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IMO Shortlist 2011, G4
WakeUp   128
N 16 minutes ago by Mathandski
Source: IMO Shortlist 2011, G4
Let $ABC$ be an acute triangle with circumcircle $\Omega$. Let $B_0$ be the midpoint of $AC$ and let $C_0$ be the midpoint of $AB$. Let $D$ be the foot of the altitude from $A$ and let $G$ be the centroid of the triangle $ABC$. Let $\omega$ be a circle through $B_0$ and $C_0$ that is tangent to the circle $\Omega$ at a point $X\not= A$. Prove that the points $D,G$ and $X$ are collinear.

Proposed by Ismail Isaev and Mikhail Isaev, Russia
128 replies
WakeUp
Jul 13, 2012
Mathandski
16 minutes ago
IMO Shortlist 2011, G2
WakeUp   29
N 17 minutes ago by AylyGayypow009
Source: IMO Shortlist 2011, G2
Let $A_1A_2A_3A_4$ be a non-cyclic quadrilateral. Let $O_1$ and $r_1$ be the circumcentre and the circumradius of the triangle $A_2A_3A_4$. Define $O_2,O_3,O_4$ and $r_2,r_3,r_4$ in a similar way. Prove that
\[\frac{1}{O_1A_1^2-r_1^2}+\frac{1}{O_2A_2^2-r_2^2}+\frac{1}{O_3A_3^2-r_3^2}+\frac{1}{O_4A_4^2-r_4^2}=0.\]

Proposed by Alexey Gladkich, Israel
29 replies
WakeUp
Jul 13, 2012
AylyGayypow009
17 minutes ago
Weird expression being integer.
MarkBcc168   24
N 42 minutes ago by AR17296174
Source: IMO Shortlist 2017 N5
Find all pairs $(p,q)$ of prime numbers which $p>q$ and
$$\frac{(p+q)^{p+q}(p-q)^{p-q}-1}{(p+q)^{p-q}(p-q)^{p+q}-1}$$is an integer.
24 replies
MarkBcc168
Jul 10, 2018
AR17296174
42 minutes ago
Find the value of n - ILL 1990 MEX1
Amir Hossein   3
N an hour ago by maromex
During the class interval, $n$ children sit in a circle and play the game described below. The teacher goes around the children clockwisely and hands out candies to them according to the following regulations: Select a child, give him a candy; and give the child next to the first child a candy too; then skip over one child and give next child a candy; then skip over two children; give the next child a candy; then skip over three children; give the next child a candy;...

