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Problem 585

Combinatorics Difficulty 2.6 Multiple choice CEMC Cayley · Canada · 2025

Lavinia has one bottle with capacity $\text{6 L}$ and another bottle with
capacity $\text{5 L}$ . Both bottles
are empty. She has a piece of paper with the following steps printed on
it:

Fill the $\text{6 L}$ bottle
completely with water.
Pour water from the $\text{6 L}$ bottle into the $\text{5 L}$ bottle until the $\text{5 L}$ bottle is full.
Empty the $\text{5 L}$
bottle.
Pour water from the $\text{6 L}$ bottle into the $\text{5 L}$ bottle until the $\text{5 L}$ bottle is full or the $\text{6 L}$ bottle is empty, whichever happens first. If the $\text{5 L}$ bottle fills, empty the $\text{5 L}$ bottle and return to the
beginning of this step.

Lavinia completes this sequence of four steps in this order a total
of $3$ times. Lavinia never spills
any water. How much water will be in the $\text{5 L}$ bottle after Lavinia is finished?

Pick one

Official solution

In step (i), Lavinia fills the $6 \text{ L}$ bottle with water. In step (ii), she fills the $5 \text{ L}$ bottle from the $6 \text{ L}$ bottle, leaving the $6 \text{ L}$ bottle with $6 \text{ L}-5 \text{ L}=1\text { L}$ of
water.

In step (iii), she empties the $5 \text{ L}$ bottle and so the $5 \text{ L}$ has no water and the $6 \text{ L}$ bottle still contains $1 \text{ L}$ .

In step (iv), she pours the $1 \text{ L}$ of water from the $6 \text{ L}$ bottle into the $5 \text{ L}$ bottle.

After completing the sequence of four steps once, the $6 \text{ L}$ bottle contains no water and
the $5 \text{ L}$ bottle contains
$1 \text{ L}$ of water.

In the chart below, we continue to track the volume of water in each of
the two bottles following each step. Each volume of water is measured in
litres.

Step:	(i)	(ii)	(iii)	(iv)
$6 \text{ L}$ bottle $6$	$6-5=1$	$1$	$0$
$5 \text{ L}$ bottle $0$	$5$	$0$	$1$

Step:	(i)	(ii)	(iii)	(iv)
$6 \text{ L}$ bottle $6$	$6-4=2$	$2$	$0$
$5 \text{ L}$ bottle $1$	$5$	$0$	$2$

Step:	(i)	(ii)	(iii)	(iv)
$6 \text{ L}$ bottle $6$	$6-3=3$	$3$	$0$
$5 \text{ L}$ bottle $2$	$5$	$0$	$3$

After Lavinia completes the sequence of four steps a total of $3$ times, the volume of water in the $5 \text{ L}$ bottle is $3 \text{ L}$ .

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.