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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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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 2025
0 replies
Another config geo with concurrent lines
a_507_bc   16
N a minute ago by Tkn
Source: BMO SL 2023 G5
Let $ABC$ be a triangle with circumcenter $O$. Point $X$ is the intersection of the parallel line from $O$ to $AB$ with the perpendicular line to $AC$ from $C$. Let $Y$ be the point where the external bisector of $\angle BXC$ intersects with $AC$. Let $K$ be the projection of $X$ onto $BY$. Prove that the lines $AK, XO, BC$ have a common point.
16 replies
a_507_bc
May 3, 2024
Tkn
a minute ago
Expressing polynomial as product of two polynomials
Sadigly   1
N 17 minutes ago by Sadigly
Source: Azerbaijan Senior NMO 2021
Define $P(x)=((x-a_1)(x-a_2)...(x-a_n))^2 +1$, where $a_1,a_2...,a_n\in\mathbb{Z}$ and $n\in\mathbb{N^+}$. Prove that $P(x)$ could be expressed as product of two non-constant polynomials with integer coefficients.
1 reply
Sadigly
Yesterday at 9:10 PM
Sadigly
17 minutes ago
Inequality
Sadigly   4
N 19 minutes ago by Adywastaken
Source: Azerbaijan Senior NMO 2019
Prove that for any $a;b;c\in\mathbb{R^+}$, we have $$(a+b)^2+(a+b+4c)^2\geq \frac{100abc}{a+b+c}$$When does the equality hold?
4 replies
Sadigly
Yesterday at 8:47 PM
Adywastaken
19 minutes ago
Geometry from Iran TST 2017
bgn   17
N 26 minutes ago by GioOrnikapa
Source: 2017 Iran TST third exam day2 p6
In triangle $ABC$ let $O$ and $H$ be the circumcenter and the orthocenter. The point $P$ is the reflection of $A$ with respect to $OH$. Assume that $P$ is not on the same side of $BC$ as $A$. Points $E,F$ lie on $AB,AC$ respectively such that $BE=PC \ ,  CF=PB$. Let $K$ be the intersection point of $AP,OH$. Prove that $\angle EKF = 90 ^{\circ}$

Proposed by Iman Maghsoudi
17 replies
bgn
Apr 27, 2017
GioOrnikapa
26 minutes ago
Function equation
hoangdinhnhatlqdqt   2
N 3 hours ago by jasperE3
Find all functions $f:\mathbb{R}\geq 0\rightarrow \mathbb{R}\geq 0$ satisfying:
$f(f(x)-x)=2x\forall x\geq 0$
2 replies
hoangdinhnhatlqdqt
Dec 17, 2017
jasperE3
3 hours ago
Compilation of functions problems
Saucepan_man02   4
N 3 hours ago by lightsbug
Could anyone post some handout/compilation of problems related to functions (difficulty similar to AIME/ARML/HMMT etc)?

