the angle bisector of $\angle BAC$, for which it suffices to prove that the distances from $G'$ to $CA$ and $AB$ are equal. Since $CF=BE$ it suffices to prove that $[G'CF]=[G'EB]$. Now, observe that$$[G'CF]=[G'CB]=[G'EB],$$because $BF\parallel BG\parallel CG'$ and $CE\parallel CG\parallel BF'$. Done.

Credits go to @magnusarg for this solution.
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Captainscrubz
78 posts
#11
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Let $P$ be the arc midpoint of $BAC$
Let an arbitrary circle from $P$ and $A$ intersect $AB$ and $AC$ at $E'$ and $F'$ respectively.
As $P$ will be the center of spiral similarity that sends $E'F'\rightarrow BC$
$\implies \angle E'PB=\angle F'PC$ and $E'P=F'P, BP=PC \implies E'B=F'C$
$\therefore E'\equiv E$ and $F'\equiv F$
If $AB<AC$ then let $H=MN\cap AC$
see that as $P$ is the center of spiral similarity that sends $NF\rightarrow MC$ and a fixed point $\implies PHMC$ is cyclic$\implies H$ is a fixed point
and $\angle NHF=\angle NPF=\angle \frac{A}{2} \implies MN \parallel A\text{ angle bisector}$
Now after the introduction of point $A'$ such that $ABA'C$ is a parallelogram the problem reduces to-
Quote:
Let $G$ be a point on the angle bisector of $A$ and let $l_1$ and $l_2$ be lines from $B$ and $C$ such that $l_1$ and $l_2$ is parallel to $AC$ and $AB$. Let $E=CG\cap l_1$ and $F=BG\cap l_2$. Prove $BE=CF$.
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