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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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Interesting inequalities
sqing   8
N 10 minutes ago by SunnyEvan
Source: Own
Let $ a,b $ be real numbers . Prove that
$$-\frac{1}{12}\leq \frac{ab+a+b+1}{(a^2+3)(b^2+3)}\leq \frac{1}{4} $$$$-\frac{5}{96}\leq \frac{ab+a+b+2}{(a^2+4)(b^2+4)}\leq \frac{1}{5} $$$$-\frac{1}{16}\leq \frac{ab+a+b+1}{(a^2+4)(b^2+4)}\leq \frac{3+\sqrt 5}{32} $$$$-\frac{1}{18}\leq \frac{ab+a+b+3}{(a^2+3)(b^2+3)}\leq \frac{2+\sqrt[3] 2+\sqrt[3] {4}}{12} $$
8 replies
sqing
Today at 4:41 AM
SunnyEvan
10 minutes ago
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4 lines concurrent
Zavyk09   0
28 minutes ago
Source: Homework
Let $ABC$ be triangle with circumcenter $(O)$ and orthocenter $H$. $BH, CH$ intersect $(O)$ again at $K, L$ respectively. Lines through $H$ parallel to $AB, AC$ intersects $AC, AB$ at $E, F$ respectively. Point $D$ such that $HKDL$ is a parallelogram. Prove that lines $KE, LF$ and $AD$ are concurrent at a point on $OH$.
0 replies
Zavyk09
28 minutes ago
0 replies
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Christmas special mock geometry olympiad 2020
Nari_Tom   1
N 43 minutes ago by Nari_Tom
Let $\triangle ABC$ be a triangle with incenter $I$, and circumcircle $\omega$. Let the circle with diameter $AI$ intersects $\omega$ at $S$ and $A$. Let $M$ be the midpoint arc $BAC$ and let $MI$ intersect $\omega$ again at a point $T$. Let the circumcircle of $\triangle MIS$ intersect the circumcircle of $\triangle BIC$ at a point $X$. Let the tangent at $S$ to $\omega$ intersect $BC$ at a point $V$. Prove that $MX$ and $VT$ intersect on $\omega$.
1 reply
Nari_Tom
an hour ago
Nari_Tom
43 minutes ago
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Prove that x1=x2=....=x2025
Rohit-2006   3
N an hour ago by kamatadu
Source: A mock
The real numbers $x_1,x_2,\cdots,x_{2025}$ satisfy,
$$x_1+x_2=2\bar{x_1}, x_2+x_3=2\bar{x_2},\cdots, x_{2025}+x_1=2\bar{x_{2025}}$$Where {$\bar{x_1},\cdots,\bar{x_{2025}}$} is a permutation of $x_1,x_2,\cdots,x_{2025}$. Prove that $x_1=x_2=\cdots=x_{2025}$
3 replies
Rohit-2006
Today at 5:22 AM
kamatadu
an hour ago
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Poly with sequence give infinitely many prime divisors
Assassino9931   2
N an hour ago by Haris1
Source: Bulgaria National Olympiad 2025, Day 1, Problem 3
Let $P(x)$ be a non-constant monic polynomial with integer coefficients and let $a_1, a_2, \ldots$ be an infinite sequence. Prove that there are infinitely many primes, each of which divides at least one term of the sequence $b_n = P(n)^{a_n} + 1$.
2 replies
Assassino9931
Yesterday at 1:51 PM
Haris1
an hour ago
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There exist N flags forming a diverse set
orl   37
N an hour ago by cherry265
Source: IMO Shortlist 2010, Combinatorics 2
On some planet, there are $2^N$ countries $(N \geq 4).$ Each country has a flag $N$ units wide and one unit high composed of $N$ fields of size $1 \times 1,$ each field being either yellow or blue. No two countries have the same flag. We say that a set of $N$ flags is diverse if these flags can be arranged into an $N \times N$ square so that all $N$ fields on its main diagonal will have the same color. Determine the smallest positive integer $M$ such that among any $M$ distinct flags, there exist $N$ flags forming a diverse set.

Proposed by Tonći Kokan, Croatia
37 replies
orl
Jul 17, 2011
cherry265
an hour ago
J
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Mansion point and incenter
Hypernova   9
N an hour ago by L13832
Source: Korea National Olympiad 2019 P6
In acute triangle $ABC$, $AB>AC$. Let $I$ the incenter, $\Omega$ the circumcircle of triangle $ABC$, and $D$ the foot of perpendicular from $A$ to $BC$. $AI$ intersects $\Omega$ at point $M(\neq A)$, and the line which passes $M$ and perpendicular to $AM$ intersects $AD$ at point $E$. Now let $F$ the foot of perpendicular from $I$ to $AD$.
Prove that $ID\cdot AM=IE\cdot AF$.
9 replies
Hypernova
Nov 16, 2019
L13832
an hour ago
J
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Number Theory Chain!
JetFire008   22
N 2 hours ago by whwlqkd
I will post a question and someone has to answer it. Then they have to post a question and someone else will answer it and so on. We can only post questions related to Number Theory and each problem should be more difficult than the previous. Let's start!

