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Difference between revisions of "2008 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===
 
(''Titu Andreescu'') Prove that for each positive integer <math>n</math>, there are pairwise relatively prime integers <math>k_0,k_1\ldots,k_n</math>, all strictly greater than 1, such that <math>k_0k_1\cdots k_n-1</math> is the product of two consecutive integers.
 
(''Titu Andreescu'') Prove that for each positive integer <math>n</math>, there are pairwise relatively prime integers <math>k_0,k_1\ldots,k_n</math>, all strictly greater than 1, such that <math>k_0k_1\cdots k_n-1</math> is the product of two consecutive integers.
  
 
[[2008 USAMO Problems/Problem 1|Solution]]
 
[[2008 USAMO Problems/Problem 1|Solution]]
  
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==Problem 2==
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===Problem 2===
 
(''Zuming Feng'') Let <math>ABC</math> be an acute, [[scalene]] triangle, and let <math>M</math>, <math>N</math>, and <math>P</math> be the midpoints of <math>\overline{BC}</math>, <math>\overline{CA}</math>, and <math>\overline{AB}</math>, respectively. Let the [[perpendicular]] [[bisect]]ors of <math>\overline{AB}</math> and <math>\overline{AC}</math> intersect ray <math>AM</math> in points <math>D</math> and <math>E</math> respectively, and let lines <math>BD</math> and <math>CE</math> intersect in point <math>F</math>, inside of triangle <math>ABC</math>. Prove that points <math>A</math>, <math>N</math>, <math>F</math>, and <math>P</math> all lie on one circle.
 
(''Zuming Feng'') Let <math>ABC</math> be an acute, [[scalene]] triangle, and let <math>M</math>, <math>N</math>, and <math>P</math> be the midpoints of <math>\overline{BC}</math>, <math>\overline{CA}</math>, and <math>\overline{AB}</math>, respectively. Let the [[perpendicular]] [[bisect]]ors of <math>\overline{AB}</math> and <math>\overline{AC}</math> intersect ray <math>AM</math> in points <math>D</math> and <math>E</math> respectively, and let lines <math>BD</math> and <math>CE</math> intersect in point <math>F</math>, inside of triangle <math>ABC</math>. Prove that points <math>A</math>, <math>N</math>, <math>F</math>, and <math>P</math> all lie on one circle.
  
 
[[2008 USAMO Problems/Problem 2|Solution]]
 
[[2008 USAMO Problems/Problem 2|Solution]]
  
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==Problem 3==
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===Problem 3===
 
(''Gabriel Carroll'') Let <math>n</math> be a positive integer. Denote by <math>S_n</math> the set of points <math>(x, y)</math> with integer coordinates such that
 
(''Gabriel Carroll'') Let <math>n</math> be a positive integer. Denote by <math>S_n</math> the set of points <math>(x, y)</math> with integer coordinates such that
 
<cmath>\left|x\right|+\left|y+\frac{1}{2}\right|<n</cmath>
 
<cmath>\left|x\right|+\left|y+\frac{1}{2}\right|<n</cmath>
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[[2008 USAMO Problems/Problem 3|Solution]]
 
[[2008 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===
 
(''Gregory Galparin'') Let <math>\mathcal{P}</math> be a [[convex polygon]] with <math>n</math> sides, <math>n\ge3</math>. Any set of <math>n-3</math> diagonals of <math>\mathcal{P}</math> that do not intersect in the interior of the polygon determine a ''triangulation'' of <math>\mathcal{P}</math> into <math>n - 2</math> triangles. If <math>\mathcal{P}</math> is regular and there is a triangulation of <math>\mathcal{P}</math> consisting of only isosceles triangles, find all the possible values of <math>n</math>.
 
(''Gregory Galparin'') Let <math>\mathcal{P}</math> be a [[convex polygon]] with <math>n</math> sides, <math>n\ge3</math>. Any set of <math>n-3</math> diagonals of <math>\mathcal{P}</math> that do not intersect in the interior of the polygon determine a ''triangulation'' of <math>\mathcal{P}</math> into <math>n - 2</math> triangles. If <math>\mathcal{P}</math> is regular and there is a triangulation of <math>\mathcal{P}</math> consisting of only isosceles triangles, find all the possible values of <math>n</math>.
  
 
[[2008 USAMO Problems/Problem 4|Solution]]
 
[[2008 USAMO Problems/Problem 4|Solution]]
  
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==Problem 5==
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===Problem 5===
 
(''Kiran Kedlaya'') Three nonnegative real numbers <math>r_1</math>, <math>r_2</math>, <math>r_3</math> are written on a blackboard. These numbers have the property that there exist integers <math>a_1</math>, <math>a_2</math>, <math>a_3</math>, not all zero, satisfying <math>a_1r_1+a_2r_2+a_3r_3=0</math>. We are permitted to perform the following operation: find two numbers <math>x</math>, <math>y</math> on the blackboard with <math>x\le y</math>, then erase <math>y</math> and write <math>y-x</math> in its place. Prove that after a finite number of such operations, we can end up with at least one <math>0</math> on the blackboard.
 
