2009 USAMO Problems
Contents
Day 1
Problem 1
Given circles and
intersecting at points
and
, let
be a line through the center of
intersecting
at points
and
and let
be a line through the center of
intersecting
at points
and
. Prove that if
and
lie on a circle then the center of this circle lies on line
.
Problem 2
Let be a positive integer. Determine the size of the largest subset of
which does not contain three elements
(not necessarily distinct) satisfying
.
Problem 3
We define a chessboard polygon to be a polygon whose sides are situated along lines of the form or
, where
and
are integers. These lines divide the interior into unit squares, which are shaded alternately grey and white so that adjacent squares have different colors. To tile a chessboard polygon by dominoes is to exactly cover the polygon by non-overlapping
rectangles. Finally, a tasteful tiling is one which avoids the two configurations of dominoes shown on the left below. Two tilings of a
rectangle are shown; the first one is tasteful, while the second is not, due to the vertical dominoes in the upper right corner.
![[asy] size(400); pathpen = linewidth(2.5); void chessboard(int a, int b, pair P){ for(int i = 0; i < a; ++i) for(int j = 0; j < b; ++j) if((i+j) % 2 == 1) fill(shift(P.x+i,P.y+j)*unitsquare,rgb(0.6,0.6,0.6)); D(P--P+(a,0)--P+(a,b)--P+(0,b)--cycle); } chessboard(2,2,(2.5,0));fill(unitsquare,rgb(0.6,0.6,0.6));fill(shift(1,1)*unitsquare,rgb(0.6,0.6,0.6)); chessboard(4,3,(6,0)); chessboard(4,3,(11,0)); MP("\mathrm{Distasteful\ tilings}",(2.25,3),fontsize(12)); /* draw lines */ D((0,0)--(2,0)--(2,2)--(0,2)--cycle); D((1,0)--(1,2)); D((2.5,1)--(4.5,1)); D((7,0)--(7,2)--(6,2)--(10,2)--(9,2)--(9,0)--(9,1)--(7,1)); D((8,2)--(8,3)); D((12,0)--(12,2)--(11,2)--(13,2)); D((13,1)--(15,1)--(14,1)--(14,3)); D((13,0)--(13,3)); [/asy]](http://latex.artofproblemsolving.com/4/b/9/4b990288de8124d702038b22187c64a70b82fd76.png)
a) Prove that if a chessboard polygon can be tiled by dominoes, then it can be done so tastefully.
b) Prove that such a tasteful tiling is unique.
Day 2
Problem 4
For let
,
, ...,
be positive real numbers such that
![$(a_1+a_2+ ... +a_n)\left( {1 \over a_1} + {1 \over a_2} + ... +{1 \over a_n} \right) \le \left(n+ {1 \over 2} \right) ^2$](http://latex.artofproblemsolving.com/5/9/6/5963cac8c96d5f27b431ca50755a0e1b60ed69c1.png)
Prove that .
Problem 5
Trapezoid , with
, is inscribed in circle
and point
lies inside triangle
. Rays
and
meet
again at points
and
, respectively. Let the line through
parallel to
intersect
and
at points
and
, respectively. Prove that quadrilateral
is cyclic if and only if
bisects
.
Problem 6
Let be an infinite, nonconstant sequence of rational numbers, meaning it is not the case that
Suppose that
is also an infinite, nonconstant sequence of rational numbers with the property that
is an integer for all
and
. Prove that there exists a rational number
such that
and
are integers for all
and
.
See also
2009 USAMO (Problems • Resources) | ||
Preceded by 2008 USAMO |
Followed by 2010 USAMO | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |