Difference between revisions of "2007 USAMO Problems"
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− | =Day 1= | + | ==Day 1== |
− | ==Problem 1== | + | ===Problem 1=== |
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. | 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. | ||
[[2007 USAMO Problems/Problem 1 | Solution]] | [[2007 USAMO Problems/Problem 1 | Solution]] | ||
− | ==Problem 2== | + | ===Problem 2=== |
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? | 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=== |
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. | 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== |
− | ==Problem 4== | + | ===Problem 4=== |
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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[[2007 USAMO Problems/Problem 4 | Solution]] | [[2007 USAMO Problems/Problem 4 | Solution]] | ||
− | ==Problem 5== | + | ===Problem 5=== |
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. | ||
[[2007 USAMO Problems/Problem 5 | Solution]] | [[2007 USAMO Problems/Problem 5 | Solution]] | ||
− | ==Problem 6== | + | ===Problem 6=== |
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 | 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> | <cmath>8P_AQ_A \cdot P_BQ_B \cdot P_CQ_C \le R^3,</cmath> | ||
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[[2007 USAMO Problems/Problem 6 | Solution]] | [[2007 USAMO Problems/Problem 6 | Solution]] | ||
− | = See | + | == See Also == |
− | + | {{USAMO newbox|year=2007|before=[[2006 USAMO]]|after=[[2008 USAMO]]}} | |
− | + | {{MAA Notice}} | |
− | {{USAMO newbox|year=2007|before=[[2006 USAMO]]|after=2008 USAMO}} |
Latest revision as of 13:42, 4 July 2013
Contents
Day 1
Problem 1
Let be a positive integer. Define a sequence by setting
and, for each
, letting
be the unique integer in the range
for which
is divisible by
. For instance, when
the obtained sequence is
. Prove that for any
the sequence
eventually becomes constant.
Problem 2
A square grid on the Euclidean plane consists of all points , where
and
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?
Problem 3
Let be a set containing
elements, for some positive integer
. Suppose that the
-element subsets of
are partitioned into two classes. Prove that there are at least
pairwise disjoint sets in the same class.
Day 2
Problem 4
An animal with cells is a connected figure consisting of
equal-sized cells.
The figure below shows an 8-cell animal.
A dinosaur is an animal with at least 2007 cells. It is said to be primitive if its cells cannot be partitioned into two or more dinosaurs. Find with proof the maximum number of cells in a primitive dinosaur.
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
cells, one with
cells is obtained by adjoining a new cell by making it adjacent to one or more existing cells.
Problem 5
Prove that for every nonnegative integer , the number
is the product of at least
(not necessarily distinct) primes.
Problem 6
Let be an acute triangle with
,
, and
being its incircle, circumcircle, and circumradius, respectively. Circle
is tangent internally to
at
and tangent externally to
. Circle
is tangent internally to
at
and tangent internally to
. Let
and
denote the centers of
and
, respectively. Define points
,
,
,
analogously. Prove that
with equality if and only if triangle
is equilateral.
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.