AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
be an acute triangle with 
. Let 
be the centroid of triangle 
and let 
be the foot of the perpendicular from 
to side 
. The median 
intersects the circumcircle 
of triangle at a second point 
. Let 
be the point where intersects . The line 
intersects the circle at a point 
, such that lies between and . Prove that the points 
and lie on a circle.
such that 
for all 
such that 
.
is written on the board. A move consists of adding either , 
or 
to the number written on the board, but only if the chosen number is coprime with the current number (for example, if the current number is 
, then in a move a player can't choose the number , but he can choose either or ). The player who first writes a perfect square on the board loses. Prove that one of the players has a winning strategy and determine who wins in the game.
identical cars on a circular track. Among all of them, they have just enough gas for one car to complete a lap. Show that there is a car that can complete a lap by collecting gas from the other cars on its way around, which doesn't have right angle, let 
be its circumcenter. Let 
and 
be orthocenters of triangle 
and 
, respectively. Let 
be circumcenter of triangle 
. If circle 
and circle 
intersect at for the second time, prove that lies on 
.
is a polynomial with real coefficients such that![\[ P(x)^2 - 1 = 4 P(x^2 - 4x + 1). \]](http://latex.artofproblemsolving.com/b/4/2/b425a938a1dbbac25e0e0793b30c9e58845d45ab.png)
Find .
and 
be altitudes in an acute triangle 
which meet at 
. Suppose that 
meets the circumcircle of at 
and 
such that lies on the shorter arc of 
and lies on the shorter arc of 
. Let 
and 
meet at 
. Show that the circumcircles of 
and 
and the line 
concur.
be a diameter of a circle 
, and let 
be the tangent at 
. Furthermore, let 
be a fixed, positive real, and consider all pairs of points 
and 
lying on , on opposite sides of , such that 
. The lines 
and 
intersect at points 
and 
, respectively. Show that all the lines 
pass through a common point.
such that
for all 
satisfying 
, your right-neighbor’s is 
, and the next person over is 
, then for every trio in the circle they see
lies inside triangle such that 
and 
. Point 
is the midpoint of segment . Point lies on segment 
with 
. Prove that 
.
be, the circumcircle of the triangle 
with 
with 
and 
it is easy to show that 
and so, we have that 
and let 
be, the midpoint of the segment 
we can say that the points 
are collinear, because of 
is the center of 
from 
we have that 
is a diameter of and from 
taken as its transversal the line segment 
applying again the Menelaos theorem,
From 
and because of 
we conclude that 
and 
we conclude that 
and the proof is completed.
is midpoint of 
and 
reflection of 
in 
then 
Since 
bisects 
is therefore equilateral and 
is on circumcircle 
of the isosceles 
Let 
cut again at 
is equilateral with 
Let 
be circumcircle with diameter of the right 
Since 
the circles 
are similar with center 
and coefficient 2. Since 
and 
it follows that 
with diameter and 
on so that 
. It is easy to see that 
. Let 
be the reflection of across . 
, and 
, so 
is equilateral, so 
is cyclic. Since 
, 
, so 
is cyclic. 
and 
, so 
, so 
must be a diameter of 
, so 
. Since 
, a homothety centered at with factor 1/2 gives 
, as desired.
meets at 
. Let 
be the mid-point of 
and is that of . Now apply cosine rules to show 
is an equilateral triangle. So 
and 
is cyclic.
, we must have 
![\[\angle EFD=\angle EFA-60^{\circ}=\angle BF'A-60^{\circ} =\angle BDA-60^{\circ}=\angle BDW=60^{\circ}-\angle DAB\]](http://latex.artofproblemsolving.com/d/1/7/d1709c4fa84c5c84fd05aabb48c335569b841098.png)
At the same time 
and 
is cyclic. So 
. Let be the circumcenter of 
and let 
. We have 
since 
, and we can compute 
. So 
is a parallelogram and is the midpoint of 
. Then we can compute that 
, implying .
and the conclusion follows.
