Difference between revisions of "2013 AIME II Problems/Problem 14"

Latest revision as of 17:41, 26 June 2024

Problem 14

For positive integers $n$ and $k$ , let $f(n, k)$ be the remainder when $n$ is divided by $k$ , and for $n > 1$ let $F(n) = \max_{\substack{1\le k\le \frac{n}{2}}} f(n, k)$ . Find the remainder when $\sum\limits_{n=20}^{100} F(n)$ is divided by $1000$ .

Solution

The Pattern

We can find that

$20\equiv 6 \pmod{7}$

$21\equiv 5 \pmod{8}$

$22\equiv 6 \pmod{8}$

$23\equiv 7 \pmod{8}$

$24\equiv 6 \pmod{9}$

$25\equiv 7 \pmod{9}$

$26\equiv 8 \pmod{9}$

Observing these and we can find that the reminders are in groups of three continuous integers, considering this is true, and we get

$99\equiv 31 \pmod{34}$

$100\equiv 32 \pmod{34}$

So the sum is $6+3\times(6+...+31)+31+32=1512$ ,it is also 17+20+23+...+95, so the answer is $\boxed{512}$ . By: Kris17

The Intuition

First, let's see what happens if we remove a restriction. Let's define $G(x)$ as

$G(x):=\max_{\substack{1\le k}} f(n, k)$

Now, if you set $k$ as any number greater than $n$ , you get n, obviously the maximum possible. That's too much freedom; let's restrict it a bit. Hence $H(x)$ is defined as

$H(x):=\max_{\substack{1\le k\le n}} f(n, k)$

Now, after some thought, we find that if we set $k=\lfloor \frac{n}{2} \rfloor+1$ we get a remainder of $\lfloor \frac{n-1}{2} \rfloor$ , the max possible. Once we have gotten this far, it is easy to see that the original equation,

$F(n) = \max_{\substack{1\le k\le \frac{n}{2}}} f(n, k)$

has a solution with $k=\lfloor \frac{n}{3} \rfloor+1$ .

$W^5$ ~Rowechen

The Proof

The solution presented above does not prove why $F(x)$ is found by dividing $x$ by $3$ . Indeed, that is the case, as rigorously shown below.

Consider the case where $x = 3k$ . We shall prove that $F(x) = f(x, k+1)$ . For all $x/2\ge n > k+1, x = 2n + q$ , where $0\le q< n$ . This is because $x < 3k + 3 < 3n$ and $x \ge 2n$ . Also, as $n$ increases, $q$ decreases. Thus, $q = f(x, n) < f(x, k+1) = k - 2$ for all $n > k+1$ . Consider all $n < k+1. f(x, k) = 0$ and $f(x, k-1) = 3$ . Also, $0 < f(x, k-2) < k-2$ . Thus, for $k > 5, f(x, k+1) > f(x, n)$ for $n < k+1$ .

Similar proofs apply for $x = 3k + 1$ and $x = 3k + 2$ . The reader should feel free to derive these proofs themself.

Generalized Solution

$Lemma:$ Highest remainder when $n$ is divided by $1\leq k\leq n/2$ is obtained for $k_0 = (n + (3 - n \pmod{3}))/3$ and the remainder thus obtained is $(n - k_0*2) = [(n - 6)/3 + (2/3)*n \pmod{3}]$ .

$Note:$ This is the second highest remainder when $n$ is divided by $1\leq k\leq n$ and the highest remainder occurs when $n$ is divided by $k_M$ = $(n+1)/2$ for odd $n$ and $k_M$ = $(n+2)/2$ for even $n$ .

Using the lemma above:

$\sum\limits_{n=20}^{100} F(n) = \sum\limits_{n=20}^{100} [(n - 6)/3 + (2/3)*n \pmod{3}]$ $= [(120*81/2)/3 - 2*81 + (2/3)*81] = 1512$

So the answer is $\boxed{512}$

Proof of Lemma: It is similar to $The Proof$ stated above.

