Difference between revisions of "2003 USAMO Problems"

Revision as of 16:56, 20 August 2008

Problems of the 2003 USAMO.

Day 1

Problem 1

Prove that for every positive integer $\displaystyle n$ there exists an $\displaystyle n$ -digit number divisible by $\displaystyle 5^n$ 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 $\displaystyle t(a_i)$ to be the number of terms in the sequence $\displaystyle A$ that precede the term $\displaystyle a_i$ and are different from $\displaystyle a_i$ . Show that, starting from any sequence $\displaystyle A$ as above, fewer than $\displaystyle n$ applications of the transformation $\displaystyle t$ lead to a sequence $\displaystyle B$ such that $\displaystyle t(B) = B$ .

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

@@ 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]]

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