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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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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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Exercise 4.3.3 Solution Intermediate Counting and Probability
ObiWanKenoblowin   0
an hour ago
What’s the motivation of splitting the set of $25$ into $5$ sets of $5$ elements. Other then the fact that $5$ is a divisor of $25,$ and that this chapter is constructive counting, this is not intuitive at all. Am i missing something?
0 replies
ObiWanKenoblowin
an hour ago
0 replies
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(a,b,c) triples
Ecrin_eren   1
N an hour ago by alexheinis
Let n be a positive integer. How many positive integer triples (a, b, c) are there such that:

a! + b! + c! = 2^n ?
1 reply
Ecrin_eren
6 hours ago
alexheinis
an hour ago
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one cyclic formed by two cyclic
CrazyInMath   15
N an hour ago by ThatApollo777
Source: EGMO 2025/3
Let $ABC$ be an acute triangle. Points $B, D, E$, and $C$ lie on a line in this order and satisfy $BD = DE = EC$. Let $M$ and $N$ be the midpoints of $AD$ and $AE$, respectively. Suppose triangle $ADE$ is acute, and let $H$ be its orthocentre. Points $P$ and $Q$ lie on lines $BM$ and $CN$, respectively, such that $D, H, M,$ and $P$ are concyclic and pairwise different, and $E, H, N,$ and $Q$ are concyclic and pairwise different. Prove that $P, Q, N,$ and $M$ are concyclic.
15 replies
+1 w
CrazyInMath
Today at 12:38 PM
ThatApollo777
an hour ago
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Vietnam Mock Test
imnotgoodatmathsorry   1
N an hour ago by vanstraelen
Second Entrance Mock test for grade 10 specialized in Mathematics at High School for Gifted Students, HNUE, Vietnam
13/4/2025

