Math 289 โ Winter 2006 โ Problem Set 8
Due March 15, 2006
1.(10) Let us denote by San intercept of axes of internal angles of a triangle ABC. The
point Slies on a line pparallel to AB. Furthermore, let Pand Qbe the intercepts of p
with the sides AC and BC respectively. Prove that
|P Q|=|AP |+|BQ|.
Here |XY |denotes a length of a line segment X Y .
2.(10) Prove that the diagonals of a quadrilateral are orthogonal if and only if its medians
have equal length. (A median of a quadrilateral connects the midpoints of two opposite
sites.)
3.(10) Find all continuous functions f:RโRwhich satisfy
f(xy) = f(x)f(y) for all x, y โR.
Are there any solutions which are not continuous?
4.(10) Let ABC be a triangle with a unit area. On the side AB there lies a point Din
one third of a distance from Ato B. Similarly, on the side BC there lies a point Ein
one quarter of a distance from Bto C. Finally, on the side CA there lies a point Fin the
middle of a segment CA. Find the area of the triangle DEF .
5.(10) Given is a unit area qudrilateral ABCD with its sides AB and CD divided to 5
equal parts and its sides BC and DA divided to 3 equal parts. Using the split points we
divide ABCD to a regular grid of 15 small quadrilaterals. Find the area of the quadrilateral
in the center of the grid.
6.(15) Find all functions f:RโRwhich for any x, y โRsatisfy
xf(y) + yf (x) = (x+y)f(x)f(y).
7.(20) Find all functions fdefined on the set of positive real numbers which take positive
real values (i.e. f:R+โR+) and satisfy the conditions
(i) f[xf(y)] = yf (x) for all positive x, y;
(ii) f(x)โ0 as xโ โ .
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