complex bashing in angles??

Given circle (O) and point P outside (O). From P draw tangents PA and PB to (O) with contact points A, B. On the opposite ray of ray BP, take point M. The circle circumscribing triangle APM intersects (O) at the second point D. Let H be the projection of B on AM. Prove that $\angle HDM$ = 2∠AMP.

3 replies

Lukariman

Yesterday at 12:43 PM

Lukariman

an hour ago

USAMO 2003 Problem 1

MithsApprentice 68

N 2 hours ago by Mamadi

Prove that for every positive integer $n$ there exists an $n$ -digit number divisible by $5^n$ all of whose digits are odd.

68 replies

MithsApprentice

Sep 27, 2005

Mamadi

2 hours ago

Centroid, altitudes and medians, and concyclic points

BR1F1SZ 3

N 2 hours ago by EeEeRUT

Source: Austria National MO Part 1 Problem 2

Let $\triangle{ABC}$ be an acute triangle with $BC > AC$ . Let $S$ be the centroid of triangle $ABC$ and let $F$ be the foot of the perpendicular from $C$ to side $AB$ . The median $CS$ intersects the circumcircle $\gamma$ of triangle $\triangle{ABC}$ at a second point $P$ . Let $M$ be the point where $CS$ intersects $AB$ . The line $SF$ intersects the circle $\gamma$ at a point $Q$ , such that $F$ lies between $S$ and $Q$ . Prove that the points $M,P,Q$ and $F$ lie on a circle.

(Karl Czakler)

3 replies

BR1F1SZ

Monday at 9:45 PM

EeEeRUT

2 hours ago

IMO Genre Predictions

ohiorizzler1434 60

N 2 hours ago by Yiyj

Everybody, with IMO upcoming, what are you predictions for the problem genres?

Personally I predict: predict

ANG GCA

60 replies

ohiorizzler1434

May 3, 2025

Yiyj

2 hours ago

square root problem

kjhgyuio 5

N 2 hours ago by Solar Plexsus

........

5 replies

kjhgyuio

May 3, 2025

Solar Plexsus

2 hours ago

Diodes and usamons

v_Enhance 47

N 2 hours ago by EeEeRUT

Source: USA December TST for the 56th IMO, by Linus Hamilton

A physicist encounters $2015$ atoms called usamons. Each usamon either has one electron or zero electrons, and the physicist can't tell the difference. The physicist's only tool is a diode. The physicist may connect the diode from any usamon $A$ to any other usamon $B$ . (This connection is directed.) When she does so, if usamon $A$ has an electron and usamon $B$ does not, then the electron jumps from $A$ to $B$ . In any other case, nothing happens. In addition, the physicist cannot tell whether an electron jumps during any given step. The physicist's goal is to isolate two usamons that she is sure are currently in the same state. Is there any series of diode usage that makes this possible?

Proposed by Linus Hamilton

47 replies

v_Enhance

Dec 17, 2014

EeEeRUT

2 hours ago

3-var inequality

sqing 1

N 3 hours ago by sqing

Source: Own

Let $a,b\geq 0 ,a^3-ab+b^3=1$ . Prove that
$\frac{1}{2}\geq \frac{a}{a^2+3 }+ \frac{b}{b^2+3} \geq \frac{1}{4}$ $\frac{1}{2}\geq \frac{a}{a^3+3 }+ \frac{b}{b^3+3} \geq \frac{1}{4}$ $\frac{1}{2}\geq \frac{a}{a^2+ab+2}+ \frac{b}{b^2+ ab+2} \geq \frac{1}{3}$ $\frac{1}{2}\geq \frac{a}{a^3+ab+2}+ \frac{b}{b^3+ ab+2} \geq \frac{1}{3}$ Let $a,b\geq 0 ,a^3+ab+b^3=3$ . Prove that
$\frac{1}{2}\geq \frac{a}{a^2+3 }+ \frac{b}{b^2+3} \geq \frac{1}{4}(\frac{1}{\sqrt[3]{3}}+\sqrt[3]{3}-1)$ $\frac{1}{2}\geq \frac{a}{a^3+3 }+ \frac{b}{b^3+3} \geq \frac{1}{2\sqrt[3]{9}}$ $\frac{1}{2}\geq \frac{a}{a^2+ab+2}+ \frac{b}{b^2+ ab+2} \geq \frac{4\sqrt[3]{3}+3\sqrt[3]{9}-6}{17}$ $\frac{1}{2}\geq \frac{a}{a^3+ab+2}+ \frac{b}{b^3+ ab+2} \geq \frac{\sqrt[3]{3}}{5}$

1 reply

sqing

3 hours ago

sqing

3 hours ago

IMO ShortList 2001, combinatorics problem 3

orl 37

N 3 hours ago by deduck

Source: IMO ShortList 2001, combinatorics problem 3, HK 2009 TST 2 Q.2

Define a $k$ -clique to be a set of $k$ people such that every pair of them are acquainted with each other. At a certain party, every pair of 3-cliques has at least one person in common, and there are no 5-cliques. Prove that there are two or fewer people at the party whose departure leaves no 3-clique remaining.

37 replies

orl

Sep 30, 2004

deduck

3 hours ago

area of O_1O_2O_3O_4 <=1, incenters of right triangles outside a square

parmenides51 2

N 3 hours ago by Solilin

Source: Thailand Mathematical Olympiad 2012 p4

Let $ABCD$ be a unit square. Points $E, F, G, H$ are chosen outside $ABCD$ so that $\angle AEB =\angle BF C = \angle CGD = \angle DHA = 90^o$ . Let $O_1, O_2, O_3, O_4$ , respectively, be the incenters of $\vartriangle ABE, \vartriangle BCF, \vartriangle CDG, \vartriangle DAH$ . Show that the area of $O_1O_2O_3O_4$ is at most $1$ .

2 replies

parmenides51

Aug 17, 2020

Solilin

3 hours ago

Geo metry

TUAN2k8 3

N 3 hours ago by TUAN2k8

Help me plss!
Given an acute triangle $ABC$ . Points $D$ and $E$ lie on segments $AB$ and $AC$ , respectively. Lines $BD$ and $CE$ intersect at point $F$ . The circumcircles of triangles $BDF$ and $CEF$ intersect at a second point $P$ . The circumcircles of triangles $ABC$ and $ADE$ intersect at a second point $Q$ . Point $K$ lies on segment $AP$ such that $KQ \perp AQ$ . Prove that triangles $\triangle BKD$ and $\triangle CKE$ are similar.

3 replies

TUAN2k8

Yesterday at 10:33 AM

TUAN2k8

3 hours ago

Functional equation of nonzero reals

proglote 8

N 4 hours ago by jasperE3

Source: Brazil MO 2013, problem #3

Find all injective functions $f\colon \mathbb{R}^* \to \mathbb{R}^*$ from the non-zero reals to the non-zero reals, such that $f(x+y) \left(f(x) + f(y)\right) = f(xy)$ for all non-zero reals $x, y$ such that $x+y \neq 0$ .

8 replies

proglote

Oct 24, 2013