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k a March Highlights and 2025 AoPS Online Class Information
jlacosta   0
Mar 2, 2025
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

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Do you have plans this summer? There are so many options to fit your schedule and goals whether attending a summer camp or taking online classes, it can be a great break from the routine of the school year. Check out our summer courses at AoPS Online, or if you want a math or language arts class that doesn’t have homework, but is an enriching summer experience, our AoPS Virtual Campus summer camps may be just the ticket! We are expanding our locations for our AoPS Academies across the country with 15 locations so far and new campuses opening in Saratoga CA, Johns Creek GA, and the Upper West Side NY. Check out this page for summer camp information.

Be sure to mark your calendars for the following events:
[list][*]March 5th (Wednesday), 4:30pm PT/7:30pm ET, HCSSiM Math Jam 2025. Amber Verser, Assistant Director of the Hampshire College Summer Studies in Mathematics, will host an information session about HCSSiM, a summer program for high school students.
[*]March 6th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar on Math Competitions from elementary through high school. Join us for an enlightening session that demystifies the world of math competitions and helps you make informed decisions about your contest journey.
[*]March 11th (Tuesday), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS Chapter Discussion MATH JAM. AoPS instructors will discuss some of their favorite problems from the MATHCOUNTS Chapter Competition. All are welcome!
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0 replies
jlacosta
Mar 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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Inspired by hunghd8
sqing   1
N an hour ago by sqing
Source: Own
Let $ a,b,c\geq 0 $ and $ a+b+c\geq 2+abc . $ Prove that
$$a^2+b^2+c^2-\frac{1}{2}a^2b^2c^2\geq 2$$$$a^2+b^2+c^2-abc-\frac{1}{2}a^2b^2c^2\geq \frac{3}{2}$$$$a^2+b^2+c^2- \frac{19}{10}abc-\frac{1}{2}a^2b^2c^2\geq -\frac{12}{25}$$$$a^2+b^2+c^2- \frac{3}{2}abc-\frac{1}{2}a^2b^2c^2\geq \frac{17\sqrt{17}-71}{16}$$
1 reply
1 viewing
sqing
Mar 21, 2025
sqing
an hour ago
J
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Integral Equations
rljmano   4
N an hour ago by rljmano
The Integral equation $\\u(x)=\int_0^1k(x,y)u(y)~dy \\ $with k and u continuous in the unit square and unit interval can have only the trivial solution. Prove this in detail. Here k(x,y)=sin(xy)
4 replies
rljmano
Mar 19, 2025
rljmano
an hour ago
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Find the constant
JK1603JK   2
N an hour ago by Quantum-Phantom
Source: unknown
Find all k such that \frac{a^{2}}{b}+\frac{b^{2}}{c}+\frac{c^{2}}{a}\le \frac{k}{abc}+3-k
holds for a,b,c>0:a+b+c=3.
2 replies
JK1603JK
Yesterday at 1:55 PM
Quantum-Phantom
an hour ago
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2 degree polynomial
PrimeSol   1
N an hour ago by PrimeSol
Let $P_{1}(x)= x^2 +b_{1}x +c_{1}, ... , P_{n}(x)=x^2+ b_{n}x+c_{n}$, $P_{i}(x)\in \mathbb{R}[x], \forall i=\overline{1,n}.$ $\forall i,j ,1 \leq i<j \leq n : P_{i}(x) \ne P_{j}(x)$.
$\forall i,j, 1\leq i<j \leq n : Q_{i,j}(x)= P_{i}(x) + P_{j}(x)$ polynomial with only one root.
$max(n)=?$
1 reply
PrimeSol
Yesterday at 6:13 AM
PrimeSol
an hour ago
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2025 Points in USATST and 2025 Sequences in Chinese TST
EthanWYX2009   1
N 2 hours ago by DottedCaculator
Source: 2025 TST-14
Let \( p_1, p_2, \cdots, p_{2025} \) be real numbers. For \( 1 \leq i \leq 2025 \), let
\[\{a_n^{(i)}\}_{n \geq 0}\]be an infinite real sequence satisfying
\[a_0^{(i)} = 0.\]It is known that:

(1)
\[a_1^{(1)}, a_1^{(2)}, \cdots, a_1^{(2025)}\]are not all zero.

