![$f\colon[0,1]\rightarrow \mathbb{R}$](http://latex.artofproblemsolving.com/f/d/b/fdb1fb0216ab9d17007b1b7b03d20d1060b88531.png)
be a continuous function. Suppose that for each 
,![$f_t\colon[0,1-t]\rightarrow\mathbb{R}$](http://latex.artofproblemsolving.com/8/5/a/85ad1acbe6c154d977d9a8554cba6d08f3e4019a.png)
,
,
.
,
is constant zero.
matrices which has only 
and 
for their entries?
is solvable over 
for every 
.
,
.
be positive integers. The square in the 
t
t
.
?
Y by
Let 
be an Euclidean 
-simplex. We will use the following notations:
-
the centre if its circumscribed hypersphere, its volume, its circumradius and its inradius, respectively.
-
the centre and the radius of the hypersphere tangent to the circumscribed sphere of 
in the vertex 
and to the hyperplane 
simultaneously.
With the above notations, the following identity holds:
by Vasile Jiglau - Romania
be an Euclidean 
-simplex. We will use the following notations:-
the centre if its circumscribed hypersphere, its volume, its circumradius and its inradius, respectively.-
the centre and the radius of the hypersphere tangent to the circumscribed sphere of 
in the vertex 
and to the hyperplane 
simultaneously.With the above notations, the following identity holds:
by Vasile Jiglau - Romania
