MathDB

2

Part of 2004 Estonia National Olympiad

Problems(4)

a_i-a_j of 5 different positive integers are different, least value of max a_i

Source: 2004 Estonia National Olympiad Final Round grade 9 p2

3/25/2020
The positive differences ai−aja_i-a_j of five different positive integers a1,a2,a3,a4,a5a_1, a_2, a_3, a_4, a_5 are all different (there are altogether 1010 such differences). Find the least possible value of the largest number among the aia_i.
minmaxDifferencealgebra
angle bisestor of <APC passes throug D, ABCD parallelogram, AM=CN,

Source: 2004 Estonia National Olympiad Final Round grade 10 p2

3/25/2020
On side, BC,ABBC, AB of a parallelogram ABCDABCD lie points M,NM,N respectively such that ∣AM∣=∣CN∣|AM| =|CN|. Let PP be the intersection of AMAM and CNCN. Prove that the angle bisector of ∠APC\angle APC passes through DD.
geometryangle bisectorparallelogramequal segments
1/OK+ 1/OJ is contant for a line passing through a point on angle bisector

Source: 2004 Estonia National Olympiad Final Round grade 11 p2

3/25/2020
Draw a line passing through a point MM on the angle bisector of the angle ∠AOB\angle AOB, that intersects OAOA and OBOB at points KK and LL respectively. Prove that the valus of the sum 1∣OK∣+1∣OL∣\frac{1}{|OK|}+\frac{1}{|OL|} does not depend on the choice of the straight line passing through MM, i.e. is defined by the size of the angle AOB and the selection of the point MM only.
Sumindependentfixedgeometryangle bisector
game with moving a button on a bar of 19 adjacent squares

Source: 2004 Estonia National Olympiad Final Round grade 12 p2

3/25/2020
Albert and Brita play a game with a bar of 1919 adjacent squares. Initially, there is a button on the middle square of the bar. At every turn Albert mentions one positive integer less than 55, and Brita moves button a number of squares in the direction of her choice - while doing so however, Brita must not move the button more than twice in one direction order. Prove that Albert can choose the numbers so that by the 1919th turn, Brita to be forced to move the button out of the bar.
combinatoricsgamegame strategy