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geometry
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Hard geometry
Lukariman   1
N 15 minutes ago by ricarlos
Given triangle ABC, any line d intersects AB at D, intersects AC at E, intersects BC at F. Let O1,O2,O3 be the centers of the circles circumscribing triangles ADE, BDF, CFE. Prove that the orthocenter of triangle O1O2O3 lies on line d.
1 reply
Lukariman
May 12, 2025
ricarlos
15 minutes ago
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Simple Geometry
AbdulWaheed   6
N 23 minutes ago by Gggvds1
Source: EGMO
Try to avoid Directed angles
Let ABC be an acute triangle inscribed in circle $\Omega$. Let $X$ be the midpoint of the arc $\overarc{BC}$ not containing $A$ and define $Y, Z$ similarly. Show that the orthocenter of $XYZ$ is the incenter $I$ of $ABC$.
6 replies
AbdulWaheed
May 23, 2025
Gggvds1
23 minutes ago
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Bosnia and Herzegovina JBMO TST 2013 Problem 4
gobathegreat   4
N 25 minutes ago by FishkoBiH
Source: Bosnia and Herzegovina Junior Balkan Mathematical Olympiad TST 2013
It is given polygon with $2013$ sides $A_{1}A_{2}...A_{2013}$. His vertices are marked with numbers such that sum of numbers marked by any $9$ consecutive vertices is constant and its value is $300$. If we know that $A_{13}$ is marked with $13$ and $A_{20}$ is marked with $20$, determine with which number is marked $A_{2013}$
4 replies
gobathegreat
Sep 16, 2018
FishkoBiH
25 minutes ago
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A geometry problem
Lttgeometry   2
N 27 minutes ago by Acrylic3491
Triangle $ABC$ has two isogonal conjugate points $P$ and $Q$. The circle $(BPC)$ intersects circle $(AP)$ at $R \neq P$, and the circle $(BQC)$ intersects circle $(AQ)$ at $S\neq Q$. Prove that $R$ and $S$ are isogonal conjugates in triangle $ABC$.
Note: Circle $(AP)$ is the circle with diameter $AP$, Circle $(AQ)$ is the circle with diameter $AQ$.
2 replies
Lttgeometry
Today at 4:03 AM
Acrylic3491
27 minutes ago
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anglechasing , circumcenter wanted
parmenides51   1
N 32 minutes ago by Captainscrubz
Source: Sharygin 2011 Final 9.2
In triangle $ABC, \angle B = 2\angle C$. Points $P$ and $Q$ on the medial perpendicular to $CB$ are such that $\angle CAP = \angle PAQ = \angle QAB = \frac{\angle A}{3}$ . Prove that $Q$ is the circumcenter of triangle $CPB$.
1 reply
parmenides51
Dec 16, 2018
Captainscrubz
32 minutes ago
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Nice FE over R+
doanquangdang   4
N an hour ago by jasperE3
Source: collect
Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ such that
\[x+f(yf(x)+1)=xf(x+y)+yf(yf(x))\]for all $x,y>0.$
4 replies
doanquangdang
Jul 19, 2022
jasperE3
an hour ago
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right triangle, midpoints, two circles, find angle
star-1ord   0
an hour ago
Source: Estonia Final Round 2025 8-3
In the right triangle $ABC$, $M$ is the midpoint of the hypotenuse $AB$. Point $D$ is chosen on the leg $BC$ so that the line segment $DM$ meets $(ACD)$ again at $K$ ($K\neq D$). Let $L$ be the reflection of $K$ in $M$. The circles $(ACD)$ and $(BCL)$ meet again at $N$ ($N\neq C$). Find the measure of $\angle KNL$.
0 replies
star-1ord
an hour ago
0 replies
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interesting functional equation
tabel   3
N an hour ago by waterbottle432
Source: random romanian contest
Determine all functions \( f : (0, \infty) \to (0, \infty) \) that satisfy the functional equation:
\[ f(f(x)(1 + y)) = f(x) + f(xy), \quad \forall x, y > 0. \]
3 replies
tabel
2 hours ago
waterbottle432
an hour ago
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Serbian selection contest for the IMO 2025 - P6
OgnjenTesic   4
N an hour ago by GreenTea2593
Source: Serbian selection contest for the IMO 2025
For an $n \times n$ table filled with natural numbers, we say it is a divisor table if:
- the numbers in the $i$-th row are exactly all the divisors of some natural number $r_i$,
- the numbers in the $j$-th column are exactly all the divisors of some natural number $c_j$,
- $r_i \ne r_j$ for every $i \ne j$.

