Triangle determined by three Simson lines

$a_n$ and $b_n$ satisfies the following recursion formulas: $a_{0}=1,$ $b_{0}=1$ , $a_{n+1}=a_{n}+b_{n}$ $\text{[math]}$ and $\text{[math]}$ $b_{n+1}=(2n+3)b_{n}+a_{n}$ . Find $\lim_{n \to \infty} \frac{a_n}{(2n-1)!!}$ $\text{[math]}$ and $\text{[math]}$ $\lim_{n \to \infty} \frac{b_n}{(2n+1)!!}.$

2 replies

Snoop76

Mar 25, 2025

maths_enthusiast_0001

an hour ago

A number theory about divisors which no one fully solved at the contest

nAalniaOMliO 21

N an hour ago by nAalniaOMliO

Source: Belarusian national olympiad 2024

Let's call a pair of positive integers $(k,n)$ interesting if $n$ is composite and for every divisor $d<n$ of $n$ at least one of $d-k$ and $d+k$ is also a divisor of $n$
Find the number of interesting pairs $(k,n)$ with $k \leq 100$
M. Karpuk

21 replies

nAalniaOMliO

Jul 24, 2024

nAalniaOMliO

an hour ago

2025 Caucasus MO Seniors P2

BR1F1SZ 3

N 2 hours ago by X.Luser

Source: Caucasus MO

Let $ABC$ be a triangle, and let $B_1$ and $B_2$ be points on segment $AC$ symmetric with respect to the midpoint of $AC$ . Let $\gamma_A$ denote the circle passing through $B_1$ and tangent to line $AB$ at $A$ . Similarly, let $\gamma_C$ denote the circle passing through $B_1$ and tangent to line $BC$ at $C$ . Let the circles $\gamma_A$ and $\gamma_C$ intersect again at point $B'$ ( $B' \neq B_1$ ). Prove that $\angle ABB' = \angle CBB_2$ .

3 replies

BR1F1SZ

Mar 26, 2025

X.Luser

2 hours ago

IMO Shortlist 2010 - Problem G1

Amir Hossein 130

N 2 hours ago by LeYohan

Let $ABC$ be an acute triangle with $D, E, F$ the feet of the altitudes lying on $BC, CA, AB$ respectively. One of the intersection points of the line $EF$ and the circumcircle is $P.$ The lines $BP$ and $DF$ meet at point $Q.$ Prove that $AP = AQ.$

Proposed by Christopher Bradley, United Kingdom

130 replies

Amir Hossein

Jul 17, 2011

LeYohan

2 hours ago

CGMO6: Airline companies and cities

v_Enhance 13

N 2 hours ago by Marcus_Zhang

Source: 2012 China Girl's Mathematical Olympiad

There are $n$ cities, $2$ airline companies in a country. Between any two cities, there is exactly one $2$ -way flight connecting them which is operated by one of the two companies. A female mathematician plans a travel route, so that it starts and ends at the same city, passes through at least two other cities, and each city in the route is visited once. She finds out that wherever she starts and whatever route she chooses, she must take flights of both companies. Find the maximum value of $n$ .

13 replies

v_Enhance

Aug 13, 2012

Marcus_Zhang

2 hours ago

nice problem

hanzo.ei 0

2 hours ago

Source: I forgot

Let triangle $ABC$ be inscribed in the circumcircle $(O)$ and circumscribed about the incircle $(I)$ , with $AB < AC$ . The incircle $(I)$ touches the sides $BC$ , $CA$ , and $AB$ at $D$ , $E$ , and $F$ , respectively. A line through $I$ , perpendicular to $AI$ , intersects $BC$ , $CA$ , and $AB$ at $X$ , $Y$ , and $Z$ , respectively. The line $AI$ meets $(O)$ at $M$ (distinct from $A$ ). The circumcircle of triangle $AYZ$ intersects $(O)$ at $N$ (distinct from $A$ ). Let $P$ be the midpoint of the arc $BAC$ of $(O)$ . The line $AI$ cuts segments $DF$ and $DE$ at $K$ and $L$ , respectively, and the tangents to the circle $(DKL)$ at $K$ and $L$ intersect at $T$ . Prove that $AT \perp BC$ .

0 replies

hanzo.ei

2 hours ago

0 replies

Find a given number of divisors of ab

proglote 9

N 2 hours ago by zuat.e

Source: Brazil MO 2013, problem #2

Arnaldo and Bernaldo play the following game: given a fixed finite set of positive integers $A$ known by both players, Arnaldo picks a number $a \in A$ but doesn't tell it to anyone. Bernaldo thens pick an arbitrary positive integer $b$ (not necessarily in $A$ ). Then Arnaldo tells the number of divisors of $ab$ . Show that Bernaldo can choose $b$ in a way that he can find out the number $a$ chosen by Arnaldo.

9 replies

proglote

Oct 24, 2013

zuat.e

2 hours ago

2025 TST 22

EthanWYX2009 1

N 3 hours ago by hukilau17

Source: 2025 TST 22

Let $A$ be a set of 2025 positive real numbers. For a subset $T \subseteq A$ , define $M_T$ as the median of $T$ when all elements of $T$ are arranged in increasing order, with the convention that $M_\emptyset = 0$ . Define
$P(A) = \sum_{\substack{T \subseteq A \\ |T| \text{ odd}}} M_T, \quad Q(A) = \sum_{\substack{T \subseteq A \\ |T| \text{ even}}} M_T.$ Find the smallest real number $C$ such that for any set $A$ of 2025 positive real numbers, the following inequality holds:
$P(A) - Q(A) \leq C \cdot \max(A),$ where $\max(A)$ denotes the largest element in $A$ .

1 reply

EthanWYX2009

5 hours ago

hukilau17

3 hours ago

Deriving Van der Waerden Theorem

Didier2 0

3 hours ago

Source: Khamovniki 2023-2024 (group 10-1)

Suppose we have already proved that for any coloring of $\Large \mathbb{N}$ in $r$ colors, there exists an arithmetic progression of size $k$ . How can we derive Van der Waerden's theorem for $W(r, k)$ from this?

0 replies

Didier2

3 hours ago

0 replies

Not so classic orthocenter problem

m4thbl3nd3r 6

N 3 hours ago by maths_enthusiast_0001

Source: own?

Let $O$ be circumcenter of a non-isosceles triangle $ABC$ and $H$ be a point in the interior of $\triangle ABC$ . Let $E,F$ be foots of perpendicular lines from $H$ to $AC,AB$ . Suppose that $BCEF$ is cyclic and $M$ is the circumcenter of $BCEF$ , $HM\cap AB=K,AO\cap BE=T$ . Prove that $KT$ bisects $EF$

6 replies

m4thbl3nd3r

Yesterday at 4:59 PM

maths_enthusiast_0001

3 hours ago

Triangle determined by three Simson lines

Strobenz 1

N Oct 19, 2018 by brokendiamond

Let $ABC$ be a triangle and let $P, Q, R$ be three points lying on the circumcircle of $ABC$ . Prove that if the three Simson lines determined by $P, Q$ and $R$ intersect at $X, Y, Z$ , then the triangle $XYZ$ is similar to the triangle $PQR$ .

I've looked into this problem since forever and I just have no idea how to prove it! Any help would really be appreciated!

1 reply