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Sharygin 2025 CR P12
Gengar_in_Galar   8
N 6 minutes ago by Kappa_Beta_725
Source: Sharygin 2025
Circles $\omega_{1}$ and $\omega_{2}$ are given. Let $M$ be the midpoint of the segment joining their centers, $X$, $Y$ be arbitrary points on $\omega_{1}$, $\omega_{2}$ respectively such that $MX=MY$. Find the locus of the midpoints of segments $XY$.
Proposed by: L Shatunov
8 replies
Gengar_in_Galar
Mar 10, 2025
Kappa_Beta_725
6 minutes ago
Sharygin 2025 CR P17
Gengar_in_Galar   6
N 10 minutes ago by Kappa_Beta_725
Source: Sharygin 2025
Let $O$, $I$ be the circumcenter and the incenter of an acute-angled scalene triangle $ABC$; $D$, $E$, $F$ be the touching points of its excircle with the side $BC$ and the extensions of $AC$, $AB$ respectively. Prove that if the orthocenter of the triangle $DEF$ lies on the circumcircle of $ABC$, then it is symmetric to the midpoint of the arc $BC$ with respect to $OI$.
Proposed by: P.Puchkov,E.Utkin
6 replies
Gengar_in_Galar
Mar 10, 2025
Kappa_Beta_725
10 minutes ago
Sharygin 2025 CR P21
Gengar_in_Galar   4
N 18 minutes ago by Kappa_Beta_725
Source: Sharygin 2025
Let $P$ be a point inside a quadrilateral $ABCD$ such that $\angle APB+\angle CPD=180^{\circ}$. Points $P_{a}$, $P_{b}$, $P_{c},$ $P_{d}$ are isogonally conjugated to $P$ with respect to the triangles $BCD$, $CDA$, $DAB$, $ABC$ respectively. Prove that the diagonals of the quadrilaterals $ABCD$ and $P_{a}P_{b}P_{c}P_{d}$ concur.
Proposed by: G.Galyapin
4 replies
Gengar_in_Galar
Mar 10, 2025
Kappa_Beta_725
18 minutes ago
Sharygin 2025 CR P18
Gengar_in_Galar   6
N 23 minutes ago by Kappa_Beta_725
Source: Sharygin 2025
Let $ABCD$ be a quadrilateral such that the excircles $\omega_{1}$ and $\omega_{2}$ of triangles $ABC$ and $BCD$ touching their sides $AB$ and $BD$ respectively touch the extension of $BC$ at the same point $P$. The segment $AD$ meets $\omega_{2}$ at point $Q$, and the line $AD$ meets $\omega_{1}$ at $R$ and $S$. Prove that one of angles $RPQ$ and $SPQ$ is right
Proposed by: I.Kukharchuk
6 replies
Gengar_in_Galar
Mar 10, 2025
Kappa_Beta_725
23 minutes ago
helpppppppp me
stupid_boiii   1
N 2 hours ago by vanstraelen
Given triangle ABC. The tangent at ? to the circumcircle(ABC) intersects line BC at point T. Points D,E satisfy AD=BD, AE=CE, and ∠CBD=∠BCE<90 ∘ . Prove that D,E,T are collinear.
1 reply
stupid_boiii
Yesterday at 4:22 AM
vanstraelen
2 hours ago
Algebra Polynomials
Foxellar   2
N 3 hours ago by Foxellar
The real root of the polynomial \( p(x) = 8x^3 - 3x^2 - 3x - 1 \) can be written in the form
\[
\frac{\sqrt[3]{a} + \sqrt[3]{b} + 1}{c},
\]where \( a, b, \) and \( c \) are positive integers. Find the value of \( a + b + c \).
2 replies
Foxellar
4 hours ago
Foxellar
3 hours ago
geometry
luckvoltia.112   0
Today at 2:29 AM
Let \( \triangle ABC \) be an acute triangle with \( AB < AC \), and its vertices lie on the circle \( (O) \). Let \( AD \) be the altitude from vertex \( A \). Let \( E \) and \( F \) be the feet of the perpendiculars from \( D \) to the lines \( AB \) and \( AC \), respectively. Let \( EF \) intersect the circle \( (O) \) again at points \( P \) and \( Q \) such that \( E \) lies between \( Q \) and \( F \). Let the lines \( AD \) and \( EF \) intersect at point \( G \). Let \( I \) be the midpoint of segment \( AD \). Let \( AO \) intersect line \( BC \) at point \( K \).
a) Prove that \( AP = AQ = AD \).
b) Prove that line \( OI \) is parallel to line \( KG \).
c)Let \( H \) be the orthocenter of triangle \( ABC \), and let \( M \) be the midpoint of segment \( BC \). $S$ is the center (HBC). Let point \( T \) lie on line \( DS \) such that ray \( KD \) is the angle bisector of \( \angle GKT \). Prove that lines \( AD \) and \( MT \) intersect at a point lying on circle \( (O) \).
0 replies
luckvoltia.112
Today at 2:29 AM
0 replies
Great Geometry with Squares on sides of triangles
SomeonecoolLovesMaths   2
N Today at 1:50 AM by happypi31415
Three squares are drawn on the sides of triangle \(ABC\) (i.e., the square on \(AB\) has \(AB\) as one of its sides and lies outside \(ABC\)). Show that the lines drawn from the vertices \(A\), \(B\), and \(C\) to the centers of the opposite squares are concurrent.

