Difference between revisions of "2007 USAMO Problems"
m (m) |
(centering, + img) |
||
Line 3: | Line 3: | ||
==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>\displaystyle a_1, a_2, a_3, \ldots</math> eventually becomes constant. |
+ | |||
[[2007 USAMO Problems/Problem 1 | Solution]] | [[2007 USAMO Problems/Problem 1 | Solution]] | ||
Line 25: | Line 26: | ||
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. | ||
− | + | <div style="text-align:center;">[[Image:2007 USAMO-4.PNG|300px]]</div> | |
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. | 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. | ||
Line 43: | Line 44: | ||
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 | ||
− | <math> | + | <div style="text-align:center;"><math> |
8P_AQ_A \cdot P_BQ_B \cdot P_CQ_C \le R^3, | 8P_AQ_A \cdot P_BQ_B \cdot P_CQ_C \le R^3, | ||
− | </math> | + | </math></div> |
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 16:27, 26 April 2007
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 it 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 |