MathDB

3

Part of 1996 Estonia National Olympiad

Problems(4)

1,000,000 piles of 1996 coins, fakes and reals

Source: 1996 Estonia National Olympiad Final Round grade 10 p3

11/4/2020
There are 1,000,0001,000,000 piles of 19961996 coins in each of them, and in one pile there are only fake coins, and in all the others - only real ones. What is the smallest weighing number that can be used to determine a heap containing counterfeit coins if the scales used have one bowl and allow weighing as much weight as desired with an accuracy of one gram, and it is also known that each counterfeit coin weighs 99 grams, and each real coin weighs 1010 grams?
combinatorics
angle chasing in a cyclic quadrilateral, diagonals and ratio of lengths related

Source: 1996 Estonia National Olympiad Final Round grade 9

3/11/2020
The vertices of the quadrilateral ABCDABCD lie on a single circle. The diagonals of this rectangle divide the angles of the rectangle at vertices AA and BB and divides the angles at vertices CC and DD in a 1:21: 2 ratio. Find angles of the quadrilateral ABCDABCD.
geometrycyclic quadrilateralratiodiagonalsangles
intersection area of equilateral after rotation around it\s center

Source: 1996 Estonia National Olympiad Final Round grade 12 p3

3/11/2020
An equilateral triangle of side1 1 is rotated around its center, yielding another equilareral triangle. Find the area of the intersection of these two triangles.
geometryrotationareaEquilateral
1992,1993, ... ,2000 form a 3x3 magic square, center number is 1996

Source: 1996 Estonia National Olympiad Final Round grade 11 p3

3/11/2020
Numbers 1992,1993,...,20001992,1993, ... ,2000 are written in a 3×33 \times 3 table to form a magic square (i.e. the sums of numbers in rows, columns and big diagonals are all equal). Prove that the number in the center is 19961996. Which numbers are placed in the corners?
algebramagic squaretable