MathDB

p1. Show that the

n \times n

determinant

\begin{vmatrix} 1+x & 1 & 1 & . & . & . & 1 \\ 1 & 1+x & 1 & . & . & . & 1 \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ 1 & 1 & . & . & . & . & 1+x \\ \end{vmatrix}

has the value zero when

x = -n

p2. Let

c > a \ge b

be the lengths of the sides of an obtuse triangle. Prove that

c^n = a^n + b^n

for no positive integer

n

.
p3. Suppose that

p_1 = p_2^2+ p_3^2 + p_4^2

, where

p_1

p_2

p_3

, and

p_4

are primes. Prove that at least one of

p_2

p_3

p_4

is equal to

3

.
p4. Suppose

X

and

Y

are points on tJhe boundary of the triangular region

ABC

such that the segment

XY

divides the region into two parts of equal area. If

XY

is the shortest such segment and

AB = 5

BC = 4

AC = 3

calculate the length of

XY

.
Hint: Of all triangles having the same area and same vertex angle the one with the shortest base is isosceles. Clearly justify all claims.
p5. Find all solutions of the following system of simultaneous equations

x + y + z = 7\,\, , \,\, x^2 + y^2 + z^2 = 31\,\,, \,\,x^3 + y^3 + z^3 = 154

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

Problem Statement