(2) For any integer \( n \geq 0 \) and any \( 1 \leq i \leq 2025 \), the following holds:

\[p_i \cdot a_n^{(i+1)} = a_{n-1}^{(i)} + a_n^{(i)} + a_{n+1}^{(i)},\]
where the sequence
\[\{a_n^{(2026)}\}\]satisfies
\[a_n^{(2026)} = a_n^{(1)}, \, n = 0, 1, 2, \cdots.\]
Prove that there exists a positive real number \( r \) such that for infinitely many positive integers \( n \),

\[\max \left\{ |a_n^{(1)}|, |a_n^{(2)}|, \cdots, |a_n^{(2025)}|\right\} \geq r.\]
1 reply
EthanWYX2009
3 hours ago
DottedCaculator
2 hours ago
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Square-free Numbers
EthanWYX2009   2
N 2 hours ago by yofro
Source: 2025 TST-13
Find all positive integers \( m \) for which there exists an infinite subset \( A \) of the positive integers such that: for any pairwise distinct positive integers \( a_1, a_2, \cdots, a_m \in A \), the sum \( a_1 + a_2 + \cdots + a_m \) and the product \( a_1a_2 \cdots a_m \) are both square-free.
2 replies
1 viewing
EthanWYX2009
3 hours ago
yofro
2 hours ago
J
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Subsets with integers
MetaphysicalWukong   0
3 hours ago
Source: Dezheng Qin, Haiyong Bao
Let A = {n ∈ ℤ ∶ n = 5q + 4 for some q ∈ ℤ} and let B = {m ∈ ℤ ∶ m = 10k − 1
for some k ∈ ℤ}. Prove that A ⊆ B.
0 replies
MetaphysicalWukong
3 hours ago
0 replies
J
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Additive Combinatorics!
EthanWYX2009   0
3 hours ago
Source: 2025 TST 15
Let \( X \) be a finite set of real numbers, \( d \) be a real number, and \(\lambda_1, \lambda_2, \cdots, \lambda_{2025}\) be 2025 non-zero real numbers. Define
\[A = \left\{ (x_1, x_2, \cdots, x_{2025}) : x_1, x_2, \cdots, x_{2025} \in X \text{ and } \sum_{i=1}^{2025} \lambda_i x_i = d \right\},\]\[B = \left\{ (x_1, x_2, \cdots, x_{2024}) : x_1, x_2, \cdots, x_{2024} \in X \text{ and } \sum_{i=1}^{2024} (-1)^i x_i = 0 \right\},\]\[C = \left\{ (x_1, x_2, \cdots, x_{2026}) : x_1, x_2, \cdots, x_{2026} \in X \text{ and } \sum_{i=1}^{2026} (-1)^i x_i = 0 \right\}.\]Show that \( |A|^2 \leq |B| \cdot |C| \).
0 replies
EthanWYX2009
3 hours ago
0 replies
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Polynomial of matrix
Mathloops   4
N 4 hours ago by ysharifi
Let A, B are two square matrices with same size.
p(x) and q(x) are real coefficients polynomial
prove:
4 replies
Mathloops
Sunday at 4:59 PM
ysharifi
4 hours ago
J
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Mixed series
Snoop76   1
N 4 hours ago by RagvaloD
$a_n$ and $b_n$ satisfies the following recursion formulas: $a_{0}=1, $ $b_{0}=1$, $ a_{n+1}=a_{n}+b_{n}$$ $ and $ $$ b_{n+1}=(2n+3)b_{n}+a_{n}$. Show that$ $ $\frac{b_n}{a_n}<2n-\frac 1 {2n}$ , $ $for $n>1$.
1 reply
Snoop76
Yesterday at 5:32 PM
RagvaloD
4 hours ago
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A problem
jokehim   3
N 4 hours ago by jokehim
Source: me
Let $a,b,c>0$ and prove $$\sqrt{\frac{a+b}{c}}+\sqrt{\frac{c+b}{a}}+\sqrt{\frac{a+c}{b}}\ge \frac{3\sqrt{6}}{2}\cdot\sqrt{\frac{3(a^3+b^3+c^3)}{(a+b+c)^3}+1}.$$