Problem 1:
1) Let $a,b$ be positive reals. Prove that: $\frac{a}{a+1} + \frac{b}{b+2} < \frac{\sqrt{a} + \sqrt{b}}{2}$
2) In a small garden there are $3$ rabbits and $3$ carrots. Each rabbit will choose randomly a carrot to eat. Find the probability of a carrot was chose by less than $2$ rabbit.
Problem 2:
1) Solve the equation system: $(x+y)(x^2+y^2)=567$ and $\sqrt{xy}(x+y)^2=243$
2) Let $a,b,c$ be positive rational numbers such that: $a+b+c=2\sqrt{abc}$
Problem 3:
Let triangle $ABC$ ($\angle A$, $\angle B$, $\angle C < 90$) with excircle $(O)$ and incircle $(I)$. Incircle $(I)$ touches $BC,CA,AB$ at $D,E,F$. The excircle with the diameter of $AI$ cuts excircle $(O)$ at $K$ ($K \neq A$). $KD$ cuts the excircle with the diameter of $AI$ at $P$ ($P \neq K$) and $AK$ cuts $BC$ at $Q$. Prove that:
1) $\Delta KEC$ ~ $\Delta KFB$ and $KD$ is the bisector of $\angle BKC$
2) $AP \bot BC$
3) $IQ$ is the tangent line of the excircle of $\Delta IBC$
Problem 4,5: (will type tomorrow)
1 reply
imnotgoodatmathsorry
5 hours ago
vanstraelen
an hour ago
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IMO ShortList 1998, geometry problem 1
orl   25
N an hour ago by cj13609517288
Source: IMO ShortList 1998, geometry problem 1
A convex quadrilateral $ABCD$ has perpendicular diagonals. The perpendicular bisectors of the sides $AB$ and $CD$ meet at a unique point $P$ inside $ABCD$. Prove that the quadrilateral $ABCD$ is cyclic if and only if triangles $ABP$ and $CDP$ have equal areas.
25 replies
orl
Oct 22, 2004
cj13609517288
an hour ago
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Mock 22nd Thailand TMO P2
korncrazy   1
N 2 hours ago by YaoAOPS
Source: Own
Let $ABC$ be a triangle with $\angle BAC=60^\circ$. Let $B'$ be the reflection of $B$ across the line $AC$ and $B'$ be the reflection of $C$ across the line $AB$. Let $B'C$ and $BC'$ intersect at $A'$. Prove that the orthocenter of triangle $ABC$ coincides with the circumcenter of triangle $A'B'C'$.
1 reply
korncrazy
2 hours ago
YaoAOPS
2 hours ago
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Mock 22nd Thailand TMO P9
korncrazy   0
2 hours ago
Source: own
Let $H_A,H_B,H_C$ be the feet of the altitudes of the triangle $ABC$ from $A,B,C$, respectively. $P$ is the point on the circumcircle of the triangle $ABC$, $H$ is the orthocenter of the triangle $ABC$, and the incircle of triangle $H_AH_BH_C$ has radius $r$. Let $T_A$ be the point such that $T_A$ and $H$ are on the opposite side of $H_BH_C$, line $T_AP$ is perpendicular to the line $H_BH_C$, and the distance from $T_A$ to line $H_BH_C$ is $r$. Define $T_B$ and $T_C$ similarly. Prove that $T_A,T_B,T_C$ are collinear.
0 replies
korncrazy
2 hours ago
0 replies
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Actually, D could be any point on the plane
Sadigly   0
3 hours ago
An arbitary point $D$ is selected on arc $BC$ not containing $A$ on $(ABC)$. $P$ and $Q$ are the reflections of point $B$ and $C$ with respect to $AD$, respectively. Circumcircles of $ABQ$ and $ACQ$ intersect at $E\neq A$. Prove that $A;D;E$ is colinear
0 replies
Sadigly
3 hours ago
0 replies
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Inequalities
sqing   2
N 4 hours ago by DAVROS
Let $ a,b,c\in [0,1] $ . Prove that
$$(a+b+c)\left(\frac{1}{a^2+3}+\frac{2}{b^2+2}+\frac{2}{c^2+2}\right)\leq \frac{19}{4}$$$$(a+b+c)\left(\frac{1}{a^2+ 4}+\frac{2}{b^2+2}+\frac{2}{c^2+2}\right)\leq \frac{23}{5}$$$$(a+b+c)\left(\frac{1}{a^2+ \frac{5}{2}}+\frac{2}{b^2+2}+\frac{2}{c^2+2}\right)\leq \frac{34}{7}$$$$(a+b+c)\left(\frac{1}{a^2+ \frac{7}{2}}+\frac{2}{b^2+2}+\frac{2}{c^2+2}\right)\leq \frac{14}{3}$$
2 replies
sqing
Yesterday at 3:33 AM
DAVROS
4 hours ago
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Geometry marathon
HoRI_DA_GRe8   844
N 5 hours ago by aidenkim119
Ok so there's been no geo marathon here for more than 2 years,so lets start one,rules remain same.
1st problem.
Let $PQRS$ be a cyclic quadrilateral with $\angle PSR=90°$ and let $H$ and $K$ be the feet of altitudes from $Q$ to the lines $PR$ and $PS$,.Prove $HK$ bisects $QS$.
P.s._eeezy ,try without ss line.
844 replies
HoRI_DA_GRe8
Sep 5, 2021
aidenkim119
5 hours ago
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Areas of triangles AOH, BOH, COH
Arne   70
N 6 hours ago by LeYohan
Source: APMO 2004, Problem 2
Let $O$ be the circumcenter and $H$ the orthocenter of an acute triangle $ABC$. Prove that the area of one of the triangles $AOH$, $BOH$ and $COH$ is equal to the sum of the areas of the other two.
70 replies
Arne
Mar 23, 2004
LeYohan
6 hours ago
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AD=BE implies ABC right
v_Enhance   113
N Today at 2:02 PM by LeYohan
Source: European Girl's MO 2013, Problem 1
The side $BC$ of the triangle $ABC$ is extended beyond $C$ to $D$ so that $CD = BC$. The side $CA$ is extended beyond $A$ to $E$ so that $AE = 2CA$. Prove that, if $AD=BE$, then the triangle $ABC$ is right-angled.
113 replies
v_Enhance
Apr 10, 2013
LeYohan
Today at 2:02 PM
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k Three concyclic quadrilaterals
Lukaluce   1
N Today at 1:16 PM by InterLoop
Source: EGMO 2025 P3
Let $ABC$ be an acute triangle. Points $B, D, E,$ and $C$ lie on a line in this order and satisfy $BD = DE = EC$. Let $M$ and $N$ be midpoints of $AD$ and $AE$, respectively. Suppose triangle $ADE$ is acute, and let $H$ be its orthocentre. Points $P$ and $Q$ lie on lines $BM$ and $CN$, respectively, such that $D, H, M,$ and $P$ are concyclic and pairwise different, and $E, H, N,$ and $Q$ are concyclic and pairwise different. Prove that $P, Q, N,$ and $M$ are concyclic. $\newline$
The orthocentre of a triangle is the point of intersection of its altitudes.
1 reply
Lukaluce
Today at 1:04 PM
InterLoop
Today at 1:16 PM
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postaffteff
JetFire008   18
N Today at 12:57 PM by Captainscrubz
Source: Internet
Let $P$ be the Fermat point of a $\triangle ABC$. Prove that the Euler line of the triangles $PAB$, $PBC$, $PCA$ are concurrent and the point of concurrence is $G$, the centroid of $\triangle ABC$.
18 replies
JetFire008
Mar 15, 2025
Captainscrubz
Today at 12:57 PM
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Inequality
JK1603JK   1
N Apr 2, 2025 by lbh_qys
Let $a,b,c\ge 0: a+b+c=3$ then prove \frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}+\frac{1}{2}\cdot\frac{abc}{ab+bc+ca}\ge \frac{5}{3}.$$
1 reply
JK1603JK
Apr 2, 2025
lbh_qys
Apr 2, 2025
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Inequality
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Reveal topic tags
inequalities
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JK1603JK
46 posts
JK1603JK
#1 h  Apr 2, 2025, 9:12 AM
Y by
Let $a,b,c\ge 0: a+b+c=3$ then prove \frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}+\frac{1}{2}\cdot\frac{abc}{ab+bc+ca}\ge \frac{5}{3}.$$
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lbh_qys
507 posts
lbh_qys
#2 h  Apr 2, 2025, 9:52 AM
Y by
JK1603JK wrote:
Let $a,b,c\ge 0: a+b+c=3$ then prove $$\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}+\frac{1}{2}\cdot\frac{abc}{ab+bc+ca}\ge \frac{5}{3}.$$

Since
\[ \sum \frac{1}{a+b} = \frac{1}{3}\sum \frac{a+b+c}{a+b} = 1 + \frac{1}{3}\sum \frac{c}{a+b}, \]and by the Cauchy–Schwarz inequality,
\[ \sum \frac{c}{a+b} \geq \frac{(\sum c)^2}{\sum c(a+b)} = \frac{9}{2\sum ab}, \]it suffices to prove that
\[ 1 + \frac{3}{2\sum ab} + \frac{abc}{2\sum ab} \geq \frac{5}{3}, \]that is,
\[ abc \geq \frac{4\sum ab - 9}{3}. \]Which is Schur’s inequality.
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