[/quote]
,
and 
with equal radii are given. Let 
,
and 
are on 
,
and 
.
as the other intersection of 
and 
.
are concurrent.
such that 
days later (but perhaps earlier, as well). Hedvig received the trophy today. Determine all integers 
for which we can be absolutely certain that she cannot receive the trophy again in 
days, given the above information.
be real numbers such that : 
. Prove that : 
and some point 
.
intersects the circumcircle of 
at 
at 
does not depend on the choice of 
grid consisting of 2025 unit squares. Within each unit square is a car, facing either up, down, left, or right. If the square in front of a car in the direction it is facing is empty, it can choose to move forward. Each car wishes to exit the 
cars from the 
on one side and 
on the opposite side. The coins are laid out in a row on the table. Peer and Solveig alternate taking turns. On his turn, Peer may turn over one or more coins, so long as he does not turn over two adjacent coins. On her turn, Solveig picks precisely two adjacent coins and turns them over. When the game begins, all the coins are showing 
for which Solveig can keep Peer from winning.
be an angle in the interval 
.
is irrational, and that 
and ![$ \cos[(k + 1)\theta ]$](http://latex.artofproblemsolving.com/b/3/a/b3a14c57a1a8b5a6585195c394181c5124f8c471.png)
are both rational for some positive integer 
,
.
be three positive reals such that 
.![\[ \frac { x^5-x^2 }{x^5+y^2+z^2} + \frac {y^5-y^2}{x^2+y^5+z^2} + \frac {z^5-z^2}{x^2+y^2+z^5} \geq 0 . \]](http://latex.artofproblemsolving.com/a/1/4/a14adf0f1e35df57c734c7df701d5a67da22dc3c.png)
be a triangle with altitude 
.
-
at 
,
and 
.
and 
on lines 
and 
such that quadrilateral 
is a rhombus.
be an integer. Two players, Alice and Bob, play the following game on the complete graph 
:
,![\[
\left( \frac{1}{4} - \varepsilon \right) n^2 + n.
\]](http://latex.artofproblemsolving.com/a/9/c/a9cca00e87c643af32eb75a2501b59060f3504fb.png)
such that the following inequality holds for all 
:![\[
ab + bc + ca \leq \frac{(a + b + c)^2}{3} + k \cdot \max \{ (a - b)^2, (b - c)^2, (c - a)^2 \} \leq a^2 + b^2 + c^2
\]](http://latex.artofproblemsolving.com/1/c/e/1ce66163068b499383adb4a4ecec86c5f4edaec3.png)
. Prove that : 
where 
are lengths of parallel sides and 
are other two sides.
Y by Adventure10
A quadrilateral 
is inscribed in a circle of center 
. It is known that the diagonals 
and 
are perpendicular. On each side we build semicircles, externally, as shown in the figure.
a) Show that the triangles
and 
have the equal areas.
b) If
cm and 
cm, determine the area of the shaded region.
is inscribed in a circle of center 
. It is known that the diagonals 
and 
are perpendicular. On each side we build semicircles, externally, as shown in the figure.a) Show that the triangles
and 
have the equal areas.b) If
cm and 
cm, determine the area of the shaded region.Attachments:
This post has been edited 3 times. Last edited by parmenides51, Jun 18, 2024, 10:55 AM
![$[AOB] = \frac{{{R^2}}}{2}\sin (A\widehat OB) = \frac{{{R^2}}}{2}\sin ({180^0} - A\widehat OB) = \frac{{{R^2}}}{2}\sin (C\widehat OD) = [COD]$](http://latex.artofproblemsolving.com/0/8/4/084797b7e25b20bc3fb82587ca3cae10b9a890bb.png)
![$[AOB]=\frac{R^2}{2}\sin\angle AOB=\frac{R^2}{2}\sin\angle COD=[COD]$](http://latex.artofproblemsolving.com/5/c/6/5c69725d4ba4a5315e4a6f97fc35610b72be0b56.png)
Let 
, 
the circle with diameter 
.![$S=\frac{1}{2}([\bigcirc AB]+[\bigcirc BC]+[\bigcirc CD]+[\bigcirc AD])-([\overarc{AB}]+[\overarc{BC}]+[\overarc{CD}]+[\overarc{DA}])=$](http://latex.artofproblemsolving.com/3/1/7/317fad572abf8f961421771e6f315fee939997ba.png)
![$=\frac{1}{2}([\bigcirc AB]+[\bigcirc BC]+[\bigcirc CD]+[\bigcirc AD])-([(ABCD)]-[ABCD])$](http://latex.artofproblemsolving.com/a/a/e/aae39e1364aea38686e91320816ea832ee883817.png)
![$[\bigcirc AB]=\pi\cdot (\frac{AB}{2})^2$](http://latex.artofproblemsolving.com/9/f/1/9f182acec2c6cf594ae0f0236242bd378f41e2fc.png)
![$AB^2=R^2+R^2-2R\cdot R\cdot\cos x=R^2(2-2\cos x)\Rightarrow [\bigcirc AB]=\frac{\pi}{2} R^2(1-\cos x)$](http://latex.artofproblemsolving.com/7/6/8/768f339b5c2f7f82c2f512c58e7dec8ff1c4f49b.png)
![$[(ABCD)]=\pi R^2$](http://latex.artofproblemsolving.com/0/8/8/08863a4cb7f91abf863d12a3d9c2e2f7b03bdb8d.png)
![$[ABCD]=\frac{AC\cdot BD}{2}=24cm^2$](http://latex.artofproblemsolving.com/0/e/0/0e0ecb23a4d816f91d00451da31f2d7e17fe08b1.png)
(Because the angle between 
)
,
,
.
