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a My Retirement & New Leadership at AoPS
rrusczyk   1346
N 16 minutes ago by KF329
I write today to announce my retirement as CEO from Art of Problem Solving. When I founded AoPS 22 years ago, I never imagined that we would reach so many students and families, or that we would find so many channels through which we discover, inspire, and train the great problem solvers of the next generation. I am very proud of all we have accomplished and I’m thankful for the many supporters who provided inspiration and encouragement along the way. I'm particularly grateful to all of the wonderful members of the AoPS Community!

I’m delighted to introduce our new leaders - Ben Kornell and Andrew Sutherland. Ben has extensive experience in education and edtech prior to joining AoPS as my successor as CEO, including starting like I did as a classroom teacher. He has a deep understanding of the value of our work because he’s an AoPS parent! Meanwhile, Andrew and I have common roots as founders of education companies; he launched Quizlet at age 15! His journey from founder to MIT to technology and product leader as our Chief Product Officer traces a pathway many of our students will follow in the years to come.

Thank you again for your support for Art of Problem Solving and we look forward to working with millions more wonderful problem solvers in the years to come.

And special thanks to all of the amazing AoPS team members who have helped build AoPS. We’ve come a long way from here:IMAGE
1346 replies
rrusczyk
Monday at 6:37 PM
KF329
16 minutes ago
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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.

Are you ready to level up with Olympiad training? Registration is open with early bird pricing available for our WOOT programs: MathWOOT (Levels 1 and 2), CodeWOOT, PhysicsWOOT, and ChemWOOT. What is WOOT? WOOT stands for Worldwide Online Olympiad Training and is a 7-month high school math Olympiad preparation and testing program that brings together many of the best students from around the world to learn Olympiad problem solving skills. Classes begin in September!

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!
[*]March 13th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar about Summer Camps at the Virtual Campus. Transform your summer into an unforgettable learning adventure! From elementary through high school, we offer dynamic summer camps featuring topics in mathematics, language arts, and competition preparation - all designed to fit your schedule and ignite your passion for learning.[/list]
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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
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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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Fiind the locus of point P
V-217   2
N 2 minutes ago by V-217
On the side $(BC)$ of the triangle $ABC$ consider a mobile point $M$. Let $B'$ the orthogonal projection of $B$ on $AM$. If the mobile points $N\in (BB'$ and $P\in (AM$ are such that $ANPC$ is a paralellogram, find the locus of point $P$ when $M$ goes through $BC$.
2 replies
V-217
Mar 22, 2025
V-217
2 minutes ago
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D1019 : Dominoes 2*1
Dattier   0
3 minutes ago
I have a 9*9 grid like this one:

IMAGE

We choose 5 white squares on the lower triangle, 5 black squares on the upper triangle and one on the diagonal, which we remove from the grid.
Like for example here:

IMAGE

Can we completely cover the grid remove from these 11 squares with 2*1 dominoes like this one:

IMAGE
0 replies
Dattier
3 minutes ago
0 replies
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Sequence with sum of zero
JSGandora   7
N 8 minutes ago by NicoN9
Source: Bay Area Math Olympiad 2004
Suppose one is given $n$ real numbers, not all zero, such that their sum is zero. Prove that one can label these numbers $a_1, a_2, \dots, a_n$ in such a manner that
\[a_1a_2 + a_2a_3 +\cdots + a_{n-1}a_n + a_na_1 < 0\]

