MathDB

p1. How many real solutions does the following system of equations have? Justify your answer.

x + y = 3

3xy -z^2 = 9

p2. After the first year the bank account of Mr. Money decreased by

25\%

, during the second year it increased by

20\%

, during the third year it decreased by

10\%

, and during the fourth year it increased by

20\%

. Does the account of Mr. Money increase or decrease during these four years and how much?
p3. Two circles are internally tangent. A line passing through the center of the larger circle intersects it at the points

A

and

D

. The same line intersects the smaller circle at the points

B

and

C

. Given that

|AB| : |BC| : |CD| = 3 : 7 : 2

, find the ratio of the radiuses of the circles.
p4. Find all integer solutions of the equation

\frac{1}{x}+\frac{1}{y}=\frac{1}{19}

p5. Is it possible to arrange the numbers

1, 2, . . . , 12

along the circle so that the absolute value of the difference between any two numbers standing next to each other would be either

3

, or

4

, or

5

? Prove your answer.
p6. Nine rectangles of the area

1

sq. mile are located inside the large rectangle of the area

5

sq. miles. Prove that at least two of the rectangles (internal rectangles of area

1

sq. mile) overlap with an overlapping area greater than or equal to

\frac19

sq. mile
PS. You should use hide for answers.

Problem Statement