Difference between revisions of "2002 USAMO Problems"

Revision as of 14:41, 28 March 2015

Problems of the 2002 USAMO.

Day 1

Problem 1

Let $S$ be a set with 2002 elements, and let $N$ be an integer with $0 \le N \le 2^{2002}$ . Prove that it is possible to color every subset of $S$ either blue or red so that the following conditions hold:

(a) the union of any two red subsets is red;

(b) the union of any two blue subsets is blue;

(c) there are exactly $N$ red subsets.

Solution

Problem 2

Let $ABC$ be a triangle such that

$\left( \cot \frac{A}{2} \right)^2 + \left( 2 \cot \frac{B}{2} \right)^2 + \left( 3 \cot \frac{C}{2} \right)^2 = \left( \frac{6s}{7r} \right)^2$ ,

where $s$ and $r$ denote its semiperimeter and inradius, respectively. Prove that triangle $ABC$ is similar to a triangle $T$ whose side lengths are all positive integers with no common divisor and determine those integers.

Solution

Problem 3

Prove that any monic polynomial (a polynomial with leading coefficient 1) of degree $n$ with real coefficients is the average of two monic polynomials of degree $n$ with $n$ real roots.

Solution

Day 2

Problem 4

Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that

$f(x^2 - y^2) = xf(x) - yf(y)$

for all pairs of real numbers $x$ and $y$ .

Solution

Problem 5

Let $a, b$ be integers greater than 2. Prove that there exists a positive integer $k$ and a finite sequence $n_1, n_2, \ldots, n_k$ of positive integers such that $n_1 = a$ , $n_k = b$ , and $n_in_{i+1}$ is divisible by $n_i + n_{i+1}$ for each $i$ ( $1 \le i \le k$ ).

Solution

Problem 6

I have an $n \times n$ sheet of stamps, from which I've been asked to tear out blocks of three adjacent stamps in a single row or column. (I can only tear along the perforations separating adjacent stamps, and each block must come out of the sheet in one piece.) Let $b(n)$ be the smallest number of blocks I can tear out and make it impossible to tear out any more blocks. Prove that there are real constants $c$ and $d$ such that

$\dfrac{1}{7} n^2 - cn \leq b(n) \leq \dfrac{1}{5} n^2 + dn$

for all $n > 0$ .

Solution

Resources

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

@@ Line 49: / Line 49: @@
 === Problem 5 ===
-Let <math>a, b </math> be integers greater than 2.  Prove that there exists a positive integer <math>k </math> and a finite sequence <math>n_1, n_2, \ldots, n_k </math> of positive integers such that <math>n_1 = a</math>, <math>n_k = b </math>, and <math>n_1n_{i+1} </math> is divisible by <math>n_i + n_{i+1} </math> for each <math>i </math> (<math> 1 \le i \le k </math>).
+Let <math>a, b </math> be integers greater than 2.  Prove that there exists a positive integer <math>k </math> and a finite sequence <math>n_1, n_2, \ldots, n_k </math> of positive integers such that <math>n_1 = a</math>, <math>n_k = b </math>, and <math>n_in_{i+1} </math> is divisible by <math>n_i + n_{i+1} </math> for each <math>i </math> (<math> 1 \le i \le k </math>).
 * [[2002 USAMO Problems/Problem 5 | Solution]]

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