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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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0 replies
jlacosta
May 1, 2025
0 replies
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Calculus, sets
wl8418   0
8 minutes ago
Is empty set a proper set of an non empty set? Why or why not? Any clarification or insight is appreciated. Thanks in advance!
0 replies
wl8418
8 minutes ago
0 replies
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Find r_1^2 + r_2^2 + r_3^2
BlackOctopus23   3
N 2 hours ago by Li0nking
Let $r_1$, $r_2$, and $r_3$ be the roots of $3x^3 - 8x^2 + 4x - 13$. Find $r_1^2 + r_2^2 + r_3^2$Solution
3 replies
BlackOctopus23
Yesterday at 1:50 AM
Li0nking
2 hours ago
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Pls help I'm stuck
UtkarshSri   2
N 2 hours ago by Li0nking
So I was solving this problem
Cbrt(x+1) - cbrt(x-1) = pow((x²-1),1/6)


Here, the square root is defined in the domain x≤-1 OR x≥1
I substituted a= cbrt(x+1) and b= cbrt(x-1)
And then the equation turned out to be
a-b= sqrt(ab)
Here, since the RHS is +ve, the LHS must be too, so a>b (Done so as to prevent extraneous roots)
Now, I can write the domain in terms of a and b as ab≥0
Squaring, we get a²+b²-3ab=0
On solving we get a/b = (3±sqrt(5))/2
Now, we know that a>b from the above condition, so the only root is a/b= (3+sqrt(5))/2 and on substituting for x, and applying componendo and dividendo after cubing, I get x= sqrt(5)/2

Now, my question is, since the equation is even i.e f(x)=f(-x), it's symmetric about the y-axis, so x= -sqrt(5)/2 must also satisfy the equation, and indeed it does, so where did I made some irreversible step that caused this loss of the root?
when I plugged the equation to wolframalpha, it showed that the "principle root" has the former solⁿ only where's for a "real valued root", it has both.I also found that on solving the equation
b-a= sqrt(ab), I get the latter root.But I mean where did that extra -ve sign come from?
I'm familiar that f(x)=sqrt(g(x)) is equivalent to f²(x)=g(x) AND f(x)≥0 AND g(x)≥0 but the g(x)≥0 condition is not necessary as f²(x) is already ≥0
and also that abs(f(x))=abs(g(x)) is equivalent to f²(x)=g²(x)

PS: Pardon me I don't know latex well, I'm tried to make this as comprehensive as possible and made sure there's no loss of equivalence in any step.
I hope someone clarifies this for me, it's really bothering me for a while.
2 replies
UtkarshSri
Yesterday at 9:57 AM
Li0nking
2 hours ago
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Geometry
MathsII-enjoy   1
N 3 hours ago by MathsII-enjoy
Given triangle $ABC$ inscribed in $(O)$, $M, N$ are respectively the midpoints of the major and minor arcs $BC$. Let $I$ be the center of the inscribed circle, $R$ be $A-mix$, $D$ is the intersection point of the line through $A$ parallel to $BC$ with $(O)$. $DI$ intersects $AR$ at $K$, take $L$ on $AK$ so that $LI//BC$. $NL$ intersects $(O)$ at $G$, $AG$ intersects $LI$ at $Z$. Prove that: $ZM$ is perpendicular to $KN$.
1 reply
MathsII-enjoy
May 23, 2025
MathsII-enjoy
3 hours ago
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Inequalities
sqing   0
4 hours ago
Let $a,b,c,d>0,a^2 + d^2-ad = (b + c)^2 $ aand $ a^2 + b^2 = c^2 + d^2.$ Prove that$$ \frac{ab+cd}{ad+bc} \geq \frac{ 4}{5}$$
0 replies
sqing
4 hours ago
0 replies
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Largest Prime Factor
P162008   3
N 4 hours ago by maromex
The largest prime factor of the sum $\sum_{k=1}^{11} k^5$ is $\lambda.$ Find the sum of the digits of $\lambda.$
3 replies
P162008
Yesterday at 12:04 AM
maromex
4 hours ago
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Inequalities
sqing   27
N 4 hours ago by sqing
Let $ a,b>0 $ . Prove that
$$ \frac{a}{a^2+a +2b+1}+ \frac{b}{b^2+2a +b+1} \leq \frac{2}{5} $$$$ \frac{a}{a^2+2a +b+1}+ \frac{b}{b^2+a +2b+1} \leq \frac{2}{5} $$
27 replies
sqing
May 13, 2025
sqing
4 hours ago
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Divisors of factorials can't be always products of consecutive integers
Johann Peter Dirichlet   0
5 hours ago
Let $M$ an even number.

