Skoliad: No. 114 - Canadian Mathematical Society

1
SKOLIAD
No. 114
Valav
Linek
Please send your solutions to problems in this Skoliad by August 1, 2009.
Solutions should be sent to Lily Yen and Mogens Hansen at the address inside
the bak over. The Skoliad setion is in transition and, unfortunately, we
have lost several of the submitted solutions to past ontests. If you have
opies of solutions that you sent to past ontests, please send them again
so that we an mention any orret solutions we reeive. (This inludes any
ontest in Skoliad appearing in or after the Marh 2008 issue of CRUX).
Our rst problem set of the year is the Math Kangaroo Contest Pratie
Set. The Kangaroo Contest is international in sope and supported in Canada
by the Canadian Mathematial Soiety and the Institute of Eletrial and
Eletronis Engineers (Northern Setion).
Our thanks go to Valeria Pandelieva, the Canadian representative of the
Kangaroo Contest, for bringing this ontest to our attention, and for making
us aware of the need for ontests and math-partiipation in the lower years
in Canada. For that reason, and also sine this ontest is straightforward
to administer (see www.mathkangaroocanada.com), we are featuring its entire
range of questions over all grades.
Finally, while it is a multiple hoie test, we ask our readers to send in
omplete solutions showing all the steps and details so that we an evaluate
the solutions and give full redit to the solvers.
Math Kangaroo Contest
Pratie Set
Part A (3 points per question)
. (Grades 3-4) In the addition example, eah letter
represents a digit. Equal digits are represented by the
same letter. Dierent digits are represented by dierent letters. Whih digit does the letter K represent?
(A) 0
(B) 1
(C) 2
(D) 8
1
+
W
O
K
O
K
O
W
(E) 9
. (Grades 5-6) Ten aterpillars, arranged in a row one behind another,
walked in the park. The length of eah aterpillar was equal to 8 m, and
the distane any two adjaent aterpillars kept for safety reasons was 2 m.
What is the total length of their row?
(A) 100 m (B) 98 m
(C) 82 m
(D) 102 m (E) 96 m
2
2
. (Grades 7-8) An ant is running along a ruler ..................................sq...q.r................................................
...
of length 10 m with a onstant speed of 1 m ...
.
1
2
3
4
5
6
7
8
9
.
per seond (see the gure). Any time when .......................................................................................
the ant reahes one of the ends of the ruler, it
turns bak and runs in the opposite diretion. It takes the ant exatly 1 seond to make a turn. The ant starts from the left end of the ruler. Nearest
whih number will it be after 2009 seonds?
(A) 1 m
(B) 2 m
(C) 3 m
(D) 4 m
(E) 5 m
3
.
....
.
4
.
....
.
.
....
.
....
.. .....
...............
.
.....
....
.
.
.
....
.
.
....
.
.
....
.
.
....
.
. (Grades 9-10) Whih of the numbers 26 , 35 , 44 , 53 , 62 is the greatest?
(A) 26
(B) 35
(C) 44
(D) 53
(E) 62
. (Grades 11-12) A deorator has prepared a mixed paint, in whih the
volumes of red and yellow olours were in the ratio 2 : 3. The resulting
olour seemed too light to him, so he added 2 L of red paint. This way, the
ratio of the volumes of the red and yellow olours hanged to 3 : 2. How
many litres of paint did the deorator use?
(B) 6 L
(C) 7 L
(D) 8 L
(E) 9 L
(A) 5 L
5
Part B (4 points per question)
6. (Grades 3-4) Two boys are playing tennis until one of them wins four
times. A tennis math annot end in a draw. What is the greatest number of
games they an play?
(B) 7
(C) 6
(D) 5
(E) 9
(A) 8
7. (Grades 5-6) In two years, my son will be twie as old as he was two years
ago. In three years, my daughter will be three times as old as she was three
years ago. Whih of the following best desribes the ages of the daughter and
the son?
(A) The son is older;
(B) The daughter is older;
(C) They are twins;
(D) The son is twie as old as the daughter;
(E) The daughter is twie as old as the son.
. (Grades 7-8) Some points are marked on a straight line so that all distanes
8
1 m, 2 m, 3 m, 4 m, 5 m, 6 m, 7 m, and 9 m are among the distanes
between these points. At least how many points are marked on the line?
(A) 4
(B) 5
(C) 6
(D) 7
(E) 8
. (Grades 9-10) Eva, Betty, Linda, and Cathy went to the inema. Sine it
was not possible to buy four seats next to eah other, they bought tikets for
seats number 7 and 8 in the 10th row and tikets for seats number 3 and 4
in the 12th row. How many seating arrangements an they hoose from, if
Cathy does not want to sit next to Betty?
