MathDB

2006 May Olympiad

Part of May Olympiad

Subcontests

(5)
5
2

triangular grid of 28 points

With 2828 points, a “triangular grid” of equal sides is formed, as shown in the figure. One operation consists of choosing three points that are the vertices of an equilateral triangle and removing these three points from the grid. If after performing several of these operations there is only one point left, in what positions can that point remain? Give all the possibilities and indicate in each case the operations carried out. Justify why the remaining point cannot be in another position. https://cdn.artofproblemsolving.com/attachments/f/c/1cedfe0e1c5086b77151538265f8e253e93d2e.gif

pieces on a 10x10 board

In some squares of a 10×1010 \times 10 board, a piece is placed in such a way that the following property is satisfied: For each square that has a piece, the number of pieces placed in the same row must be greater than or equal to the number of pieces placed in the same column. How many tiles can there be on the board? Give all chances.
1
2

(a+1)/b and (b+1)/a are naturals

Determine all pairs of natural numbers aa and bb such that a+1b\frac{a+1}{b} and b+1a\frac{b+1}{a} they are natural numbers.

digits callendar date reflected at a mirror

A digital calendar displays the date: day, month, and year, with 22 digits for the day, 22 digits for the month, and 22 digits for the year. For example, 01−01−0101-01-01 is January 11, 20012001 and 05−25−2305-25-23 is May 2525, 20232023. In front of the calendar is a mirror. The digits of the calendar are as in the figure https://cdn.artofproblemsolving.com/attachments/c/5/a08a4e34071fff4d33b95b23690254f55b33e1.gif
If 0,1,2,50, 1, 2, 5, and 88 are reflected, respectively, in 0,1,5,20, 1, 5, 2, and 88, and the other digits lose meaning when reflected, determine how many days of the century, when reflected in the mirror, also correspond to a date.
3
2

6 positive integers wanted

Write a positive integer in each box so that: All six numbers are different. The sum of the six numbers is 100100. If each number is multiplied by its neighbor (in a clockwise direction) and the six results of those six multiplications are added, the smallest possible value is obtained. Explain why a lower value cannot be obtained. https://cdn.artofproblemsolving.com/attachments/7/1/6fdadd6618f91aa03cdd6720cc2d6ee296f82b.gif

red points among 20 collinear points

There are 2020 collinear points, separated by the same distance: .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .   .. \,\,\, . \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\, . Miguel has to paint three or more of these points red, in such a way that the red points are separated by the same distance and it is impossible to paint exactly one more point red without violating the previous condition. Determine in how many ways Miguel can do his homework.
2
2
4
2

find area of a triangle created by diagonals of a trapezoid

Let ABCDABCD be a trapezoid of bases ABAB and CDCD . Let OO be the intersection point of the diagonals ACAC and BDBD. If the area of the triangle ABCABC is 150150 and the area of the triangle ACDACD is 120120, calculate the area of the triangle BCOBCO.

colored prism 6x5x5 into 150 cubes 1x1x1

With 150150 white cubes of 1×1×11 \times 1 \times 1 a prism of 6×5×56 \times 5 \times 5 is assembled, its six faces are painted blue and then the prism is disassembled. Lucrecia must build a new prism, without holes, exclusively using cubes that have at least one blue face and so that the faces of Lucrecia's prism are all completely blue. Give the dimensions of the prism with the largest volume that Lucrecia can assemble.