Can we multiply expectation? In other words, is $E(XY)=E(X)E(Y)$ valid? I feel that this is valid for independent events $X$ and $Y$ but may not be for dependent events. In this case would $E(a_ia_{i+1})=E(a_i)E(a_{i+1})$? If that is the case the problem becomes trivialized.
7 replies
JSGandora
Apr 2, 2013
NicoN9
8 minutes ago
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Inspired by IMO 1984
sqing   3
N 22 minutes ago by sqing
Source: Own
Let $ a,b,c\geq 0 $ and $a^2+b^2+ ab +24abc\geq\frac{81}{64}$. Prove that
$$a+b+\frac{9}{5}c\geq\frac{9}{8}$$$$a+b+\frac{3}{2}c\geq \frac{9}{8}\sqrt [3]{\frac{3}{2}}-\frac{3}{16}$$$$a+b+\frac{8}{5}c\geq \frac{9\sqrt [3]{25}-4}{20}$$Let $ a,b,c\geq 0 $ and $ a^2+b^2+ ab +18abc\geq\frac{343}{324} $. Prove that
$$a+b+\frac{6}{5}c\geq\frac{7\sqrt 7}{18}$$$$a+b+\frac{27}{25}c\geq\frac{35\sqrt [3]5-9}{50}$$
3 replies
1 viewing
sqing
5 hours ago
sqing
22 minutes ago
J
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tangent circles
george_54   0
32 minutes ago
$ABC$ is a triangle with circumcenter $(\Omega)$ and $(\omega)$ is a circle tangent to $BC$ and internally to $(\Omega).$ The tangent
from $A$ to $(\omega)$ intersects $(\Omega)$ again at $D.$ If $T, P$ are the contact points of $(\omega)$ with $BC, AD$ respectively, prove that $CT=AC\cdot PD+DC\cdot PA.$
0 replies
george_54
32 minutes ago
0 replies
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2025 Caucasus MO Seniors P6
BR1F1SZ   1
N 34 minutes ago by pco
Source: Caucasus MO
It is known that from segments of lengths $a$, $b$ and $c$, a triangle can be formed. Can it happen that from segments of lengths$$\sqrt{a^2 + bc}, \quad \sqrt{b^2 + ca}, \quad \sqrt{c^2 + ab}$$an obtuse triangle can be formed?
1 reply
BR1F1SZ
Today at 12:48 AM
pco
34 minutes ago
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Inspired by IMO 1984
sqing   0
39 minutes ago
Source: Own
Let $ a,b,c\geq 0 $ and $a+b+c=1$. Prove that
$$a^2+b^2+c+ ab +9abc\leq 1$$$$ a^2+b^2+c +ab+10 abc\leq\frac{28}{27}$$$$a^2+b^2+c+ ab +\frac{19}{2}abc\leq\frac{55}{54}$$
0 replies
2 viewing
sqing
39 minutes ago
0 replies
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Integers with determinant \pm 1
anantmudgal09   32
N an hour ago by anudeep
Source: INMO 2021 Problem 1
Suppose $r \ge 2$ is an integer, and let $m_1, n_1, m_2, n_2, \dots, m_r, n_r$ be $2r$ integers such that $$\left|m_in_j-m_jn_i\right|=1$$for any two integers $i$ and $j$ satisfying $1 \le i<j \le r$. Determine the maximum possible value of $r$.