Find the value of $n$ for which the teacher can ensure that every child get at least one candy eventually (maybe after many circles).
3 replies
Amir Hossein
Sep 18, 2010
maromex
an hour ago
Self-evident inequality trick
Lukaluce   14
N an hour ago by sqing
Source: 2025 Junior Macedonian Mathematical Olympiad P4
Let $x, y$, and $z$ be positive real numbers, such that $x^2 + y^2 + z^2 = 3$. Prove the inequality
\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]When does the equality hold?
14 replies
+1 w
Lukaluce
May 18, 2025
sqing
an hour ago
maximum value
moldovan   4
N an hour ago by MathIQ.
Source: Austria 1989
Natural numbers $ a \le b \le c \le d$ satisfy $ a+b+c+d=30$. Find the maximum value of the product $ P=abcd.$
4 replies
moldovan
Jul 10, 2009
MathIQ.
an hour ago
R to R, with x+f(xy)=f(1+f(y))x
NicoN9   5
N an hour ago by jasperE3
Source: Own.
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that\[
x+f(xy)=f(1+f(y))x
\]for all $x, y\in \mathbb{R}$.
5 replies
NicoN9
May 11, 2025
jasperE3
an hour ago
Chain of floors
Assassino9931   1
N 2 hours ago by pi_quadrat_sechstel
Source: Vojtech Jarnik IMC 2025, Category I, P2
Determine all real numbers $x>1$ such that
\[ \left\lfloor\frac{n+1}{x}\right\rfloor = n - \left\lfloor \frac{n}{x} \right\rfloor + \left \lfloor \frac{\left \lfloor \frac{n}{x} \right\rfloor}{x}\right \rfloor - \left \lfloor \frac{\left \lfloor \frac{\left\lfloor \frac{n}{x} \right\rfloor}{x} \right\rfloor}{x}\right \rfloor + \cdots \]for any positive integer $n$.
1 reply
Assassino9931
May 2, 2025
pi_quadrat_sechstel
2 hours ago
2015 Taiwan TST Round 2 Quiz 1 Problem 2
wanwan4343   8
N 2 hours ago by Want-to-study-in-NTU-MATH
Source: 2015 Taiwan TST Round 2 Quiz 1 Problem 2
Let $\omega$ be the incircle of triangle $ABC$ and $\omega$ touches $BC$ at $D$. $AD$ meets $\omega$ again at $L$. Let $K$ be $A$-excenter, and $M,N$ be the midpoint of $BC,KM$, respectively. Prove that $B,C,N,L$ are concyclic.
8 replies
wanwan4343
Jul 12, 2015
Want-to-study-in-NTU-MATH
2 hours ago
Inspired by Butterfly
sqing   2
N 2 hours ago by sqing
Source: Own
Let $ a,b,c>0. $ Prove that
$$a^2+b^2+c^2+ab+bc+ca+abc-3(a+b+c) \geq 34-14\sqrt 7$$$$a^2+b^2+c^2+ab+bc+ca+abc-\frac{433}{125}(a+b+c) \geq \frac{2(57475-933\sqrt{4665})}{3125} $$
2 replies
sqing
Today at 8:52 AM
sqing
2 hours ago
Triangle form by perpendicular bisector
psi241   52
N 2 hours ago by ihatemath123
Source: IMO Shortlist 2018 G5
Let $ABC$ be a triangle with circumcircle $\Omega$ and incentre $I$. A line $\ell$ intersects the lines $AI$, $BI$, and $CI$ at points $D$, $E$, and $F$, respectively, distinct from the points $A$, $B$, $C$, and $I$. The perpendicular bisectors $x$, $y$, and $z$ of the segments $AD$, $BE$, and $CF$, respectively determine a triangle $\Theta$. Show that the circumcircle of the triangle $\Theta$ is tangent to $\Omega$.
52 replies
psi241
Jul 17, 2019
ihatemath123
2 hours ago
Regional Olympiad - FBH 2018 Grade 9 Problem 3
gobathegreat   5
N 2 hours ago by justaguy_69
Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2018
Let $p$ and $q$ be prime numbers such that $p^2+pq+q^2$ is perfect square. Prove that $p^2-pq+q^2$ is prime
5 replies
gobathegreat
Sep 18, 2018
justaguy_69
2 hours ago
Parallel condition and isogonal
ItzsleepyXD   1
N Apr 30, 2025 by moony_
Source: Own , Mock Thailand Mathematic Olympiad P5
Let $ABC$ be triangle and point $D$ be $A-$ altitude of $\triangle ABC$ .
Let $E,F$ be a point on $AC$ and $AB$ such that $DE\parallel AB$ and $DF\parallel AC$ . Point $G$ is the intersection of $(AEF)$ and $(ABC)$ . Point $P$ be intersection of $(ADG)$ and $BC$ . Line $GD$ intersect circumcircle of $\triangle ABC$ again at $Q$ .
Prove that
(a) $\angle BAP = \angle QAC$ .
(b) $AQ$ bisect $BC$ .
1 reply
ItzsleepyXD
Apr 30, 2025
moony_
Apr 30, 2025
Parallel condition and isogonal
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G H BBookmark kLocked kLocked NReply
Source: Own , Mock Thailand Mathematic Olympiad P5
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ItzsleepyXD
147 posts
#1
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Let $ABC$ be triangle and point $D$ be $A-$ altitude of $\triangle ABC$ .
Let $E,F$ be a point on $AC$ and $AB$ such that $DE\parallel AB$ and $DF\parallel AC$ . Point $G$ is the intersection of $(AEF)$ and $(ABC)$ . Point $P$ be intersection of $(ADG)$ and $BC$ . Line $GD$ intersect circumcircle of $\triangle ABC$ again at $Q$ .
Prove that
(a) $\angle BAP = \angle QAC$ .
(b) $AQ$ bisect $BC$ .
Z K Y
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moony_
22 posts
#2 • 1 Y
Y by Funcshun840
cool ratio lemma... >w<
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