2018 BAMO

Part of BAMO Contests

Subcontests

(7)

1

2018 BAMO 5 dissect regular n-gon in integer-ratio right triangles

To dissect a polygon means to divide it into several regions by cutting along finitely many line segments. For example, the diagram below shows a dissection of a hexagon into two triangles and two quadrilaterals: https://cdn.artofproblemsolving.com/attachments/0/a/378e477bcbcec26fc90412c3eada855ae52b45.png An integer-ratio right triangle is a right triangle whose side lengths are in an integer ratio. For example, a triangle with sides

3,4,5

is an integer-ratio right triangle, and so is a triangle with sides

\frac52 \sqrt3 ,6\sqrt3, \frac{13}{2} \sqrt3

. On the other hand, the right triangle with sides

\sqrt2 ,\sqrt5, \sqrt7

is not an integer-ratio right triangle. Determine, with proof, all integers

n

for which it is possible to completely dissect a regular

n

-sided polygon into integer-ratio right triangles.

1

2018 BAMO 4 a/b+b/c+d/a is integer => abc perfect cube

(a) Find two quadruples of positive integers

(a,b, c,n)

, each with a different value of

n

3

\frac{a}{b} +\frac{b}{c} +\frac{c}{a} = n

(b) Show that if

a,b, c

are nonzero integers such that

\frac{a}{b} +\frac{b}{c} +\frac{c}{a}

is an integer, then

abc

is a perfect cube. (A perfect cube is a number of the form

n^3

n

is an integer.)

1

2018 BAMO E/3 2000 numbers, 1 ot -1 around a circle

2002

numbers, each equal to

1

-1

, are written around a circle. For every two adjacent numbers, their product is taken; it turns out that the sum of all

2002

such products is negative. Prove that the sum of the original numbers has absolute value less than or equal to

1000

. (The absolute value of

x

is usually denoted by

|x|

. It is equal to

x

x \ge 0

-x

x < 0

|6| = 6, |0| = 0

|-7| = 7

1

2018 BAMO D/2 parallel lines, tangents, P_7 coincides with P_1

P_1, P_2, P_3

P_4

be arranged around a circle in that order. (One possible example is drawn in Diagram 1.) Next draw a line through

P_4

P_1P_2

, intersecting the circle again at

P_5

. (If the line happens to be tangent to the circle, we simply take

P_5 =P_4

, as in Diagram 2. In other words, we consider the second intersection to be the point of tangency again.) Repeat this process twice more, drawing a line through

P_5

P_2P_3

, intersecting the circle again at

P_6

, and finally drawing a line through

P_6

P_3P_4

, intersecting the circle again at

P_7

P_7

is the same point as

P_1

. https://cdn.artofproblemsolving.com/attachments/5/7/fa8c1b88f78c09c3afad2c33b50c2be4635a73.png

1

2018 BAMO C/1 square-friendly integers, m^2+18m+c is a perfect square,

c

is square-friendly if it has the following property: For every integer

m

m^2+18m+c

is a perfect square. (A perfect square is a number of the form

n^2

n

is an integer. For example,

49 = 7^2

is a perfect square while

46

is not a perfect square. Further, as an example,

6

is not square-friendly because for

m = 2

(2)2 +(18)(2)+6 = 46

46

is not a perfect square.) In fact, exactly one square-friendly integer exists. Show that this is the case by doing the following: (a) Find a square-friendly integer, and prove that it is square-friendly. (b) Prove that there cannot be two different square-friendly integers.

1

2018 BAMO B sum of perimeters of rectangles by 2 lines in a square

A square with sides of length

1

cm is given. There are many different ways to cut the square into four rectangles. Let

S

be the sum of the four rectangles’ perimeters. Describe all possible values of

S

with justification.

1

2018 BAMO A 25 people of different heights in 5x5 grid, taller?

Twenty-five people of different heights stand in a

5\times 5

grid of squares, with one person in each square. We know that each row has a shortest person, suppose Ana is the tallest of these five people. Similarly, we know that each column has a tallest person, suppose Bev is the shortest of these five people. Assuming Ana and Bev are not the same person, who is taller: Ana or Bev? Prove that your answer is always correct.