and 
. Let 
intersect the circumcircle 
again at . Also, let the circle with diameter 
intersect again at 
. Let be the reflection of over 
. We see that lies on 
and 
is a 30, 60, 90 triangle with right angle at 
. Thus there is a homothety centered at that takes 
. is the midpoint of , then the intersection of 
with closer to 
is the midpoint of . So point lies on , and the perpendicular condition is prove.
is a homogenized coordinate while 
is not. be the point on for which 
is an equilateral triangle. We will use as our reference triangle instead of 
, with 
, 
and 
. Then it is easy to see 
, since is the midpoint of 
.
implies that lies on the circumcircle 
. Thus, if 
, then 
. Now, we can calculate the points and , giving
Now, we can calculate the displacement vectors 
and 
. We get
Finally, it remains to show that the vectors are perpendicular, which happens iff![\[
\big((3v-2)w+(3w-2)v\big)+\big((3u+1)w+(3w-2)(u-1)\big)+\big((3u+1)v+(3v-2)(u-1)\big)=0
\]](http://latex.artofproblemsolving.com/c/3/6/c360ea105ffd16a0a69bc2775d38fb38ba849fd5.png)
by the perpendicularity criterion. But the LHS is equal to![\[ 6(vw+uw+uv)-4(u+v+w)+4 = 0-4+4=0, \]](http://latex.artofproblemsolving.com/3/c/d/3cdd2c6748bd819c11f70c38ffe4cceaacddc1ee.png)
and we are done.
lies on the perpendicular bisector of . be the midpoint of , 
.
intersects at . with scale . 
, 
, 
, 
and 
. It now suffices to prove that lies on on 
.
and 
. lies on . Since![\[CD\cdot CD' = 2CD^2 = \tfrac{2}{3} AC^2 = AC\cdot AF'\]](http://latex.artofproblemsolving.com/c/f/a/cfafa60b9a99ea7c47039b70f47083549882260c.png)
Now, note that 
is an equilateral triangle, since, 
and 
. lies on since 
.
. Then 
and 
are tangent to each other and 
is their common tangent.
was the circumcenter of 
are tangent to the 
, thus 
is symmedian of 
and subsequently median of 
, thus, if 
was the midpoint of , then 
(actually 
is a parallelogram, thus 
). But from 
and 
we get 
, hence 
and 
is 
right-angled.
be the midpoint of and let be the point on 
such that 
. Since 
, 
is tangent to 
. Since 
, quadrilateral is cyclic and 
. Taking a homothety at by scale factor 
, we find that 
. Thus, quadrilateral is cyclic and 
, as desired.
be the midpoint of and let 
be the foot of perpendicular from to , then,
Hence, 
, Let be the reflection of in , then, 
is the angle bisector of 
, hence now, 
, hence, is cyclic 
, since, 
, hence, 
and 
, now, 
, therefore, 
is cyclic, 
![[asy]
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draw((5.3383604845092725,-1.492379251671836)--(13.630832927947854,-6.173292884194681), linewidth(1) + wrwrwr);
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draw((14.3,-2)--(13.630832927947854,-6.173292884194681), linewidth(1) + dtsfsf);
/* dots and labels */
dot((9.48,2.12),dotstyle);
label("B", (9.56,2.32), NE * labelscalefactor);
dot((2.64,-6.28),dotstyle);
label("A", (2.72,-6.08), NE * labelscalefactor);
dot((19.12,-6.12),dotstyle);
label("$C$", (19.2,-5.92), NE * labelscalefactor);
dot((10.833811978464832,-1.4426337818774861),linewidth(4pt) + dotstyle);
label("$D$", (10.92,-1.28), NE * labelscalefactor);
dot((5.3383604845092725,-1.492379251671836),linewidth(4pt) + dotstyle);
label("$O$", (5.42,-1.34), NE * labelscalefactor);
dot((13.630832927947854,-6.173292884194681),linewidth(4pt) + dotstyle);
label("$F$", (13.72,-6.02), NE * labelscalefactor);
dot((14.3,-2),linewidth(4pt) + dotstyle);
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[/asy]](http://latex.artofproblemsolving.com/5/4/f/54fb3e0da95d45d090376e6b8517748ceeac625d.png)
and 
. We know that:
This means that 
. Note that:
Hence, 
. Since 
and 
with 
, 
. Now, assume WLOG 
(if this is not the case, we can always scale the diagram). Then, 
. Given that 
, we can deduce that 
. Hence, 
, so is a parallelogram. By Stewart’s theorem:
Note that the intersection of and is the midpoint of ( is a parallelogram). The homothety that takes to (from ) also takes the midpoint of to , so the distance from the midpoint of to is 
. It follows that 
is a right triangle.
