MathDB

2005 May Olympiad

Part of May Olympiad

Subcontests

(5)
5
2

battleship revisited

The enemy ship has landed on a 9×99\times 9 board that covers exactly 55 squares of the board, like this: https://cdn.artofproblemsolving.com/attachments/2/4/ae5aa95f5bb5e113fd5e25931a2bf8eb872dbe.png The ship is invisible. Each defensive missile covers exactly one square, and destroys the ship if it hits one of the 55 squares that it occupies. Determine the minimum number of missiles needed to destroy the enemy ship with certainty .

1-7 at each box of 7x7 board

a) In each box of a 7×77\times 7 board one of the numbers is written: 1,2,3,4,5,61, 2, 3, 4, 5, 6 or 77 of so that each number is written in seven different boxes. Is it possible that in no row and no column are consecutive numbers written?
b) In each box of a 5×55\times 5 board one of the numbers is written: 1,2,3,41, 2, 3, 4 or 55 of so that each one is written in five different boxes. Is it possible that in no row and in no column are consecutive numbers written?
1
2

smallest 3-digit number wanted , product of two 2-digit numbers

Find the smallest 33-digit number that is the product of two 22-digit numbers , so that the seven digits of these three numbers are all different.

6 geo figures and 6 colors

On the blackboard were six figures: a circle, a triangle, a square, a trapezoid, a pentagon and a hexagon, painted in six colors: blue, white, red, yellow, green and brown. Each figure had only one color and all the figures were of different colors. The next day he wondered what color each figure was. Paul replied: “The circle was red, the triangle was blue, the square was white, the trapezoid was green, the pentagon was brown, and the hexagon was yellow. Sofía answered: “The circle was yellow, the triangle was green, the square was red, the trapezoid was blue, the pentagon was brown, and the hexagon was white.” Pablo was wrong three times and Sofia twice, and it is known that the pentagon was brown. Determine if it is possible to know with certainty what the color of each of the figures was.
4
2

12 men and 12 women at the dance

At a dance there are 1212 men, numbered 11 to 1212, and 1212 women numbered 11 to 1212. Each man is assigned a “secret friend” among the 1111 others. They all danced all the pieces. In the first piece each man danced with the woman who has the same number. From then on, each man danced the new piece with the woman who had danced the piece earlier with his secret friend. In the third piece the couples were: https://cdn.artofproblemsolving.com/attachments/c/d/f5ea0931e5751739c1ba556f84ab5736f2d11a.png Find the number of each man's secret friend.

paper constructions, equilateral + rectangle (May Olympiad 2005 L1)

There are two paper figures: an equilateral triangle and a rectangle. The height of rectangle is equal to the height of the triangle and the base of the rectangle is equal to the base of the triangle. Divide the triangle into three parts and the rectangle into two, using straight cuts, so that with the five pieces can be assembled, without gaps or overlays, a equilateral triangle. To assemble the figure, each part can be rotated and / or turned around.
2
2

Remainder and divison

Gonçalo writes in a board four of the the following numbers 0,1,2,3,40, 1, 2, 3, 4, he can repeat numbers. Nicolas can realize the following operation: change one number of the board, by the remainder(in the division by 55) of the product of others two numbers of the board. Nicolas wins if all the four numbers are equal, determine if Gonçalo can choose numbers such that Nicolas will never win.

6 consecutive autodivi integers wanted

An integer is called autodivi if it is divisible by the two-digit number formed by its last two digits (tens and units). For example, 7801378013 is autodivi as it is divisible by 1313, 85178517 is autodivi since it is divisible by 1717. Find 66 consecutive integers that are autodivi and that have the digits of the units, tens and hundreds other than 00.
3
2

Angles of the triangle

In a triangle ABCABC with AB=ACAB = AC, let MM be the midpoint of CBCB and let DD be a point in BCBC such that ∠BAD=∠BAC6\angle BAD = \frac{\angle BAC}{6}. The perpendicular line to ADAD by CC intersects ADAD in NN where DN=DMDN = DM. Find the angles of the triangle BACBAC.

assign 0 and 1 at 99 collinear points

A segment ABAB of length 100100 is divided into 100100 little segments of length 11 by 9999 intermediate points. Endpoint AA is assigned 00 and endpoint BB is assigned 11. Gustavo assigns each of the 9999 intermediate points a 00 or a 11, at his choice, and then color each segment of length 11 blue or red, respecting the following rule: The segments that have the same number at their ends are red, and the segments that have different numbers at their ends are blue. Determine if Gustavo can assign the 00's and 11's so as to get exactly 3030 blue segments. And 3535 blue segments? (In each case, if the answer is yes, show a distribution of 00's and 11's, and if the answer is no, explain why).