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Difference between revisions of "1976 IMO Problems/Problem 3"

(got to there, but I am stuck.)
 
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== Solution ==
 
== Solution ==
−
The first statement tells us that the box has integer dimensions. Let the dimensions be <math>a</math>, <math>b</math>, and <math>c</math>, where they're each greater than 1. The second statement tells us that if <math>\lfloor \dfrac{a}{\sqrt[3]{2}} \rfloor *\lfloor \dfrac{b}{\sqrt[3]{2}} \rfloor *\lfloor \dfrac{c}{\sqrt[3]{2}} \rfloor</math> boxes with side length <math>\sqrt[3]{2}</math> are put into the box, that takes up <math>\dfrac{2abc}{5}</math> units of area. Thus <math>\lfloor \dfrac{a}{\sqrt[3]{2}} \rfloor *\lfloor \dfrac{b}{\sqrt[3]{2}} \rfloor *\lfloor \dfrac{c}{\sqrt[3]{2}} \rfloor=\dfrac{abc}{5}</math>. Now the range of values that the LHS can take up is <math>(\dfrac{a}{\sqrt[3]{2}}-1) (\dfrac{b}{\sqrt[3]{2}}-1) (\dfrac{c}{\sqrt[3]{2}}-1)</math> to <math>\dfrac{abc}{2}</math>, exclusive, since the fractions are not integers. Therefore we must find all <math>a</math>, <math>b</math>, and <math>c</math> such that
+
We name a,b,c the sides of the parallelepiped, which are positive integers. We also put
 +
<cmath>
 +
\begin{align*}
 +
x &= \left\lfloor\frac{a}{\sqrt[3]{2}}\right\rfloor \\
 +
y &= \left\lfloor\frac{b}{\sqrt[3]{2}}\right\rfloor \\
 +
z &= \left\lfloor\frac{c}{\sqrt[3]{2}}\right\rfloor \\
 +
\end{align*}
 +
</cmath>
 +
It is clear that <math>xyz</math> is the maximal number of cubes with sides of length <math>\sqrt[3]{2}</math> that
 +
can be put into the parallelepiped with sides parallels to the sides of the box.
 +
Hence the corresponding volume is <math>V_2=2\cdot xyz</math>. We need <math>V_2=0.4\cdot V_1=0.4\cdot abc</math>,
 +
hence <cmath> \frac ax\cdot \frac by\cdot \frac cz=5\ \ \ \ \ \ \ \ (1)</cmath>
 +
We give the  values of <math>x</math> and <math>a/x</math> for <math>a=1,\dots ,8</math>. The same table is valid for <math>b,y</math> and <math>c,z</math>.
 +
<cmath> \begin{tabular}{|c|c|c|}
 +
  \hline
 +
  a & x & a/x \\ \hline
 +
  1 & 0 & - \\ \hline
 +
  2 & 1 & 2 \\ \hline
 +
  3 & 2 &  3/2 \\ \hline
 +
  4 & 3 & 4/3 \\ \hline
 +
  5 & 3 & 5/3 \\ \hline
 +
  6 & 4 & 3/2 \\ \hline
 +
  7 & 5 & 7/5 \\ \hline
 +
  8 & 6 &  4/3 \\
 +
  \hline
 +
\end{tabular} </cmath>
 +
By simple inspection we obtain two solutions of <math>(1)</math>: <math>\{a,b,c\}=\{2,5,3\}</math> and <math>\{a,b,c\}=\{2,5,6\}</math>.
 +
We now show that they are the only solutions.
  
