Proving the line is indeed a radical axis

by azzam2912, May 17, 2025, 6:23 AM

Given an acute triangle ABC with altitudes AD, BE, and CF intersecting at point H. Let O be the center of the circumcircle of triangle ABC. The Tangents to the circumcircle of triangle ABC from points B and C intersect at point T. Let K and L be reflections of point O on lines AB and AC respectively. The circumcircles of triangle DFK and DEL intersect a second time at point P. Prove that points P, D, and T are collinear.

geometry

power of a point

radical axis

A point on BC

by jayme, May 17, 2025, 6:08 AM

Dear Mathlinkers,

1. ABC a triangle
2. 0 the circumcircle
3. D the pole of BC wrt 0
4. B', C' the symmetrics of B, C wrt AC, AB
5. 1b, 1c the circumcircles of the triangles BB'D, CC'D
6. T the second point of intersection of the tangent to 1c at D with 1b.

Prove : B, C and T are collinear.

Sincerely
Jean-Louis

geometry

Balkan Mathematical Olympiad

by ABCD1728, May 17, 2025, 5:44 AM

Can anyone provide the PDF version of the book "Balkan Mathematical Olympiads" by Mircea Becheanu and Bogdan Enescu (published by XYZ press in 2014), thanks!

BMO

IMO

Find all p(x) such that p(p) is a power of 2

by truongphatt2668, May 15, 2025, 1:05 PM

Find all polynomial $P(x) \in \mathbb{R}[x]$ such that:
$P(p_i) = 2^{a_i}$ with $p_i$ is an $i$ th prime and $a_i$ is an arbitrary positive integer.

algebra

polynomial

A sharp one with 3 var

by mihaig, May 13, 2025, 7:20 PM

Let $a,b,c\geq0$ satisfying
$\left(a+b+c-2\right)^2+8\leq3\left(ab+bc+ca\right).$ Prove
$ab+bc+ca+abc\geq4.$

inequalities

Interesting problem from a friend

by v4913, Nov 25, 2023, 12:49 AM

Let the incircle $(I)$ of $\triangle{ABC}$ touch $BC$ at $D$ , $ID \cap (I) = K$ , let $\ell$ denote the line tangent to $(I)$ through $K$ . Define $E, F \in \ell$ such that $\angle{EIF} = 90^{\circ}, EI, FI \cap (AEF) = E', F'$ . Prove that the circumcenter $O$ of $\triangle{ABC}$ lies on $E'F'$ .

geometry

incircle

Circumcenter

Inequality on APMO P5

by Jalil_Huseynov, May 17, 2022, 6:50 PM

Let $a,b,c,d$ be real numbers such that $a^2+b^2+c^2+d^2=1$ . Determine the minimum value of $(a-b)(b-c)(c-d)(d-a)$ and determine all values of $(a,b,c,d)$ such that the minimum value is achived.

inequalities

APMO

APMO 2022

Problem 60:

by henderson, Jan 25, 2017, 6:06 PM

ghghghhghg

geometry

angle bisector

parallelogram

IMO 2016 Problem 2

by shinichiman, Jul 11, 2016, 6:38 AM

Find all integers $n$ for which each cell of $n \times n$ table can be filled with one of the letters $I,M$ and $O$ in such a way that:

in each row and each column, one third of the entries are $I$ , one third are $M$ and one third are $O$ ; and
in any diagonal, if the number of entries on the diagonal is a multiple of three, then one third of the entries are $I$ , one third are $M$ and one third are $O$ .

Note. The rows and columns of an $n \times n$ table are each labelled $1$ to $n$ in a natural order. Thus each cell corresponds to a pair of positive integer $(i,j)$ with $1 \le i,j \le n$ . For $n>1$ , the table has $4n-2$ diagonals of two types. A diagonal of first type consists all cells $(i,j)$ for which $i+j$ is a constant, and the diagonal of this second type consists all cells $(i,j)$ for which $i-j$ is constant.

This post has been edited 2 times. Last edited by shinichiman, Jul 11, 2016, 6:40 AM

3 numbers have their fractional parts lying in the interval

by orl, Aug 10, 2008, 1:01 AM

Let $a, b, c$ be positive integers satisfying the conditions $b > 2a$ and $c > 2b.$ Show that there exists a real number $\lambda$ with the property that all the three numbers $\lambda a, \lambda b, \lambda c$ have their fractional parts lying in the interval $\left(\frac {1}{3}, \frac {2}{3} \right].$

This post has been edited 1 time. Last edited by Amir Hossein, May 11, 2011, 10:11 AM
Reason: Fixed, thanks math154!

IMO ShortList 2002, algebra problem 3

by orl, Sep 28, 2004, 1:19 PM

Let $P$ be a cubic polynomial given by $P(x)=ax^3+bx^2+cx+d$ , where $a,b,c,d$ are integers and $a\ne0$ . Suppose that $xP(x)=yP(y)$ for infinitely many pairs $x,y$ of integers with $x\ne y$ . Prove that the equation $P(x)=0$ has an integer root.