Collinearity with orthocenter

by Retemoeg, Mar 30, 2025, 4:42 PM

Given scalene triangle $ABC$ with circumcenter $(O)$ . Let $H$ be a point on $(BOC)$ such that $\angle AOH = 90^{\circ}$ . Denote $N$ the point on $(O)$ satisfying $AN \parallel BC$ . If $L$ is the projection of $H$ onto $BC$ , show that $LN$ passes through the orthocenter of $\triangle ABC$ .

geometry

Easy problem

by Hip1zzzil, Mar 30, 2025, 1:18 PM

$(C,M,S)$ is a pair of real numbers such that

$2C+M+S-2C^{2}-2CM-2MS-2SC=0$
$C+2M+S-3M^{2}-3CM-3MS-3SC=0$
$C+M+2S-4S^{2}-4CM-4MS-4SC=0$

Find $2C+3M+4S$ .

This post has been edited 1 time. Last edited by Hip1zzzil, Yesterday at 1:20 PM
Reason: Rr

algebra

Polynomials and their shift with all real roots and in common

by Assassino9931, Mar 30, 2025, 1:12 PM

We call two non-constant polynomials friendly if each of them has only real roots, and every root of one polynomial is also a root of the other. For two friendly polynomials $P(x), Q(x)$ and a constant $C \in \mathbb{R}, C \neq 0$ , it is given that $P(x) + C$ and $Q(x) + C$ are also friendly polynomials. Prove that $P(x) \equiv Q(x)$ .

This post has been edited 1 time. Last edited by Assassino9931, Yesterday at 1:13 PM

algebra

polynomial

Another "OR" FE problem

by pokmui9909, Mar 29, 2025, 10:16 AM

Let $\mathbb{R}$ be the set of real numbers. Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ that satisfy the following condition. Here, $f^{100}(x)$ is the function obtained by composing $f(x)$ $100$ times, that is, $(\underbrace{f \circ f \circ \cdots \circ f}_{100 \ \text{times}})(x).$

(Condition) For all $x, y \in \mathbb{R}$ , $f(x + f^{100}(y)) = x + y \ \ \ \text{or} \ \ \ f(f^{100}(x) + y) = x + y$

This post has been edited 1 time. Last edited by pokmui9909, Saturday at 10:25 AM

algebra

function

functional equation

Incenter and midpoint geom

by sarjinius, Jul 17, 2024, 12:41 PM

Let $ABC$ be a triangle with $AB < AC < BC$ . Let the incenter and incircle of triangle $ABC$ be $I$ and $\omega$ , respectively. Let $X$ be the point on line $BC$ different from $C$ such that the line through $X$ parallel to $AC$ is tangent to $\omega$ . Similarly, let $Y$ be the point on line $BC$ different from $B$ such that the line through $Y$ parallel to $AB$ is tangent to $\omega$ . Let $AI$ intersect the circumcircle of triangle $ABC$ at $P \ne A$ . Let $K$ and $L$ be the midpoints of $AC$ and $AB$ , respectively.
Prove that $\angle KIL + \angle YPX = 180^{\circ}$ .

Proposed by Dominik Burek, Poland

This post has been edited 4 times. Last edited by sarjinius, Jul 17, 2024, 4:20 PM

parallel tangent line

Orthocenter madness once again!

by MathLuis, Oct 22, 2023, 10:58 PM

Let $ABC$ be an acute triangle with orthocenter $H$ . Points $A_1$ , $B_1$ , $C_1$ are chosen in the interiors of sides $BC$ , $CA$ , $AB$ , respectively, such that $\triangle A_1B_1C_1$ has orthocenter $H$ . Define $A_2 = \overline{AH} \cap \overline{B_1C_1}$ , $B_2 = \overline{BH} \cap \overline{C_1A_1}$ , and $C_2 = \overline{CH} \cap \overline{A_1B_1}$ .

Prove that triangle $A_2B_2C_2$ has orthocenter $H$ .

Ankan Bhattacharya

This post has been edited 2 times. Last edited by v_Enhance, Oct 22, 2023, 11:44 PM

The reflection of AD intersect (ABC) lies on (AEF)

by alifenix-, Jan 27, 2020, 5:00 PM

Let $ABC$ be a triangle. Distinct points $D$ , $E$ , $F$ lie on sides $BC$ , $AC$ , and $AB$ , respectively, such that quadrilaterals $ABDE$ and $ACDF$ are cyclic. Line $AD$ meets the circumcircle of $\triangle ABC$ again at $P$ . Let $Q$ denote the reflection of $P$ across $BC$ . Show that $Q$ lies on the circumcircle of $\triangle AEF$ .

Proposed by Ankan Bhattacharya

This post has been edited 2 times. Last edited by alifenix-, Jan 27, 2020, 7:03 PM

geometry

Angle Chasing

auyesl

Number of times to do Euclidean GCD.

by MarkBcc168, Jul 10, 2018, 11:21 AM

Let $p$ be an odd prime number and $\mathbb{Z}_{>0}$ be the set of positive integers. Suppose that a function $f:\mathbb{Z}_{>0}\times\mathbb{Z}_{>0}\to\{0,1\}$ satisfies the following properties:

$f(1,1)=0$ .
$f(a,b)+f(b,a)=1$ for any pair of relatively prime positive integers $(a,b)$ not both equal to 1;
$f(a+b,b)=f(a,b)$ for any pair of relatively prime positive integers $(a,b)$ .

Prove that
$\sum_{n=1}^{p-1}f(n^2,p) \geqslant \sqrt{2p}-2.$

This post has been edited 4 times. Last edited by MarkBcc168, Feb 8, 2020, 2:01 PM

IMO Shortlist

number theory

Euclidean algorithm

Similar triangles and complementary angles

by math154, Jul 2, 2012, 3:16 AM

Let $\triangle ABC$ be an acute triangle with circumcenter $O$ such that $AB<AC$ , let $Q$ be the intersection of the external bisector of $\angle A$ with $BC$ , and let $P$ be a point in the interior of $\triangle ABC$ such that $\triangle BPA$ is similar to $\triangle APC$ . Show that $\angle QPA + \angle OQB = 90^{\circ}$ .

Alex Zhu.

USAMO 1995

by paul_mathematics, Dec 31, 2004, 1:01 PM

Given a nonisosceles, nonright triangle ABC, let O denote the center of its circumscribed circle, and let $A_1$ , $B_1$ , and $C_1$ be the midpoints of sides BC, CA, and AB, respectively. Point $A_2$ is located on the ray $OA_1$ so that $OAA_1$ is similar to $OA_2A$ . Points $B_2$ and $C_2$ on rays $OB_1$ and $OC_1$ , respectively, are defined similarly. Prove that lines $AA_2$ , $BB_2$ , and $CC_2$ are concurrent, i.e. these three lines intersect at a point.

perpendicular bisector

Submit

done!
$~~~~$
by pupitrethebean, Mar 19, 2025, 2:05 PM
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pupitre

by Retemoeg, Mar 30, 2025, 4:42 PM

by Hip1zzzil, Mar 30, 2025, 1:18 PM

by Assassino9931, Mar 30, 2025, 1:12 PM

by pokmui9909, Mar 29, 2025, 10:16 AM

by sarjinius, Jul 17, 2024, 12:41 PM

by MathLuis, Oct 22, 2023, 10:58 PM

by alifenix-, Jan 27, 2020, 5:00 PM

by MarkBcc168, Jul 10, 2018, 11:21 AM

by math154, Jul 2, 2012, 3:16 AM

by paul_mathematics, Dec 31, 2004, 1:01 PM