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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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[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 2, 2025
0 replies
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JEE Related ig?
mikkymini2   6
N 14 minutes ago by Idiot_of_the64squares
Hey everyone,

Just wanted to see if there are any other JEE aspirants on this forum currently prepping for it[mention year if you can]

I am actually entering 10th this year and have decided to try for it...So this year is just going to go in me strengthening my math (IOQM level (heard its enough till Mains part, so will start from there) for the problem solving part, and learn some topics from 11th and 12th as well)

It would be great to connect with others who are going through the same thing - share study strategies, tips, resources, discuss, and maybe even form study groups(not sure how to tho :maybe: ) and motivate each other ig?. :D
So yea, cya later
6 replies
mikkymini2
Yesterday at 2:54 PM
Idiot_of_the64squares
14 minutes ago
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Inequalities
sqing   1
N 40 minutes ago by sqing
Let $ a,b,c,d\geq 0 ,a-b+d=21 $ and $ a+3b+4c=101 $. Prove that
$$ - \frac{1681}{3}\leq ab - cd \leq 820$$$$ - \frac{16564}{9}\leq ac -bd \leq 420$$$$ - \frac{10201}{48}\leq ad- bc \leq\frac{1681}{3}$$
1 reply
sqing
Today at 3:53 AM
sqing
40 minutes ago
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Factorise (x+1)(x+2)(x+3)(x+4)-3
Idiot_of_the64squares   3
N an hour ago by Idiot_of_the64squares
On expansion the expression becomes:
$ x^4+10x^3+35x^2 +50x+ 21 $
I cannot solve it further
3 replies
Idiot_of_the64squares
2 hours ago
Idiot_of_the64squares
an hour ago
J
H
Inequalities
sqing   8
N 2 hours ago by sqing
Let $ a,b,c $ be real numbers so that $ a+2b+3c=2 $ and $ 2ab+6bc+3ca =1. $ Show that
$$-\frac{1}{6} \leq ab-bc+ ca\leq \frac{1}{2}$$$$\frac{5-\sqrt{61}}{9} \leq a-b+c\leq \frac{5+\sqrt{61}}{9} $$
8 replies
sqing
Apr 9, 2025
sqing
2 hours ago
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No more topics!
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KSEA NMSC Mock Contest Group B (Last Problem)
Shiyul   5
N Apr 5, 2025 by Shiyul
Let $a_n$ be a sequence defined by $a_n = a^2 + 1$. Then the product of four consecutive terms in $a_n$ can be written as the product of two terms in $a_n$. Find $p + q$ if $(a_(11))(a_(12))(a_(13))(a_(14)) = (a_p)(a_q)$.
5 replies
Shiyul
Apr 4, 2025
Shiyul
Apr 5, 2025
J
KSEA NMSC Mock Contest Group B (Last Problem)
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Reveal topic tags
Sequences and Series
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G H BBookmark kLocked kLocked NReply
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Shiyul
17 posts
Shiyul
#1 h  Apr 4, 2025, 3:09 PM
Y by
Let $a_n$ be a sequence defined by $a_n = a^2 + 1$. Then the product of four consecutive terms in $a_n$ can be written as the product of two terms in $a_n$. Find $p + q$ if $(a_(11))(a_(12))(a_(13))(a_(14)) = (a_p)(a_q)$.
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Lankou
1384 posts
Lankou
#2 h  Apr 4, 2025, 4:08 PM
Y by
Do you mean $a_n=n^2+1$ instead?
in this case $(p,q)\in \{(3, 7697), (133,183)\}$
This post has been edited 1 time. Last edited by Lankou, Apr 4, 2025, 6:19 PM
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Shiyul
17 posts
Shiyul
#3 h  Apr 4, 2025, 6:29 PM
Y by
Yes. I just realized that the answer key was wrong. Can you tell me how you arrived at your answer?
This post has been edited 1 time. Last edited by Shiyul, Apr 4, 2025, 6:32 PM
Reason: Answer key was wrong
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Giant_PT
23 posts
Giant_PT
#4 h  Apr 4, 2025, 10:00 PM
Y by
In this contest, would you have to submit all possible answers or just one of the answers? I mean $(p,q)=(133,183),(183,133)$ one is pretty easy to find because
$$(11^2+1)(12^2+1)=132^2+11^2+12^2+1=133^2+1, (13^2+1)(14^2+1)=182^2+13^2+14^2+1=183^2+1$$I don't see the how you should get the other solutions.
This post has been edited 1 time. Last edited by Giant_PT, Apr 4, 2025, 10:01 PM
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Shiyul
17 posts
Shiyul
#5 h  Apr 5, 2025, 3:24 AM
Y by
In the contest, you have to find the sum, so the order doesn’t matter. Thanks for your reply, you are KOOL
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Shiyul
17 posts
Shiyul
#6 h  Apr 5, 2025, 3:26 AM
Y by
Also, I just realized the answer key was right
I just got mandela effect’d
Also I have this competition tomorrow, so thanks for your help, you will never be forgotten
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