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Difference between revisions of "2007 USAMO Problems"

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=Day 1=
 
=Day 1=
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==Problem 1==
 
==Problem 1==
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Let <math>n</math> be a positive integer. Define a sequence by setting <math>a_1=n</math> and, for each <math>k>1</math>, letting <math>a_k</math> be the unique integer in the range <math>0\le a_k\le k-1</math> for which <math>a_1+a_2+\cdots+a_k</math> is divisible by <math>k</math>. For instance, when <math>n=9</math> the obtained sequence is <math>9, 1, 2, 0, 3, 3, 3, \ldots</math>.  Prove that for any <math>n</math> the sequence <math>a_1, a_2, a_3, \ldots</math> eventually becomes constant.
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Let <math>n</math> be a positive integer. Define a sequence by setting <math>a_1 = n</math> and, for each <math>k>1</math>, letting <math>a_k</math> be the unique integer in the range <math>0 \le a_k \le k-1</math> for which <math>a_1 + a_2 + \cdots + a_k</math> is divisible by <math>k</math>. For instance, when <math>n=9</math> the obtained sequence is <math>9, 1, 2, 0, 3, 3, 3, \ldots</math>.  Prove that for any <math>n</math> the sequence <math>\displaystyle a_1, a_2, a_3, \ldots</math> eventually becomes constant.
 
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[[2007 USAMO Problems/Problem 1 | Solution]]
 
[[2007 USAMO Problems/Problem 1 | Solution]]
  
 
==Problem 2==
 
==Problem 2==
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A square grid on the Euclidean plane consists of all points <math>(m,n)</math>, where <math>m</math> and <math>n</math> are integers. Is it possible to cover all grid points by an infinite family of discs with non-overlapping interiors if each disc in the family has radius at least 5?
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A square grid on the Euclidean plane consists of all points <math>(m,n)</math>, where <math>m</math> and <math>n</math> are integers. Is it possible to cover all grid points by an infinite family of discs with non-overlapping interiors if each disc in the family has radius at least 5?
 
  
 
[[2007 USAMO Problems/Problem 2 | Solution]]
 
[[2007 USAMO Problems/Problem 2 | Solution]]
  
 
==Problem 3==
 
==Problem 3==
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Let <math>S</math> be a set containing <math>n^2+n-1</math> elements, for some positive integer <math>n</math>. Suppose that the <math>n</math>-element subsets of <math>S</math> are partitioned into two classes. Prove that there are at least <math>n</math> pairwise disjoint sets in the same class.
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Let <math>S</math> be a set containing <math>n^2+n-1</math> elements, for some positive integer <math>n</math>. Suppose that the <math>n</math>-element subsets of <math>S</math> are partitioned into two classes. Prove that there are at least <math>n</math> pairwise disjoint sets in the same class.
 
  
 
[[2007 USAMO Problems/Problem 3 | Solution]]
 
[[2007 USAMO Problems/Problem 3 | Solution]]
  
 
=Day 2=
 
=Day 2=
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==Problem 4==
 
==Problem 4==
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An ''animal'' with <math>n</math> ''cells'' is a connected figure consisting of <math>n</math> equal-sized cells.<math>{}^1</math> The figure below shows an 8-cell animal.
 
An ''animal'' with <math>n</math> ''cells'' is a connected figure consisting of <math>n</math> equal-sized cells.<math>{}^1</math> The figure below shows an 8-cell animal.
  
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<div style="text-align:center;">[[Image:2007 USAMO-4.PNG|300px]]</div>
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[[Image:2007 USAMO-4.PNG|300px|center]]
  
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A ''dinosaur'' is an animal with at least 2007 cells. It is said to be ''primitive'' it its cells cannot be partitioned into two or more dinosaurs. Find with proof the maximum number of cells in a primitive dinosaur.
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A ''dinosaur'' is an animal with at least 2007 cells. It is said to be ''primitive'' it its cells cannot be partitioned into two or more dinosaurs. Find with proof the maximum number of cells in a primitive dinosaur.
  
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<small><math>{}^1</math>Animals are also called ''polyominoes''. They can be defined inductively. Two cells are ''adjacent'' if they share a complete edge. A single cell is an animal, and given an animal with <math>n</math> cells, one with <math>n+1</math> cells is obtained by adjoining a new cell by making it adjacent to one or more existing cells.</small>
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<small><math>{}^1</math>Animals are also called ''polyominoes''. They can be defined inductively. Two cells are ''adjacent'' if they share a complete edge. A single cell is an animal, and given an animal with <math>n</math> cells, one with <math>n+1</math> cells is obtained by adjoining a new cell by making it adjacent to one or more existing cells.</small>
  
 
[[2007 USAMO Problems/Problem 4 | Solution]]
 
[[2007 USAMO Problems/Problem 4 | Solution]]
  
 
==Problem 5==
 
==Problem 5==
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Prove that for every nonnegative integer <math>n</math>, the number <math>7^{7^n}+1</math> is the product of at least <math>2n+3</math> (not necessarily distinct) primes.
 
