MathDB

p1.

x,y,z

are required to be non-negative whole numbers, find all solutions to the pair of equations

x+y+z=40

2x + 4y + 17z = 301.

p2. Let

P

be a point lying between the sides of an acute angle whose vertex is

O

. Let

A,B

be the intersections of a line passing through

P

with the sides of the angle. Prove that the triangle

AOB

has minimum area when

P

bisects the line segment

AB

.
p3. Find all values of

x

for which

|3x-2|+|3x+1|=3

.
p4. Prove that

x^2+y^2+z^2

cannot be factored in the form

(ax + by + cz) (dx + ey + fz),

a, b, c, d, e, f

real.
p5. Let

f(x)

be a continuous function for all real values of

x

such that

f(a)\le f(b)

whenever

a\le b

. Prove that, for every real number

r

, the equation

x + f(x) = r

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

Problem Statement