over be 
, the midpoint of be and the point at which the circumcircle of 
intersects be .
; since lies on , 
and we have that 
, so 
.
, lies on the circumcircle of . Now, consider the homothety sending the circumcircle of 
to the circumcircle of 
centered at . It has scale factor , so it maps to , and hence 
is cyclic. This gives 
as desired.
was not on segment . Then I misread that is outside . What am I doing?![[asy]
size(250);
pair A = origin, C = (3, 0), D = (3/2, sqrt(3)/2);
pair X = extension(C, D, (0, 0), (0, 50));
pair B = circumcenter(A, D, X)+abs(circumcenter(A, D, X)-A)*dir(87);
draw(A--C--D--B--cycle);
pair EE = (B+C)/2;
pair F = C*2/3;
draw(B--C);
draw(D--EE--F--cycle, dashed);
pair M = (D+F)/2;
draw(EE--M);
draw(EE--circumcenter(A, B, D)--F);
draw(circumcenter(A, B, D)--B);
draw(circumcenter(A, B, D)--C, dashed);
draw(circumcenter(A, B, D)--D--A);
dot("$O$", circumcenter(A, B, D), SW);
dot("$A$", A, SW);
dot("$C$", C, SE);
dot("$B$", B, N);
dot("$F$", F, S);
dot("$M$", M, SW);
dot("$E$", EE, NE);
dot(D);
label("$D$", D+(0, 0.05), N);
[/asy]](http://latex.artofproblemsolving.com/8/9/3/8932564ef0e570bbdd036744e74282df905f03b6.png)
be the midpoint of . It suffices to show that 
.
. From the Law of Cosines, we obtain 
so it suffices to show that 
.
is 1 by the Extended Law of Sines, so compute 
. Furthermore, since 
and 
, is a parallelogram. It follows that there is a homothety at taking to 
with ratio 
, and hence is the midpoint of 
. with ratio 
taking 
to 
. It follows that 
for all choices of , and as desired. 
be the reflection of over and be the reflection of over and be the midpoint of . It is trivial that 
. Now if 
then 
and 
and since 
, we have 
is a 30-60-90 triangle.
is cyclic and that 
. In addition we get 
. This means 
but since 
, we have that lies on 
.
from sends 
to 
respectively, so they are all cyclic. And thus ![\[\angle DEF=180-\angle DMF=90\]](http://latex.artofproblemsolving.com/a/b/c/abcdc9443899373a5b61bad6d6faa48f4e97f868.png)
so we are done.
outside triangle 
such that 
and 
. Consider the circle centered at with radius 
, which we let equal 1. Denote this circle as 
. We then consider the configuration on the complex plane, letting be 
(or the center/origin), making the unit circle. Since , we know that lies on .
, 
, 
, and 
. We will keep these as reference, as these will be useful later on., it suffices to show that 
. We substitute the values above that were kept as reference to obtain that
We know that 
, 
, 
, and 
from substitution. Therefore, we get the following:
Combining denominators and simplifying, the numerator becomes
It turns out that this expression is nice AF and everything CANCELS OUT, so therefore 
, as desired.
. Note that the midpoint of AC=M lies on since ADC is isosceles, 
is the midpoint of CD since CFD is isosceles (proof: DMC is 30-60-90, MF/FC=(1/6)/(1/3)=1/2=MD/DC implies DF bisects angle MDC which is 60 deg), and F is the midpoint of 
with C, since APD=120 means DPF=60, whence FPD is equilateral implies FP=FD=FC; in particular, (ADP) goes to (MNF), with A going to E by homothety at C; in particular, since B lies on (ADP), E lies on (DNFM), as desired. 
