MathDB

p1. Consider the system of simultaneous equations

(x+y)(x+z)=a^2

(x+y)(y+z)=b^2

(x+z)(y+z)=c^2

, where

abc \ne 0

. Find all solutions

(x,y,z)

in terms of

a

b

, and

c

.
p2. Shown in the figure is a triangle

PQR

upon whose sides squares of areas

13

25

, and

36

sq. units have been constructed. Find the area of the hexagon

ABCDEF

. https://cdn.artofproblemsolving.com/attachments/b/6/ab80f528a2691b07430d407ff19b60082c51a1.png
p3. Suppose

p,q

, and

r

are positive integers no two of which have a common factor larger than

1

. Suppose

P,Q

, and

R

are positive integers such that

\frac{P}{p}+\frac{Q}{q}+\frac{R}{r}

is an integer. Prove that each of

P/p

Q/q

, and

R/r

is an integer.
p4. An isosceles tetrahedron is a tetrahedron in which opposite edges are congruent. Prove that all face angles of an isosceles tetrahedron are acute angles. https://cdn.artofproblemsolving.com/attachments/7/7/62c6544b7c3651bba8a9d210cd0535e82a65bd.png
p5. Suppose that

p_1

p_2

p_3

and

p_4

are the centers of four non-overlapping circles of radius

1

in a plane and that,

p

is any point in that plane. Prove that

\overline{p_1p}^2+\overline{p_2p}^2+\overline{p_3p}^2+\overline{p_4p}^2 \ge 6.

PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement