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Incircle of a triangle is tangent to (ABC)
amar_04   11
N 33 minutes ago by Nari_Tom
Source: XVII Sharygin Correspondence Round P18
Let $ABC$ be a scalene triangle, $AM$ be the median through $A$, and $\omega$ be the incircle. Let $\omega$ touch $BC$ at point $T$ and segment $AT$ meet $\omega$ for the second time at point $S$. Let $\delta$ be the triangle formed by lines $AM$ and $BC$ and the tangent to $\omega$ at $S$. Prove that the incircle of triangle $\delta$ is tangent to the circumcircle of triangle $ABC$.
11 replies
amar_04
Mar 2, 2021
Nari_Tom
33 minutes ago
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Two very hard parallel
jayme   3
N 35 minutes ago by jayme
Source: own inspired by EGMO
Dear Mathlinkers,

1. ABC a triangle
2. D, E two point on the segment BC so that BD = DE= EC
3. M, N the midpoint of ED, AE
4. H the orthocenter of the acutangle triangle ADE
5. 1, 2 the circumcircle of the triangle DHM, EHN
6. P, Q the second point of intersection of 1 and BM, 2 and CN
7. U, V the second points of intersection of 2 and MN, PQ.

Prove : UV is parallel to PM.

Sincerely
Jean-Louis
3 replies
jayme
Yesterday at 12:46 PM
jayme
35 minutes ago
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Inequality with n-gon sides
mihaig   3
N 36 minutes ago by mihaig
Source: VL
If $a_1,a_2,\ldots, a_n~(n\geq3)$ are are the lengths of the sides of a $n-$gon such that
$$\sum_{i=1}^{n}{a_i}=1,$$then
$$(n-2)\left[\sum_{i=1}^{n}{\frac{a_i^2}{(1-a_i)^2}}-\frac n{(n-1)^2}\right]\geq(2n-1)\left(\sum_{i=1}^{n}{\frac{a_i}{1-a_i}}-\frac n{n-1}\right)^2.$$
When do we have equality?