Thanks..
4 replies
Saucepan_man02
May 7, 2025
lightsbug
3 hours ago
How many nonnegative integers
Darealzolt   1
N 5 hours ago by elizhang101412
How many nonnegative integers can be written in the form
\[
a_7 \cdot 3^7 + a_6 \cdot 3^6 + a_5 \cdot 3^5 + a_4 \cdot 3^4 + a_3 \cdot 3^3 + a_2 \cdot 3^2 + a_1 \cdot 3^1 + a_0 \cdot 3^0
\]where \( a_i \in \{-1, 0, 1\} \) for \( 0 \le i \le 7 \)?
1 reply
Darealzolt
5 hours ago
elizhang101412
5 hours ago
How much sides does M and N have
Darealzolt   0
5 hours ago
Two regular polygons have \( m \) sides and \( n \) sides, respectively. The total number of sides is 33, and the total number of diagonals is 243. What are the values of \( m \) and \( n \)?
0 replies
Darealzolt
5 hours ago
0 replies
PIE practice
Serengeti22   0
Today at 3:20 AM
Does anybody know any good problems to practice PIE that range from mid-AMC10/12 level - early AIME level for pracitce.
0 replies
Serengeti22
Today at 3:20 AM
0 replies
Square number
linkxink0603   5
N Today at 1:44 AM by linkxink0603
Find m is positive interger such that m^4+3^m is square number
5 replies
linkxink0603
May 9, 2025
linkxink0603
Today at 1:44 AM
Functions
Entrepreneur   5
N Today at 12:33 AM by RandomMathGuy500
Let $f(x)$ be a polynomial with integer coefficients such that $f(0)=2020$ and $f(a)=2021$ for some integer $a$. Prove that there exists no integer $b$ such that $f(b) = 2022$.
5 replies
Entrepreneur
Aug 18, 2023
RandomMathGuy500
Today at 12:33 AM
Logarithmic function
jonny   2
N Yesterday at 11:09 PM by KSH31415
If $\log_{6}(15) = a$ and $\log_{12}(18)=b,$ Then $\log_{25}(24)$ in terms of $a$ and $b$
2 replies
jonny
Jul 15, 2016
KSH31415
Yesterday at 11:09 PM
book/resource recommendations
walterboro   0
Yesterday at 8:57 PM
hi guys, does anyone have book recs (or other resources) for like aime+ level alg, nt, geo, comb? i want to learn a lot of theory in depth
also does anyone know how otis or woot is like from experience?
0 replies
walterboro
Yesterday at 8:57 PM
0 replies
Engineers Induction FTW
RP3.1415   11
N Yesterday at 6:53 PM by Markas
Define a sequence as $a_1=x$ for some real number $x$ and \[ a_n=na_{n-1}+(n-1)(n!(n-1)!-1) \]for integers $n \geq 2$. Given that $a_{2021} =(2021!+1)^2 +2020!$, and given that $x=\dfrac{p}{q}$, where $p$ and $q$ are positive integers whose greatest common divisor is $1$, compute $p+q.$
11 replies
RP3.1415
Apr 26, 2021
Markas
Yesterday at 6:53 PM
Area due to polygon rolling on polygon
Vrangr   5
N Aug 5, 2018 by Tsukuyomi
Source: Sharygin 2018 grade 9, P8
Consider a fixed regular $n$-gon of unit side. When a second regular $n$-gon of unit size rolls around the first one, one of its vertices successively pinpoints the vertices of a closed broken line $\kappa$ as in the figure.

IMAGE

Let $A$ be the area of a regular $n$-gon of unit side, and let $B$ be the area of a regular $n$-gon of unit circumradius. Prove that the area enclosed by $\kappa$ equals $6A-2B$.
5 replies
Vrangr
Aug 2, 2018
Tsukuyomi
Aug 5, 2018
Area due to polygon rolling on polygon
G H J
G H BBookmark kLocked kLocked NReply
Source: Sharygin 2018 grade 9, P8
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Vrangr
1600 posts
#1 • 3 Y
Y by samrocksnature, Adventure10, Mango247
Consider a fixed regular $n$-gon of unit side. When a second regular $n$-gon of unit size rolls around the first one, one of its vertices successively pinpoints the vertices of a closed broken line $\kappa$ as in the figure.

[asy]
int n=9;
draw(polygon(n));
for (int i = 0; i<n;++i) {
 draw(reflect(dir(360*i/n + 90), dir(360*(i+1)/n + 90))*polygon(n), dashed+linewidth(0.4));
 draw(reflect(dir(360*i/n + 90),dir(360*(i+1)/n + 90))*(0,1)--reflect(dir(360*(i-1)/n + 90),dir(360*i/n + 90))*(0,1), linewidth(1.2));
}
[/asy]