Kris17

Video Solution

https://youtu.be/mQ4_1dp8Wm8?si=Ae5HAc0cZQAjdtWl

~MathProblemSolvingSkills.com

@@ Line 4: / Line 4: @@
 ==Solution==
-===Easy solution without strict proof===
+===The Pattern===
 We can find that
@@ Line 28: / Line 28: @@
 <math>100\equiv 32 \pmod{34}</math>
-So the sum is <math>5+3\times(6+...+31)+32\times 2=1512</math>, so the answer is <math>\boxed{512}</math>.
+So the sum is <math>6+3\times(6+...+31)+31+32=1512</math>,it is also 17+20+23+...+95, so the answer is <math>\boxed{512}</math>.
+By: Kris17
+===The Intuition===
+First, let's see what happens if we remove a restriction. Let's define <math>G(x)</math> as
+<math>G(x):=\max_{\substack{1\le k}} f(n, k)</math>
+Now, if you set <math>k</math> as any number greater than <math>n</math>, you get n, obviously the maximum possible. That's too much freedom; let's restrict it a bit. Hence <math>H(x)</math> is defined as
+<math>H(x):=\max_{\substack{1\le k\le n}} f(n, k)</math>
+Now, after some thought, we find that if we set <math>k=\lfloor \frac{n}{2} \rfloor+1</math> we get a remainder of <math>\lfloor \frac{n-1}{2} \rfloor</math>, the max possible. Once we have gotten this far, it is easy to see that the original equation,
+<math>F(n) = \max_{\substack{1\le k\le \frac{n}{2}}} f(n, k)</math>
+has a solution with <math>k=\lfloor \frac{n}{3} \rfloor+1</math>.
+<math>W^5</math>~Rowechen
+===The Proof===
+The solution presented above does not prove why <math>F(x)</math> is found by dividing <math>x</math> by <math>3</math>. Indeed, that is the case, as rigorously shown below.
+Consider the case where <math>x = 3k</math>. We shall prove that <math>F(x) = f(x, k+1)</math>.
+For all <math>x/2\ge n > k+1, x = 2n + q</math>, where <math>0\le q< n</math>. This is because <math>x < 3k + 3 < 3n</math> and <math>x \ge 2n</math>. Also, as <math>n</math> increases, <math>q</math> decreases. Thus, <math>q = f(x, n) < f(x, k+1) = k - 2</math> for all <math>n > k+1</math>.
+Consider all <math>n < k+1. f(x, k) = 0</math> and <math>f(x, k-1) = 3</math>. Also, <math>0 < f(x, k-2) < k-2</math>. Thus, for <math>k > 5, f(x, k+1) > f(x, n)</math> for <math>n < k+1</math>.
+Similar proofs apply for <math>x = 3k + 1</math> and <math>x = 3k + 2</math>. The reader should feel free to derive these proofs themself.
+===Generalized Solution===
+<math>Lemma:</math> Highest remainder when <math>n</math> is divided by <math>1\leq k\leq n/2</math> is obtained for <math>k_0 = (n + (3 - n \pmod{3}))/3</math> and the remainder thus obtained is <math>(n - k_0*2) = [(n - 6)/3 + (2/3)*n \pmod{3}]</math>.
+<math>Note:</math> This is the second highest remainder when <math>n</math> is divided by <math>1\leq k\leq n</math> and the highest remainder occurs when <math>n</math> is divided by <math>k_M</math> = <math>(n+1)/2</math> for odd <math>n</math> and  <math>k_M</math> = <math>(n+2)/2</math> for even <math>n</math>.
+Using the lemma above:
+<math>\sum\limits_{n=20}^{100} F(n) = \sum\limits_{n=20}^{100} [(n - 6)/3 + (2/3)*n \pmod{3}] </math>
+<math>= [(120*81/2)/3 - 2*81 + (2/3)*81]
+= 1512</math>
+So the answer is <math>\boxed{512}</math>
+Proof of Lemma: It is similar to <math>The Proof</math> stated above.
+Kris17
+==Video Solution==
+https://youtu.be/mQ4_1dp8Wm8?si=Ae5HAc0cZQAjdtWl
+~MathProblemSolvingSkills.com
 ==See Also==
 {{AIME box|year=2013|n=II|num-b=13|num-a=15}}
 {{MAA Notice}}

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