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Part of 2021 Caucasus Mathematical Olympiad

Problems(2)

Source: VI Caucasus Mathematical Olympiad

a

b

c

be real numbers such that

a^2+b=c^2

b^2+c=a^2

c^2+a=b^2

. Find all possible values of

abc

algebraCaucasusvalue

Numbers that are greater than all its right\left neighbours

Source: VI Caucasus Mathematical Olympiad

Integers from 1 to 100 are placed in a row in some order. Let us call a number large-right, if it is greater than each number to the right of it; let us call a number large-left, is it is greater than each number to the left of it. It appears that in the row there are exactly

k

large-right numbers and exactly

k

large-left numbers. Find the maximal possible value of

k