2002 OIM Problems/Problem 5: Difference between revisions

Latest revision as of 17:23, 5 May 2025

Problem

The sequence of real numbers $a1, a2, \cdots$ is defined as:

$a_1 = 56, a_{n+1} = a_n - \frac{1}{a_n}$

for every integer $n \ge 1$ .

Prove that there exists an integer $k$ , $1 \le k \le 2002$ , such that $a_k < 0$ .

~translated into English by Tomas Diaz. orders@tomasdiaz.com

Solution

Notice that every time we apply the recursion, we are essentially subtracting $\frac{1}{a_n}$ from the current term. Clearly, the sequence is decreasing, so for all $n$ , $a_n\le56$ , so $\frac{1}{a_n}\ge\frac{1}{56}$ . Then, after every application of the recursion, the value of $a_i$ will decrease by at least $\frac{1}{56}$ ; clearly, it will eventually reach negative numbers.

~ eevee9406

@@ Line 1: / Line 1: @@
 == Problem ==
-In the square <math>ABCD</math>, let <math>P</math> and <math>Q</math> be points belonging to the sides <math>BC</math> and <math>CD</math> respectively, different from the ends, such that <math>BP = CQ</math>. Points <math>X</math> and <math>Y</math> are considered, belonging to the segments <math>AP</math> and <math>AQ</math> respectively. Show that, whatever <math>X</math> and <math>Y</math>, there exists a triangle whose sides have the lengths of the segments <math>BX</math>, <math>XY</math>, and <math>DY</math>.
+The sequence of real numbers <math>a1, a2, \cdots</math> is defined as:
-~translated into English by Tomas Diaz. ~orders@tomasdiaz.com
+<cmath>a_1 = 56, a_{n+1} = a_n - \frac{1}{a_n}</cmath>
+for every integer <math>n \ge 1</math>.
+Prove that there exists an integer <math>k</math>, <math>1 \le k \le 2002</math>, such that <math>a_k < 0</math>.
+~translated into English by Tomas Diaz. orders@tomasdiaz.com
 == Solution ==
-{{solution}}
+Notice that every time we apply the recursion, we are essentially subtracting <math>\frac{1}{a_n}</math> from the current term. Clearly, the sequence is decreasing, so for all <math>n</math>, <math>a_n\le56</math>, so <math>\frac{1}{a_n}\ge\frac{1}{56}</math>. Then, after every application of the recursion, the value of <math>a_i</math> will decrease by at least <math>\frac{1}{56}</math>; clearly, it will eventually reach negative numbers.
+~ [https://artofproblemsolving.com/wiki/index.php/User:Eevee9406 eevee9406]
 == See also ==
 https://www.oma.org.ar/enunciados/ibe18.htm

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2002 OIM Problems/Problem 5: Difference between revisions

Latest revision as of 17:23, 5 May 2025

Problem

Solution

See also