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Contests & Programs AMC and other contests, summer programs, etc.

AMC and other contests, summer programs, etc.

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Contests & Programs AMC and other contests, summer programs, etc.

AMC and other contests, summer programs, etc.

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centroid wanted, point that minimizes sum of squares of distances from sides

parmenides51 1

N an hour ago by SuperBarsh

Source: Oliforum Contest V 2017 p9 https://artofproblemsolving.com/community/c2487525_oliforum_contes

Given a triangle $ABC$ , let $P$ be the point which minimizes the sum of squares of distances from the sides of the triangle. Let $D, E, F$ the projections of $P$ on the sides of the triangle $ABC$ . Show that $P$ is the barycenter of $DEF$ .

(Jack D’Aurizio)

1 reply

parmenides51

Sep 29, 2021

SuperBarsh

an hour ago

Strictly monotone polynomial with an extra condition

Popescu 11

N an hour ago by Iveela

Source: IMSC 2024 Day 2 Problem 2

Let $\mathbb{R}_{>0}$ be the set of all positive real numbers. Find all strictly monotone (increasing or decreasing) functions $f:\mathbb{R}_{>0} \to \mathbb{R}$ such that there exists a two-variable polynomial $P(x, y)$ with real coefficients satisfying
$f(xy)=P(f(x), f(y))$ for all $x, y\in\mathbb{R}_{>0}$ .

Proposed by Navid Safaei, Iran

11 replies

Popescu

Jun 29, 2024

Iveela

an hour ago

Hard geometry

Lukariman 1

N an hour ago by ricarlos

Given triangle ABC, any line d intersects AB at D, intersects AC at E, intersects BC at F. Let O1,O2,O3 be the centers of the circles circumscribing triangles ADE, BDF, CFE. Prove that the orthocenter of triangle O1O2O3 lies on line d.

1 reply

Lukariman

May 12, 2025

ricarlos

an hour ago

Russian Diophantine Equation

LeYohan 1

N an hour ago by Natrium

Source: Moscow, 1963

Find all integer solutions to

$\frac{xy}{z} + \frac{xz}{y} + \frac{yz}{x} = 3$ .

1 reply

LeYohan

Yesterday at 2:59 PM

Natrium

an hour ago

Simple Geometry

AbdulWaheed 6

N an hour ago by Gggvds1

Source: EGMO

Try to avoid Directed angles
Let ABC be an acute triangle inscribed in circle $\Omega$ . Let $X$ be the midpoint of the arc $\overarc{BC}$ not containing $A$ and define $Y, Z$ similarly. Show that the orthocenter of $XYZ$ is the incenter $I$ of $ABC$ .

6 replies

AbdulWaheed

May 23, 2025

Gggvds1

an hour ago

Bosnia and Herzegovina JBMO TST 2013 Problem 4

gobathegreat 4

N an hour ago by FishkoBiH

Source: Bosnia and Herzegovina Junior Balkan Mathematical Olympiad TST 2013

It is given polygon with $2013$ sides $A_{1}A_{2}...A_{2013}$ . His vertices are marked with numbers such that sum of numbers marked by any $9$ consecutive vertices is constant and its value is $300$ . If we know that $A_{13}$ is marked with $13$ and $A_{20}$ is marked with $20$ , determine with which number is marked $A_{2013}$

4 replies

gobathegreat

Sep 16, 2018

FishkoBiH

an hour ago

A geometry problem

Lttgeometry 2

N an hour ago by Acrylic3491

Triangle $ABC$ has two isogonal conjugate points $P$ and $Q$ . The circle $(BPC)$ intersects circle $(AP)$ at $R \neq P$ , and the circle $(BQC)$ intersects circle $(AQ)$ at $S\neq Q$ . Prove that $R$ and $S$ are isogonal conjugates in triangle $ABC$ .
Note: Circle $(AP)$ is the circle with diameter $AP$ , Circle $(AQ)$ is the circle with diameter $AQ$ .

2 replies

Lttgeometry

Today at 4:03 AM

Acrylic3491

an hour ago

anglechasing , circumcenter wanted

parmenides51 1

N an hour ago by Captainscrubz

Source: Sharygin 2011 Final 9.2

In triangle $ABC, \angle B = 2\angle C$ . Points $P$ and $Q$ on the medial perpendicular to $CB$ are such that $\angle CAP = \angle PAQ = \angle QAB = \frac{\angle A}{3}$ . Prove that $Q$ is the circumcenter of triangle $CPB$ .

1 reply

parmenides51

Dec 16, 2018

Captainscrubz

an hour ago

Nice FE over R+

doanquangdang 4

N 2 hours ago by jasperE3

Source: collect

Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ such that
$x+f(yf(x)+1)=xf(x+y)+yf(yf(x))$ for all $x,y>0.$

4 replies

doanquangdang

Jul 19, 2022

jasperE3

2 hours ago

right triangle, midpoints, two circles, find angle

star-1ord 0

2 hours ago

Source: Estonia Final Round 2025 8-3

In the right triangle $ABC$ , $M$ is the midpoint of the hypotenuse $AB$ . Point $D$ is chosen on the leg $BC$ so that the line segment $DM$ meets $(ACD)$ again at $K$ ( $K\neq D$ ). Let $L$ be the reflection of $K$ in $M$ . The circles $(ACD)$ and $(BCL)$ meet again at $N$ ( $N\neq C$ ). Find the measure of $\angle KNL$ .

0 replies

star-1ord

2 hours ago

0 replies

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