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Three concyclic quadrilaterals
Lukaluce   1
N 7 minutes ago by InterLoop
Source: EGMO 2025 P3
Let $ABC$ be an acute triangle. Points $B, D, E,$ and $C$ lie on a line in this order and satisfy $BD = DE = EC$. Let $M$ and $N$ be midpoints of $AD$ and $AE$, respectively. Suppose triangle $ADE$ is acute, and let $H$ be its orthocentre. Points $P$ and $Q$ lie on lines $BM$ and $CN$, respectively, such that $D, H, M,$ and $P$ are concyclic and pairwise different, and $E, H, N,$ and $Q$ are concyclic and pairwise different. Prove that $P, Q, N,$ and $M$ are concyclic. $\newline$
The orthocentre of a triangle is the point of intersection of its altitudes.
1 reply
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Lukaluce
19 minutes ago
InterLoop
7 minutes ago
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Hard number theory
truongngochieu   0
8 minutes ago
Find all integers $a,b$ such that $a^2+a+1=7^b$
0 replies
truongngochieu
8 minutes ago
0 replies
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GCD of sums of consecutive divisors
Lukaluce   1
N 9 minutes ago by Marius_Avion_De_Vanatoare
Source: EGMO 2025 P1
For a positive integer $N$, let $c_1 < c_2 < ... < c_m$ be all the positive integers smaller than $N$ that are coprime to $N$. Find all $N \ge 3$ such that
\[gcd(N, c_i + c_{i + 1}) \neq 1\]for all $1 \le i \le m - 1$.
1 reply
+4 w
Lukaluce
20 minutes ago
Marius_Avion_De_Vanatoare
9 minutes ago
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Arithmetic means as terms of a sequence
Lukaluce   0
20 minutes ago
Source: EGMO 2025 P2
An infinite increasing sequence $a_1 < a_2 < a_3 < ...$ of positive integers is called central if for every positive integer $n$, the arithmetic mean of the first $a_n$ terms of the sequence is equal to $a_n$. Show that there exists an infinite sequence $b_1, b_2, b_3, ...$ of positive integers such that for every central sequence $a_1, a_2, a_3, ...$, there are infinitely many positive integers $n$ with $a_n = b_n$.
0 replies
Lukaluce
20 minutes ago
0 replies
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pairwise coprime sum gcd
InterLoop   1
N 24 minutes ago by MaxSze
Source: EGMO 2025/1
For a positive integer $N$, let $c_1 < c_2 < \dots < c_m$ be all the positive integers smaller than $N$ that are coprime to $N$. Find all $N \ge 3$ such that
$$\gcd(N, c_i + c_{i+1}) \neq 1$$for all $1 \le i \le m - 1$.
1 reply
InterLoop
an hour ago
MaxSze
24 minutes ago
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postaffteff
JetFire008   18
N 26 minutes ago by Captainscrubz
Source: Internet
Let $P$ be the Fermat point of a $\triangle ABC$. Prove that the Euler line of the triangles $PAB$, $PBC$, $PCA$ are concurrent and the point of concurrence is $G$, the centroid of $\triangle ABC$.
18 replies
1 viewing
JetFire008
Mar 15, 2025
Captainscrubz
26 minutes ago
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Similarity
AHZOLFAGHARI   17
N 34 minutes ago by ariopro1387
Source: Iran Second Round 2015 - Problem 3 Day 1
Consider a triangle $ABC$ . The points $D,E$ are on sides $AB,AC$ such that $BDEC$ is a cyclic quadrilateral. Let $P$ be the intersection of $BE$ and $CD$. $H$ is a point on $AC$ such that $\angle PHA = 90^{\circ}$. Let $M,N$ be the midpoints of $AP,BC$. Prove that: $ ACD \sim MNH $.
17 replies
AHZOLFAGHARI
May 7, 2015
ariopro1387
34 minutes ago
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one cyclic formed by two cyclic
CrazyInMath   0
an hour ago
Source: EGMO 2025/3
Let $ABC$ be an acute triangle. Points $B, D, E$, and $C$ lie on a line in this order and satisfy $BD = DE = EC$. Let $M$ and $N$ be the midpoints of $AD$ and $AE$, respectively. Suppose triangle $ADE$ is acute, and let $H$ be its orthocentre. Points $P$ and $Q$ lie on lines $BM$ and $CN$, respectively, such that $D, H, M,$ and $P$ are concyclic and pairwise different, and $E, H, N,$ and $Q$ are concyclic and pairwise different. Prove that $P, Q, N,$ and $M$ are concyclic.
0 replies
+12 w
CrazyInMath
an hour ago
0 replies
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A problem with non-negative a,b,c
KhuongTrang   3
N an hour ago by KhuongTrang
Source: own
Problem. Let $a,b,c$ be non-negative real variables with $ab+bc+ca\neq 0.$ Prove that$$\color{blue}{\sqrt{\frac{8a^{2}+\left(b-c\right)^{2}}{\left(b+c\right)^{2}}}+\sqrt{\frac{8b^{2}+\left(c-a\right)^{2}}{\left(c+a\right)^{2}}}+\sqrt{\frac{8c^{2}+\left(a-b\right)^{2}}{\left(a+b\right)^{2}}}\ge \sqrt{\frac{18(a^{2}+b^{2}+c^{2})}{ab+bc+ca}}.}$$Equality holds iff $(a,b,c)\sim(t,t,t)$ or $(a,b,c)\sim(t,t,0)$ where $t>0.$
3 replies
KhuongTrang
Mar 4, 2025
KhuongTrang
an hour ago
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Number Theory Chain!
JetFire008   52
N an hour ago by Anto0110
I will post a question and someone has to answer it. Then they have to post a question and someone else will answer it and so on. We can only post questions related to Number Theory and each problem should be more difficult than the previous. Let's start!