−
<cmath>(\dfrac{a}{\sqrt[3]{2}}-1) (\dfrac{b}{\sqrt[3]{2}}-1) (\dfrac{c}{\sqrt[3]{2}}-1)<\dfrac{abc}{5}<\dfrac{abc}{2}.</cmath>
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We can assume <math>\frac ax\ge \frac by \ge \frac cz</math>. So necessarily <math>\frac ax\ge \sqrt[3]{5}</math>. Note that
 +
the definition of <math>x</math> implies <cmath> x< a/\sqrt[3]2 < x+1,</cmath>
 +
hence <cmath>\sqrt[3]2< a/x < \sqrt[3]2(1+\frac 1x)</cmath>
 +
If <math>a\ge 4</math> then <math>x\ge 3</math> and <math>\frac ax<\sqrt[3]2(1+\frac 1x)\le \sqrt[3]2(\frac 43)<\sqrt[3]5</math>
 +
since <math>2\cdot \frac {4^3}{3^3}<5</math>. So we have only left the cases <math>a=2</math> and <math>a=3</math>. But for <math>a=3</math>
 +
we have <math>a/x=3/2<\sqrt[3]5</math> and so necessarily <math>a=2</math> and <math>a/x=2</math>.
 +
It follows
 +
<cmath>    \frac by \cdot \frac cz =\frac 52 \ \ \ \ \ \ (2)
 +
</cmath>
  
−
The <math>\dfrac{abc}{2}</math> is redundant, so we can eliminate that. We simplify the LHS:
 
  
−
<cmath>\dfrac{(a-\sqrt[3]{2})(b-\sqrt[3]{2})(c-\sqrt[3]{2})}{2}<\dfrac{abc}{5}</cmath>
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Note that the definitions of <math>y,z</math> imply <cmath> y< b/\sqrt[3]2 < y+1,\ \
 +
\textrm{and} \ \ z< c/\sqrt[3]2 < z+1.\ \ \ \ (3)</cmath>
 +
Moreover we have from (2) and from <math>b/y\ge c/z</math> that
 +
<cmath>
 +
    \frac by \ge \sqrt{5/2}\ \ \ \ \ (4)
 +
</cmath>
  
−
<cmath>\dfrac{abc-\sqrt[3]{2}ab-\sqrt[3]{2}bc-\sqrt[3]{2}ac+\sqrt[3]{4}a+\sqrt[3]{4}b+\sqrt[3]{4}c-2}{2}<\dfrac{abc}{5}</cmath>
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If <math>b=2</math> then <math>b/y=2</math> and we would have <math>c/z=5/4<\sqrt[3]2</math>, which contradicts <math>(3)</math>.
  
−
<cmath>3abc-5\sqrt[3]{2}ab -5\sqrt[3]{2}bc -5\sqrt[3]{2}ac+5\sqrt[3]{4}a+5\sqrt[3]{4}b+5\sqrt[3]{4}c-10<0</cmath>
+
On the other hand, if <math>b>5</math> then <math>y>4</math> and <math>\frac by<\sqrt[3]2(1+\frac 1y)\le \sqrt[3]2(\frac 54)<\sqrt{5/2}</math>
−
 
+
since <math>2^2\cdot \frac {5^6}{4^6}<\frac{5^3}{2^3}</math> as <math>5^3<2^7</math>. So we have only left the
−
{{incomplete|solution}}
+
cases <math>b=3,4,5</math>. But for <math>b=3</math> we have <math>b/y=3/2<\sqrt{5/2}</math> and for <math>b=4</math> we have
 +
<math>b/y=4/3<\sqrt{5/2}</math> and so necessarily <math>b=5</math> and <math>b/y=5/3</math> (<math>>\sqrt{5/2}</math>)
  
 +
So we arrive finally at <math>a=2,b=5</math> and <math>c/z=3/2</math>. If <math>c\ge 8</math> then <math>z\ge 6</math> and <math>\frac cz<\sqrt[3]2(1+\frac 1z)\le \sqrt[3]2(\frac 76)<\frac 32</math> since <math>2\cdot \frac {7^3}{6^3}<\frac{3^3}{2^3}</math>. On the other hand, for <math>c\le 7</math> there are the only two possible values <math>c=3</math> and <math>c=6</math> which yield the known solutions.
 
== See also ==
 
== See also ==
 
{{IMO box|year=1976|num-b=2|num-a=4}}
 
{{IMO box|year=1976|num-b=2|num-a=4}}

Latest revision as of 16:27, 29 January 2021

Problem

A box whose shape is a parallelepiped can be completely filled with cubes of side $1.$ If we put in it the maximum possible number of cubes, each of volume $2$, with the sides parallel to those of the box, then exactly $40$ percent from the volume of the box is occupied. Determine the possible dimensions of the box.

