[/quote]
,
s.t 
be positive real numbers . Prove that :
and 
be isogonal conjugates with respect to 
.
and 
be their respective pedal triangles. Let![\[ X_1=P_2Q_3\cap P_3Q_2,\quad X_2=P_1Q_3\cap P_3Q_1,\quad X_3=P_1Q_2\cap P_2Q_1 \]](http://latex.artofproblemsolving.com/7/f/e/7fe3ec4f69ec074635b2202daad9371e3bb8c72d.png)
Prove that the points 
,
and 
lie on the line 
.
,
,
be real numbers. Prove that![\[ab + bc + ca + \max\{|a - b|, |b - c|, |c - a|\} \le 1 + \frac{1}{3} (a + b + c)^2.\]](http://latex.artofproblemsolving.com/3/4/9/3499fc076778ca2d6b4ba57c37460d45da166b8c.png)
be integer numbers satisfy that: 
.
?
of an isosceles right-angled triangle 
such 
and 
are marked, such that 
.
.
.
be its circumcircle, 
its orthocenter, and 
the foot of the altitude from 
.
be the midpoint of 
.
such that 
and let 
.
,
,
and 
are tangent to each other.
be positive real numbers. Find the minimum value of![\[ T = (x+y+z+t)^2.\left[\dfrac{1}{x(y+z+t)}+\dfrac{1}{y(z+t+x)}+\dfrac{1}{z(t+x+y)}+\dfrac{1}{t(x+y+z)}\right] \]](http://latex.artofproblemsolving.com/c/a/c/cac0967ebdf30c1f79a1ec3e468cc035402333a7.png)
,
,
,
be the diametrically opposite point of 
and 
intersect the incircle again at 
and 
,
,
,
are concurrent, i.e., the lines intersect at the same point.
and 
be two similar right-angled triangles (in the same plane) with 
such that 
.
,
.
is written on the blackboard. Every time, Carmela can erase the number 
on the black board and replace it with a new number 
,
is a perfect square. Prove or disprove that all positive integers 
can be written exactly once on the blackboard.
,
such that 
lie on 
be its center. Let 
be some point lying on the circumcircle of 
.
is tangent to 
is tangent to 
and 
,
meet the circumcircle of 
and 
.
,
.
and 
exist on 
and 
are tangent to 
,
,
,
.
at the orthocenter of 
,
be a point lying on 
such that 
.
.
be its incircle, and let 
be the midpoint of minor arc 
on the circumcircle of 
at points 
,
and 
.
and 
denote the 
,
,
and 
,
and 
.
and 
are tangent to each other.
and 
be circles tangent to 
meets 
.
similarly.
and 
.
and 
.
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Hello AoPS!
Like each of the past few IMOs, there will be another livestream hosted by former USA IMO team members, attempting the 2021 IMO Problems! This year's stream will be hosted by James Lin and Edward Wan, with special guests Mihir Singhal and Zack Chroman, and proctored by Michael Ren!
We will start Tuesday, July 20 at 8 PM EDT, over at https://www.twitch.tv/jayliner66. We will solve the problems in order, with the ambitious plan to complete all six problems by the end of the night. If you can, please join us live to see us struggle and to cheer us on -- and if not, the broadcast will be archived for the sixty more days (no guarantees as I'm unfamiliar with streaming on Twitch). Finally, good luck to all of the competitors at this year's IMO -- especially team USA!
Edit: The time has been pushed back to 8 PM EDT to align with the official release of the IMO Problems.
Like each of the past few IMOs, there will be another livestream hosted by former USA IMO team members, attempting the 2021 IMO Problems! This year's stream will be hosted by James Lin and Edward Wan, with special guests Mihir Singhal and Zack Chroman, and proctored by Michael Ren!
We will start Tuesday, July 20 at 8 PM EDT, over at https://www.twitch.tv/jayliner66. We will solve the problems in order, with the ambitious plan to complete all six problems by the end of the night. If you can, please join us live to see us struggle and to cheer us on -- and if not, the broadcast will be archived for the sixty more days (no guarantees as I'm unfamiliar with streaming on Twitch). Finally, good luck to all of the competitors at this year's IMO -- especially team USA!
Edit: The time has been pushed back to 8 PM EDT to align with the official release of the IMO Problems.
This post has been edited 2 times. Last edited by mcdonalds106_7, Jul 19, 2021, 2:39 AM
Come back now to watch day 2, hopefully with less technical difficulties.
