MathDB

2013 ToT Spring Senior A p6 sums related to 5 real numbers

Source:

3/5/2020

There are five distinct real positive numbers. It is known that the total sum of their squares and the total sum of their pairwise products are equal. (a) Prove that we can choose three numbers such that it would not be possible to make a triangle with sides' lengths equal to these numbers. (b) Prove that the number of such triples is at least six (triples which consist of the same numbers in different order are considered the same).

algebraProductSum

2013 ToT Spring Junior A p6 peprendicular wanted and given, incircle

Source:

3/4/2020

Let

ABC

be a right-angled triangle,

I

its incenter and

B_0, A_0

points of tangency of the incircle with the legs

AC

and

BC

respectively. Let the perpendicular dropped to

AI

from

A_0

and the perpendicular dropped to

BI

from

B_0

meet at point

P

. Prove that the lines

CP

and

AB

are perpendicular.

perpendicularincirclegeometry

2013 ToT Fall Junior A p6 3n+1 is a prime number

Source:

3/22/2020

The number

1- \frac12 +\frac13-\frac14+...+\frac{1}{2n-1}-\frac{1}{2n}

is represented as an irreducible fraction. If

3n+1

is a prime number, prove that the numerator of this fraction is a multiple of

3n + 1

.

primeFractiondividesdivisornumber theoryprime numbers

6

Problems(3)