(A) 24
(B) 20
(C) 16
(D) 12
(E) 8
9
3
10.
(Grades 11-12) Triangle ABC is
isoseles with BC = AC . The segments
DE , F G, HI , KL, M N , OP , and XY
divide the sides AC and CB into equal
parts. Find XY , if AB = 40 m.
(A) 38 m
(B) 35 m
(C) 33 m
(D) 30 m
(E) 27 m
C
...
... ....
D ..................................... E
F .......................................................... G
H ............................................................................. I
K .................................................................................................. L
M ....................................................................................................................... N
O ............................................................................................................................................ P
X ................................................................................................................................................................ Y
A ........................................................................................................................................................................... B
Part C (5 points per question)
. (Grades 3-4) Matt and Nik onstruted
two buildings, shown in the gures, using idential ubes. Matt's building weighs 200 g, and
Nik's building weighs 600 g. How many ubes
from Nik's building are hidden and annot be
seen in the gure?
(B) 2
(C) 3
(A) 1
(D) 4
(E) 5
11
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.. .. .. .....................
...........................................................
................................................
.. .. .. .. .. ...
................................................
............
................................
. .
..............................
.........................................
............................
Nik's
building
Matt's
building
12. (Grades 5-6) Consider all four-digit numbers divisible by 6 whose digits
are in inreasing order, from left to right. What is the hundreds digit of the
largest suh number?
(A) 7
(B) 6
(C) 5
(D) 4
(E) 3
. (Grades 7-8) A square of side length 3 is divided
by several segments into polygons as shown in the gure. What perent of the area of the original square is
the area of the shaded gure?
(A) 30%
(B) 33 13 %
(C) 35%
(E) 50%
(D) 40%
13
.........1..........................2.................
.. ...................................................................................................................... ..... 1
. ........................................................................................................ .
2 .. ............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.. ................................................................................................................................................................................................................................................................................................................................. ...
......................................................................................................................................................................................................................................................................................................................................................................................................... .. 2
. . .................................................................................................................................................................................................................... ....
.
1 ....
........................................................
2
1
14. (Grades 9-10) A boy always tells the truth on Thursdays and Fridays,
always tells lies on Tuesdays, and tells either truth or lies on the rest of the
days of the week. Every day he was asked what his name was and six times in
a row he gave the following answers: John, Bob, John, Bob, Pit, Bob. What
did he answer on the seventh day?
(A) John
(B) Bob
(C) Pit
(D) Kate
(E) Not enough information to deide
. (Grades 11-12) An equilateral triangle and a irle
M are insribed in a irle K , as shown in the gure.
What is the ratio of the area of K to the area of M ?
(A) 8 : 1
(B) 10 : 1
(C) 12 : 1
(D) 14 : 1
(E) 16 : 1
15
....
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...... K
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..... ... ........
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....M ....
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4
Conours Math Kangaroo
Feuille d'entra^
nement
Partie A (3 points par question)
. (Classes 3-4) Dans l'exemple d'addition idessus, haque lettre dierente
represente
un hire
dierent.
Quel hire la lettre K represente-t-elle
?
O
K
O
1
(A) 0
(B) 1
(C) 2
+
W
(D) 8
K
O
W
(E) 9
. (Classes 5-6) Dix henilles se promenaient a la le indienne dans un par.
Chaque henille mesurait 8 m et, pour des raisons de seurit
e,
elles gardaient une distane de 2 m entre haune d'elles. Quelle etait
la longueur
totale de leur ortege
?
2
(A) 100 m
(B) 98 m
(C) 82 m
(D) 102 m
(E) 96 m
. (Classes 7-8) Une fourmi ourt le long ..................................sq...q.r................................................
...
d'une regle
de 10 m de longueur, a la vitesse ...
.
1
2
3
4
5
6
7
8
9
.
onstante de 1 m a la seonde (voir la gure). .......................................................................................
Chaque fois qu'elle atteint une extremit
e,
elle
ourt dans la diretion opposee
et elle met exatement 1 seonde pour hanger de diretion. La fourmi part de l'extremit
e gauhe de la regle.
Pres
de
quel hire sera-telle apres
2009 seondes ?
3
.....
.
(A) 1 m
4
(B) 2 m
(C) 3 m
.....
.
(D) 4 m
.....
.
...
.. ......
..............
....
.....
.
.
.....
.
.....
.
.....
.
.....