Solution

We name a,b,c the sides of the parallelepiped, which are positive integers. We also put \begin{align*} x &= \left\lfloor\frac{a}{\sqrt[3]{2}}\right\rfloor \\ y &= \left\lfloor\frac{b}{\sqrt[3]{2}}\right\rfloor \\ z &= \left\lfloor\frac{c}{\sqrt[3]{2}}\right\rfloor \\ \end{align*} It is clear that $xyz$ is the maximal number of cubes with sides of length $\sqrt[3]{2}$ that can be put into the parallelepiped with sides parallels to the sides of the box. Hence the corresponding volume is $V_2=2\cdot xyz$. We need $V_2=0.4\cdot V_1=0.4\cdot abc$, hence \[\frac ax\cdot \frac by\cdot \frac cz=5\ \ \ \ \ \ \ \ (1)\] We give the values of $x$ and $a/x$ for $a=1,\dots ,8$. The same table is valid for $b,y$ and $c,z$. \[\begin{tabular}{|c|c|c|} \hline a & x & a/x \\ \hline 1 & 0 & - \\ \hline 2 & 1 & 2 \\ \hline 3 & 2 & 3/2 \\ \hline 4 & 3 & 4/3 \\ \hline 5 & 3 & 5/3 \\ \hline 6 & 4 & 3/2 \\ \hline 7 & 5 & 7/5 \\ \hline 8 & 6 & 4/3 \\ \hline \end{tabular}\] By simple inspection we obtain two solutions of $(1)$: $\{a,b,c\}=\{2,5,3\}$ and $\{a,b,c\}=\{2,5,6\}$. We now show that they are the only solutions.

We can assume $\frac ax\ge \frac by \ge \frac cz$. So necessarily $\frac ax\ge \sqrt[3]{5}$. Note that the definition of $x$ implies \[x< a/\sqrt[3]2 < x+1,\] hence \[\sqrt[3]2< a/x < \sqrt[3]2(1+\frac 1x)\] If $a\ge 4$ then $x\ge 3$ and $\frac ax<\sqrt[3]2(1+\frac 1x)\le \sqrt[3]2(\frac 43)<\sqrt[3]5$ since $2\cdot \frac {4^3}{3^3}<5$. So we have only left the cases $a=2$ and $a=3$. But for $a=3$ we have $a/x=3/2<\sqrt[3]5$ and so necessarily $a=2$ and $a/x=2$. It follows \[\frac by \cdot \frac cz =\frac 52 \ \ \ \ \ \ (2)\]


Note that the definitions of $y,z$ imply \[y< b/\sqrt[3]2 < y+1,\ \ \textrm{and} \ \ z< c/\sqrt[3]2 < z+1.\ \ \ \ (3)\] Moreover we have from (2) and from $b/y\ge c/z$ that \[\frac by \ge \sqrt{5/2}\ \ \ \ \ (4)\]

If $b=2$ then $b/y=2$ and we would have $c/z=5/4<\sqrt[3]2$, which contradicts $(3)$.

On the other hand, if $b>5$ then $y>4$ and $\frac by<\sqrt[3]2(1+\frac 1y)\le \sqrt[3]2(\frac 54)<\sqrt{5/2}$ since $2^2\cdot \frac {5^6}{4^6}<\frac{5^3}{2^3}$ as $5^3<2^7$. So we have only left the cases $b=3,4,5$. But for $b=3$ we have $b/y=3/2<\sqrt{5/2}$ and for $b=4$ we have $b/y=4/3<\sqrt{5/2}$ and so necessarily $b=5$ and $b/y=5/3$ ($>\sqrt{5/2}$)

So we arrive finally at $a=2,b=5$ and $c/z=3/2$. If $c\ge 8$ then $z\ge 6$ and $\frac cz<\sqrt[3]2(1+\frac 1z)\le \sqrt[3]2(\frac 76)<\frac 32$ since $2\cdot \frac {7^3}{6^3}<\frac{3^3}{2^3}$. On the other hand, for $c\le 7$ there are the only two possible values $c=3$ and $c=6$ which yield the known solutions.

See also

1976 IMO (Problems) • Resources
Preceded by
Problem 2 		1 • 2 • 3 • 4 • 5 • 6 Followed by
Problem 4
All IMO Problems and Solutions
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