Show that $\frac{n!}{M^2}$ is not the product of consecutive integers for infinitely many naturals $n$.
0 replies
Johann Peter Dirichlet
5 hours ago
0 replies
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IOQM P22 2024
SomeonecoolLovesMaths   3
N Yesterday at 10:51 PM by SomeonecoolLovesMaths
In a triangle $ABC$, $\angle BAC = 90^{\circ}$. Let $D$ be the point on $BC$ such that $AB + BD = AC + CD$. Suppose $BD : DC = 2:1$. if $\frac{AC}{AB} = \frac{m + \sqrt{p}}{n}$, Where $m,n$ are relatively prime positive integers and $p$ is a prime number, determine the value of $m+n+p$.
3 replies
SomeonecoolLovesMaths
Sep 8, 2024
SomeonecoolLovesMaths
Yesterday at 10:51 PM
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AP calc?
Thayaden   30
N Yesterday at 9:53 PM by Pengu14
How are we all feeling on AP calc guys?
30 replies
Thayaden
May 20, 2025
Pengu14
Yesterday at 9:53 PM
J
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Calculate the radius of a circle using sidelengths.
richminer   0
Yesterday at 6:17 PM
Given triangle ABC with incircle (I), with D being the touchpoint of (I) and BC. Let M be the tangent point of the A-Mixtilinear circle (internally tangent). A' is the reflection of A through I. Calculate the radius of the circle (MDA') using the side lengths of the triangle ABC.
0 replies
richminer
Yesterday at 6:17 PM
0 replies
J
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Number of real roots
girishpimoli   0
Yesterday at 5:35 PM
Number of real roots of

$\displaystyle 2\sin(\theta)\cos(3\theta)\sin(5\theta)=-1$
0 replies
girishpimoli
Yesterday at 5:35 PM
0 replies
J
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Factorization Ex.28a Q30
Obvious_Wind_1690   1
N Yesterday at 4:43 PM by Lankou
Please help with factorization. Given is the question


\begin{align*} a(a+1)x^2+(a+b)xy-b(b-1)y^2\\ \end{align*}
And the given answer is


\begin{align*} [(a+1)x-(b-1)y][ax+by]\\ \end{align*}
But I am unable to reach the answer.
1 reply
Obvious_Wind_1690
Yesterday at 4:17 AM
Lankou
Yesterday at 4:43 PM
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Polynomials
P162008   4
N Yesterday at 4:19 PM by HAL9000sk
If $f(x)$ is a polynomial function such that $f(x) = x\sqrt{1 + (x + 1)\sqrt{1 + (x + 2)\sqrt{1 + (x + 3)\sqrt{1 + \cdots}}}}$ then

A) Degree of $f(x)$ must be greater than $2$

B) $f(-2) = 0$

C) $\sum_{r=1}^{5} \frac{1}{f(r)} = \frac{25}{42}$

D) $\sum_{r=1}^{n} \frac{1}{f(r)} = \frac{n(3n + 5)}{4(n+1)(n+2)}$
4 replies
P162008
Monday at 11:18 PM
HAL9000sk
Yesterday at 4:19 PM
J
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J
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Calculus BC help
needcalculusasap45   7
N Apr 20, 2025 by ehz2701
So basically, I have the AP Calculus BC exam in less than a month, and I have only covered until Unit 6 or 7 of the cirriculum. I am self studying this course (no teacher) and have not had much time to study bc of 6 other APs. I need to finish 8, 9, and 10 in less than 2 weeks. What can I do ? I would appreciate any help or resources anyone could provide. Could I just learn everything from barrons and princeton? Also, I have not taken AP Calculus AB before.

7 replies
needcalculusasap45
Apr 19, 2025
ehz2701
Apr 20, 2025
J
Calculus BC help
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Reveal topic tags
calculus
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needcalculusasap45
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needcalculusasap45
#1 h  Apr 19, 2025, 1:51 PM
Y by
So basically, I have the AP Calculus BC exam in less than a month, and I have only covered until Unit 6 or 7 of the cirriculum. I am self studying this course (no teacher) and have not had much time to study bc of 6 other APs. I need to finish 8, 9, and 10 in less than 2 weeks. What can I do ? I would appreciate any help or resources anyone could provide. Could I just learn everything from barrons and princeton? Also, I have not taken AP Calculus AB before.
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needcalculusasap45
6 posts
needcalculusasap45
#2 h  Apr 19, 2025, 9:10 PM
Y by
bump....
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SmartGroot
2792 posts
SmartGroot
#3 h  Apr 19, 2025, 9:12 PM
Y by
needcalculusasap45 wrote:
So basically, I have the AP Calculus BC exam in less than a month, and I have only covered until Unit 6 or 7 of the cirriculum. I am self studying this course (no teacher) and have not had much time to study bc of 6 other APs. I need to finish 8, 9, and 10 in less than 2 weeks. What can I do ? I would appreciate any help or resources anyone could provide. Could I just learn everything from barrons and princeton? Also, I have not taken AP Calculus AB before.

