Geometry hard

by Lukariman, May 7, 2025, 2:14 PM

Given triangle ABC inscribed in circle (O). The bisector of angle A intersects (O) at D. Let M, N be the midpoints of AB, AC respectively. OD intersects BC at P and AD intersects MN at S. The circle circumscribed around triangle MPS intersects BC at Q different from P. Prove that QA is tangent to (O).

This post has been edited 1 time. Last edited by Lukariman, 19 minutes ago

geometry

Great similarity

by steven_zhang123, May 7, 2025, 2:13 PM

As shown in the figure, there are two points $D$ and $E$ outside triangle $ABC$ such that $\angle DAB = \angle CAE$ and $\angle ABD + \angle ACE = 180^{\circ}$ . Connect $BE$ and $DC$ , which intersect at point $O$ . Let $AO$ intersect $BC$ at point $F$ . Prove that $\angle ACE = \angle AFC$ .

Attachments:

This post has been edited 1 time. Last edited by steven_zhang123, 19 minutes ago

geometry

similar triangles

similarity

Integer polynomial w factorials

by Solilin, May 7, 2025, 2:12 PM

Let $a_1, a_2, ..., a_{2012}$ be pairwise distinct integers. Show that the equation $(x -a_1)(x - a_2)...(x - a_{2012}) = (1006!)^2$ has at most one integral solution.

3-var inequality

by sqing, May 7, 2025, 12:43 PM

Let $a,b>0$ and $\frac{1}{a^2+3}+ \frac{1}{b^2+ 3} \leq \frac{1}{2} .$ Prove that
$a^2+ab+b^2\geq 3$ $a^2-ab+b^2 \geq 1$ Let $a,b>0$ and $\frac{1}{a^3+3}+ \frac{1}{b^3+ 3}\leq \frac{1}{2} .$ Prove that
$a^3+ab+b^3 \geq 3$ $a^3-ab+b^3\geq 1$

This post has been edited 1 time. Last edited by sqing, 2 hours ago

inequalities proposed

inequalities

Coloring plane in black

by Ryan-asadi, May 7, 2025, 10:55 AM

..........

This post has been edited 1 time. Last edited by Ryan-asadi, 2 hours ago

combinatorics

Iranians playing with cards module a prime number.

by Ryan-asadi, May 7, 2025, 10:41 AM

.........

This post has been edited 1 time. Last edited by Ryan-asadi, 2 hours ago

number theory

prime numbers

An analytic sequence

by Ryan-asadi, May 7, 2025, 10:10 AM

..........

This post has been edited 1 time. Last edited by Ryan-asadi, 2 hours ago

algebra

IMO Genre Predictions

by ohiorizzler1434, May 3, 2025, 6:51 AM

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

Personally I predict: predict

B.Stat & B.Math 2022 - Q8

by integrated_JRC, May 8, 2022, 11:53 AM

Find the minimum value of $\big|\sin x+\cos x+\tan x+\cot x+\sec x+\operatorname{cosec}x\big|$ for real numbers $x$ not multiple of $\frac{\pi}{2}$ .

isi

Indian Statistical Institute

AD=BE implies ABC right

by v_Enhance, Apr 10, 2013, 3:12 PM

The side $BC$ of the triangle $ABC$ is extended beyond $C$ to $D$ so that $CD = BC$ . The side $CA$ is extended beyond $A$ to $E$ so that $AE = 2CA$ . Prove that, if $AD=BE$ , then the triangle $ABC$ is right-angled.

Submit

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91 shouts

math_exploration

by Lukariman, May 7, 2025, 2:14 PM

by steven_zhang123, May 7, 2025, 2:13 PM

by Solilin, May 7, 2025, 2:12 PM

by sqing, May 7, 2025, 12:43 PM

by Ryan-asadi, May 7, 2025, 10:55 AM

by Ryan-asadi, May 7, 2025, 10:41 AM

by Ryan-asadi, May 7, 2025, 10:10 AM

by ohiorizzler1434, May 3, 2025, 6:51 AM

by integrated_JRC, May 8, 2022, 11:53 AM

by v_Enhance, Apr 10, 2013, 3:12 PM