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IMO Shortlist 2012, Number Theory 5
lyukhson   32
N 19 minutes ago by awesomeming327.
Source: IMO Shortlist 2012, Number Theory 5
For a nonnegative integer $n$ define $\operatorname{rad}(n)=1$ if $n=0$ or $n=1$, and $\operatorname{rad}(n)=p_1p_2\cdots p_k$ where $p_1<p_2<\cdots <p_k$ are all prime factors of $n$. Find all polynomials $f(x)$ with nonnegative integer coefficients such that $\operatorname{rad}(f(n))$ divides $\operatorname{rad}(f(n^{\operatorname{rad}(n)}))$ for every nonnegative integer $n$.
32 replies
1 viewing
lyukhson
Jul 29, 2013
awesomeming327.
19 minutes ago
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A weird problem
jayme   2
N 2 hours ago by lolsamo
Dear Mathlinkers,

1. ABC a triangle
2. 0 the circumcircle
3. I the incenter
4. 1 a circle passing througn B and C
5. X, Y the second points of intersection of 1 wrt BI, CI
6. 2 the circumcircle of the triangle XYI
7. M, N the symetrics of B, C wrt XY.

Question : if 2 is tangent to 0 then, 2 is tangent to MN.

Sincerely
Jean-Louis
2 replies
jayme
Today at 6:52 AM
lolsamo
2 hours ago
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Channel name changed
Plane_geometry_youtuber   10
N 2 hours ago by Yiyj
Hi,

Due to the search handle issue in youtube. My channel is renamed to Olympiad Geometry Club. And the new link is as following:

https://www.youtube.com/@OlympiadGeometryClub

Recently I introduced the concept of harmonic bundle. I will move on to the conjugate median soon. In the future, I will discuss more than a thousand theorems on plane geometry and hopefully it can help to the students preparing for the Olympiad competition.

Please share this to the people may need it.

Thank you!
10 replies
Plane_geometry_youtuber
Yesterday at 9:31 PM
Yiyj
2 hours ago
J
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Euler line of incircle touching points /Reposted/
Eagle116   6
N 5 hours ago by pigeon123
Let $ABC$ be a triangle with incentre $I$ and circumcentre $O$. Let $D,E,F$ be the touchpoints of the incircle with $BC$, $CA$, $AB$ respectively. Prove that $OI$ is the Euler line of $\vartriangle DEF$.
6 replies
Eagle116
Apr 19, 2025
pigeon123
5 hours ago
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Parallel lines on a rhombus
buratinogigle   1
N 6 hours ago by Giabach298
Source: Own, Entrance Exam for Grade 10 Admission, HSGS 2025
Given the rhombus $ABCD$ with its incircle $\omega$. Let $E$ and $F$ be the points of tangency of $\omega$ with $AB$ and $AC$ respectively. On the edges $CB$ and $CD$, take points $G$ and $H$ such that $GH$ is tangent to $\omega$ at $P$. Suppose $Q$ is the intersection point of the lines $EG$ and $FH$. Prove that two lines $AP$ and $CQ$ are parallel or coincide.
1 reply
buratinogigle
Today at 3:17 PM
Giabach298
6 hours ago
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Orthocenter lies on circumcircle
whatshisbucket   90
N 6 hours ago by bjump
Source: 2017 ELMO #2
Let $ABC$ be a triangle with orthocenter $H,$ and let $M$ be the midpoint of $\overline{BC}.$ Suppose that $P$ and $Q$ are distinct points on the circle with diameter $\overline{AH},$ different from $A,$ such that $M$ lies on line $PQ.$ Prove that the orthocenter of $\triangle APQ$ lies on the circumcircle of $\triangle ABC.$

