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IMO Shortlist 2012, Geometry 8
lyukhson   33
N 16 minutes ago by awesomeming327.
Source: IMO Shortlist 2012, Geometry 8
Let $ABC$ be a triangle with circumcircle $\omega$ and $\ell$ a line without common points with $\omega$. Denote by $P$ the foot of the perpendicular from the center of $\omega$ to $\ell$. The side-lines $BC,CA,AB$ intersect $\ell$ at the points $X,Y,Z$ different from $P$. Prove that the circumcircles of the triangles $AXP$, $BYP$ and $CZP$ have a common point different from $P$ or are mutually tangent at $P$.

Proposed by Cosmin Pohoata, Romania
33 replies
lyukhson
Jul 29, 2013
awesomeming327.
16 minutes ago
IMO 2012 P5
mathmdmb   123
N an hour ago by SimplisticFormulas
Source: IMO 2012 P5
Let $ABC$ be a triangle with $\angle BCA=90^{\circ}$, and let $D$ be the foot of the altitude from $C$. Let $X$ be a point in the interior of the segment $CD$. Let $K$ be the point on the segment $AX$ such that $BK=BC$. Similarly, let $L$ be the point on the segment $BX$ such that $AL=AC$. Let $M$ be the point of intersection of $AL$ and $BK$.

Show that $MK=ML$.

Proposed by Josef Tkadlec, Czech Republic
123 replies
mathmdmb
Jul 11, 2012
SimplisticFormulas
an hour ago
Fixed line
TheUltimate123   14
N an hour ago by amirhsz
Source: ELMO Shortlist 2023 G4
Let \(D\) be a point on segment \(PQ\). Let \(\omega\) be a fixed circle passing through \(D\), and let \(A\) be a variable point on \(\omega\). Let \(X\) be the intersection of the tangent to the circumcircle of \(\triangle ADP\) at \(P\) and the tangent to the circumcircle of \(\triangle ADQ\) at \(Q\). Show that as \(A\) varies, \(X\) lies on a fixed line.

Proposed by Elliott Liu and Anthony Wang
14 replies
TheUltimate123
Jun 29, 2023
amirhsz
an hour ago
Computing functions
BBNoDollar   7
N 2 hours ago by ICE_CNME_4
Let $f : [0, \infty) \to [0, \infty)$, $f(x) = \dfrac{ax + b}{cx + d}$, with $a, d \in (0, \infty)$, $b, c \in [0, \infty)$. Prove that there exists $n \in \mathbb{N}^*$ such that for every $x \geq 0$
\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\](For $n \in \mathbb{N}^*$ and $x \geq 0$, the notation $f_n(x)$ represents $\underbrace{(f \circ f \circ \dots \circ f)}_{n \text{ times}}(x)$. )
7 replies
BBNoDollar
May 18, 2025
ICE_CNME_4
2 hours ago
RMO 2024 Q2
SomeonecoolLovesMaths   14
N 2 hours ago by Adywastaken
Source: RMO 2024 Q2
For a positive integer $n$, let $R(n)$ be the sum of the remainders when $n$ is divided by $1,2, \cdots , n$. For example, $R(4) = 0 + 0 + 1 + 0 = 1,$ $R(7) = 0 + 1 + 1 + 3 + 2 + 1 + 0 = 8$. Find all positive integers such that $R(n) = n-1$.
14 replies
1 viewing
SomeonecoolLovesMaths
Nov 3, 2024
Adywastaken
2 hours ago
Decimal functions in binary
Pranav1056   3
N 2 hours ago by ihategeo_1969
Source: India TST 2023 Day 3 P1
Let $\mathbb{N}$ be the set of all positive integers. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that $f(x) + y$ and $f(y) + x$ have the same number of $1$'s in their binary representations, for any $x,y \in \mathbb{N}$.
3 replies
Pranav1056
Jul 9, 2023
ihategeo_1969
2 hours ago
Beautiful numbers in base b
v_Enhance   21
N 2 hours ago by Martin2001
Source: USEMO 2023, problem 1
A positive integer $n$ is called beautiful if, for every integer $4 \le b \le 10000$, the base-$b$ representation of $n$ contains the consecutive digits $2$, $0$, $2$, $3$ (in this order, from left to right). Determine whether the set of all beautiful integers is finite.

