[/quote]
,
if within that number exists a digit 
that appears exactly 
times in that number, hence find the number of 
be real numbers, such that![\[
a^2+b^2+c^2+abc=5
\]](http://latex.artofproblemsolving.com/7/c/8/7c8044347b7e25cb21dbef42c65ec351cefd4699.png)
![\[
a+b+c=3
\]](http://latex.artofproblemsolving.com/6/4/a/64a3034ae02dd28aa6220ee5c37a22d608074173.png)
Prove that atleast one of the numbers 
.
![\[\angle PAD:\angle PBA:\angle DPA=1:2:3=\angle CBP:\angle BAP:\angle BPC\]](http://latex.artofproblemsolving.com/9/3/8/9388a91325842f1ec67a4693f0f10af46ed66c6d.png)
Prove that the following three lines meet in a point: the internal bisectors of angles 
and 
and the perpendicular bisector of segment 
.
with 
.
of 
and the length of 
is 
intersects 
and 
.
.
midpoint of side 
in an isosceles triangle 
with 
.
on a smaller arc 
of circumcircle of triangle 
.
point inside of angle 
such that 
and 
.
does not depend on choice of 
)
.
is chosen on arc 
,
intersect 
,
.
is a parallelogram. Point 
,
.
at 
,
intersects 
at 
.
intersects 
at points 
,
and 
respectively.
.
.
?
?
be the incircle of the triangle 
.
be the circumcircle of the triangle 
.
intersect at the point 
be the intersection of the common tangents of the circles 
is tangent to the circumcircle of the triangle 
is the foot of the altitude from 
.
and 
lie on the circles 
and 
,
and 
.
and 
be vertices of regular pentagon inscribed in a circle whose radius is 
and center is at 
.
is a parallelogram. Suppose that 
.
a convex quadrilateral such that each of the segments 
and 
is tangent to 
be the circumcircle of the triangle 
.
beyond 
beyond 
meets 
and 
beyond ![\[A D+D T+T X+X A=C D+D Y+Y Z+Z C.\]](http://latex.artofproblemsolving.com/8/2/6/8261276053071cc6cc06dda824a883426bf1d4ac.png)
.
are 
is the center of the square. Prove that points 
Y by Adventure10
Let 
be a square of side paper 
and 
a point on side 
. By folding the paper along the 
line, point 
determines the point 
, as seen in the figure. The line 
cuts the side 
at 
. Calculate the perimeter of the triangle 
be a square of side paper 
and 
a point on side 
. By folding the paper along the 
line, point 
determines the point 
, as seen in the figure. The line 
cuts the side 
at 
. Calculate the perimeter of the triangle 
This post has been edited 3 times. Last edited by parmenides51, Jun 3, 2024, 5:14 PM
.
.
.
.
.
