Integral-Summation Duality

by Mathandski, Apr 28, 2025, 7:58 PM

Given a continuous function $f$ such that $f(2x) = 3 f(x)$ and $\int_0^1 f(x) \, dx = 1$ , evaluate $\int_1^2 f(x) \, dx$ .

function

algebra

functional equation

Summation

Integral

Interesting number theory

by giangtruong13, Apr 28, 2025, 4:15 PM

Let $a,b$ be integer numbers $\geq 3$ satisfy that: $a^2=b^3+ab$ . Prove that:
a) $a,b$ are even
b) $4b+1$ is a perfect square number
c) $a$ can’t be any power $\geq 1$ of a positive integer number

number theory

amazing balkan combi

by egxa, Apr 27, 2025, 1:57 PM

There are $n$ cities in a country, where $n \geq 100$ is an integer. Some pairs of cities are connected by direct (two-way) flights. For two cities $A$ and $B$ we define:

$(i)$ A $\emph{path}$ between $A$ and $B$ as a sequence of distinct cities $A = C_0, C_1, \dots, C_k, C_{k+1} = B$ , $k \geq 0$ , such that there are direct flights between $C_i$ and $C_{i+1}$ for every $0 \leq i \leq k$ ;
$(ii)$ A $\emph{long path}$ between $A$ and $B$ as a path between $A$ and $B$ such that no other path between $A$ and $B$ has more cities;
$(iii)$ A $\emph{short path}$ between $A$ and $B$ as a path between $A$ and $B$ such that no other path between $A$ and $B$ has fewer cities.
Assume that for any pair of cities $A$ and $B$ in the country, there exist a long path and a short path between them that have no cities in common (except $A$ and $B$ ). Let $F$ be the total number of pairs of cities in the country that are connected by direct flights. In terms of $n$ , find all possible values $F$

Proposed by David-Andrei Anghel, Romania.

This post has been edited 6 times. Last edited by egxa, Yesterday at 10:59 PM

Combi

combinatorics

Arbitrary point on BC and its relation with orthocenter

by falantrng, Apr 27, 2025, 11:47 AM

In an acute-angled triangle $ABC$ , $H$ be the orthocenter of it and $D$ be any point on the side $BC$ . The points $E, F$ are on the segments $AB, AC$ , respectively, such that the points $A, B, D, F$ and $A, C, D, E$ are cyclic. The segments $BF$ and $CE$ intersect at $P.$ $L$ is a point on $HA$ such that $LC$ is tangent to the circumcircle of triangle $PBC$ at $C.$ $BH$ and $CP$ intersect at $X$ . Prove that the points $D, X,$ and $L$ lie on the same line.

Proposed by Theoklitos Parayiou, Cyprus

This post has been edited 1 time. Last edited by falantrng, Yesterday at 4:38 PM

geometry

BMO

BMO 2025

by GreekIdiot, Apr 27, 2025, 11:39 AM

Does anyone have the problems? They should have finished by now.

contest

BMO

Trillium geometry

by Assassino9931, Feb 3, 2023, 10:04 PM

The angle bisectors at $A$ and $C$ in a non-isosceles triangle $ABC$ with incenter $I$ intersect its circumcircle $k$ at $A_0$ and $C_0$ , respectively. The line through $I$ , parallel to $AC$ , intersects $A_0C_0$ at $P$ . Prove that $PB$ is tangent to $k$ .

function

by CarlFriedrichGauss-1777, Jun 4, 2021, 1:40 PM

Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that:
$f(2021+xf(y))=yf(x+y+2021)$

function

Similarity through arc midpoint in right triangle

by cjquines0, May 26, 2017, 11:15 AM

Let $\omega$ be the circumcircle of right-angled triangle $ABC$ ( $\angle A = 90^{\circ}$ ). The tangent to $\omega$ at point $A$ intersects the line $BC$ at point $P$ . Suppose that $M$ is the midpoint of the minor arc $AB$ , and $PM$ intersects $\omega$ for the second time in $Q$ . The tangent to $\omega$ at point $Q$ intersects $AC$ at $K$ . Prove that $\angle PKC = 90^{\circ}$ .

Proposed by Davood Vakili

This post has been edited 1 time. Last edited by cjquines0, May 26, 2017, 11:15 AM

geometry

circumcircle

Quadratic system

by juckter, Jun 22, 2014, 4:27 PM

Let $n$ be a positive integer. Find all real solutions $(a_1, a_2, \dots, a_n)$ to the system:

$a_1^2 + a_1 - 1 = a_2$ $a_2^2 + a_2 - 1 = a_3$ $\hspace*{3.3em} \vdots$ $a_{n}^2 + a_n - 1 = a_1$

This post has been edited 1 time. Last edited by juckter, Dec 5, 2016, 2:01 AM

Can this sequence be bounded?

by darij grinberg, Jan 19, 2005, 11:00 AM

Let $a_0$ , $a_1$ , $a_2$ , ... be an infinite sequence of real numbers satisfying the equation $a_n=\left|a_{n+1}-a_{n+2}\right|$ for all $n\geq 0$ , where $a_0$ and $a_1$ are two different positive reals.

Can this sequence $a_0$ , $a_1$ , $a_2$ , ... be bounded?

Proposed by Mihai Bălună, Romania

This post has been edited 1 time. Last edited by djmathman, Sep 27, 2015, 2:12 PM