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Τρίτη 14 Ιανουαρίου 2025

Junior Balkan Mathematical Olympiad 2001 [Shortlists & Solutions]

  1. Find the positive integers n that are not divisible by 3 if the number 2n210+2133 is a perfect cube.
  2. Let Pn (n=3,4,5,6,7) be the set of positive integers nk+nl+nm, where k,l,m are positive integers. Find n such that a) In the set Pn there are infinitely many squares. b) In the set Pn there are no squares.
  3. Find all the three-digit numbers abc such that the 6003-digit number abcabcabc is divisible by 91.
  4. The discriminant of the equation x2ax+b=0 is the square of a rational number and a and b are integers. Prove that the roots of the equation are integers.
  5. Let xk=k(k+1)2 for all integers k1. Prove that for any integer n10, between the numbers A=x1+x2++xn1 and B=A+xn there is at least one square.
  6. Find all integers x and y such that x3±y3=2001p, where p is prime.
  7. Prove that there are are no positive integers x and y such that x5+y5+1=(x+2)5+(y3)5.
  8. Prove that no three points with integer coordinates can be the vertices of an equilateral triangle.
  9. Consider a convex quadrilateral ABCD with AB=CD and BAC=30. Prove that if ADC=150 then BCA=ACD.
  10. A triangle ABC is inscribed in the circle C(O,R). Let α<1 be the ratio of the radii of the circles tangent to C, and both of the rays (AB and (AC. The numbers β<1 and γ<1 are defined analogously. Prove that α+β+γ=1.
  11. Consider a triangle ABC with AB=AC, and D the foot of the altitude from the vertex A. The point E lies on the side AB such that ACE=ECB=18. If AD=3, find the length of the segment CE.
  12. Consider the triangle ABC with A=90 and BC. A circle C(O,R) passes through B and C and intersects the sides AB and AC at D and E, respectively. Let S be the foot of the perpendicular from A to BC and let K be the intersection point of AS with the segment DE. If M is the midpoint of BC. Prove that AKOM is a parallelogram.
  13. At a conference there are n mathematicians. Each of them knows exactly k fellow mathematicians. Find the smallest value of k such that there are at least three mathematicians that are acquainted each with the other two. Find the smallest value of k such that there (always) exists 3 mathematicians X,Y,Z such that X and Y know each other, X and Z know each other and Y and Z know each other.