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Difference between revisions of "2009 USAMO Problems"

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=Day 1=
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==Day 1==
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==Problem 1==
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===Problem 1===
 
Given circles <math>\omega_1</math> and <math>\omega_2</math> intersecting at points <math>X</math> and <math>Y</math>, let <math>\ell_1</math> be a line through the center of <math>\omega_1</math> intersecting <math>\omega_2</math> at points <math>P</math> and <math>Q</math> and let <math>\ell_2</math> be a line through the center of <math>\omega_2</math> intersecting <math>\omega_1</math> at points <math>R</math> and <math>S</math>.  Prove that if <math>P, Q, R</math> and <math>S</math> lie on a circle then the center of this circle lies on line <math>XY</math>.
 
Given circles <math>\omega_1</math> and <math>\omega_2</math> intersecting at points <math>X</math> and <math>Y</math>, let <math>\ell_1</math> be a line through the center of <math>\omega_1</math> intersecting <math>\omega_2</math> at points <math>P</math> and <math>Q</math> and let <math>\ell_2</math> be a line through the center of <math>\omega_2</math> intersecting <math>\omega_1</math> at points <math>R</math> and <math>S</math>.  Prove that if <math>P, Q, R</math> and <math>S</math> lie on a circle then the center of this circle lies on line <math>XY</math>.
  
 
[[2009 USAMO Problems/Problem 1|Solution]]
 
[[2009 USAMO Problems/Problem 1|Solution]]
  
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==Problem 2==
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===Problem 2===
 
Let <math>n</math> be a positive integer.  Determine the size of the largest subset of <math>\{ - n, - n + 1, \ldots , n - 1, n\}</math> which does not contain three elements <math>a, b, c</math> (not necessarily distinct) satisfying <math>a + b + c = 0</math>.
 
Let <math>n</math> be a positive integer.  Determine the size of the largest subset of <math>\{ - n, - n + 1, \ldots , n - 1, n\}</math> which does not contain three elements <math>a, b, c</math> (not necessarily distinct) satisfying <math>a + b + c = 0</math>.
  
 
[[2009 USAMO Problems/Problem 2|Solution]]
 
[[2009 USAMO Problems/Problem 2|Solution]]
  
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==Problem 3==
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===Problem 3===
 
We define a ''chessboard polygon'' to be a polygon whose sides are situated along lines of the form <math>x = a</math> or <math>y = b</math>, where <math>a</math> and <math>b</math> 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 <math>1 \times 2</math> rectangles. Finally, a ''tasteful tiling'' is one which avoids the two configurations of dominoes shown on the left below. Two tilings of a <math>3 \times 4</math> rectangle are shown; the first one is tasteful, while the second is not, due to the vertical dominoes in the upper right corner.
 
We define a ''chessboard polygon'' to be a polygon whose sides are situated along lines of the form <math>x = a</math> or <math>y = b</math>, where <math>a</math> and <math>b</math> 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 <math>1 \times 2</math> rectangles. Finally, a ''tasteful tiling'' is one which avoids the two configurations of dominoes shown on the left below. Two tilings of a <math>3 \times 4</math> rectangle are shown; the first one is tasteful, while the second is not, due to the vertical dominoes in the upper right corner.
  
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[[2009 USAMO Problems/Problem 3|Solution]]
 
[[2009 USAMO Problems/Problem 3|Solution]]
  
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=Day 2=
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==Day 2==
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==Problem 4==
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===Problem 4===
 
For <math>n \ge 2</math> let <math>a_1</math>, <math>a_2</math>, ..., <math>a_n</math> be positive real numbers such that  
 
For <math>n \ge 2</math> let <math>a_1</math>, <math>a_2</math>, ..., <math>a_n</math> be positive real numbers such that  
 
<center><math> (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 </math></center>
 
<center><math> (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 </math></center>
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Prove that max <math>(a_1, a_2, ... ,a_n) \le  4\, \text{min}\, (a_1, a_2, ... , a_n)</math>.
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Prove that <math>\text{max}\, (a_1, a_2, ... ,a_n) \le  4\, \text{min}\, (a_1, a_2, ... , a_n)</math>.
  
 
[[2009 USAMO Problems/Problem 4|Solution]]
 
[[2009 USAMO Problems/Problem 4|Solution]]
  
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==Problem 5==
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===Problem 5===
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Trapezoid <math>ABCD</math>, with <math>\overline{AB}||\overline{CD}</math>, is inscribed in circle <math>\omega</math> and point <math>G</math> lies inside triangle <math>BCD</math>.  Rays <math>AG</math> and <math>BG</math> meet <math>\omega</math> again at points <math>P</math> and <math>Q</math>, respectively.  Let the line through <math>G</math> parallel to <math>\overline{AB}</math> intersects <math>\overline{BD}</math> and <math>\overline{BC}</math> at points <math>R</math> and <math>S</math>, respectively.  Prove that quadrilateral <math>PQRS</math> is cyclic if and only if <math>\overline{BG}</math> bisects <math>\angle CBD</math>.
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Trapezoid <math>ABCD</math>, with <math>\overline{AB}||\overline{CD}</math>, is inscribed in circle <math>\omega</math> and point <math>G</math> lies inside triangle <math>BCD</math>.  Rays <math>AG</math> and <math>BG</math> meet <math>\omega</math> again at points <math>P</math> and <math>Q</math>, respectively.  Let the line through <math>G</math> parallel to <math>\overline{AB}</math> intersect <math>\overline{BD}</math> and <math>\overline{BC}</math> at points <math>R</math> and <math>S</math>, respectively.  Prove that quadrilateral <math>PQRS</math> is cyclic if and only if <math>\overline{BG}</math> bisects <math>\angle CBD</math>.
  