Let $A$ be the area of a regular $n$-gon of unit side, and let $B$ be the area of a regular $n$-gon of unit circumradius. Prove that the area enclosed by $\kappa$ equals $6A-2B$.
This post has been edited 6 times. Last edited by Vrangr, Aug 3, 2018, 8:11 PM
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math_pi_rate
1218 posts
#2 • 2 Y
Y by Adventure10, Mango247
5 students solved it during the contest
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math_pi_rate
1218 posts
#3 • 2 Y
Y by Adventure10, Mango247
The oficial solution used the fact that the rolling polygon is the reflection of the fixed polygon in each side. Thus the point marked must be the reflection of one vertex of the fixed polygon in each side of the fixed polygon, and we had to calculate the area of the broken line formed by these reflections. Now the rest is not so difficult. All one has to do is divide this region inside the broken line into a number of triangles such that these triangles can be combined to give the required answer.
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Mindstormer
102 posts
#4 • 2 Y
Y by Adventure10, Mango247
This one looks like a nice and hard problem problem indeed. I wonder if the limiting case when $n \to \infty$ can be used to find the area of a cardioid?
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Vrangr
1600 posts
#5 • 2 Y
Y by Adventure10, Mango247
Mindstormer wrote:
This one looks like a nice and hard problem problem indeed.
The problem isn't that hard, it's nice though. I'll post a solution soon.
Mindstormer wrote:
I wonder if the limiting case when $n \to \infty$ can be used to find the area of a cardioid?
That's the first thing I thought of when I saw the problem :rotfl:
And yes, the limiting case does give the area of a cardoid. Just have to changing the regular $n$-gon of unit side to unit radius and need to scale the final area appropriately.
This post has been edited 2 times. Last edited by Vrangr, Aug 4, 2018, 3:42 PM
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Tsukuyomi
31 posts
#6 • 3 Y
Y by guptaamitu1, Adventure10, Mango247
Label the vertices of the regular $n$-gon as $A_0A_1\cdots A_{n-1}$. WLOG let $A_0$ be the initial vertex of the rolling polygon. When the rolling polygon has reached the side $\overline{A_jA_{j+1}}$, the moving vertex is placed on the $j$th vertex of this rolling $n$-gon from $A_j$ ( anti-clockwise). Call this vertex $B_j$. Since, at this moment, the two polygons are reflections about $\overline{A_jA_{j+1}}$ the point $B_j$ is the reflection of $A_0$ over $\overline{A_jA_{j+1}}$. Hence we can write

\begin{align*}
[\kappa] &=\sum_{j=0}^{n-1}\left( [A_jA_0A_{j+1}]+[A_jB_jA_{j+1}]\right)+\sum_{j=0}^{n-1}[B_jA_{j+1}B_{j+1}]
\end{align*}
Notice that $[A_jA_0A_{j+1}]=[A_jB_jA_{j+1}]$ and $B_jA_{j+1}=B_{j+1}A_{j+1}=A_0A_{j+1}$ as they are reflections over $A_jA_{j+1}$ and $\angle{B_jA_{j+1}B_{j+1}}=\dfrac{4\pi}{n}.$ Thus we have

\begin{align*}
[\kappa] &=2A+\dfrac{1}{2}\sum_{j=1}^{n-1}\overline{A_0A_j}^2\sin\dfrac{4\pi}{n}.
\end{align*}By similarity we have $B=4A\sin^2\dfrac{\pi}{n}$ and using $A=\dfrac{n}{4}\cot\dfrac{\pi}{n}$, it suffices to prove that

\begin{align*}
\dfrac{1}{2}\sum_{j=1}^{n-1}\overline{A_0A_j}^2\sin\dfrac{4\pi}{n}=4A-2B &=\left(4-8\sin^2{\dfrac{\pi}{n}}\right)A \\
\sum_{j=1}^{n-1}\overline{A_0A_j}^2\cos\dfrac{2\pi}{n}\sin\dfrac{2\pi}{n} &=4\cos{\dfrac{2\pi}{n}}A \\
\sum_{j=1}^{n-1}\overline{A_0A_j}^2\sin{\dfrac{2\pi}{n}} &=4A.
\end{align*}
Assign the complex number $A_k=r\omega^k$ where $\omega$ is a $n$-th root of unity and $r=\dfrac{1}{2\sin{\dfrac{\pi}{n}}}$. Then we have

\begin{align*}
\sum_{j=1}^{n-1}\overline{A_0A_j}^2 &=\sum_{j=1}^{n-1}|r-r\omega^j|^2 \\
&=r^2\sum_{j=1}^{n-1}\left(1-\omega^j\right)\left(1-\overline{\omega}^j\right) \\
&=r^2\sum_{j=1}^{n-1}\left(1-\omega^{j}-\overline{\omega}^j+1\right)\\
&=2nr^2,
\end{align*}since $\sum \omega^j=\sum \overline{\omega}^j=0$. Comparing the terms, it is easy to see that $nr^2\sin{\dfrac{2\pi}{n}}=2A$, giving us our desired result.
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