Question 1
22 replies
JetFire008
Apr 7, 2025
whwlqkd
2 hours ago
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Connecting chaos in a grid
Assassino9931   1
N 2 hours ago by internationalnick123456
Source: Bulgaria National Olympiad 2025, Day 1, Problem 2
Exactly \( n \) cells of an \( n \times n \) square grid are colored black, and the remaining cells are white. The cost of such a coloring is the minimum number of white cells that need to be recolored black so that from any black cell \( c_0 \), one can reach any other black cell \( c_k \) through a sequence \( c_0, c_1, \ldots, c_k \) of black cells where each consecutive pair \( c_i, c_{i+1} \) are adjacent (sharing a common side) for every \( i = 0, 1, \ldots, k-1 \). Let \( f(n) \) denote the maximum possible cost over all initial colorings with exactly \( n \) black cells. Determine a constant $\alpha$ such that
\[ \frac{1}{3}n^{\alpha} \leq f(n) \leq 3n^{\alpha} \]for any $n\geq 100$.
1 reply
Assassino9931
Yesterday at 1:50 PM
internationalnick123456
2 hours ago
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Hard inequality
JK1603JK   0
2 hours ago
Source: unknown
Prove $$\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\le \sqrt{2\left(4\sqrt{2}-3\right)\left(a+b+c\right)+12\left(3-2\sqrt{2}\right)\frac{ab+bc+ca}{a+b+c}},\ \ \forall a,b,c\ge 0: a+b+c>0.$$Does EV theorem work?
0 replies
JK1603JK
2 hours ago
0 replies
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ACP = PCB = 8, PBC = 11 find PAC
kamatadu   2
N 2 hours ago by ND_
Source: RSM mock
Consider $\triangle ABC$. Let $P$ be an interior point such that $\angle ACP = \angle PCB = 8^{\circ}$, $\angle PBC = 11^{\circ}$ and $\angle ABP = 30^{\circ}$. Find $\angle PAC$.
2 replies
kamatadu
5 hours ago
ND_
2 hours ago
J
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interesting inequality
pennypc123456789   0
3 hours ago
Let \( a,b,c \) be real numbers satisfying \( a+b+c = 3 \) . Find the maximum value of
\[P = \dfrac{a(b+c)}{a^2+2bc+3} + \dfrac{b(a+c) }{b^2+2ca +3 } + \dfrac{c(a+b)}{c^2+2ab+3}.\]
0 replies
pennypc123456789
3 hours ago
0 replies
J
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Interesting Property of McCay Cubic
kaede_Arcadia   0
3 hours ago
Source: Own
Property: Given a $ \triangle ABC $ with orthocenter $H$, isogonal conjugate $(P,Q)$ lies on the McCay cubic. Let $P_aP_bP_c$ be the pedal triangle of $P$ wrt $ \triangle ABC $ and let $Q_B = BQ \cap CA ,Q_C = CQ \cap AB$. Let $Y,Z$ be the second intersection of $\odot (P_aP_bP_c)$ with $\odot (BP_bQ_B), \odot (CP_cQ_C)$ and let $X$ be the Poncelet point of $ABCP$. Then the lines $BY,CZ,PQ,XH$ are concurrent.

Proof: Let $S = BY \cap CZ$ and let $J,K$ be the second intersection of $\odot (P_aP_bP_c)$ with $BY,CZ$
Now, we use two basic lemmas that follows :

Lemma 1 (well-known): Let $Q_aQ_bQ_c$ be the pedal triangle of $Q$ wrt $ \triangle ABC $. Then $\measuredangle PAQ = \measuredangle (\odot (P_aP_bP_c), \overline{P_aQ_a})$.

Lemma 2 (well-known): Let $P_AP_BP_C$ be the circumcevian triangle of $P$ wrt $ \triangle ABC $. Then $\triangle P_AP_BP_C$ and $\triangle P_aP_bP_c$ are homothetic.

Back to main problem, from the Reim's theorem, we know that $BQ_B \parallel JQ_b$. On the other hand, from the Lemma 1, we see that $BP \parallel JP_b$. Therefore from the Lemma 2, we see that $S,P_b,P_B$ and $S,P_c,P_C$, respectively, are collinear and $S$ is the insimilicenter of $\odot (ABC)$ and $\odot (P_aP_bP_c)$, so $S \in PQ$.
Also, from the Third Fontene's theorem, we know that the nine-point circle $\omega$ of $ \triangle ABC $ is tangent to $\odot (P_aP_bP_c)$ at $X$. Hence by applying Monge-D'Alembert's theorem to $\odot (ABC), \odot (P_aP_bP_c), \omega$, we see that $S,H,X$ are collinear.
0 replies
kaede_Arcadia
3 hours ago
0 replies
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Chinese Girls Mathematical Olympiad 2015, Problem 7
sqing   7
N 3 hours ago by IndexLibrorumProhibitorum
Source: China Shenzhen ,13 Aug 2015
Let $x_1,x_2,\cdots,x_n \in(0,1)$ , $n\geq2$. Prove that$$\frac{\sqrt{1-x_1}}{x_1}+\frac{\sqrt{1-x_2}}{x_2}+\cdots+\frac{\sqrt{1-x_n}}{x_n}<\frac{\sqrt{n-1}}{x_1 x_2 \cdots x_n}.$$
7 replies
sqing
Aug 13, 2015
IndexLibrorumProhibitorum
3 hours ago
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right triangle from BB,CA,DC if AD = BC, <ADB +<ACB= < CAB + <DBA= 30^o
parmenides51   0
Jul 6, 2020
Source: All-Siberian Open School Olympiad 2018-19 8.4
In the convex quadrilateral $ABCD$, it is known that $AD = BC$ and $\angle ADB + \angle ACB = \angle CAB + \angle DBA = 30^o$. Prove that from segments $DB, CA$ and $DC$ you can make a right triangle.
0 replies
parmenides51
Jul 6, 2020
0 replies
J
right triangle from BB,CA,DC if AD = BC, <ADB +<ACB= < CAB + <DBA= 30^o
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parmenides51
30630 posts
parmenides51
#1 h  Jul 6, 2020, 6:45 AM
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In the convex quadrilateral $ABCD$, it is known that $AD = BC$ and $\angle ADB + \angle ACB = \angle CAB + \angle DBA = 30^o$. Prove that from segments $DB, CA$ and $DC$ you can make a right triangle.
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