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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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inequalities
pennypc123456789   2
N 22 minutes ago by arqady
If $a,b,c$ are positive real numbers, then
$$ \frac{a + b}{a + 7b + c} + \dfrac{b + c}{b + 7c + a}+\dfrac{c + a}{c + 7a + b} \geq \dfrac{2}{3}$$
we can generalize this problem
2 replies
pennypc123456789
an hour ago
arqady
22 minutes ago
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Coaxal Circles
fattypiggy123   29
N 27 minutes ago by sttsmet
Source: China TSTST Test 2 Day 1 Q3
Let $ABCD$ be a quadrilateral and let $l$ be a line. Let $l$ intersect the lines $AB,CD,BC,DA,AC,BD$ at points $X,X',Y,Y',Z,Z'$ respectively. Given that these six points on $l$ are in the order $X,Y,Z,X',Y',Z'$, show that the circles with diameter $XX',YY',ZZ'$ are coaxal.
29 replies
fattypiggy123
Mar 13, 2017
sttsmet
27 minutes ago
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q(x) to be the product of all primes less than p(x)
orl   16
N 39 minutes ago by Maximilian113
Source: IMO Shortlist 1995, S3
For an integer $x \geq 1$, let $p(x)$ be the least prime that does not divide $x$, and define $q(x)$ to be the product of all primes less than $p(x)$. In particular, $p(1) = 2.$ For $x$ having $p(x) = 2$, define $q(x) = 1$. Consider the sequence $x_0, x_1, x_2, \ldots$ defined by $x_0 = 1$ and \[ x_{n+1} = \frac{x_n p(x_n)}{q(x_n)} \] for $n \geq 0$. Find all $n$ such that $x_n = 1995$.
16 replies
orl
Aug 10, 2008
Maximilian113
39 minutes ago
J
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Inspired by JK1603JK
sqing   0
an hour ago
Source: Own
Let $ a,b,c\geq 0 $ and $ ab+bc+ca=3. $ Prove that
$$ (a+b+c-3)(12-5abc)\ge 2(a-b)(b-c)(c-a)$$$$6(a+b+c-3)(5-2abc)\ge 5(a-b)(b-c)(c-a)$$$$2(a+b+c-3)(9-5abc)\ge 3(a-b)(b-c)(c-a)$$$$3(a+b+c-3)(14-5abc)\ge 7(a-b)(b-c)(c-a)$$
0 replies
1 viewing
sqing
an hour ago
0 replies
J
No more topics!
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Trigonometry Problems Collection
Amir Hossein   24
N Apr 13, 2017 by Gaurav123
See the attachment. :)

This is a nice collection of trigonometry problems. I searched but didn't find a nice collection about trigonometry, so searched trigonometry problems posted on AoPS and made this file. :)
24 replies
Amir Hossein
Feb 16, 2011
Gaurav123
Apr 13, 2017
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Trigonometry Problems Collection
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Reveal topic tags
trigonometry
Problem Sets
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Amir Hossein
5452 posts
Amir Hossein
#1 h  Feb 16, 2011, 7:57 PM • 28 Y
Y by ShahinH, lafius, ahaanomegas, fireonice18, StudentIV, AdithyaBhaskar, m1234567, Taekyukim02, Severus, Uros, mathleticguyyy, fjm30, hansduran0123, Adventure10, and 14 other users
See the attachment. :)

This is a nice collection of trigonometry problems. I searched but didn't find a nice collection about trigonometry, so searched trigonometry problems posted on AoPS and made this file. :)
Attachments:
Trigonometry Problems - Amir Hossein Parvardi.pdf (87kb)
50TRIG.tex (10kb)
This post has been edited 1 time. Last edited by Amir Hossein, Nov 21, 2017, 9:49 AM
Reason: Added the TeX file.
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Potla
1886 posts
Potla
#2 h  Feb 16, 2011, 8:09 PM • 3 Y
Y by Adventure10, Mango247, and 1 other user
As usual, you are really amazing, Amir! Thanks a lot for this file, I am sure it will be helpful in our school curriculum and many competitive exams!

Again, thanks a lot for posting this useful file. :)
This post has been edited 1 time. Last edited by Potla, Feb 17, 2011, 3:48 AM
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mahanmath
1354 posts
mahanmath
#3 h  Feb 16, 2011, 8:46 PM • 4 Y
Y by Adventure10, Mango247, and 2 other users
Many thanks Amir :o !
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sindennisz
152 posts
sindennisz
#4 h  Feb 16, 2011, 10:24 PM • 3 Y
Y by Adventure10, Mango247, and 1 other user
Never seen a better collection....thank you!