Proposed by Michael Ren
90 replies
whatshisbucket
Jun 26, 2017
bjump
6 hours ago
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Hagge-like circles, Jerabek hyperbola, Lemoine cubic
kosmonauten3114   0
Today at 4:05 PM
Source: My own
Let $\triangle{ABC}$ be a scalene oblique triangle with circumcenter $O$ and orthocenter $H$, and $P$ ($\neq \text{X(3), X(4)}$, $\notin \odot(ABC)$) a point in the plane.
Let $\triangle{A_1B_1C_1}$, $\triangle{A_2B_2C_2}$ be the circumcevian triangles of $O$, $P$, respectively.
Let $\triangle{P_AP_BP_C}$ be the pedal triangle of $P$ with respect to $\triangle{ABC}$.
Let $A_1'$ be the reflection in $P_A$ of $A_1$. Define $B_1'$, $C_1'$ cyclically.
Let $A_2'$ be the reflection in $P_A$ of $A_2$. Define $B_2'$, $C_2'$ cyclically.
Let $O_1$, $O_2$ be the circumcenters of $\triangle{A_1'B_1'C_1'}$, $\triangle{A_2'B_2'C_2'}$, respectively.

Prove that:
1) $P$, $O_1$, $O_2$ are collinear if and only if $P$ lies on the Jerabek hyperbola of $\triangle{ABC}$.
2) $H$, $O_1$, $O_2$ are collinear if and only if $P$ lies on the Lemoine cubic (= $\text{K009}$) of $\triangle{ABC}$.
0 replies
kosmonauten3114
Today at 4:05 PM
0 replies
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Incenter perpendiculars and angle congruences
math154   84
N Today at 4:00 PM by zuat.e
Source: ELMO Shortlist 2012, G3
$ABC$ is a triangle with incenter $I$. The foot of the perpendicular from $I$ to $BC$ is $D$, and the foot of the perpendicular from $I$ to $AD$ is $P$. Prove that $\angle BPD = \angle DPC$.

Alex Zhu.
84 replies
math154
Jul 2, 2012
zuat.e
Today at 4:00 PM
J
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Tangency of circles with "135 degree" angles
Shayan-TayefehIR   4
N Today at 3:56 PM by Mysteriouxxx
Source: Iran Team selection test 2024 - P12
For a triangle $\triangle ABC$ with an obtuse angle $\angle A$ , let $E , F$ be feet of altitudes from $B , C$ on sides $AC , AB$ respectively. The tangents from $B , C$ to circumcircle of triangle $\triangle ABC$ intersect line $EF$ at points $K , L$ respectively and we know that $\angle CLB=135$. Point $R$ lies on segment $BK$ in such a way that $KR=KL$ and let $S$ be a point on line $BK$ such that $K$ is between $B , S$ and $\angle BLS=135$. Prove that the circle with diameter $RS$ is tangent to circumcircle of triangle $\triangle ABC$.

