Difference between revisions of "1989 USAMO Problems/Problem 1"

Revision as of 18:39, 11 April 2007

Problem

For each positive integer $n$ , let

$S_n = 1 + \frac 12 + \frac 13 + \cdots + \frac 1n$

$T_n = S_1 + S_2 + S_3 + \cdots + S_n$

$U_n = \frac{T_1}{2} + \frac{T_2}{3} + \frac{T_3}{4} + \cdots + \frac{T_n}{n+1}$ .

Find, with proof, integers $0 < a,\ b,\ c,\ d < 1000000$ such that $\displaystyle T_{1988} = a S_{1989} - b$ and $\displaystyle U_{1988} = c S_{1989} - d$ .

Solution

Let us prove that $\displaystyle T_{n-1} = nS_n - n$ . Expanding:

$\left(1\right) + \left(1 + \frac 12\right) + \ldots + \left(1 + \frac 12 + \frac 13 + \ldots + \frac 1n\right) = n\left(\sum_{i=1}^n \frac 1i\right) - n \displaystyle$

Grouping like terms, there are $n-1$ $\displaystyle 1$ s, $n-2$ $\frac 12$ s, and so on:

$\left(\sum_{i=1}^{n-1} \frac 1i \cdot (n - i)\right) = n\left(\sum_{i=1}^n \frac 1i\right) - n$

$\left(\sum_{I=1}^{n-1} \frac ni\right) - (n - 1) =\left(\sum_{i=1}^n \frac ni\right) - n$

$-(n-1) = n \cdot \frac 1n - n \Longrightarrow 1 - n = 1 - n$

which completes our proof. Thus, for $\displaystyle n = 1989$ , we have that $\displaystyle T_{1988} = 1989 S_{1989} - 1989$ , and so $a = b = 1989 \displaystyle$ .

For the second part, use our previously derived identity to determine $\displaystyle U_{n-1}$ in terms of $\displaystyle S_n$ . The problem simplifies to:

$U_{n-1} = \frac{2 S_2 - 2}{2} + \frac{3 S_3 - 3}{3} + \ldots + \frac{nS_{n} - n}{n}$

$=S_2 + S_3 + \ldots S_{n} - (n - 1) + \left(S_1 - S_1\right)$

$\displaystyle = T_{n-1} + S_n - (n-1)$

$\displaystyle = \left(nS_n - n\right) + S_n - n + 1 - S_1$

$\displaystyle = (n + 1)S_n - 2n$

Thus, we have $\displaystyle U_{n-1} = (n+1)S_n - 2n$ . For $\displaystyle n = 1989$ , we get that $\displaystyle c = 1990$ and $d = 2 \cdot 1989 = 3978$

@@ Line 11: / Line 11: @@
 == Solution ==
-{{solution}}
+Let us prove that <math>\displaystyle T_{n-1} = nS_n - n</math>. Expanding:
+:<math>\left(1\right) + \left(1 + \frac 12\right) + \ldots + \left(1 + \frac 12 + \frac 13 + \ldots + \frac 1n\right) = n\left(\sum_{i=1}^n \frac 1i\right) - n \displaystyle</math>
+Grouping like terms, there are <math>n-1</math> <math>\displaystyle 1</math>s, <math>n-2</math> <math>\frac 12</math>s, and so on:
+:<math>\left(\sum_{i=1}^{n-1} \frac 1i \cdot (n - i)\right) = n\left(\sum_{i=1}^n \frac 1i\right) - n</math>
+:<math>\left(\sum_{I=1}^{n-1} \frac ni\right) - (n - 1)  =\left(\sum_{i=1}^n \frac ni\right) - n</math>
+:<math>-(n-1) = n \cdot \frac 1n - n \Longrightarrow 1 - n = 1 - n</math>
+which completes our proof. Thus, for <math>\displaystyle n = 1989</math>, we have that <math>\displaystyle T_{1988} = 1989 S_{1989} - 1989</math>, and so <math>a = b = 1989 \displaystyle</math>.
+For the second part, use our previously derived identity to determine <math>\displaystyle U_{n-1}</math> in terms of <math>\displaystyle S_n</math>. The problem simplifies to:
+:<math>U_{n-1} = \frac{2 S_2 - 2}{2} +  \frac{3 S_3 - 3}{3}  + \ldots + \frac{nS_{n} - n}{n}</math>
+:<math>=S_2 + S_3 + \ldots S_{n} - (n - 1) + \left(S_1 - S_1\right)</math>
+:<math>\displaystyle = T_{n-1} + S_n - (n-1)</math>
+:<math>\displaystyle = \left(nS_n - n\right) + S_n - n + 1 - S_1</math>
+:<math>\displaystyle = (n + 1)S_n - 2n</math>
+Thus, we have <math>\displaystyle U_{n-1} = (n+1)S_n - 2n</math>. For <math>\displaystyle n = 1989</math>, we get that <math>\displaystyle c = 1990</math> and <math>d = 2 \cdot 1989 = 3978</math>
 == See also ==

1989 USAMO (Problems • Resources)
Preceded by First question	Followed by Problem 2
1 • 2 • 3 • 4 • 5
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Difference between revisions of "1989 USAMO Problems/Problem 1"

Revision as of 18:39, 11 April 2007

Problem

Solution

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