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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
stuck on a system of recurrence sequence
Nonecludiangeofan   1
N 5 minutes ago by pco
Please guys help me solve this nasty problem that i've been stuck for the past month:
Let \( (a_n) \) and \( (b_n) \) be two sequences defined by:
\[
a_{n+1} = \frac{1 + a_n + a_n b_n}{b_n} \quad \text{and} \quad b_{n+1} = \frac{1 + b_n + a_n b_n}{a_n}
\]for all \( n \ge 0 \), with initial values \( a_0 = 1 \) and \( b_0 = 2 \).

Prove that:
\[
a_{2024} < 5.
\]
(btw am still not comfortable with system of recurrence sequences)
1 reply
Nonecludiangeofan
Yesterday at 10:32 PM
pco
5 minutes ago
Number Theory
MuradSafarli   4
N 9 minutes ago by mdnajibl477
find all natural numbers \( (a, b) \) such that the following equation holds:

\[
7^a + 1 = 2b^2
\]
4 replies
MuradSafarli
Yesterday at 7:55 PM
mdnajibl477
9 minutes ago
euler-totient function
Laan   0
32 minutes ago
Proof that there are infinitely many positive integers $n$ such that
$\varphi(n)<\varphi(n+1)<\varphi(n+2)$
0 replies
Laan
32 minutes ago
0 replies
Inspired by JK1603JK
sqing   0
32 minutes ago
Source: Own
Let $ a,b,c\geq 0 $ and $ ab+bc+ca=2. $ Prove that
$$ \frac{a+b+c-3abc}{a^2b+b^2c+c^2a}\geq\frac{1}{2}$$$$ \frac{a+b+c-3abc-2}{a^2b+b^2c+c^2a}\geq\frac{1-\sqrt 6}{2}$$$$  \frac{a+b+c-3abc-1 }{a^2b+b^2c+c^2a} \geq\frac{2-\sqrt 6}{4}$$$$ \frac{a+b+c-\frac{1}{6}abc-2}{a^2b+b^2c+c^2a}\geq\frac{13}{9}-\sqrt {\frac{3}{2}}$$$$ \frac{a+b+c-abc-2}{a^2b+b^2c+c^2a}\geq\frac{7-3\sqrt 6}{6}$$
0 replies
+2 w
sqing
32 minutes ago
0 replies
No more topics!
2002!\cdot k without certain digits
nguyentrang   3
N May 2, 2008 by mazur89
Is there a positive integer $ k$ such that none of the digits $ 3,4,5,6$ appear in the decimal representation of the number $ 2002!\cdot k$?
3 replies
nguyentrang
Apr 28, 2008
mazur89
May 2, 2008
2002!\cdot k without certain digits
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G H BBookmark kLocked kLocked NReply
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nguyentrang
274 posts
#1 • 3 Y
Y by Adventure10, Mango247, and 1 other user
Is there a positive integer $ k$ such that none of the digits $ 3,4,5,6$ appear in the decimal representation of the number $ 2002!\cdot k$?
Z K Y
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mazur89
19 posts
#2 • 2 Y
Y by Adventure10, Mango247
Yes. There is also a positive integer $ k$ such that none of the digits $ 2,3,4,5,6,7,8,9$ appear in the decimal representation of the number $ 2002!\cdot k$ because it has the form $ 11...100...0$.
Z K Y
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nguyentrang
274 posts
#3 • 2 Y
Y by Adventure10, Mango247
Please show the proof..
Z K Y
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mazur89
19 posts
#4 • 4 Y
Y by Adventure10, Mango247, Mango247, Mango247
Well, in the sequence $ 1, 11, 111, 1111, ...$ there are two numbers $ k, l$ such that $ k\equiv l(mod2002!)$. Their difference is divisible by $ 2002!$ and has the form $ 11...100...0$.
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