MathDB

2

Part of 1997 All-Russian Olympiad

Problems(6)

polygon invariant under a 90^\circ rotation

Source: ARMO 1997, 9.2

4/20/2013
Given a convex polygon M invariant under a 90∘90^\circ rotation, show that there exist two circles, the ratio of whose radii is 2\sqrt2, one containing M and the other contained in M. A. Khrabrov
invariantgeometrygeometric transformationrotationratiogeometry proposed
33 students

Source: ARMO 1997, 9.6

4/20/2013
A class consists of 33 students. Each student is asked how many other students in the class have his first name, and how many have his last name. It turns out that each number from 0 to 10 occurs among the answers. Show that there are two students in the class with the same first and last name. A. Shapovalov
combinatorics proposedcombinatorics
n\times n square grid is rolled into a cylinder

Source: ARMO 1997, 10.2

4/20/2013
An n×nn\times n square grid (n⩾3n\geqslant 3) is rolled into a cylinder. Some of the cells are then colored black. Show that there exist two parallel lines (horizontal, vertical or diagonal) of cells containing the same number of black cells. E. Poroshenko
combinatorics proposedcombinatorics
Show that $O$;$D$;$K$ are collinear

Source: ARMO 1997, 10.6

4/20/2013
A circle centered at OO and inscribed in triangle ABCABC meets sides ACAC;ABAB;BCBC at KK;MM;NN, respectively. The median BB1BB_1 of the triangle meets MNMN at DD. Show that OO;DD;KK are collinear. M. Sonkin
geometry proposedgeometry
white hat, black hat or a red hat

Source: ARMO 1997, 11.2

4/20/2013
The Judgment of the Council of Sages proceeds as follows: the king arranges the sages in a line and places either a white hat, black hat or a red hat on each sage's head. Each sage can see the color of the hats of the sages in front of him, but not of his own hat or of the hats of the sages behind him. Then one by one (in an order of their choosing), each sage guesses a color. Afterward, the king executes those sages who did not correctly guess the color of their own hat. The day before, the Council meets and decides to minimize the number of executions. What is the smallest number of sages guaranteed to survive in this case? K. Knop
P.S. Of course, the sages hear the previous guesses.
See also http://www.artofproblemsolving.com/Forum/viewtopic.php?f=42&t=530552
probabilitycombinatorics proposedcombinatorics
polygon, a line $l$ and a point $P$ on $l$

Source: ARMO 1997, 11.6

4/20/2013
We are given a polygon, a line ll and a point PP on ll in general position: all lines containing a side of the polygon meet ll at distinct points diering from PP. We mark each vertex of the polygon the sides meeting which, extended away from the vertex, meet the line ll on opposite sides of PP. Show that PP lies inside the polygon if and only if on each side of ll there are an odd number of marked vertices. O. Musin
geometry proposedgeometry