MathDB

p1. One side of a triangle is equal to one third of the sum of the other two sides. Show that the angle opposite the first side is the smallest of the angles of the triangle.
p2. A very vain mathematician's apprentice claimed that he could write any integer positive as a product of fractions of the form

\frac{2q-1}{q}

with

q> 0

integer.

\bullet

Prove what said the apprentice is wrong.

\bullet

Tell how you have written the number

49

.
p3. Determine the digits that have been omitted in the multiplication:https://cdn.artofproblemsolving.com/attachments/1/b/eb9a15ba0c019b3a8d909eed7f2f84428a4ca5.png
p4. What fractions should be removed from the sum

\frac12 + \frac13 + \frac14 + \frac16 + \frac18 + \frac{1}{10} + \frac{1}{12}

so that the sum is

1

? Give all the possibilities and explain why there are no more.
p5. Let

P

be a point on side

BC

of a triangle

ABC

. The parallel through

P

AB

intersects at side

AC

at point

Q

, and the parallel through

P

AC

intersects

AB

at point

R

. The ratio between the areas of the triangles

RBP

and

QBC

k^2

. Determine the ratio of the areas of the triangles

ARQ

and

ABC

.
p6. In how many ways is it possible to rearrange the word MATEMATICO so that there are no two adjacent equal letters?
p7. Set

A

has

5

different numbers. If we do the sum of each pair of numbers from

A

10

results are obtained:

1977

;

1982

1983

1984

1985

1990

1993

1994

1999

and

2001

. What are those

5

numbers?

Problem Statement