Oleg Kryzhanovsky
21 replies
v_Enhance
Oct 21, 2023
Martin2001
2 hours ago
Polynomial method of moving points
MathHorse   6
N 2 hours ago by Potyka17
Two Hungarian math olympians achieved significant breakthroughs in the field of polynomial moving points. Their main results are summarised in the attached pdf. Check it out!
6 replies
MathHorse
Jun 30, 2023
Potyka17
2 hours ago
Intertwined numbers
miiirz30   2
N 2 hours ago by Gausikaci
Source: 2025 Euler Olympiad, Round 2
Let a pair of positive integers $(n, m)$ that are relatively prime be called intertwined if among any two divisors of $n$ greater than $1$, there exists a divisor of $m$ and among any two divisors of $m$ greater than $1$, there exists a divisor of $n$. For example, pair $(63, 64)$ is intertwined.

a) Find the largest integer $k$ for which there exists an intertwined pair $(n, m)$ such that the product $nm$ is equal to the product of the first $k$ prime numbers.
b) Prove that there does not exist an intertwined pair $(n, m)$ such that the product $nm$ is the product of $2025$ distinct prime numbers.
c) Prove that there exists an intertwined pair $(n, m)$ such that the number of divisors of $n$ is greater than $2025$.

Proposed by Stijn Cambie, Belgium
2 replies
miiirz30
Yesterday at 10:12 AM
Gausikaci
2 hours ago
Geometry
shactal   0
2 hours ago
Two intersecting circles $C_1$ and $C_2$ have a common tangent that meets $C_1$ in $P$ and $C_2$ in $Q$. The two circles intersect at $M$ and $N$ where $N$ is closer to $PQ$ than $M$ . Line $PN$ meets circle $C_2$ a second time in $R$. Prove that $MQ$ bisects angle $\widehat{PMR}$.
0 replies
shactal
2 hours ago
0 replies
Inspired by 2025 KMO
sqing   1
N 2 hours ago by sqing
Source: Own
Let $ a,b,c,d  $ be real numbers satisfying $ a+b+c+d=0 $ and $ a^2+b^2+c^2+d^2= 6 .$ Prove that $$ -\frac{3}{4} \leq abcd\leq\frac{9}{4}$$Let $ a,b,c,d  $ be real numbers satisfying $ a+b+c+d=6 $ and $ a^2+b^2+c^2+d^2= 18 .$ Prove that $$ -\frac{9(2\sqrt{3}+3)}{4} \leq abcd\leq\frac{9(2\sqrt{3}-3)}{4}$$
1 reply
sqing
3 hours ago
sqing
2 hours ago
Tangents forms triangle with two times less area
NO_SQUARES   1
N Apr 23, 2025 by Luis González
Source: Kvant 2025 no. 2 M2831
Let $DEF$ be triangle, inscribed in parabola. Tangents in points $D,E,F$ forms triangle $ABC$. Prove that $S_{DEF}=2S_{ABC}$. ($S_T$ is area of triangle $T$).
From F.S.Macaulay's book «Geometrical Conics», suggested by M. Panov
1 reply
NO_SQUARES
Apr 23, 2025
Luis González
Apr 23, 2025
Tangents forms triangle with two times less area
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Source: Kvant 2025 no. 2 M2831
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NO_SQUARES
1133 posts
#1 • 5 Y
Y by ehuseyinyigit, buratinogigle, kiyoras_2001, GeoKing, nguyenducmanh2705
Let $DEF$ be triangle, inscribed in parabola. Tangents in points $D,E,F$ forms triangle $ABC$. Prove that $S_{DEF}=2S_{ABC}$. ($S_T$ is area of triangle $T$).
From F.S.Macaulay's book «Geometrical Conics», suggested by M. Panov
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Luis González
4149 posts
#2 • 1 Y
Y by NO_SQUARES
Lemma: Let $P$ be any point on the Steiner circumellipse of $\triangle {ABC}$ and let $\triangle DEF$ be the cevian triangle of $P$ WRT $\triangle{ABC}.$ Then $[DEF]=2 \cdot [ABC].$

Proof: Let $\triangle XYZ$ be the antimedial triangle of $\triangle {ABC}$ ($X,Y,Z$ against $A,B,C$). The Steiner circumellipse $\mathcal{S}$ of $\triangle {ABC}$ is obviously the Steiner inellipse of $\triangle XYZ$ tangent to its sides at $A,B,C$ $\Longrightarrow$ $X,E,F$ are collinear on the polar of $D$ WRT $\mathcal{S}$ and similarly $Y,F,D$ and $Z,D,E$ are respectively collinear. Thus by Pappus theorem for the hexagon $EAFZXY,$ it follows that $FZ \cap EY$ is at infinity, i.e. $FZ \parallel EY$ and similarly we get $DX \parallel FZ$ $\Longrightarrow$ $DX \parallel EY \parallel FZ$ $\Longrightarrow$ $[DEF]=[DYZ]=[CYZ]=2\cdot [ABC]. \ \blacksquare$

Back to the problem. Let $P \equiv AD \cap BE \cap CF$ be the perspector of the parabola. Since the center of any inconic is the isotomcomplement of its perspector (in the parabola case at infinity), then it follows then that the isotomic conjugate of $P$ WRT $\triangle {ABC}$ must be on the line at infinity $\Longrightarrow$ $P$ is on the Steiner circumellipse of $\triangle {ABC}$ and the result follows from the lemma.
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