6

Part of 2023 Tuymaada Olympiad

Problems(2)

Source: 2023 Tuymaada Junior P6

<span class='latex-italic'>Euclidean step</span>

transforms a pair

(a, b)

of positive integers,

a > b

(b, r)

r

is the remainder when a is divided by

b

. Let us call the

<span class='latex-italic'>complexity</span>

(a, b)

the number of Euclidean steps needed to transform it to a pair of the form

(s, 0)

. Prove that if

ad - bc = 1

, then the complexities of

(a, b)

(c, d)

differ at most by

2

number theoryEuclidean algorithm

Inequality on a plane

Source: Tuymaada 2023 Senior P6

n

segments with lengths

a_1, a_2, \dots , a_n

are drawn. Every ray beginning at the point

O

meets at least one of the segments. Let

h_i

be the distance from

O

i

-th segment (not the line!) Prove the inequality

\frac{a_1}{h_1}+\frac{a_2}{h_2} + \ldots + \frac{a_i}{h_i} \geqslant 2 \pi.

inequalitiesgeometryPlaneTuymaadaGeometry inequality