Prove that for every nonnegative integer <math>n</math>, the number <math>7^{7^n}+1</math> is the product of at least <math>2n+3</math> (not necessarily distinct) primes.
  
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==Problem 6==
 
==Problem 6==
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Let <math>ABC</math> be an acute triangle with <math>\omega</math>, <math>\Omega</math>, and <math>R</math> being its incircle, circumcircle, and circumradius, respectively. Circle <math>\omega_A</math> is tangent internally to <math>\Omega</math> at <math>A</math> and tangent externally to <math>\omega</math>. Circle <math>\Omega_A</math> is tangent internally to <math>\Omega</math> at <math>A</math> and tangent internally to <math>\omega</math>. Let <math>P_A</math> and <math>Q_A</math> denote the centers of <math>\omega_A</math> and <math>\Omega_A</math>, respectively.  Define points <math>P_B</math>, <math>Q_B</math>, <math>P_C</math>, <math>Q_C</math> analogously.  Prove that
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Let <math>ABC</math> be an acute triangle with <math>\omega</math>, <math>\Omega</math>, and <math>R</math> being its incircle, circumcircle, and circumradius, respectively. Circle <math>\omega_A</math> is tangent internally to <math>\Omega</math> at <math>A</math> and tangent externally to <math>\omega</math>. Circle <math>\Omega_A</math> is tangent internally to <math>\Omega</math> at <math>A</math> and tangent internally to <math>\omega</math>. Let <math>P_A</math> and <math>Q_A</math> denote the centers of <math>\omega_A</math> and <math>\Omega_A</math>, respectively.  Define points <math>P_B</math>, <math>Q_B</math>, <math>P_C</math>, <math>Q_C</math> analogously.  Prove that
+
<cmath>8P_AQ_A \cdot P_BQ_B \cdot P_CQ_C \le R^3,</cmath>
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<div style="text-align:center;"><math>
 
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8P_AQ_A \cdot P_BQ_B \cdot P_CQ_C \le R^3,
 
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</math></div>
 
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with equality if and only if triangle <math>ABC</math> is equilateral.
 
with equality if and only if triangle <math>ABC</math> is equilateral.
  

Revision as of 00:45, 3 May 2008

Day 1

Problem 1

Let $n$ be a positive integer. Define a sequence by setting $a_1=n$ and, for each $k>1$, letting $a_k$ be the unique integer in the range $0\le a_k\le k-1$ for which $a_1+a_2+\cdots+a_k$ is divisible by $k$. For instance, when $n=9$ the obtained sequence is $9, 1, 2, 0, 3, 3, 3, \ldots$. Prove that for any $n$ the sequence $a_1, a_2, a_3, \ldots$ eventually becomes constant.

Solution

Problem 2

A square grid on the Euclidean plane consists of all points $(m,n)$, where $m$ and $n$ are integers. Is it possible to cover all grid points by an infinite family of discs with non-overlapping interiors if each disc in the family has radius at least 5?

Solution

Problem 3

Let $S$ be a set containing $n^2+n-1$ elements, for some positive integer $n$. Suppose that the $n$-element subsets of $S$ are partitioned into two classes. Prove that there are at least $n$ pairwise disjoint sets in the same class.

Solution

Day 2

Problem 4

An animal with $n$ cells is a connected figure consisting of $n$ equal-sized cells.${}^1$ The figure below shows an 8-cell animal.

2007 USAMO-4.PNG

A dinosaur is an animal with at least 2007 cells. It is said to be primitive it its cells cannot be partitioned into two or more dinosaurs. Find with proof the maximum number of cells in a primitive dinosaur.

${}^1$Animals are also called polyominoes. They can be defined inductively. Two cells are adjacent if they share a complete edge. A single cell is an animal, and given an animal with $n$ cells, one with $n+1$ cells is obtained by adjoining a new cell by making it adjacent to one or more existing cells.

Solution

Problem 5

Prove that for every nonnegative integer $n$, the number $7^{7^n}+1$ is the product of at least $2n+3$ (not necessarily distinct) primes.

Solution

Problem 6

Let $ABC$ be an acute triangle with $\omega$, $\Omega$, and $R$ being its incircle, circumcircle, and circumradius, respectively. Circle $\omega_A$ is tangent internally to $\Omega$ at $A$ and tangent externally to $\omega$. Circle $\Omega_A$ is tangent internally to $\Omega$ at $A$ and tangent internally to $\omega$. Let $P_A$ and $Q_A$ denote the centers of $\omega_A$ and $\Omega_A$, respectively. Define points $P_B$, $Q_B$, $P_C$, $Q_C$ analogously. Prove that \[8P_AQ_A \cdot P_BQ_B \cdot P_CQ_C \le R^3,\] with equality if and only if triangle $ABC$ is equilateral.

Solution

See also

2007 USAMO (Problems • Resources)
Preceded by
2006 USAMO 		Followed by
2008 USAMO
1 • 2 • 3 • 4 • 5 • 6
All USAMO Problems and Solutions
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