 
[[2009 USAMO Problems/Problem 5|Solution]]
 
[[2009 USAMO Problems/Problem 5|Solution]]
  
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==Problem 6==
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===Problem 6===
 
Let <math>s_1, s_2, s_3, \ldots</math> be an infinite, nonconstant sequence of rational numbers, meaning it is not the case that <math>s_1 = s_2 = s_3 = \cdots.</math>  Suppose that <math>t_1, t_2, t_3, \ldots</math> is also an infinite, nonconstant sequence of rational numbers with the property that <math>(s_i - s_j)(t_i - t_j)</math> is an integer for all <math>i</math> and <math>j</math>.  Prove that there exists a rational number <math>r</math> such that <math>(s_i - s_j)r</math> and <math>(t_i - t_j)/r</math> are integers for all <math>i</math> and <math>j</math>.
 
Let <math>s_1, s_2, s_3, \ldots</math> be an infinite, nonconstant sequence of rational numbers, meaning it is not the case that <math>s_1 = s_2 = s_3 = \cdots.</math>  Suppose that <math>t_1, t_2, t_3, \ldots</math> is also an infinite, nonconstant sequence of rational numbers with the property that <math>(s_i - s_j)(t_i - t_j)</math> is an integer for all <math>i</math> and <math>j</math>.  Prove that there exists a rational number <math>r</math> such that <math>(s_i - s_j)r</math> and <math>(t_i - t_j)/r</math> are integers for all <math>i</math> and <math>j</math>.
  
 
[[2009 USAMO Problems/Problem 6|Solution]]
 
[[2009 USAMO Problems/Problem 6|Solution]]
  
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= See also =
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== See Also ==
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*[[USAMO Problems and Solutions]]
 
−
 
 
 
{{USAMO newbox|year=2009|before=[[2008 USAMO]]|after=[[2010 USAMO]]}}
 
{{USAMO newbox|year=2009|before=[[2008 USAMO]]|after=[[2010 USAMO]]}}

Latest revision as of 17:16, 17 September 2012

Day 1

Problem 1

Given circles $\omega_1$ and $\omega_2$ intersecting at points $X$ and $Y$, let $\ell_1$ be a line through the center of $\omega_1$ intersecting $\omega_2$ at points $P$ and $Q$ and let $\ell_2$ be a line through the center of $\omega_2$ intersecting $\omega_1$ at points $R$ and $S$. Prove that if $P, Q, R$ and $S$ lie on a circle then the center of this circle lies on line $XY$.

Solution

Problem 2

Let $n$ be a positive integer. Determine the size of the largest subset of $\{ - n, - n + 1, \ldots , n - 1, n\}$ which does not contain three elements $a, b, c$ (not necessarily distinct) satisfying $a + b + c = 0$.

Solution

Problem 3

We define a chessboard polygon to be a polygon whose sides are situated along lines of the form $x = a$ or $y = b$, where $a$ and $b$ 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 $1 \times 2$ rectangles. Finally, a tasteful tiling is one which avoids the two configurations of dominoes shown on the left below. Two tilings of a $3 \times 4$ 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]

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.

Solution

Day 2

Problem 4

For $n \ge 2$ let $a_1$, $a_2$, ..., $a_n$ be positive real numbers such that

$(a_1+a_2+ ... +a_n)\left( {1 \over a_1} + {1 \over a_2} + ... +{1 \over a_n} \right) \le \left(n+ {1 \over 2} \right) ^2$

Prove that $\text{max}\, (a_1, a_2, ... ,a_n) \le 4\, \text{min}\, (a_1, a_2, ... , a_n)$.

Solution

Problem 5

Trapezoid $ABCD$, with $\overline{AB}||\overline{CD}$, is inscribed in circle $\omega$ and point $G$ lies inside triangle $BCD$. Rays $AG$ and $BG$ meet $\omega$ again at points $P$ and $Q$, respectively. Let the line through $G$ parallel to $\overline{AB}$ intersect $\overline{BD}$ and $\overline{BC}$ at points $R$ and $S$, respectively. Prove that quadrilateral $PQRS$ is cyclic if and only if $\overline{BG}$ bisects $\angle CBD$.

Solution

Problem 6

Let $s_1, s_2, s_3, \ldots$ be an infinite, nonconstant sequence of rational numbers, meaning it is not the case that $s_1 = s_2 = s_3 = \cdots.$ Suppose that $t_1, t_2, t_3, \ldots$ is also an infinite, nonconstant sequence of rational numbers with the property that $(s_i - s_j)(t_i - t_j)$ is an integer for all $i$ and $j$. Prove that there exists a rational number $r$ such that $(s_i - s_j)r$ and $(t_i - t_j)/r$ are integers for all $i$ and $j$.

Solution

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
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