- Find the positive integers
that are not divisible by if the number is a perfect cube. - Let
be the set of positive integers , where are positive integers. Find such that a) In the set there are infinitely many squares. b) In the set there are no squares. - Find all the three-digit numbers
such that the -digit number is divisible by . - The discriminant of the equation
is the square of a rational number and and are integers. Prove that the roots of the equation are integers. - Let
for all integers . Prove that for any integer , between the numbers and there is at least one square. - Find all integers
and such that , where is prime. - Prove that there are are no positive integers
and such that - Prove that no three points with integer coordinates can be the vertices of an equilateral triangle.
- Consider a convex quadrilateral
with and . Prove that if then . - A triangle
is inscribed in the circle . Let be the ratio of the radii of the circles tangent to , and both of the rays and . The numbers and are defined analogously. Prove that - Consider a triangle
with , and the foot of the altitude from the vertex . The point lies on the side such that . If , find the length of the segment . - Consider the triangle
with and . A circle passes through and and intersects the sides and at and , respectively. Let be the foot of the perpendicular from to and let be the intersection point of with the segment . If is the midpoint of . Prove that is a parallelogram. - At a conference there are
mathematicians. Each of them knows exactly fellow mathematicians. Find the smallest value of such that there are at least three mathematicians that are acquainted each with the other two. Find the smallest value of such that there (always) exists mathematicians such that and know each other, and know each other and and know each other.
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Τρίτη 14 Ιανουαρίου 2025
Junior Balkan Mathematical Olympiad 2001 [Shortlists & Solutions]
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