(''Kiran Kedlaya'') Three nonnegative real numbers <math>r_1</math>, <math>r_2</math>, <math>r_3</math> are written on a blackboard. These numbers have the property that there exist integers <math>a_1</math>, <math>a_2</math>, <math>a_3</math>, not all zero, satisfying <math>a_1r_1+a_2r_2+a_3r_3=0</math>. We are permitted to perform the following operation: find two numbers <math>x</math>, <math>y</math> on the blackboard with <math>x\le y</math>, then erase <math>y</math> and write <math>y-x</math> in its place. Prove that after a finite number of such operations, we can end up with at least one <math>0</math> on the blackboard.
  
 
[[2008 USAMO Problems/Problem 5|Solution]]
 
[[2008 USAMO Problems/Problem 5|Solution]]
  
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==Problem 6==
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===Problem 6===
 
(''[[Sam Vandervelde]]'') At a certain mathematical conference, every pair of mathematicians are either friends or strangers. At mealtime, every participant eats in one of two large dining rooms. Each mathematician insists upon eating in a room which contains an even number of his or her friends. Prove that the number of ways that the mathematicians may be split between the two rooms is a power of two (i.e., is of the form <math>2^k</math> for some positive integer <math>k</math>).
 
(''[[Sam Vandervelde]]'') At a certain mathematical conference, every pair of mathematicians are either friends or strangers. At mealtime, every participant eats in one of two large dining rooms. Each mathematician insists upon eating in a room which contains an even number of his or her friends. Prove that the number of ways that the mathematicians may be split between the two rooms is a power of two (i.e., is of the form <math>2^k</math> for some positive integer <math>k</math>).
  
 
[[2008 USAMO Problems/Problem 6|Solution]]
 
[[2008 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]]
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{{USAMO newbox|year=2008|before=[[2007 USAMO Problems|2007 USAMO]]|after=[[2009 USAMO Problems|2009 USAMO]]}}
  
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{{USAMO newbox|year=2008|before=[[2007 USAMO]]|after=2009 USAMO}}
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{{MAA Notice}}

Latest revision as of 20:48, 12 August 2014

Day 1

Problem 1

(Titu Andreescu) Prove that for each positive integer $n$, there are pairwise relatively prime integers $k_0,k_1\ldots,k_n$, all strictly greater than 1, such that $k_0k_1\cdots k_n-1$ is the product of two consecutive integers.

Solution

Problem 2

(Zuming Feng) Let $ABC$ be an acute, scalene triangle, and let $M$, $N$, and $P$ be the midpoints of $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$, respectively. Let the perpendicular bisectors of $\overline{AB}$ and $\overline{AC}$ intersect ray $AM$ in points $D$ and $E$ respectively, and let lines $BD$ and $CE$ intersect in point $F$, inside of triangle $ABC$. Prove that points $A$, $N$, $F$, and $P$ all lie on one circle.

Solution

Problem 3

(Gabriel Carroll) Let $n$ be a positive integer. Denote by $S_n$ the set of points $(x, y)$ with integer coordinates such that \[\left|x\right|+\left|y+\frac{1}{2}\right|<n\] A path is a sequence of distinct points $(x_1,y_1),(x_2,y_2),\ldots,(x_\ell,y_\ell)$ in $S_n$ such that, for $i=2,\ldots,\ell$, the distance between $(x_i,y_i)$ and $(x_{i-1},y_{i-1})$ is $1$ (in other words, the points $(x_i,y_i)$ and $(x_{i-1},y_{i-1})$ are neighbors in the lattice of points with integer coordinates). Prove that the points in $S_n$ cannot be partitioned into fewer than $n$ paths (a partition of $S_n$ into $m$ paths is a set $\mathcal{P}$ of $m$ nonempty paths such that each point in $S_n$ appears in exactly one of the $m$ paths in $\mathcal{P}$).

Solution

Day 2

Problem 4

(Gregory Galparin) Let $\mathcal{P}$ be a convex polygon with $n$ sides, $n\ge3$. Any set of $n-3$ diagonals of $\mathcal{P}$ that do not intersect in the interior of the polygon determine a triangulation of $\mathcal{P}$ into $n - 2$ triangles. If $\mathcal{P}$ is regular and there is a triangulation of $\mathcal{P}$ consisting of only isosceles triangles, find all the possible values of $n$.

Solution

Problem 5

(Kiran Kedlaya) Three nonnegative real numbers $r_1$, $r_2$, $r_3$ are written on a blackboard. These numbers have the property that there exist integers $a_1$, $a_2$, $a_3$, not all zero, satisfying $a_1r_1+a_2r_2+a_3r_3=0$. We are permitted to perform the following operation: find two numbers $x$, $y$ on the blackboard with $x\le y$, then erase $y$ and write $y-x$ in its place. Prove that after a finite number of such operations, we can end up with at least one $0$ on the blackboard.

Solution

Problem 6

(Sam Vandervelde) At a certain mathematical conference, every pair of mathematicians are either friends or strangers. At mealtime, every participant eats in one of two large dining rooms. Each mathematician insists upon eating in a room which contains an even number of his or her friends. Prove that the number of ways that the mathematicians may be split between the two rooms is a power of two (i.e., is of the form $2^k$ for some positive integer $k$).

Solution

See Also

2008 USAMO (Problems • Resources)
Preceded by
2007 USAMO 		Followed by
2009 USAMO
1 • 2 • 3 • 4 • 5 • 6
All USAMO Problems and Solutions

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