(V. Cîrtoaje and L. Giugiuc, 2021)
3 replies
mihaig
Feb 25, 2022
mihaig
36 minutes ago
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Advanced topics in Inequalities
va2010   23
N 37 minutes ago by Novmath
So a while ago, I compiled some tricks on inequalities. You are welcome to post solutions below!
23 replies
1 viewing
va2010
Mar 7, 2015
Novmath
37 minutes ago
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JBMO TST Bosnia and Herzegovina 2022 P3
Motion   7
N 43 minutes ago by cafer2861
Source: JBMO TST Bosnia and Herzegovina 2022
Let $ABC$ be an acute triangle. Tangents on the circumscribed circle of triangle $ABC$ at points $B$ and $C$ intersect at point $T$. Let $D$ and $E$ be a foot of the altitudes from $T$ onto $AB$ and $AC$ and let $M$ be the midpoint of $BC$. Prove:
A) Prove that $M$ is the orthocenter of the triangle $ADE$.
B) Prove that $TM$ cuts $DE$ in half.
7 replies
Motion
May 21, 2022
cafer2861
43 minutes ago
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hard problem
Cobedangiu   5
N an hour ago by arqady
Let $a,b,c>0$ and $a+b+c=3$. Prove that:
$\dfrac{4}{a+b}+\dfrac{4}{b+c}+\dfrac{4}{c+a} \le \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+3$
5 replies
Cobedangiu
Yesterday at 1:51 PM
arqady
an hour ago
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density over modulo M
SomeGuy3335   3
N 2 hours ago by ja.
Let $M$ be a positive integer and let $\alpha$ be an irrational number. Show that for every integer $0\leq a < M$, there exists a positive integer $n$ such that $M \mid \lfloor{n \alpha}\rfloor-a$.
3 replies
SomeGuy3335
Apr 20, 2025
ja.
2 hours ago
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Diophantine equation !
ComplexPhi   5
N 2 hours ago by aops.c.c.
Source: Romania JBMO TST 2015 Day 1 Problem 4
Solve in nonnegative integers the following equation :
$$21^x+4^y=z^2$$
5 replies
ComplexPhi
May 14, 2015
aops.c.c.
2 hours ago
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Combo problem
soryn   0
3 hours ago
The school A has m1 boys and m2 girls, and ,the school B has n1 boys and n2 girls. Each school is represented by one team formed by p students,boys and girls. If f(k) is the number of cases for which,the twice schools has,togheter k girls, fund f(k) and the valute of k, for which f(k) is maximum.
0 replies
soryn
3 hours ago
0 replies
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Parity and sets
betongblander   7
N 3 hours ago by ihategeo_1969
Source: Brazil National Olympiad 2020 5 Level 3
Let $n$ and $k$ be positive integers with $k$ $\le$ $n$. In a group of $n$ people, each one or always
speak the truth or always lie. Arnaldo can ask questions for any of these people
provided these questions are of the type: “In set $A$, what is the parity of people who speak to
true? ”, where $A$ is a subset of size $ k$ of the set of $n$ people. The answer can only
be “$even$” or “$odd$”.
a) For which values of $n$ and $k$ is it possible to determine which people speak the truth and
which people always lie?
b) What is the minimum number of questions required to determine which people
speak the truth and which people always lie, when that number is finite?
7 replies
betongblander
Mar 18, 2021
ihategeo_1969
3 hours ago
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Mount Inequality erupts on a sequence :o
GrantStar   88
N 3 hours ago by Nari_Tom
Source: 2023 IMO P4
Let $x_1,x_2,\dots,x_{2023}$ be pairwise different positive real numbers such that
\[a_n=\sqrt{(x_1+x_2+\dots+x_n)\left(\frac{1}{x_1}+\frac{1}{x_2}+\dots+\frac{1}{x_n}\right)}\]is an integer for every $n=1,2,\dots,2023.$ Prove that $a_{2023} \geqslant 3034.$
88 replies
GrantStar
Jul 9, 2023
Nari_Tom
3 hours ago
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nice problem
hanzo.ei   2
N Mar 30, 2025 by Lil_flip38
Source: I forgot
Let triangle $ABC$ be inscribed in the circumcircle $(O)$ and circumscribed about the incircle $(I)$, with $AB < AC$. The incircle $(I)$ touches the sides $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. The line $AI$ cuts segments $DF$ and $DE$ at $K$ and $L$, respectively, and the tangents to the circle $(DKL)$ at $K$ and $L$ intersect at $T$. Prove that $AT \perp BC$.
2 replies
hanzo.ei
Mar 29, 2025
Lil_flip38
Mar 30, 2025
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nice problem
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Reveal topic tags
geometry unsolved
geometry
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G H BBookmark kLocked kLocked NReply
Source: I forgot
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hanzo.ei
20 posts
hanzo.ei
#1 h  Mar 29, 2025, 5:58 PM • 1 Y
Y by cubres
Let triangle $ABC$ be inscribed in the circumcircle $(O)$ and circumscribed about the incircle $(I)$, with $AB < AC$. The incircle $(I)$ touches the sides $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. The line $AI$ cuts segments $DF$ and $DE$ at $K$ and $L$, respectively, and the tangents to the circle $(DKL)$ at $K$ and $L$ intersect at $T$. Prove that $AT \perp BC$.
This post has been edited 1 time. Last edited by hanzo.ei, Apr 7, 2025, 3:09 PM
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hanzo.ei
20 posts
hanzo.ei
#2 h  Mar 30, 2025, 2:21 PM
Y by
bump!!!!
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Lil_flip38
53 posts
Lil_flip38
#3 h  Mar 30, 2025, 7:48 PM • 2 Y
Y by hanzo.ei, X.Luser
I dont understand why there are so many unnecessary points defined, but oh well
Let \(S\) be the midpoint of \(BC\), \(R\) be the midpoint of \(AH\) where \(H\) is the foot of the altitude, \(AI\cap BC=P,\), let \(V\) be the point on \(I\) such that \(SV=SD\). Let \(AV\cap BC=Q\). It is well known that \(\angle AVD=90^\circ\) Let \(U=AI\cap DV\), and \(AT\cap BC = H'\)
We first claim that \(V\) lies on \((DKL)\), and that the center is the midpoint of \(BC\). First, consider inversion about \(I\). \(K,L\) are inverses, because \(L\) is mapped to \(AI\cap (CDEI)\) which is \(K\) by Iran lemma. This implies that \((DKL)\) is orthogonal to \(I\), so the center of \(DKL\) lies on \(BC\) as \(ID\perp BC\). Also by Iran lemma, \(BL\perp AI, CK\perp AI\implies BL\parallel CK\). So if we now consider the perpendicular bisector of \(KL\) it passes through \(S\). Thus, \(S\) is the center of \((DKL)\). Now, by definition, \(V\) lies on this circle aswell. Again, it is well known that \(Q\) is the \(A\)-extouch point, so it also lies on \((DKL)\). Note that \((L,K;D,V)=-1\), so \(T\) lies on \(DV\), and \((T, O;D,V)=-1\). Now, as
\[-1=(T,O;D,V) \overset{A}{=}(H',P;D,Q)\]and as its well known that \(R,I,Q\) lie on a line, we also have:
\[-1=(H,A;\infty_{\perp BC},R)\overset{I}{=}(H,P;D,Q)\]It follows that \(H=H'\) as desired.
I believe that most of the well known lemmas i stated throughout the solution are in EGMO chapter 4
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