IMO 2014/1

Let $a_0 < a_1 < a_2 \ldots$ be an infinite sequence of positive integers. Prove that there exists a unique integer $n \geq 1$ such that

$a_n < \frac{a_0 + a_1 + \ldots + a_n } { n} \leq a_{n+1} .$

#Algebra

Easy Math Editor

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Comments

Let's call $S_k = a_0 + a_1 + ... + a_k$ and suppose $\frac { { S }_{ k } }{ k } >{ a }_{ k+1 } \; \forall m \leq k$ Then we have $k\cdot { S }_{ k }+{ S }_{ k }>k\cdot { a }_{ k+1 }+k\cdot{ S }_{ k }\quad \Leftrightarrow \quad \frac { { S }_{ k } }{ k } >\frac { { S }_{ k+1 } }{ k+1 }\\ \quad \Leftrightarrow \quad \frac { { S }_{ k+1 } }{ k+1 } <\frac { { S }_{ k } }{ k } <\frac { { S }_{ k-1 } }{ k-1 } <...<{ S }_{ 1 }={ a }_{ 0 }+{ a }_{ 1 }$ Hence, we prove by induction $2(a_{ 0 }+{ a }_{ 1 })>{ a }_{ 0 }+{ a }_{ 1 }+{ a }_{ 2 }\quad \Leftrightarrow \quad { a }_{ 0 }+{ a }_{ 1 }>{ a }_{ 2 }\\ k{ a }_{ k+1 }<{ S }_{ k }<k(a_{ 0 }+{ a }_{ 1 })\quad \Leftrightarrow \quad { a }_{ k+1 }<(a_{ 0 }+{ a }_{ 1 })$ Because $a_{ 0 }+{ a }_{ 1 }$ is a sum of integer number, there must be a unique number in the sequence for which $a_{q+1} \geq a_{ 0 }+{ a }_{ 1 }$ and $a_{q} < a_{ 0 }+{ a }_{ 1 }$ . We now show that $q$ is the required number. We have alredy proved that $\frac { { S }_{ q } }{q } \leq { a }_{ q+1 }$ by contraddiction! Thus we have only to show that ${ a }_{ q }<\frac { S_{ q } }{ q } \quad \Leftrightarrow \quad q{ \cdot a }_{ q }-a_{ q }<S_{ q }-a_{ q }\quad \Leftrightarrow \quad { a }_{ q }<\frac { S_{ q-1 } }{ q-1 }$ And because $S_{ q-1 }(q-1)+{ a }_{ q }(q-1)<S_{ q-1 }+S_{ q-1 }(q-1)\quad \Leftrightarrow \quad \frac { S_{ q } }{ q } <\frac { S_{ q-1 } }{ q-1 }\\ \quad \Leftrightarrow \quad \frac { S_{ q } }{ q } <\frac { S_{ q-1 } }{ q-1 } <\frac { S_{ q-2 } }{ q-2 } <...<S_{ 1 }={ a }_{ 0 }+{ a }_{ 1 }$ Excactly as done beofre we must have $a_q < a_0 + a_1$ . That is true. And we are done.

Andrea Gallese - 6 years, 11 months ago

Beautiful solution

Scrub Lord - 4 years ago

Question 1 was way easier than I expected it to be. Bummed that there were no proper number theory problems.

Mursalin Habib - 6 years, 11 months ago

In recent years, they have made question 1 (and sometimes 4) pretty easy and direct to approach, so that all of the participants (who come from a wide range of ability level) have something to do, and have a chance of geting a HM.

Calvin Lin Staff - 6 years, 11 months ago

This was indeed very nice, but also pretty easy, even for an IMO P1. I have posted my solution in AoPS.

http://www.artofproblemsolving.com/Forum/viewtopic.php?p=3542324&#p3542324

Sreejato Bhattacharya - 6 years, 11 months ago

Here's a conjecture:

The last possible $n$ that satisfies $a_n < \dfrac{a_0+a_1+\cdots +a_n}{n}$ also satisfies $\dfrac{a_0+a_1+\cdots +a_n}{n}\le a_{n+1}$ . This is just a guess though.

These later posts will be what I have came up with.

Daniel Liu - 6 years, 11 months ago

Update: letting $a_0=x$ and $a_1=y$ , we find that in order for $a_k < \dfrac{a_0+a_1+\cdots +a_k}{k}$ , we must have $a_k < x+y$ . However, since $x+y$ is finite, and the sequence is strictly increasing, we must have that there is a maximum $k$ that satisfies this condition.

So let $k$ be the maximum such that it still satisfies the condition. Thus, $a_{k+1} \ge x+y$ . However, $\dfrac{a_0+a_1+\cdots +a_k}{k} < \dfrac{x+y+(x+y)+\cdots +(x+y)}{k}=x+y$ . Thus, $\dfrac{a_0+a_1+\cdots +a_k}{k} < a_{k+1}$ .

This seems really sketchy though. Something seems wrong... one of my inequalities was strict when forming the right inequality, but the inequality in the problem is not strict...

EDIT: found out why mine was strict and the problem's wasn't. As long as the maximum $k$ satisfies $k\ge 2$ , then the upper bound inequality is strict, just like my result. However, the special case of the maximum $k$ is $k=1$ yields a non-strict inequality.

TO DO: prove uniqueness of satisfying both inequalities.

Daniel Liu - 6 years, 11 months ago

I realize that I have to tighten the bound for each $a_k < x+y$ to something more strict, or something like that in order to prove the uniqueness of that one $k$ that satisfies both inequalities. I feel like I'm almost there, but just can't get over the last little bump. I need to somehow use the $a_k < a_{k+1}$ to my advantage...

Daniel Liu - 6 years, 11 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$