, 
, 
, 
, and 
where 
. We see that 
so we must have that 
. Since , 
, as desired.
. Therefore, the circle with diameter passes through the midpoints of and . Thus, taking a homothety with scale factor from will take this circle to the circle with , , and the reflection of over . Therefore, it suffices to show that is on this circle as well. But this is just because 
. 
. Moreover, let be the circumcenter of , and let intersect again at point 
. It is clear that 
, which implies that 
. is a parallelogram, so denoting the midpoint of 
as , we find that is also the midpoint of 
. Thus, 
, with scale factor 
:![\[ME = \frac{1}{2} OB = 1 = MD = MF, \]](http://latex.artofproblemsolving.com/e/5/a/e5a77be2d7c43a1947c02228e90521d46dd5fc71.png)
is a right triangle.
. So 
also passes through the midpoints of and . So a homothety at with scale factor finishes. was literally only used once.
be the point on the line parallel to 
through and which lies on the circle centered at with radius 
and so we will consider 
as our reference triangle and will be its center. Furthermore, let 
.
is a parallelogram and quadrilateral 
is an isosceles trapezoid with radius 
, which sends to and as 
, it sends to , hence 
and is a parallelogram.
, hence is an isosceles trapezoid. be the reflection of across , therefore 
.
is a cyclic quadrilateral centered at 
and we claim is tangent to it.
, consequently 
, hence is the center of .
is cyclic, which follows from 
, proving our claim.
, which combined with yields that 
is the side bisector of , hence 
, as desired. is almost the center of (we just have to get rid of the angle 
), which is the motivation to add . Besides that, the condition of 
is only used when proving 
, that is the parallelogram and isosceles trapezoid are also true when 
is arbitrary. Moreover, if is the point satisfying , will always be a cyclic quadrilateral with center , however won't satisfy 
.
Prove that
Let 
Prove that![$$ (a+b)(a+1)(b+1) \leq \frac{3}{2}+\sqrt[3]{6}+\sqrt[3]{36}$$](http://latex.artofproblemsolving.com/f/5/e/f5e68204507aa9c3e71c28844bbee3a804546b94.png)
and 
.
lies inside triangle 
such that 
and 
.
is the midpoint of segment 
.
.
.
is the midpoint of segment 
.
,
.
.
and 
.
is inscribed, which equivalent to prove 
is inscribed :
.
is cyclic, which is equivalent to proving that 
.
homothety centered at 
.
implies 
is fixed with only 
.
are concyclic 
are concyclic 
at 
and WLOG 
We know 
is cyclic since 
It is also a trapezoid because 
Thus, 
and 
with scale factor 
Thus, 
meaning that 
which implies that 
![[asy]
unitsize(1cm);
pair A, B, C, D, E, F;
D=(0,0);
A=(-3sqrt(3),-3);
C=(3sqrt(3),-3);
B=(-2sqrt(3),2sqrt(3));
E=(B+C)/2;
F=(2C+A)/3;
draw(circle((-2sqrt(3),0),2sqrt(3)));
draw(circle((D+F)/2,distance(D,F)/2));
draw(F--E--D--A--C--D--B--A--B--C);
dot("$O$", (-2sqrt(3),0));
label("$A$", A, SW);
label("$B$", B, N);
label("$C$", C, SE);
label("$D$", D, W);
label("$E$", E, N);
label("$F$", F, SE);
[/asy]](http://latex.artofproblemsolving.com/9/f/3/9f329ffad335791c51a5dd6b72c02e4edb4e1725.png)
,
.
and 
,
is the circle with diameter 
.
and 
.
is equilateral.
is a 30-30-120 triangle, so 
and 
.
,![\[\angle DEF = \angle XBY = 180 - \angle YAX = 180-60-30 = 90. \quad \blacksquare\]](http://latex.artofproblemsolving.com/8/b/5/8b589864bb85beac2918cfb1a5443bea8f5acc92.png)
,
,
is free.
,
and 
,
and 
.
,
,
,