be a circle inscribed inside a rhombus 
.
and 
be variable points on 
and 
respectively, such as 
is always the tangent line to 
is the same.
are constructed on diameter 
of a semicircle of radius 
.
into four line segments of equal length, as shown. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicircles?
be 
,
.
?
through 
,
such that 
is also prime.
vertices of an octagon each red, black, and white, such that no two adjacent sides are the same color?
denote the set of positive integers. Find all functions 
such that for positive integers 
and 
![\[f(a^2 + b^2) = f(a)f(b) \text{ and } f(a^2) = f(a)^2.\]](http://latex.artofproblemsolving.com/c/1/f/c1f8ffe04cfcc45b498f3931a4796d5c56dc04d0.png)
be an integer. Rowan and Colin play a game on an 
grid of squares, where each square is colored either red or blue. Rowan is allowed to permute the rows of the grid and Colin is allowed to permute the columns. A grid coloring is orderly if: [list] [*]no matter how Rowan permutes the rows of the coloring, Colin can then permute the columns to restore the original grid coloring; and [*]no matter how Colin permutes the columns of the coloring, Rowan can then permute the rows to restore the original grid coloring. [/list] In terms of 
,
is selected inside acute 
so that 
and 
.
is chosen on ray 
so that 
.
be the midpoint of 
.
.
that satisfy![\[
f(x^2-y)+2yf(x)=f(f(x))+f(y)
\]](http://latex.artofproblemsolving.com/f/b/b/fbb693770cd90d8a144026608074122e54da5802.png)
for all 
.
Y by
Exercise P10. (4 points) Given triangle ABC. Let D and E be the midpoints of AB and CD respectively. Suppose angle ACD = 2 angle DEB. Prove that 2 .angle AED = angle DCB +180°.
Exercise P11. Given rectangle ABCD and a point P in the plane. Let X, Y, Z, W, S and T be the orthogonal projections of P onto the lines AB, BC, CD, DA, AC and BD respectively. The perpendicular bisectors of XY and W Z intersect at Q. The perpendicular bisectors of Y Z and XW intersect at R. Prove that QR is parallel to ST.
Exercise P11. Given rectangle ABCD and a point P in the plane. Let X, Y, Z, W, S and T be the orthogonal projections of P onto the lines AB, BC, CD, DA, AC and BD respectively. The perpendicular bisectors of XY and W Z intersect at Q. The perpendicular bisectors of Y Z and XW intersect at R. Prove that QR is parallel to ST.
This post has been edited 2 times. Last edited by Math2030, Mar 12, 2025, 3:45 PM
;
.
.
intersects the perpendicular in the point 
.
intersects the perpendicular in the point 
.
.
intersects the perpendicular bisector
in the point ![$Q(\frac{b^{2}(c-q)-c(cq+p^{2}-q^{2})}{2[b(c-q)-cp]},\frac{b^{2}p-b(c^{2}+p^{2}-q^{2})+c^{2}q}{2[cp-b(c-q)]})$](http://latex.artofproblemsolving.com/f/e/9/fe9315b264f43e2907c88eb4f2415ea27666dcce.png)
.
intersects the perpendicular bisector
in the point 
.
.