Proposed by B Sury
32 replies
anantmudgal09
Mar 7, 2021
anudeep
an hour ago
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Long polynomial factorization
wassupevery1   0
an hour ago
Source: 2025 Vietnam IMO TST - Problem 6
For each prime $p$ of the form $4k+3$ with $k \in \mathbb{Z}^+$, consider the polynomial $$Q(x)=px^{2p} - x^{2p-1} + p^2x^{\frac{3p+1}{2}}+2(p^2+1)x^p + p^2 x^{\frac{p-1}{2}} -x + p.$$Determine all ordered pairs of polynomials $f, g$ with integer coefficients such that $Q(x)=f(x)g(x)$.
0 replies
wassupevery1
an hour ago
0 replies
J
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Good set of cells
wassupevery1   0
an hour ago
Source: 2025 Vietnam IMO TST - Problem 5
There is an $n \times n$ grid which has rows and columns numbered from $1$ to $n$; the cell at row $i$ and column $j$ is denoted as the cell at $(i, j)$. A subset $A$ of the cells is called good if for any two cells at $(x_1, y), (x_2, y)$, the cells $(u, v)$ satisfying $x_1 < u \leq x_2, v<y$ or $x_1 \leq u < x_2, v>y$ are not in $A$. Determine the minimal number of good sets such that they are pairwise disjoint and every cell of the board belongs to exactly one good set.
0 replies
wassupevery1
an hour ago
0 replies
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Find the value
sqing   2
N an hour ago by sqing
Source: Hunan changsha 2025
Let $ a,b,c $ be real numbers such that $ abc\neq 0,2a-b+c= 0 $ and $ a-2b-c=0. $ Find the value of $\frac{a^2+b^2+c^2}{ab+bc+ca}.$
Let $ a,b,c $ be real numbers such that $ abc\neq 0,a+2b+3c= 0 $ and $ 2a+3b+4c=0. $ Find the value of $\frac{ab+bc+ca}{a^2+b^2+c^2}.$
2 replies
sqing
3 hours ago
sqing
an hour ago
J
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Binomial Non-divisibility
wassupevery1   0
an hour ago
Source: 2025 Vietnam IMO TST - Problem 4
Find all positive integers $k$ for which there are infinitely many positive integers $n$ such that $\binom{(2025+k)n}{2025n}$ is not divisible by $kn+1$.
0 replies
wassupevery1
an hour ago
0 replies
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Inspired by IMO 1984
sqing   2
N an hour ago by sqing
Source: Own
Let $ a,b,c\geq 0 $ and $a+b+c=1$. Prove that
$$a^2+b^2+ ab +24abc\leq\frac{81}{64}$$Equality holds when $a=b=\frac{3}{8},c=\frac{1}{4}.$
$$a^2+b^2+ ab +18abc\leq\frac{343}{324}$$Equality holds when $a=b=\frac{7}{18},c=\frac{2}{9}.$
2 replies
sqing
5 hours ago
sqing
an hour ago
J
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2025 Caucasus MO Juniors P1
BR1F1SZ   1
N an hour ago by pco
Source: Caucasus MO
Anya and Vanya’s houses are located on the straight road. The distance between their houses is divided by a shop and a school into three equal parts. If Anya and Vanya leave their houses at the same time and walk towards each other, they will meet near the shop. If Anya rides a scooter, then her speed will increase by $150\,\text{m/min}$, and they will meet near the school. Find Vanya’s speed of walking.
1 reply
BR1F1SZ
Today at 12:54 AM
pco
an hour ago
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Prove that P1(x), P2(x) ,... Pn(x) = k has no root
truongphatt2668   2
N Mar 23, 2025 by truongphatt2668
Let $n \in \mathbb{N}^*$ and $P_1(x),P_2(x), \ldots P_n(x) \in \mathbb{Z}[x]$ such that $\mathrm{deg} P_i = 2, \forall i = \overline{1,n}$. Prove that exists many $k \in \mathbb{N}$ such that every equation: $P_i(x) = k, \forall i = \overline{1,n}$ has no real roots
2 replies
truongphatt2668
Mar 23, 2025
truongphatt2668
Mar 23, 2025
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Prove that P1(x), P2(x) ,... Pn(x) = k has no root
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Reveal topic tags
polynomial
algebra
number theory
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truongphatt2668
297 posts
truongphatt2668
#1 h  Mar 23, 2025, 2:26 AM
Y by
Let $n \in \mathbb{N}^*$ and $P_1(x),P_2(x), \ldots P_n(x) \in \mathbb{Z}[x]$ such that $\mathrm{deg} P_i = 2, \forall i = \overline{1,n}$. Prove that exists many $k \in \mathbb{N}$ such that every equation: $P_i(x) = k, \forall i = \overline{1,n}$ has no real roots
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removablesingularity
560 posts
removablesingularity
#2 h  Mar 23, 2025, 5:26 AM
Y by
If $P_i = a_ix^2+b_ix+c_i$ then $P_i\left(x\right) = k$ has discriminant $b_i^2 - 4a_i\left(c_i - k\right),i=1,2,\cdots,n$. The equations has no real roots if $b_i^2-4a_ic_i+4a_i.k = b_i^2 - 4a_i\left(c_i - k\right) < 0 \iff -4a_i.k>b_i^2 - 4a_i.c_i$. Then for natural numbers $k$ with $k >\frac{4a_i.c_i -b_i^2}{4a_i},i=1,2,\cdots,n$ all the equations $P_i(x) = k$ has no real roots.
This post has been edited 1 time. Last edited by removablesingularity, Mar 23, 2025, 4:51 PM
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truongphatt2668
297 posts
truongphatt2668
#3 h  Mar 23, 2025, 4:02 PM
Y by
removablesingularity wrote:
If $P_i = a_ix^2+b_ix+c_i$ then $P_i\left(x\right) = k$ has discriminant $b_i^2 - 4a_i\left(c_i - k\right),i=1,2,\cdots,n$. The equations has no real roots if $=b_i^2-4a_ic_i+4a_i.k = b_i^2 - 4a_i\left(c_i - k\right) < 0 \iff -4a_i.k>b_i^2 - 4a_i.c_i$. Then for natural numbers $k$ with $k >\frac{4a_i.c_i -b_i^2}{4a_i},i=1,2,\cdots,n$ all the equations $P_i(x) = k$ has no real roots.

Is it that easy?
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