MathDB

7.1 Construct a trapezoid given four sides.
7.2 Prove that

(1 + x + x^2 + ...+ x^{100})(1 + x^{102}) - 102x^{101} \ge 0 .

7.3 In a quadrilateral

ABCD

M

is the midpoint of AB,

N

is the midpoint of

CD

. Lines

AD

and BC intersect

MN

at points

P

and

Q

, respectively. Prove that if

\angle BQM = \angle APM

, then

BC=AD

. https://cdn.artofproblemsolving.com/attachments/a/2/1c3cbc62ee570a823b5f3f8d046da9fbb4b0f2.png
7.4 / 6.4 Each of the eight given different natural numbers less than

16

. Prove that among their pairwise differences there is at least at least three are the same.
7.5 / 8.4 An entire arc of circle is drawn through the vertices

A

and

C

of the rectangle

ABCD

lying inside the rectangle. Draw a line parallel to

AB

intersecting

BC

at point

P

AD

at point

Q

, and the arc

AC

at point

R

so that the sum of the areas of the figures

AQR

and

CPR

is the smallest. https://cdn.artofproblemsolving.com/attachments/1/4/9b5a594f82a96d7eff750e15ca6801a5fc0bf1.png
7.6 / 6.5 The distance AB is 100 km. From A and B , cyclists simultaneously ride towards each other at speeds of 20 km/h and 30 km/hour accordingly. Together with the first A, a fly flies out with speed 50 km/h, she flies until she meets the cyclist from B, after which she turns around and flies back until she meets the cyclist from A, after which turns around, etc. How many kilometers will the fly fly in the direction from A to B until the cyclists meet?
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988083_1967_leningrad_math_olympiad]here.

Problem Statement