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A party is attended by boys and girls. Each person attending the party knows exactly 3 boys and exactly 7 girls among the other people. Prove that the number of all the people attending the party is divisible by 20.
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k a March Highlights and 2025 AoPS Online Class Information
jlacosta 0
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!
[*]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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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
stuck on a system of recurrence sequence
Nonecludiangeofan 0
Please guys help me solve this nasty problem that i've been stuck for the past month:
Let
and 
be two sequences defined by:
![\[
a_{n+1} = \frac{1 + a_n + a_n b_n}{b_n} \quad \text{and} \quad b_{n+1} = \frac{1 + b_n + a_n b_n}{a_n}
\]](//latex.artofproblemsolving.com/a/a/0/aa0a5c5acedc990eb345fb50a8395db05307a4e7.png)
for all 
, with initial values 
and 
.
Prove that:
![\[
a_{2024} < 5.
\]](//latex.artofproblemsolving.com/f/c/a/fca6e6630f33429b5a7ea867d639a2e29ff881f2.png)
(btw am still not comfortable with system of recurrence sequences)
Let
and 
be two sequences defined by:![\[
a_{n+1} = \frac{1 + a_n + a_n b_n}{b_n} \quad \text{and} \quad b_{n+1} = \frac{1 + b_n + a_n b_n}{a_n}
\]](http://latex.artofproblemsolving.com/a/a/0/aa0a5c5acedc990eb345fb50a8395db05307a4e7.png)
for all 
, with initial values 
and 
.Prove that:
![\[
a_{2024} < 5.
\]](http://latex.artofproblemsolving.com/f/c/a/fca6e6630f33429b5a7ea867d639a2e29ff881f2.png)
(btw am still not comfortable with system of recurrence sequences)
0 replies
A huge group of children compare their heights
Tintarn 5
N
by InCtrl
Source: All-Russian MO 2024 9.8
children, no two of the same height, lined up. Let us call a pair of different children 
good if between them there is no child whose height is greater than the height of one of 
and 
, but less than the height of the other. What is the greatest number of good pairs that could be formed? (Here, and 
are considered the same pair.)Proposed by I. Bogdanov
5 replies
Iran Inequality
mathmatecS 15
N
by Marcus_Zhang
Source: Iran 1998
When 
satisfy 
prove in equality.
satisfy 
prove in equality.
15 replies
Inequality involving x, y and z
cefer 46
N
by Baimukh
Source: Balkan MO 2012 - Problem 2
Prove that
![\[\sum_{cyc}(x+y)\sqrt{(z+x)(z+y)} \geq 4(xy+yz+zx),\]](//latex.artofproblemsolving.com/0/4/7/047adb616a6be9f9ed52f2385fa592fb5464e420.png)
for all positive real numbers
and 
.
![\[\sum_{cyc}(x+y)\sqrt{(z+x)(z+y)} \geq 4(xy+yz+zx),\]](http://latex.artofproblemsolving.com/0/4/7/047adb616a6be9f9ed52f2385fa592fb5464e420.png)
for all positive real numbers
and 
.
46 replies
Diferential ecuation from physics
QQQ43 1
N
by QQQ43
Find all functions f:R -> R such that :
f''(x)+f'(x)*b+cos(f(x))*c=a ; where a,b,c are constants in R
f'(0)=0
f(0)=0
f''(x)+f'(x)*b+cos(f(x))*c=a ; where a,b,c are constants in R
f'(0)=0
f(0)=0
1 reply
ISI 2024 P1
MrOreoJuice 7
N
by Levieee
Find, with proof, all possible values of 
such that
![\[\lim_{n \to \infty} \left( \frac{1 + 2^{1/3} + 3^{1/3} + \dots + n^{1/3}}{n^t} \right ) = c\]](//latex.artofproblemsolving.com/e/1/a/e1a87bd7bee41a7d656e2ff067a87dff0c9929ec.png)
for some real 
. Also find the corresponding values of 
.
such that![\[\lim_{n \to \infty} \left( \frac{1 + 2^{1/3} + 3^{1/3} + \dots + n^{1/3}}{n^t} \right ) = c\]](http://latex.artofproblemsolving.com/e/1/a/e1a87bd7bee41a7d656e2ff067a87dff0c9929ec.png)
for some real 
. Also find the corresponding values of 
.
7 replies
1 viewing
Differentiation Marathon!
LawofCosine 186
N
by LawofCosine
Hello, everybody!
This is a differentiation marathon. It is just like an ordinary marathon, where you can post problems and provide solutions to the problem posted by the previous user. You can only post differentiation problems (not including integration and differential equations) and please don't make it too hard!
Have fun!
(Sorry about the bad english)
This is a differentiation marathon. It is just like an ordinary marathon, where you can post problems and provide solutions to the problem posted by the previous user. You can only post differentiation problems (not including integration and differential equations) and please don't make it too hard!
Have fun!
(Sorry about the bad english)
186 replies
IMC 1994 D2 P1
j___d 12
N
by mqoi_KOLA
Let ![$f\in C^1[a,b]$](//latex.artofproblemsolving.com/8/4/8/848da12534d564560fe1cb016142159234be98b4.png)
, 
and suppose that 
, 
, is such that
for all ![$x\in [a,b]$](//latex.artofproblemsolving.com/7/8/d/78d9dcd969a2c45f7cc9234aa1377b6ec1c2d5d3.png)
. Is it true that 
for all ?
![$f\in C^1[a,b]$](http://latex.artofproblemsolving.com/8/4/8/848da12534d564560fe1cb016142159234be98b4.png)
, 
and suppose that 
, 
, is such that
for all ![$x\in [a,b]$](http://latex.artofproblemsolving.com/7/8/d/78d9dcd969a2c45f7cc9234aa1377b6ec1c2d5d3.png)
. Is it true that 
for all ?
12 replies
Solve the following Limit
deepthinka 1
N
by HacheB2031
Solve:
\lim_{ x \to \frac{\pi}{2}^+ } tan(x)
NB:The calculus textbook I'm reading gives the answer
as as ( -\infty ) and not '0.027'.
( The textbook doesn't provide any algebraic justification
for this answer, it just plots the graphs.
But i'll like a Clear algebraic explanation
)
\lim_{ x \to \frac{\pi}{2}^+ } tan(x)
NB:The calculus textbook I'm reading gives the answer
as as ( -\infty ) and not '0.027'.
( The textbook doesn't provide any algebraic justification
for this answer, it just plots the graphs.
But i'll like a Clear algebraic explanation
)
1 reply
why cl(W) cap X is compact confusion
enter16180 1
N
by Tip_pay
can someone say here why 
is compact?
is compact?
1 reply
Problem 2
blug 1
N
by aidan0626
Source: Polish Junior Math Olympiad Finals 2025
A party is attended by boys and girls. Each person attending the party knows exactly 3 boys and exactly 7 girls among the other people. Prove that the number of all the people attending the party is divisible by 20.
1 reply
Problem 2
G
H
G
H
Source: Polish Junior Math Olympiad Finals 2025
The post below has been deleted. Click to close.
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This post has been deleted. Click here to see post.
boys and 
girls. Each boy knows 
girls, so there are 
such edges. Each girl knows 
boys, so there are 
edges. Therefore, we have 
This means that 
and 
for a positive integer 
However, remember that we know that the number of boys and girls are even, so we have 
for some positive integer 
and 
,
,N Quick Reply
G
H