3 replies
jokehim
Mar 1, 2025
jokehim
4 hours ago
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2025 BAMO Problem D/2
BR1F1SZ   4
N 5 hours ago by cosinesine
Let $\mathcal{S}$ be a finite, nonempty set of points in the plane such that, for every point $A$ in $\mathcal{S}$, there exist points $B,C$ in $\mathcal{S}$ (distinct from $A$) such that $\angle BAC = 125^\circ$. What is the smallest possible number of points in $\mathcal{S}$?
4 replies
BR1F1SZ
Yesterday at 7:00 PM
cosinesine
5 hours ago
J
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An integral
gaussiemann144   0
Yesterday at 8:15 PM
Given $\alpha, \beta$-
$\alpha = \int_0^1 xe^{\frac{x^2-1}{2}} \cos(x) dx \quad \beta = \int_1^{3/2} e^{2(x^2-2x)} \sqrt{1-\cos(4x-4)} dx$
Find- $$\frac{\alpha - \cos(1) + e^{-1/2}}{\beta}$$
0 replies
gaussiemann144
Yesterday at 8:15 PM
0 replies
J
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Putnam 1987 B1
TwinPrime   2
N Yesterday at 6:40 PM by Sagnik123Biswas
Evaluate
\[ \int_2^4 \frac{\sqrt{\ln(9-x)}\,dx}{\sqrt{\ln(9-x)}+\sqrt{\ln(x+3)}}. \]
2 replies
TwinPrime
Aug 5, 2019
Sagnik123Biswas
Yesterday at 6:40 PM
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J
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Parametric to cartesian planes
MetaphysicalWukong   2
N Sunday at 3:46 PM by vanstraelen
Source: Jiamiao Fan
Find cartesian equations for the planes below. with steps
2 replies
MetaphysicalWukong
Sunday at 6:17 AM
vanstraelen
Sunday at 3:46 PM
J
Parametric to cartesian planes
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Reveal topic tags
analytic geometry
linear algebra
L
G H BBookmark kLocked kLocked NReply
Source: Jiamiao Fan
The post below has been deleted. Click to close.
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MetaphysicalWukong
243 posts
MetaphysicalWukong
#1 h  Sunday at 6:17 AM
Y by
Find cartesian equations for the planes below. with steps
Attachments:
This post has been edited 1 time. Last edited by MetaphysicalWukong, Sunday at 6:18 AM
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MetaphysicalWukong
243 posts
MetaphysicalWukong
#2 h  Sunday at 6:48 AM
Y by
bumping Hu Ningyang
This post has been edited 1 time. Last edited by MetaphysicalWukong, Sunday at 6:48 AM
Z K Y
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vanstraelen
8937 posts
vanstraelen
#3 h  Sunday at 3:46 PM
Y by
$\left\{\begin{array}{lll} x=0+k \cdot 1+l \cdot 3 \\ y=-3+k \cdot (-1)+l \cdot (-2) \\ z=2+ k \cdot 1 + l \cdot (-1)\end{array}\right.$
$\left\{\begin{array}{lll} x=k \cdot 1+l \cdot 3 \\ y+3=k \cdot (-1)+l \cdot (-2) \\ z-2=k \cdot 1 + l \cdot (-1)\end{array}\right.$

System of three equations and two unknowns,
$\left|\begin{array}{lll} x & 1 & 3 \\ y+3 & -1 & -2 \\ z-2 & 1 & -1 \end{array}\right|=0$.
Sarrus:$x-2(z-2)+3(y+3)+2x+3(z-2)+y+3=0$,
$3x+4(y+3)+z-2=0$,
$3x+4y+z+10=0$.
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