nice problem

by hanzo.ei, Mar 29, 2025, 5:58 PM

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 $AT \perp BC$ .

geometry unsolved

geometry

2025 Caucasus MO Seniors P2

by BR1F1SZ, Mar 26, 2025, 12:39 AM

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 $\angle ABB' = \angle CBB_2$ .

geometry

Nordic 2025 P3

by anirbanbz, Mar 25, 2025, 12:36 PM

Let $ABC$ be an acute triangle with orthocenter $H$ and circumcenter $O$ . Let $E$ and $F$ be points on the line segments $AC$ and $AB$ respectively such that $AEHF$ is a parallelogram. Prove that $\vert OE \vert = \vert OF \vert$ .

NMC

geometry

Hard limits

by Snoop76, Mar 25, 2025, 9:13 AM

$a_n$ 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 $\text{[math]}$ $\lim_{n \to \infty} \frac{b_n}{(2n+1)!!}.$

series

limit

double factorial

D1018 : Can you do that ?

by Dattier, Mar 24, 2025, 6:01 AM

We can find $A,B,C$ , such that $\gcd(A,B)=\gcd(C,A)=\gcd(A,2)=1$ and $\forall n \in \mathbb N^*, (C^n \times B \mod A) \mod 2=0$ .

For example :

$C=20$
$A=47650065401584409637777147310342834508082136874940478469495402430677786194142956609253842997905945723173497630499054266092849839$

$B=238877301561986449355077953728734922992395532218802882582141073061059783672634737309722816649187007910722185635031285098751698$

Can you find $A,B,C$ such that $A>3$ is prime, $C,B \in (\mathbb Z/A\mathbb Z)^*$ with $o(C)=(A-1)/2$ and $\forall n \in \mathbb N^*, (C^n \times B \mod A) \mod 2=0$ ?

arithmetic

number theory

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

by nAalniaOMliO, Jul 24, 2024, 7:40 PM

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

This post has been edited 1 time. Last edited by nAalniaOMliO, Jul 27, 2024, 10:08 AM

number theory

f( - f (x) - f (y))= 1 -x - y , in Zxz

by parmenides51, Nov 22, 2020, 1:16 PM

Find all functions $f: Z \to Z$ that satisfy $f(-f (x) - f (y))= 1 -x - y$ for all $x, y \in Z$

This post has been edited 1 time. Last edited by parmenides51, Nov 22, 2020, 1:21 PM

functional equation

functional equation in Z

functional

algebra

Find a given number of divisors of ab

by proglote, Oct 24, 2013, 9:51 PM

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.

algebra

polynomial

combinatorics proposed

combinatorics

CGMO6: Airline companies and cities

by v_Enhance, Aug 13, 2012, 10:47 PM

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 $n$ .

induction

combinatorics unsolved

combinatorics

IMO Shortlist 2010 - Problem G1

by Amir Hossein, Jul 17, 2011, 2:10 AM

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

geometry

circumcircle

geometric transformation

reflection

IMO Shortlist

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I'm sorry, I cannot make a post about the "performance" you mentioned, ohiorizzler1434.
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are you a chat gpt
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I already have a post on the Collatz Conjecture, but I'll make another, better one soon.
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Your blog looks skibidi ohio! Please make a post about the collatz conjecture next, with a full solution!
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I get emails every post you make. Also, third post!?
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Maybe conic sections?
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Does anyone have some ideas for me to write about?
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That's nice to know. I'm also learning new, interesting things on here myself, too.
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I think it would also be cool if you did a problem of the day every day, as I see from today's problem.
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42 shouts

Pi in the Sky

by hanzo.ei, Mar 29, 2025, 5:58 PM

by BR1F1SZ, Mar 26, 2025, 12:39 AM

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