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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.

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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.

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[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.
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[*]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!
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0 replies
jlacosta
Mar 2, 2025
0 replies
J
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Nice function question
srnjbr   2
N 5 minutes ago by pco
Find all functions f:R+--R+ such that for all a,b>0, f(af(b)+a)(f(bf(a))+a)=1
2 replies
srnjbr
Today at 4:28 AM
pco
5 minutes ago
J
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Inequality with real numbers
JK1603JK   2
N 15 minutes ago by SunnyEvan
Source: unknown
Let a,b,c are real numbers. Prove that (a^3+b^3+c^3+3abc)^4+(a+b+c)^3(a+b-c)^3(-a+b+c)^3(a-b+c)^3>=0
2 replies
1 viewing
JK1603JK
4 hours ago
SunnyEvan
15 minutes ago
J
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Mathhhhh
mathbetter   10
N 19 minutes ago by togrulhamidli2011
Three turtles are crawling along a straight road heading in the same
direction. "Two other turtles are behind me," says the first turtle. "One turtle is
behind me and one other is ahead," says the second. "Two turtles are ahead of me
and one other is behind," says the third turtle. How can this be possible?
10 replies
mathbetter
Thursday at 11:21 AM
togrulhamidli2011
19 minutes ago
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SONG circle?
YaoAOPS   1
N an hour ago by bin_sherlo
Source: own?
Let triangle $ABC$ have incenter $I$ and intouch triangle $DEF$. Let the circumcircle of $ABC$ intersect $(AEF)$ at $S$ and have center $O$. Let $N$ be the midpoint of arc $BAC$ on the circumcircle. Suppose quadrilateral $SONG$ is cyclic such that $X = SN \cap OG$ lies on $BC$. Show that $\angle XGD = 90^\circ$.
1 reply
YaoAOPS
4 hours ago
bin_sherlo
an hour ago
J
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Equation with complex numbers on the unit circle
Tintarn   9
N Today at 4:30 AM by Fibonacci_math
Source: IMC 2024, Problem 1
Determine all pairs $(a,b) \in \mathbb{C} \times \mathbb{C}$ of complex numbers satisfying $|a|=|b|=1$ and $a+b+a\overline{b} \in \mathbb{R}$.
9 replies
Tintarn
Aug 7, 2024
Fibonacci_math
Today at 4:30 AM
J
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numerical analysis
ay19bme   1
N Today at 3:05 AM by YuLuo
...............
1 reply
ay19bme
Yesterday at 4:48 PM
YuLuo
Today at 3:05 AM
J
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distribution function
We2592   1
N Yesterday at 9:55 PM by alexheinis
Q)The distribution function $F(x)$ of a variate $X$ is defined as follows:
\[ F(x) = \begin{cases} A, & -\infty < x < -1, \\ B, & -1 \leq x < 0, \\ C, & 0 \leq x < 2, \\ D, & 2 \leq x < \infty. \end{cases} \]
where $A,B,C,D$ are constants. Determine the values of $A,B,C,D$ it being given that $P(X=0)=\frac{1}{6}$ and $P(X>1)=\frac{2}{3}$

