MathDB

p1. Solve for

x

and

y

\frac{1}{x^2}+\frac{1}{xy}=\frac{1}{9}

and

\frac{1}{y^2}+\frac{1}{xy}=\frac{1}{16}

p2. Find positive integers

p

and

q

, with

q

as small as possible, such that

\frac{7}{10} <\frac{p}{q} <\frac{11}{15}

.
p3. Define

a_1 = 2

and

a_{n+1} = a^2_n -a_n + 1

for all positive integers

n

. If

i > j

, prove that

a_i

and

a_j

have no common prime factor.
p4. A number of points are given in the interior of a triangle. Connect these points, as well as the vertices of the triangle, by segments that do not cross each other until the interior is subdivided into smaller disjoint regions that are all triangles. It is required that each of the givien points is always a vertex of any triangle containing it. Prove that the number of these smaller triangular regions is always odd.
p5. In triangle

ABC

, let

\angle ABC=\angle ACB=40^o

is extended to

D

such that

AD=BC

. Prove that

\angle BCD=10^o

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Problem Statement