Proposed by Mehran Talaei
4 replies
Shayan-TayefehIR
May 19, 2024
Mysteriouxxx
Today at 3:56 PM
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Line bisects a segment
buratinogigle   1
N Today at 3:41 PM by cj13609517288
Source: Own, Entrance Exam for Grade 10 Admission, HSGS 2025
Let $ABC$ be a triangle with $AB = AC$. A circle $(O)$ is tangent to sides $AC$ and $AB$, and $O$ is the midpoint of $BC$. Points $E$ and $F$ lie on sides $AC$ and $AB$, respectively, such that segment $EF$ is tangent to circle $(O)$ at point $P$. Let $H$ and $K$ be the orthocenters of triangles $OBF$ and $OCE$, respectively. Prove that line $OP$ bisects segment $HK$.
1 reply
buratinogigle
Today at 3:08 PM
cj13609517288
Today at 3:41 PM
J
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Three collinear points
buratinogigle   1
N Today at 2:26 PM by Giabach298
Source: Own, Entrance Exam for Grade 10 Admission, HSGS 2025
Let $ABC$ be a triangle with points $E$ and $F$ lying on rays $AC$ and $AB$, respectively, such that $AE = AF$. On the line $EF$, choose points $M$ and $N$ such that $CM \perp CA$ and $BN \perp BA$. Let $K$ and $L$ be the feet of the perpendiculars from $M$ and $N$ to line $BC$, respectively. Let $J$ be the intersection point of lines $LF$ and $KE$. Prove that the reflection of $J$ over line $EF$ lies on the line connecting $A$ and the circumcenter of triangle $ABC$.
1 reply
buratinogigle
Today at 2:21 PM
Giabach298
Today at 2:26 PM
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Find the value
sqing   9
N Apr 20, 2025 by sqing
Source: 2025 Tsinghua University
Let $A= \lim_{n\to\infty}\tan^n\left(\frac{\pi}{4}+\frac{1}{n}\right) . $ Find the value of $[100A] .$
9 replies
sqing
Apr 20, 2025
sqing
Apr 20, 2025
J
Find the value
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Source: 2025 Tsinghua University
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sqing
42565 posts
sqing
#1 h  Apr 20, 2025, 9:57 AM
Y by
Let $A= \lim_{n\to\infty}\tan^n\left(\frac{\pi}{4}+\frac{1}{n}\right) . $ Find the value of $[100A] .$
This post has been edited 2 times. Last edited by sqing, Apr 20, 2025, 1:20 PM
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pco
23515 posts
pco
#3 h  Apr 20, 2025, 12:41 PM
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sqing wrote:
Let $A=\lim_{n\to\infty}(\frac{\pi}{4}+\frac{1}{n})^n . $ Find the value of $[100A] .$
$A=0$ and so $\boxed{\lfloor 100A\rfloor=0}$
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Quantum-Phantom
276 posts
Quantum-Phantom
#4 h  Apr 20, 2025, 1:11 PM
Y by
It should be
\[A=\lim_{n\to\infty}\tan^n\left(\frac{\pi}{4}+\frac{1}{n}\right).\]
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sqing
42565 posts
sqing
#5 h  Apr 20, 2025, 1:19 PM
Y by
Thank you very much.
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sqing
42565 posts
sqing
#6 h  Apr 20, 2025, 1:20 PM
Y by
Sorry.Thank pco.
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SunnyEvan
145 posts
SunnyEvan
#7 h  Apr 20, 2025, 1:24 PM
Y by
Quantum-Phantom wrote:
It should be
\[A=\lim_{n\to\infty}\tan^n\left(\frac{\pi}{4}+\frac{1}{n}\right).\]

$$ A=\lim_{n\to\infty}\tan^n\left(\frac{\pi}{4}+\frac{1}{n}\right) =e^{\lim_{n\to\infty}n ln\tan(\frac{\pi}{4}+\frac{1}{n})} $$let $t=\frac{1}{n}$
$$ e^{\lim_{n\to\infty}n ln\tan(\frac{\pi}{4}+\frac{1}{n})} = e^{\lim_{t\to 0} \frac{ln\tan(\frac{\pi}{4}+t)}{t}} = e^{\lim_{t\to 0} \frac{1}{tan(\frac{\pi}{4}+t)}(sec(\frac{\pi}{4}+t))^2}= e^{\lim_{t\to 0} \frac{1}{sin(\frac{\pi}{4}+t)cos(\frac{\pi}{4}+t)}}=e^2$$$[100A] =738$
This post has been edited 3 times. Last edited by SunnyEvan, Apr 20, 2025, 1:53 PM
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sqing
42565 posts
sqing
#9 h  Apr 20, 2025, 1:48 PM
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Prove that $$\lim_{n\to\infty}\tan^n\left(\frac{\pi}{4}+\frac{1}{n}\right) = e^2$$
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sqing
42565 posts
sqing
#10 h  Apr 20, 2025, 1:57 PM
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sqing wrote:
Prove that $$\lim_{n\to\infty}\tan^n\left(\frac{\pi}{4}+\frac{1}{n}\right) = e^2$$
https://artofproblemsolving.com/community/c7h1643781p10364775
https://math.stackexchange.com/questions/1750591/limit-of-tan-function?noredirect=1
*
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SunnyEvan
145 posts
SunnyEvan
#11 h  Apr 20, 2025, 2:00 PM
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sqing wrote:
sqing wrote:
Prove that $$\lim_{n\to\infty}\tan^n\left(\frac{\pi}{4}+\frac{1}{n}\right) = e^2$$
https://artofproblemsolving.com/community/c7h1643781p10364775
https://math.stackexchange.com/questions/1750591/limit-of-tan-function?noredirect=1
*

Thank you very much !@Sqing . :-D Hospital always help with lots of problems .
This post has been edited 1 time. Last edited by SunnyEvan, Apr 20, 2025, 2:02 PM
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sqing
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sqing
#12 h  Apr 20, 2025, 2:56 PM
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Thanks.
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