"6 other AP's" ---> you're taking 7 APs in one year?

also are you a junior or something else
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needcalculusasap45
6 posts
needcalculusasap45
#4 h  Apr 19, 2025, 9:19 PM
Y by
SmartGroot wrote:
needcalculusasap45 wrote:
So basically, I have the AP Calculus BC exam in less than a month, and I have only covered until Unit 6 or 7 of the cirriculum. I am self studying this course (no teacher) and have not had much time to study bc of 6 other APs. I need to finish 8, 9, and 10 in less than 2 weeks. What can I do ? I would appreciate any help or resources anyone could provide. Could I just learn everything from barrons and princeton? Also, I have not taken AP Calculus AB before.

"6 other AP's" ---> you're taking 7 APs in one year?

also are you a junior or something else
yes to both
Z K Y
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SmartGroot
2792 posts
SmartGroot
#5 h  Apr 19, 2025, 9:51 PM
Y by
needcalculusasap45 wrote:
SmartGroot wrote:
needcalculusasap45 wrote:
So basically, I have the AP Calculus BC exam in less than a month, and I have only covered until Unit 6 or 7 of the cirriculum. I am self studying this course (no teacher) and have not had much time to study bc of 6 other APs. I need to finish 8, 9, and 10 in less than 2 weeks. What can I do ? I would appreciate any help or resources anyone could provide. Could I just learn everything from barrons and princeton? Also, I have not taken AP Calculus AB before.

"6 other AP's" ---> you're taking 7 APs in one year?

also are you a junior or something else
yes to both

you take 7-8 APs throughout your whole high school experience... not in a single year... how do you get your extracurriculars in?
This post has been edited 1 time. Last edited by SmartGroot, Apr 19, 2025, 9:52 PM
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needcalculusasap45
6 posts
needcalculusasap45
#6 h  Apr 19, 2025, 9:57 PM
Y by
Quote:
you take 7-8 APs throughout your whole high school experience... not in a single year... how do you get your extracurriculars in?

bro thats not the point here

i just need the best strat and resources that i could learn stuff asap
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SmartGroot
2792 posts
SmartGroot
#7 h  Apr 19, 2025, 11:21 PM
Y by
needcalculusasap45 wrote:
Quote:
you take 7-8 APs throughout your whole high school experience... not in a single year... how do you get your extracurriculars in?

bro thats not the point here

i just need the best strat and resources that i could learn stuff asap

Alright, understood, it's just that in order to get genuine advice, the question you ask needs to be credible. I'm not sure what your situation is but it's not ideal and risky to cram 7 AP classes/tests in one year. In these situations, you can cram but it's not going to be expected to get the result you desire. I have never heard of anyone who is taking seven AP exams in one year of high school.
This post has been edited 2 times. Last edited by SmartGroot, Apr 19, 2025, 11:23 PM
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ehz2701
766 posts
ehz2701
#8 h  Apr 20, 2025, 1:42 AM
Y by
needcalculusasap45 wrote:
So basically, I have the AP Calculus BC exam in less than a month, and I have only covered until Unit 6 or 7 of the cirriculum. I am self studying this course (no teacher) and have not had much time to study bc of 6 other APs. I need to finish 8, 9, and 10 in less than 2 weeks. What can I do ? I would appreciate any help or resources anyone could provide. Could I just learn everything from barrons and princeton? Also, I have not taken AP Calculus AB before.

I would say the most confusing portion is series. Thee are excellent stuff online, but you should focus on learning some of the convergence tests out there. I guarantee you on the FRQ there will be a series problem. Also, guaranteed rate problem and explaining the meaning kind of question. There are practice tests available online, so learn the form they want the answer in. Like, if $P(t)$ is the amount of bananas Kyle currently has in his garage (where $t$ is in hours), then what does $P’’(21)=-1.235$ mean in this context with units or something.

(At time $t=21$ hours, the rate at which the amount of Bananas Kyle has is decreasing by $1.235$ bananas per hour per hour)
This post has been edited 2 times. Last edited by ehz2701, Apr 20, 2025, 1:46 AM
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