Difference between revisions of "2003 USAMO Problems"
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=== Problem 1 === | === Problem 1 === | ||
− | Prove that for every positive integer <math> | + | Prove that for every positive integer <math>n </math> there exists an <math>n </math>-digit number divisible by <math>5^n </math> all of whose digits are odd. |
* [[2003 USAMO Problems/Problem 1 | Solution]] | * [[2003 USAMO Problems/Problem 1 | Solution]] | ||
Line 29: | Line 29: | ||
</math> | </math> | ||
</center> | </center> | ||
− | by setting <math> | + | by setting <math>t(a_i) </math> to be the number of terms in the sequence <math>A </math> that precede the term <math>a_i </math> and are different from <math>a_i </math>. Show that, starting from any sequence <math>A </math> as above, fewer than <math>n </math> applications of the transformation <math>t </math> lead to a sequence <math>B </math> such that <math>t(B) = B </math>. |
* [[2003 USAMO Problems/Problem 3 | Solution]] | * [[2003 USAMO Problems/Problem 3 | Solution]] | ||
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=== Problem 4 === | === Problem 4 === | ||
+ | Let <math>ABC</math> be a triangle. A circle passing through <math>A</math> and <math>B</math> intersects segments <math>AC</math> and <math>BC</math> at <math>D</math> and <math>E</math>, respectively. Lines <math>AB</math> and <math>DE</math> intersect at <math>F</math>, while lines <math>BD</math> and <math>CF</math> intersect at <math>M</math>. Prove that <math>MF = MC</math> if and only if <math>MB\cdot MD = MC^2</math>. | ||
* [[2003 USAMO Problems/Problem 4 | Solution]] | * [[2003 USAMO Problems/Problem 4 | Solution]] | ||
=== Problem 5 === | === Problem 5 === | ||
+ | Let <math>a</math>, <math>b</math>, <math>c</math> be positive real numbers. Prove that | ||
+ | <center><math>\dfrac{(2a + b + c)^2}{2a^2 + (b + c)^2} + \dfrac{(2b + c + a)^2}{2b^2 + (c + a)^2} + \dfrac{(2c + a + b)^2}{2c^2 + (a + b)^2} \le 8.</math></center> | ||
* [[2003 USAMO Problems/Problem 5 | Solution]] | * [[2003 USAMO Problems/Problem 5 | Solution]] | ||
=== Problem 6 === | === Problem 6 === | ||
+ | At the vertices of a regular hexagon are written six nonnegative integers whose sum is 2003. Bert is allowed to make moves of the following form: he may pick a vertex and replace the number written there by the absolute value of the difference between the numbers written at the two neighboring vertices. Prove that Bert can make a sequence of moves, after which the number 0 appears at all six vertices. | ||
* [[2003 USAMO Problems/Problem 6 | Solution]] | * [[2003 USAMO Problems/Problem 6 | Solution]] | ||
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* [http://www.unl.edu/amc/e-exams/e8-usamo/e8-1-usamoarchive/2003-ua/03usamo-test.shtml 2003 USAMO Problems and Solutions] | * [http://www.unl.edu/amc/e-exams/e8-usamo/e8-1-usamoarchive/2003-ua/03usamo-test.shtml 2003 USAMO Problems and Solutions] | ||
* [http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=27&year=2003 2003 USAMO Problems on the Resources page] | * [http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=27&year=2003 2003 USAMO Problems on the Resources page] | ||
+ | {{USAMO newbox|year=2003|before=[[2002 USAMO]]|after=[[2004 USAMO]]}} | ||
+ | {{MAA Notice}} |
Latest revision as of 09:21, 14 May 2021
Contents
Day 1
Problem 1
Prove that for every positive integer there exists an
-digit number divisible by
all of whose digits are odd.
Problem 2
A convex polygon in the plane is dissected into smaller convex polygons by drawing all of its diagonals. The lengths of all sides and all diagonals of the polygon
are rational numbers. Prove that the lengths of all sides of all polygons in the dissection are also rational numbers.
Problem 3
Let . For every sequence of integers
satisfying , for
, define another sequence
by setting to be the number of terms in the sequence
that precede the term
and are different from
. Show that, starting from any sequence
as above, fewer than
applications of the transformation
lead to a sequence
such that
.
Day 2
Problem 4
Let be a triangle. A circle passing through
and
intersects segments
and
at
and
, respectively. Lines
and
intersect at
, while lines
and
intersect at
. Prove that
if and only if
.
Problem 5
Let ,
,
be positive real numbers. Prove that
![$\dfrac{(2a + b + c)^2}{2a^2 + (b + c)^2} + \dfrac{(2b + c + a)^2}{2b^2 + (c + a)^2} + \dfrac{(2c + a + b)^2}{2c^2 + (a + b)^2} \le 8.$](http://latex.artofproblemsolving.com/d/2/b/d2be8552ac3b2dcfb8d235a80ddc4d812b2f2155.png)
Problem 6
At the vertices of a regular hexagon are written six nonnegative integers whose sum is 2003. Bert is allowed to make moves of the following form: he may pick a vertex and replace the number written there by the absolute value of the difference between the numbers written at the two neighboring vertices. Prove that Bert can make a sequence of moves, after which the number 0 appears at all six vertices.
Resources
- USAMO Problems and Solutions
- 2003 USAMO Problems and Solutions
- 2003 USAMO Problems on the Resources page
2003 USAMO (Problems • Resources) | ||
Preceded by 2002 USAMO |
Followed by 2004 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.