MathDB

p1. Find the sum of the

120

numbers

12345

12354

12435

...

54321

, which are obtained by swapping the digits

1, 2, 3, 4

5

p2. Find all naturals

n

such that

\frac{n + 81}{2n-5}

is a natural number.
p3. On each point on the number line, that has the form

\frac{a}{b}

(where

a, b

are comprime integers) we draw a circle tangent to the line at that point, with radius

\frac{1}{2b^2}

. All circles are located in the same half plane. Show that these circles are not secant, and that the circles corresponding to

\frac{a}{c}

and

\frac{b}{d}

are tangent if and only if

ad- bc =\pm 1

.https://cdn.artofproblemsolving.com/attachments/6/d/117cd6c918a64cf17338e587c2e200d76bb073.png
p4. Find the minimum number of cuts that need to be made to fully divide, into cubes of

1

cm of edge, a wooden cube of

4

cm of edge using a saw that allows only horizontal and vertical cuts, and allows reordering the pieces after each cut.
p5. The figure shows three circles of radius

1

, which are tangent in pairs. Find the area of the region among the three circles (gray in the figure). https://cdn.artofproblemsolving.com/attachments/1/0/e041bf5b4e224f19179fcac2f10e3b474acd31.png
p6. Find all pairs of rational numbers

x

and

y

that solve the equation:

y^2-(x-1) x^2 = 0

p7. A child receives an amount of money. This money can be received in three different ways :

\bullet

1

dollar in advance and the rest in monthly installments of

39

dollars.

\bullet

10

dollars in advance and the rest in monthly installments of

11

dollars.

\bullet

2

dollars in advance and the rest in monthly installments of

7

dollars. Determine the minimum amount to receive.

Problem Statement