Επιστροφή στην Κεντρική Σελίδα.
Source: Croatian National Competition 1993, grade 11
Problem: A set of circles in the plane with the sum of diameters equal to 41 can be completely covered with a circle of diameter 10. Prove that there exists a line which intersects more than 4 of the circles from the given set.
Problem 2
Source: Mathematical Digest 100
Problem: Given collinear points A, B and C such that B lies on the segment AC and AB=2 * BC, and the point D such that DBA=60 degrees and DCA=45 degrees, determine the angle ADB.
Note: A purely geomatrical solution is preferrable.
Problem 3
Source: New Zealand IMO Training
Problem: A duck is located in the center of a circular pond. A leg injury prevents the duck from taking off fom water. A hungry fox is at the edge of the pond and can 4 times faster than the duck can swim. Can the duck escape swim to the edge of the pond and not be caught by the fox?
Notes:
i. For which values of fox/duck speed ratio can the duck escape?
ii. Provided the duck can escape, which path should it take to maximize its distance from the fox at the moment it reaches the edge of the pond?
Problem 4
Source: Croatian National Competition 1995, grade 10
Problem: Given a triangle ABC, let h_a, h_b and h_c be its respective heights, and D, E and F intersections of angle bisectors of ABC with respective opposite edges. If d_a, d_b and d_c are distances of D, E, F from AB, CB, CA respectively, prove that it holds:
d_a/h_a + d_b/h_b + d_c/h_c >= 3/2
Problem 5
Source: Croatian National Competition 1995, grade 9
Problem: Prove that there exists a number divisible by 1999 ending in 1995 in decimal system.
Problem 6
Source: Croatian National Competition 1995, grade 10
Problem: On a 56 km long train line there are 11 stations A_1, A_2, ..., A_11. Distances of the form d(A_i, A_(i+2)) (i=1, 2, ..., 9) are not greater than 12 km, and distances of the form d(A_i, A_(i+3)) (i=1, 2, ..., 8) are not smaller than 17 km. How big is the distance d(A_2, A_7)?
Επιστροφή στην Κεντρική Σελίδα.