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Geometry hard
Lukariman   0
19 minutes ago
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).
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Lukariman
19 minutes ago
0 replies
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Great similarity
steven_zhang123   0
20 minutes ago
Source: a friend
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 $\angle ACE = \angle AFC$.
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+2 w
steven_zhang123
20 minutes ago
0 replies
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Integer polynomial w factorials
Solilin   0
20 minutes ago
Source: 9th Thailand MO
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.
0 replies
Solilin
20 minutes ago
0 replies
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Coloring plane in black
Ryan-asadi   2
N 30 minutes ago by Primeniyazidayi
Source: Iran Team Selection Test - P3
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2 replies
Ryan-asadi
4 hours ago
Primeniyazidayi
30 minutes ago
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B.Stat & B.Math 2022 - Q8
integrated_JRC   6
N an hour ago by Titeer_Bhar
Source: Indian Statistical Institute (ISI) - B.Stat & B.Math Entrance 2022
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 $\frac{\pi}{2}$.
6 replies
integrated_JRC
May 8, 2022
Titeer_Bhar
an hour ago
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AD=BE implies ABC right
v_Enhance   117
N an hour ago by cj13609517288
Source: European Girl's MO 2013, Problem 1
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.
117 replies
v_Enhance
Apr 10, 2013
cj13609517288
an hour ago
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IMO Genre Predictions
ohiorizzler1434   64
N an hour ago by ariopro1387
Everybody, with IMO upcoming, what are you predictions for the problem genres?


Personally I predict: predict
64 replies
ohiorizzler1434
May 3, 2025
ariopro1387
an hour ago
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3-var inequality
sqing   1
N 2 hours ago by sqing
Source: Own
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
$$a^3+ab+b^3 \geq 3$$$$ a^3-ab+b^3\geq 1 $$
1 reply
sqing
2 hours ago
sqing
2 hours ago
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Iranians playing with cards module a prime number.
Ryan-asadi   2
N 2 hours ago by AshAuktober
Source: Iranian Team Selection Test - P2
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2 replies
Ryan-asadi
4 hours ago
AshAuktober
2 hours ago
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An analytic sequence
Ryan-asadi   1
N 2 hours ago by AshAuktober
Source: Iran Team Selection Test - P1
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1 reply
Ryan-asadi
4 hours ago
AshAuktober
2 hours ago
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Point moving towards vertices and changing plans again and again
Miquel-point   0
Apr 6, 2025
Source: Romanian IMO TST 1981, Day 3 P6
In the plane of traingle $ABC$ we consider a variable point $M$ which moves on line $MA$ towards $A$. Halfway there, it stops and starts moving in a straight line line towards $B$. Halfway there, it stops and starts moving in a straight line towards $C$, and halfway there it stops and starts moving in a straight line towards $A$, and so on. Show that $M$ will get as close as we want to the vertices of a fixed triangle with area $\text{area}(ABC)/7$.
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Miquel-point
Apr 6, 2025
0 replies
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Point moving towards vertices and changing plans again and again
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Source: Romanian IMO TST 1981, Day 3 P6
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Miquel-point
478 posts
Miquel-point
#1 h  Apr 6, 2025, 7:47 PM • 1 Y
Y by PikaPika999
In the plane of traingle $ABC$ we consider a variable point $M$ which moves on line $MA$ towards $A$. Halfway there, it stops and starts moving in a straight line line towards $B$. Halfway there, it stops and starts moving in a straight line towards $C$, and halfway there it stops and starts moving in a straight line towards $A$, and so on. Show that $M$ will get as close as we want to the vertices of a fixed triangle with area $\text{area}(ABC)/7$.
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