how to solve
1 reply
We2592
Yesterday at 1:23 PM
alexheinis
Yesterday at 9:55 PM
J
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Complex roots
Sarbajit10598   7
N Yesterday at 5:57 PM by quasar_lord
Source: C.M.I entrance exam 2019
2.(a)Count the number of all the complex roots $\omega$ of the equation $z^{2019}-1=0$ which follows $$|\omega+1|\geq \sqrt{2+\sqrt{2}}$$(b) Solve for real $x$
$$\frac{8^x+27^x}{12^x+18^x}=\frac{7}{6}$$
7 replies
Sarbajit10598
May 15, 2019
quasar_lord
Yesterday at 5:57 PM
J
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CMI Entrance 19#3
bubu_2001   12
N Yesterday at 5:17 PM by quasar_lord
Evaluate $\int_{ 0 }^{ \infty } ( 1 + x^2 )^{-( m + 1 )} \mathrm{d}x$ where $m \in \mathbb{N} $
12 replies
bubu_2001
Oct 31, 2019
quasar_lord
Yesterday at 5:17 PM
J
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analysis
ay19bme   2
N Yesterday at 3:27 PM by ay19bme
..........
2 replies
ay19bme
Thursday at 8:06 PM
ay19bme
Yesterday at 3:27 PM
J
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Integrate the reciprocal of a geometric series
IHaveNoIdea010   0
Yesterday at 2:31 PM
Determine the exact value of $$\int_{0}^{\infty} \frac{1}{\sum_{n=0}^{10} x^n} \,dx$$
0 replies
IHaveNoIdea010
Yesterday at 2:31 PM
0 replies
J
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Do these have a closed form?
Entrepreneur   0
Yesterday at 2:17 PM
Source: Own
$$\int_0^\infty\frac{t^{n-1}}{(t+\alpha)^2+m^2}dt.$$$$\int_0^\infty\frac{e^{nt}}{(t+\alpha)^2+m^2}dt.$$$$\int_0^\infty\frac{dx}{(1+x^a)^m(1+x^b)^n}.$$
0 replies
Entrepreneur
Yesterday at 2:17 PM
0 replies
J
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more rafinament limits integrals
teomihai   6
N Yesterday at 2:11 PM by teomihai
Let$ f:[0,1]->R$ continously function with $f'$ bounded
Find $$\lim_{n \rightarrow \infty}n(n\int_{0}^{1}x^{n}f(x)dx-f(1))$$
6 replies
teomihai
Yesterday at 4:55 AM
teomihai
Yesterday at 2:11 PM
J
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Derivative of Normalization Map has null space of dimension 1
myth17   4
N Yesterday at 1:43 PM by myth17
Let $f(\vec{x}) = \frac{\vec{x}}{||\vec{x}||}$ be defined on $\mathbb{R}^n \setminus \{\vec{0}\}$. Show that the dimension of the kernel of $Df_{\vec{x}}$ for any $\vec{x} \in \mathbb{R}^n \setminus \{\vec{0}\}$ is $1$.
4 replies
myth17
Thursday at 5:16 PM
myth17
Yesterday at 1:43 PM
J
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J
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number theory
MuradSafarli   9
N Mar 18, 2025 by GreekIdiot
Find all natural numbers \( k \) such that

\[ 4k^3 + 4k + 1 \]
is a perfect square.
9 replies
MuradSafarli
Mar 14, 2025
GreekIdiot
Mar 18, 2025
J
number theory
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Reveal topic tags
number theory
L
G H BBookmark kLocked kLocked NReply
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MuradSafarli
60 posts
MuradSafarli
#1 h  Mar 14, 2025, 6:05 AM
Y by
Find all natural numbers \( k \) such that

\[ 4k^3 + 4k + 1 \]
is a perfect square.
Z K Y
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Tuvshuu
8 posts
Tuvshuu
#2 h  Mar 14, 2025, 11:43 AM
Y by
I think only solutions are 1, 3
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CHESSR1DER
34 posts
CHESSR1DER
#3 h  Mar 14, 2025, 2:06 PM • 1 Y
Y by MuradSafarli
Let $n^2 = 4k^3+4k+1$. Then $(n-2k-1)(n+2k+1) = 4k^2(k-1)$. LHS $\equiv n^2-1 (mod k)$ and LHS $\equiv n^2-1 (mod( k-1))$.
So $k(k-1) | n^2-1$.
$(k-1)|4(k^2+1)$ . But since $(k^2+1;k-1) \leq 2$ and $(4;k-1) \leq 4$. So $ k-1 \leq 8$
Checking values from 1 to 9 gives us 1,3.
Edit: Solution is wrong.
This post has been edited 1 time. Last edited by CHESSR1DER, Mar 14, 2025, 3:58 PM
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internationalnick123456
107 posts
internationalnick123456
#4 h  Mar 14, 2025, 2:31 PM
Y by
@above taking modulo $k-1$ only gives us $LHS\equiv n^2 - 9\pmod{k-1}$
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Nuran2010
49 posts
Nuran2010
#5 h  Mar 14, 2025, 4:56 PM
Y by
MuradSafarli wrote:
Find all natural numbers \( k \) such that

\[ 4k^3 + 4k + 1 \]
is a perfect square.[/quote
Asan-asan suallari paylasib ozunu biabir eleme :(
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MuradSafarli
60 posts
MuradSafarli
#6 h  Mar 14, 2025, 5:08 PM • 1 Y
Y by Sadigly
Go and respect yourself! NURAN
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fathermather_AZE
2 posts
fathermather_AZE
#7 h  Mar 14, 2025, 7:19 PM
Y by
MuradSafarli wrote:
Go and respect yourself! NURAN

agree
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iniffur
531 posts
iniffur
#8 h  Mar 14, 2025, 11:48 PM
Y by
Partial

Click to reveal hidden text
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Sadigly
119 posts
Sadigly
#9 h  Mar 18, 2025, 6:59 AM • 1 Y
Y by MuradSafarli
MuradSafarli wrote:
Go and respect yourself! NURAN

Cox.qozal.buyurdunuz.qozal.insan
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GreekIdiot
112 posts
GreekIdiot
#10 h  Mar 18, 2025, 10:44 PM
Y by
Try bounding
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