MathDB

1

game in equilateral grid

Bob proposes the following game to Johanna. The board in the figure is an equilateral triangle subdivided in turn into

256

small equilateral triangles, one of which is painted in black. Bob chooses any point inside the board and places a small token. Johanna can make three types of plays. Each of them consists of choosing any of the

3

vertices of the board and move the token to the midpoint between the current position of the tile and the chosen vertex. In the second figure we see an example of a move in which Johana chose vertex

A

. Johanna wins if she manages to place her piece inside the triangle black. Prove that Johanna can always win in at most

4

moves.
[asy] unitsize(8 cm);
pair A, B, C; int i;
A = dir(60); C = (0,0); B = (1,0);
fill((6/16*(1,0) + 1/16*dir(60))--(7/16*(1,0) + 1/16*dir(60))--(6/16*(1,0) + 2/16*dir(60))--cycle, gray(0.7)); draw(A--B--C--cycle);
for (i = 1; i <= 15; ++i) { draw(interp(A,B,i/16)--interp(A,C,i/16)); draw(interp(B,C,i/16)--interp(B,A,i/16)); draw(interp(C,A,i/16)--interp(C,B,i/16)); }
label("

A

", A, N); label("

B

", B, SE); label("

C

", C, SW); [/asy]
[asy] unitsize(8 cm);
pair A, B, C, X, Y, Z; int i;
A = dir(60); C = (0,0); B = (1,0); X = 9.2/16*(1,0) + 3.3/16*dir(60); Y = (A + X)/2; Z = rotate(60,X)*(Y);
fill((6/16*(1,0) + 1/16*dir(60))--(7/16*(1,0) + 1/16*dir(60))--(6/16*(1,0) + 2/16*dir(60))--cycle, gray(0.7)); draw(A--B--C--cycle);
for (i = 1; i <= 15; ++i) { draw(interp(A,B,i/16)--interp(A,C,i/16)); draw(interp(B,C,i/16)--interp(B,A,i/16)); draw(interp(C,A,i/16)--interp(C,B,i/16)); }
draw(A--X, dotted); draw(arc(Z,abs(X - Y),-12,40), Arrow(6));
label("

A

", A, N); label("

B

", B, SE); label("

C

", C, SW); dot(A); dot(X); dot(Y); [/asy]

4

1

31 guests around a round table

31

guests at a party sit in equally spaced chairs around a round table , but they have not noticed that there are cards with the names of the guests on the stalls. (a) Assuming they have been so unlucky that no one is in the room which corresponds to him, show that it is possible to get at least two people to stay in their correct position, without anyone getting up from their seat, turning the table. (b) Show a configuration where exactly one guest is in his assigned place and where in no way that the table is turned it is possible to achieve that at least two remain right.

3

1

2player game, n-> n+1 or n-> n/2

Two players, Aurelio and Bernardo, play the following game. Aurelio begins by writing the number

1

. Next it is Bernardo's turn, who writes number

2

. From then on, each player chooses whether to add

1

to the number just written by the previous player, or whether multiply that number by

2

. Then write the result and it's the other player's turn. The first player to write a number greater than

2007

loses the game. Determine if one of the players can ensure victory no matter what the other does.

1

horses moves in 16x16 board

On a chessboard of

16 \times 16

squares, a "horse" moves making only movements of two types: from each square you can move either two squares to the right and one up, or two boxes up and one to the right. Determine in how many ways different the horse can move from the lower left square of the board to the top right box.

6

1

<MTB - <CTM does not depend on X, where X lies on circumcircle

Given an

\triangle ABC

isoceles with base

BC

we note with

M

the midpoint of said base. Let

X

be any point on the shortest arc

AM

of the circumcircle of

\triangle ABM

and let

T

be a point on the inside

\angle BMA

such that

\angle TMX = 90^o

and

TX = BX

. Show that

\angle MTB - \angle CTM

does not depend on

X

.

2

1

circle with min area that contains a given triangle

Given a

\triangle ABC

, determine which is the circle with the smallest area that contains it.

2007 Chile National Olympiad

Subcontests

game in equilateral grid

31 guests around a round table

2player game, n-> n+1 or n-> n/2

horses moves in 16x16 board

<MTB - <CTM does not depend on X, where X lies on circumcircle

circle with min area that contains a given triangle

2007 Chile National Olympiad

Subcontests

game in equilateral grid

31 guests around a round table

2player game, n-&gt; n+1 or n-&gt; n/2

horses moves in 16x16 board

&lt;MTB - &lt;CTM does not depend on X, where X lies on circumcircle

circle with min area that contains a given triangle

2player game, n-> n+1 or n-> n/2

<MTB - <CTM does not depend on X, where X lies on circumcircle