IMAGE
2 replies
SomeonecoolLovesMaths
Yesterday at 9:44 PM
happypi31415
Today at 1:50 AM
A suspcious assumption
NamelyOrange   2
N Today at 1:30 AM by maromex
Let $a,b,c,d$ be positive integers. Maximize $\max(a,b,c,d)$ if $a+b+c+d=a^2-b^2+c^2-d^2=2012$.
2 replies
NamelyOrange
Yesterday at 1:53 PM
maromex
Today at 1:30 AM
n is divisible by 5
spiralman   1
N Yesterday at 8:42 PM by KSH31415
n is an integer. There are n integers such that they are larger or equal to 1, and less or equal to 6. Sum of them is larger or equal to 4n, while sum of their square is less or equal to 22n. Prove n is divisible by 5.
1 reply
spiralman
Wednesday at 7:38 PM
KSH31415
Yesterday at 8:42 PM
Monochromatic Triangle
FireBreathers   1
N Yesterday at 8:08 PM by KSH31415
We are given in points in a plane and we connect some of them so that 10n^2 + 1 segments are drawn. We color these segments in 2 colors. Prove that we can find a monochromatic triangle.
1 reply
FireBreathers
Yesterday at 2:28 PM
KSH31415
Yesterday at 8:08 PM
how difficult are these problems
rajukaju   1
N Yesterday at 7:28 PM by Shan3t
I can solve only the first 4 problems of the last general round of the HMMT competition: https://hmmt-archive.s3.amazonaws.com/tournaments/2024/nov/gen/problems.pdf

As a prediction, would this mean I am good enough to qualify for AIME? How does the difficulty compare?

1 reply
rajukaju
Yesterday at 6:43 PM
Shan3t
Yesterday at 7:28 PM
Maximum value of function (with two variables)
Saucepan_man02   1
N Yesterday at 1:39 PM by Saucepan_man02
If $f(\theta) = \min(|2x-7|+|x-4|+|x-2 -\sin \theta|)$, where $x, \theta \in \mathbb R$, then maximum value of $f(\theta)$.
1 reply
Saucepan_man02
Yesterday at 1:25 PM
Saucepan_man02
Yesterday at 1:39 PM
It is given that $M=1+\frac12+\frac13+\frac14+\cdots+\frac{1}{23}=\frac{n}{23!},
Vulch   3
N Yesterday at 11:58 AM by mohabstudent1
It is given that $M=1+\frac12+\frac13+\frac14+\cdots+\frac{1}{23}=\frac{n}{23!},$ where $n$ is a natural number.What is the remainder when $n$ is divided by $13?$
3 replies
Vulch
Apr 9, 2025
mohabstudent1
Yesterday at 11:58 AM
Polygons Which Don't Fit
somebodyyouusedtoknow   1
N Apr 27, 2025 by kiyoras_2001
Source: San Diego Honors Math Contest 2025 Part II, Problem 1
Let $P_1,P_2,\ldots,P_n$ be polygons, no two of which are similar. Show that there are polygons $Q_1,Q_2,\ldots,Q_n$ where $Q_i$ is similar to $P_i$ so that for no $i \neq j$ does $Q_i$ contain a polygon that's congruent to $Q_j$.

Note. Here, the word "contain" means for the construction we have, we cannot select a size for $Q_j$ so that $Q_j$ is wholly contained in $Q_i$, and so it does not intersect the edges of $Q_i$ at all.
1 reply
somebodyyouusedtoknow
Apr 26, 2025
kiyoras_2001
Apr 27, 2025
Polygons Which Don't Fit
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Source: San Diego Honors Math Contest 2025 Part II, Problem 1
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somebodyyouusedtoknow
259 posts
#1
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Let $P_1,P_2,\ldots,P_n$ be polygons, no two of which are similar. Show that there are polygons $Q_1,Q_2,\ldots,Q_n$ where $Q_i$ is similar to $P_i$ so that for no $i \neq j$ does $Q_i$ contain a polygon that's congruent to $Q_j$.

Note. Here, the word "contain" means for the construction we have, we cannot select a size for $Q_j$ so that $Q_j$ is wholly contained in $Q_i$, and so it does not intersect the edges of $Q_i$ at all.
This post has been edited 1 time. Last edited by somebodyyouusedtoknow, Apr 26, 2025, 11:41 PM
Reason: Added clarity for the notes
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kiyoras_2001
678 posts
#2
Y by
Induct on $n$. The base case $n=2$ is obvious; for fixed $Q_1=P_1$ consider the largest copy of $P_2$ inside it and slightly enlarge it to obtain $Q_2$.

Now the step $n\to n+1$. That is, given $Q_1, \ldots, Q_n$ with $Q_i \not\subset Q_j$ for all $i\ne j$ we seek for appropriate $Q_{n+1}$.

For each $i\in [n]:= \{1, \ldots, n\}$ define $R_i^-$ as the largest copy of $P_{n+1}$ inside $Q_i$. As well as define $R_i^+$ as the smallest copy of $P_{n+1}$ containing $Q_i$.

We want to show that there is a copy of $P_{n+1}$ which is larger than $R_i^-$ and smaller than $R_i^+$ for all $i\in [n]$. That is, we need to prove that $\max_{i\in [n]} R_i^- \subsetneq \min_{i\in [n]}R_i^+$. Suppose not, then there are $i\ne j$ such that $Q_i \subseteq R_i^+ \subseteq R_j^- \subseteq Q_j$. This contradiction proves that we can select $Q_{n+1}$ appropriately.
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