A prime number $p$ is given. Determine the smallest natural number $n$, divisible by $p$, such that there exists an $n \times n$ divisor table, or prove that such $n$ does not exist.

Proposed by Pavle Martinović
4 replies
OgnjenTesic
May 22, 2025
GreenTea2593
an hour ago
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pairs (m, n) such that a fractional expression is an integer
cielblue   2
N 2 hours ago by cielblue
Find all pairs $(m,\ n)$ of positive integers such that $\frac{m^3-mn+1}{m^2+mn+2}$ is an integer.
2 replies
cielblue
Yesterday at 8:38 PM
cielblue
2 hours ago
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Sociable set of people
jgnr   23
N 2 hours ago by quantam13
Source: RMM 2012 day 1 problem 1
Given a finite number of boys and girls, a sociable set of boys is a set of boys such that every girl knows at least one boy in that set; and a sociable set of girls is a set of girls such that every boy knows at least one girl in that set. Prove that the number of sociable sets of boys and the number of sociable sets of girls have the same parity. (Acquaintance is assumed to be mutual.)

(Poland) Marek Cygan
23 replies
jgnr
Mar 3, 2012
quantam13
2 hours ago
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two 3D problems in one day
egxa   1
N Apr 18, 2025 by sami1618
Source: All Russian 2025 11.2
A right prism \(ABCA_1B_1C_1\) is given. It is known that triangles \(A_1BC\), \(AB_1C\), \(ABC_1\), and \(ABC\) are all acute. Prove that the orthocenters of these triangles, together with the centroid of triangle \(ABC\), lie on the same sphere.
1 reply
egxa
Apr 18, 2025
sami1618
Apr 18, 2025
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two 3D problems in one day
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Source: All Russian 2025 11.2
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egxa
211 posts
egxa
#1 h  Apr 18, 2025, 9:44 AM • 2 Y
Y by buratinogigle, Miquel-point
A right prism \(ABCA_1B_1C_1\) is given. It is known that triangles \(A_1BC\), \(AB_1C\), \(ABC_1\), and \(ABC\) are all acute. Prove that the orthocenters of these triangles, together with the centroid of triangle \(ABC\), lie on the same sphere.
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sami1618
913 posts
sami1618
#2 h  Apr 18, 2025, 2:41 PM
Y by
Let $H_A$, $H_B$, $H_C$, and $H$ be the orthocenters of triangles $A_1BC$, $AB_1C$, $ABC_1$, and $ABC$, respectively. Let $G$ be the centroid of triangle $ABC$.

Claim. The point $H_A$ is the foot of $H$ onto the plane $A_1BC$.
Proof. Let $H'$ be the foot of $H$ onto the plane $A_1BC$. Notice that $BH$ is perpendicular to plane $ACC_1A_1$. Thus $BH\perp A_1C$, which implies $BH'\perp A_1C$. Similarly, $CH'\perp A_1B$. Thus $H'=H_A$, as required.

An identical result holds for $H_B$ and $H_C$. Notice the centroids of triangles $A_1BC$, $AB_1C$, and $ABC_1$ all coincide at some point $X$. Since $G$ is the foot of $X$ onto the plane $ABC$, it follows that $\angle HGX=90^{\circ}$. By the claim, we also get that $\angle HH_AX=HH_BX=HH_CX=90^{\circ}$. Thus all five points lie on the sphere with diameter $HX$, concluding the proof.
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