MathDB

p1. Ana is thinking of two numbers

a

b

and Beto tries to guess them. Beto asks for a clue and Ana tells him: "Just knowing that they comply

a^2 + a = b^2 + 3b + 2

you will be able to determine them." After listening to the clue, Beto is able to deduce the values of

a

and

b

. Determine these values.
p2. In the following figure the circles have a radius of

1

cm and

GH = 5

cm. Determine the length of the sides of the rectangle. https://cdn.artofproblemsolving.com/attachments/a/4/6c79ac93ba8e2f7c94045c54cd5f3d25162051.png
p3. There are three gears in the position shown in the figure. It is known that they have an entire perimeter

P_1

P_2

and

P_3

(from left to right) and that if the first is given a complete turn, then one turn complete turn to the second and finally a complete turn to the third (all clockwise), the three gears remain in their original position. Determine the possible values of

(P_1, P_2, P_3)

. https://cdn.artofproblemsolving.com/attachments/2/e/fa7cf45b5419acec4a64fcba0d08087ddbaa23.png
p4. In a regular pyramid-shaped container with a square base,

120

^3

of water is placed. If the pyramid is placed with the base on the ground, the water covers up to

5

cm in height and if placed upside down, The water covers

10

cm in height. Determine the side of the base and the height of the container.
p5. A frog is at the origin of the plane. In the second

k

, the frog decides which direction walk (up, down, left or right) and walk

2

units. Prove that (a) The frog can reach

(4, 0)

and

(4, 4)

. (b) There is a path that the frog can take to pass through all the points of the shape

(4a, 4b)

with

a

b

integers. (c) The frog can reach any integer coordinate point on the plane.
PS. You should use hide for answers.

Problem Statement