Difference between revisions of "2003 USAMO Problems"

Latest revision as of 08:21, 14 May 2021

Problems of the 2003 USAMO.

Day 1

Problem 1

Prove that for every positive integer $n$ there exists an $n$ -digit number divisible by $5^n$ all of whose digits are odd.

Solution

Problem 2

A convex polygon $\mathcal{P}$ 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 $\mathcal{P}$ are rational numbers. Prove that the lengths of all sides of all polygons in the dissection are also rational numbers.

Solution

Problem 3

Let $n \neq 0$ . For every sequence of integers

$A = a_0,a_1,a_2,\dots, a_n$

satisfying $0 \le a_i \le i$ , for $i=0,\dots,n$ , define another sequence

$t(A)= t(a_0), t(a_1), t(a_2), \dots, t(a_n)$

by setting $t(a_i)$ to be the number of terms in the sequence $A$ that precede the term $a_i$ and are different from $a_i$ . Show that, starting from any sequence $A$ as above, fewer than $n$ applications of the transformation $t$ lead to a sequence $B$ such that $t(B) = B$ .

Solution

Day 2

Problem 4

Let $ABC$ be a triangle. A circle passing through $A$ and $B$ intersects segments $AC$ and $BC$ at $D$ and $E$ , respectively. Lines $AB$ and $DE$ intersect at $F$ , while lines $BD$ and $CF$ intersect at $M$ . Prove that $MF = MC$ if and only if $MB\cdot MD = MC^2$ .

Solution

Problem 5

Let $a$ , $b$ , $c$ be positive real numbers. Prove that

$\dfrac{(2a + b + c)^2}{2a^2 + (b + c)^2} + \dfrac{(2b + c + a)^2}{2b^2 + (c + a)^2} + \dfrac{(2c + a + b)^2}{2c^2 + (a + b)^2} \le 8.$

Solution

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.

Solution

Resources

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.

@@ Line 5: / Line 5: @@
 === Problem 1 ===
-Prove that for every positive integer <math> \displaystyle n </math> there exists an <math> \displaystyle n </math>-digit number divisible by <math> \displaystyle 5^n </math> all of whose digits are odd.
+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]]
@@ Line 29: / Line 29: @@
 </math>
 </center>
-by setting <math> \displaystyle t(a_i) </math> to be the number of terms in the sequence <math> \displaystyle A </math> that precede the term <math> \displaystyle a_i </math> and are different from <math> \displaystyle a_i </math>. Show that, starting from any sequence <math> \displaystyle A </math> as above, fewer than <math> \displaystyle n </math> applications of the transformation <math> \displaystyle t </math> lead to a sequence <math> \displaystyle B </math> such that <math> \displaystyle t(B) = B </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]]
@@ Line 36: / Line 36: @@
 === 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]]
 === 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]]
 === 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]]
@@ Line 52: / Line 56: @@
 * [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]
+{{USAMO newbox|year=2003|before=[[2002 USAMO]]|after=[[2004 USAMO]]}}
+{{MAA Notice}}

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