sindennisz
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1=2
4594 posts
1=2
#5 h  Feb 17, 2011, 1:00 AM • 2 Y
Y by Adventure10 and 1 other user
Thanks, I needed this! :D
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Goutham
3130 posts
Goutham
#6 h  Feb 17, 2011, 2:05 AM • 2 Y
Y by Adventure10 and 1 other user
Thanks a lot, Amir.
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AndrewTom
12750 posts
AndrewTom
#7 h  Feb 17, 2011, 8:08 AM • 3 Y
Y by Adventure10, Mango247, and 1 other user
Many thanks, amparvardi.
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goodar2006
1347 posts
goodar2006
#8 h  Feb 17, 2011, 12:19 PM • 3 Y
Y by Adventure10, Mango247, and 1 other user
Thanks Amir. Nice job! :)
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sayak38kolay
195 posts
sayak38kolay
#9 h  Feb 17, 2011, 5:45 PM • 2 Y
Y by Adventure10 and 1 other user
Thanks a lot amparvardi...
:lol:
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omar123
2 posts
omar123
#10 h  Feb 17, 2011, 6:52 PM • 2 Y
Y by Adventure10 and 1 other user
Thanks a lot amparvardi for this nice collection
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applepi2000
2226 posts
applepi2000
#11 h  Feb 17, 2011, 8:55 PM • 1 Y
Y by Adventure10
This is incredible! Thanks so much, ampvardi!
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Jason27603
629 posts
Jason27603
#12 h  Feb 26, 2011, 5:12 PM • 3 Y
Y by Adventure10, Mango247, and 1 other user
Thank you!
Do you have a solutions file for this?
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Amir Hossein
5452 posts
Amir Hossein
#13 h  Feb 26, 2011, 5:16 PM • 3 Y
Y by Adventure10, Mango247, and 1 other user
Thank you all for your appreciation. :)
Jason27603 wrote:
Thank you!
Do you have a solutions file for this?

I just searched and found them on AoPS, but some of the problems does not have solutions on the site, so you may post them on forums (after searching, of course).
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siddharthanand
282 posts
siddharthanand
#14 h  Mar 13, 2011, 12:26 PM • 3 Y
Y by Adventure10, Mango247, and 1 other user
amparvardi wrote:
I just searched and found them on AoPS, but some of the problems does not have solutions on the site, so you may post them on forums (after searching, of course).

hey amir , problems after 40 are exclusive to series marathon and the rest questions are brilliant . hoping to find some solutions other than the link $!$
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Amir Hossein
5452 posts
Amir Hossein
#15 h  Mar 13, 2011, 12:35 PM • 3 Y
Y by Adventure10, Mango247, and 1 other user
If you're interested to see some solutions, here you can post your solutions to the problems :).
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applepi2000
2226 posts
applepi2000
#16 h  Mar 13, 2011, 4:57 PM • 2 Y
Y by Adventure10, Mango247
3
More later.
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Amir Hossein
5452 posts
Amir Hossein
#17 h  Mar 13, 2011, 5:39 PM • 3 Y
Y by Adventure10, Mango247, and 1 other user
@applepi2000: but no, $\sqrt{\cos^2 4 \theta} = \pm \cos 4 \theta.$
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applepi2000
2226 posts
applepi2000
#18 h  Mar 19, 2011, 10:26 PM • 1 Y
Y by Adventure10
Sorry, forgot that case -_-

Anyway, I'll try other ones then:
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mrmohamed
169 posts
mrmohamed
#19 h  Mar 25, 2011, 8:17 AM • 2 Y
Y by Adventure10, Mango247
amparvardi wrote:
See the attachment. :)

This is a nice collection of trigonometry problems. I searched but didn't find a nice collection about trigonometry, so searched trigonometry problems posted on AoPS and made this file. :)
many thanks for you Amir great post
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Ligouras
826 posts
Ligouras
#20 h  Apr 2, 2011, 3:19 PM • 1 Y
Y by Adventure10
Thanks a lot, Amir
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brandbest1
259 posts
brandbest1
#21 h  Dec 31, 2013, 5:14 AM • 1 Y
Y by Adventure10
HUGE BUMP

Any solutions to 1 and 2?
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LzuTao
6 posts
LzuTao
#22 h  Aug 20, 2015, 3:28 PM • 2 Y
Y by Adventure10, Mango247
Amir Hossein wrote:
See the attachment. :)

This is a nice collection of trigonometry problems. I searched but didn't find a nice collection about trigonometry, so searched trigonometry problems posted on AoPS and made this file. :)

Thanks Amir, but where did you find them? I have solutions for some problems.
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nikolapavlovic
1246 posts
nikolapavlovic
#23 h  Aug 20, 2015, 4:09 PM • 2 Y
Y by Adventure10, Mango247
21
very nice article
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AdithyaBhaskar
652 posts
AdithyaBhaskar
#24 h  Aug 20, 2015, 4:58 PM • 2 Y
Y by Adventure10, Mango247
Nice job! Thanks. :D
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Gaurav123
1 post
Gaurav123
#25 h  Apr 13, 2017, 8:39 PM • 1 Y
Y by Adventure10
Can I get a book in pdf , what was available in your data base.
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