Question 1
52 replies
JetFire008
Apr 7, 2025
Anto0110
an hour ago
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Convex quad
MithsApprentice   81
N an hour ago by LeYohan
Source: USAMO 1993
Let $\, ABCD \,$ be a convex quadrilateral such that diagonals $\, AC \,$ and $\, BD \,$ intersect at right angles, and let $\, E \,$ be their intersection. Prove that the reflections of $\, E \,$ across $\, AB, \, BC, \, CD, \, DA \,$ are concyclic.
81 replies
MithsApprentice
Oct 27, 2005
LeYohan
an hour ago
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Ratios in a right triangle
PNT   1
N Apr 6, 2025 by Mathzeus1024
Source: Own.
Let $ABC$ be a right triangle in $A$ with $AB<AC$. Let $M$ be the midpoint of $AB$ and $D$ a point on $AC$ such that $DC=DB$. Let $X=(BDC)\cap MD$.
Compute in terms of $AB,BC$ and $AC$ the ratio $\frac{BX}{DX}$.
1 reply
PNT
Jun 9, 2023
Mathzeus1024
Apr 6, 2025
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Ratios in a right triangle
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Reveal topic tags
geometry
right triangle
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Source: Own.
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PNT
320 posts
PNT
#1 h  Jun 9, 2023, 10:58 PM
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Let $ABC$ be a right triangle in $A$ with $AB<AC$. Let $M$ be the midpoint of $AB$ and $D$ a point on $AC$ such that $DC=DB$. Let $X=(BDC)\cap MD$.
Compute in terms of $AB,BC$ and $AC$ the ratio $\frac{BX}{DX}$.
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Mathzeus1024
806 posts
Mathzeus1024
#2 h  Apr 6, 2025, 1:15 PM
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In the $xy-$plane let $A(0,0); B(0,c); C(b,0); M(0,c/2)$ (with $b>c$) be defined for right $\Delta ABC$. Now, let $D(t,0)$ lie on the same line as $AC$ and to the left of $\Delta ABC$. If $|DB|=|DC|$, then:

$\sqrt{t^2+c^2}=b+t \Rightarrow t^2+c^2 = t^2+2bt+b^2 \Rightarrow t = -\frac{b^2-c^2}{2b}$ (i)

Extending the line $MD$ to $BC$ results in the point $X = MD \cap \Delta BDC$, or:

$MD: y - c/2 = \frac{c/2}{\frac{b^2-c^2}{2b}}x \Rightarrow y = \frac{bc}{b^2-c^2}x + \frac{c}{2}$ (ii);

$BC: y = -\frac{c}{b}x + c$ (iii);

or $X(x,y) = \left(\frac{(b^2-c^2)b}{2b^2-c^2}, \frac{b^2c}{2b^2-c^2}\right)$. This leads to the computations of $|BX|, |DX|$:

$|BX| = \sqrt{\left(0-\frac{(b^2-c^2)b}{2b^2-c^2}\right)^2 + \left(c-\frac{b^2c}{2b^2-c^2}\right)^2} = \left(\frac{b^2-c^2}{2b^2-c^2}\right)\sqrt{b^2+c^2}$ (iv);

$|DX| = \sqrt{ \left( -\frac{b^2-c^2}{2b} - \frac{(b^2-c^2)b}{2b^2-c^2} \right)^{2} + \left(0 - \frac{b^{2}c}{2b^2-c^2} \right)^{2} } = \frac{\sqrt{4b^{6}c^{2} +(4b^2-c^2)^{2}(b^2-c^2)^{2} } } {2b(2b^2-c^2)}$ (v);

of which:

$\frac{|BX|}{|DX|} = \frac{2b(b^2-c^2)\sqrt{b^2+c^2}}{\sqrt{4b^{6}c^{2}+(4b^2-c^2)^{2}(b^2-c^2)^{2}}} = \textcolor{red}{\frac{2|AC||BC|(|AC|^2-|AB|^2)}{\sqrt{4|AC|^{6}|AB|^{2} +(4|AC|^2-|AB|^2)^{2}(|AC|^2-|AB|^2)^{2}}}}$.
This post has been edited 7 times. Last edited by Mathzeus1024, Apr 6, 2025, 1:45 PM
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