.
(E) 5 m
. (Classes 9-10) Lequel des nombres 26 , 35 , 44 , 53 , 62 est-il le plus grand ?
(A) 26
(B) 35
(C) 44
(D) 53
(E) 62
. (Classes 11-12) Un deorateur
a prepar
e un melange
de peinture ou les
volumes des ouleurs rouge et jaune etaient
dans un rapport de 2 : 3. Trouvant le melange
trop lair, il ajouta 2 L de peinture rouge. Le rapport des
volumes des ouleurs rouge et jaune devint alors de 3 : 2. Combien de litres
de peinture le deorateur
a-t-il utilise ?
5
(A) 5 L
(B) 6 L
(C) 7 L
(D) 8 L
(E) 9 L
Partie B (4 points par question)
. (Classes 3-4) Deux garons jouent au tennis jusqu'a e que l'un d'eux gagne
quatre fois. Un math de tennis ne peut nir en un pointage nul. Quel est le
plus grand nombre de jeux qu'ils peuvent jouer ?
6
(A) 8
(B) 7
(C) 6
(D) 5
(E) 9
5
. (Classes 5-6) Dans deux ans, mon ls aura deux fois l'^age qu'il avait il y a
deux ans. Dans trois ans, ma lle aura trois fois l'^age qu'elle avait il y a trois
ans. Quelle reponse
derit-elle
le mieux l'^age de la lle et du ls ?
(A) Le ls est plus a^ ge ;
(B) La lle est plus a^ gee
;
(C) Ils sont des jumeaux ; (D) Le ls est deux fois plus a^ ge que la lle ;
(E) La lle est deux fois plus a^ gee
que le ls.
7
. (Classes 7-8) Sur une droite on marque des points de sorte que toutes
les distanes de 1 m, 2 m, 3 m, 4 m, 5 m, 6 m, 7 m et 9 m gurent
parmi les distanes entre es points. Combien y a-t-il au minimum de points
marques
sur ette droite ?
(A) 4
(B) 5
(C) 6
(D) 7
(E) 8
8
9. (Classes 9-10) Liliane, Niole, Katia et Charlotte sont allees
au inema.
Comme il n'etait
pas possible d'aheter quatre plaes ensemble, elles ont
et d'autres
ahete des billets pour les sieges
numero
7 et 8 dans la 10e -rangee
pour les sieges
numero
3 et 4 dans la 12e -rangee.
De ombien de manieres
peuvent-elles hoisir de s'asseoir, si Charlotte ne veut pas e^ tre assise a ot
^ e
de Niole ?
(B) 20
(C) 16
(D) 12
(E) 8
(A) 24
. (Classes 11-12) Soit ABC un triangle
isoele
ave BC = AC . Les segments
DE , F G, HI , KL, M N , OP et XY
divisent les ot
^ es
AC et CB en parties
egales.
Trouver XY si AB = 40 m.
(B) 35 m
(A) 38 m
(C) 33 m
(D) 30 m
(E) 27 m
10
C
..
.... ....
D ..................................... E
F ......................................................... G
H .............................................................................. I
K .................................................................................................. L
M ...................................................................................................................... N
O ............................................................................................................................................ P
X ................................................................................................................................................................. Y
A .......................................................................................................................................................................... B
Partie C (5 points par question)
. (Classes 3-4) En utilisant des ubes identiques, Mathieu et Niolas ont onstruit deux
b^atiments, omme illustres
dans les gures. Le
b^atiment de Mathieu pese
200 g et elui de Niolas 600 g. Combien de ubes du b^atiment de
Niolas sont-ils ahes
et ne peuvent e^ tre vus
dans la gure ?
(B) 2
(C) 3
(A) 1
(D) 4
(E) 5
11
..........
.......................
......................................
.. .. .. .....................
...........................................................
................................................
.. .. .. .. .. ...
................................................
b^atiment
de Niolas
............
................................
.
..............................
.........................................
............................
b^atiment
de Mathieu
. (Classes 5-6) On onsidere
tous les nombres de quatre hires, divisibles
par 6 et dont les hires, lus de gauhe a droite, vont en ordre roissant. Quel
est le hire des entaines dans le plus grand de es nombres ?
(A) 7
(B) 6
(C) 5
(D) 4
(E) 3
12
6
. (Classes 7-8) On divise un arre de ot
^ e 3 en polygones ave plusieurs segments omme indique dans
la gure. Quel est le perentage de l'aire de la gure
ombree
par rapport a elle du arre ?
(A) 30%
(B) 33 13 %
(C) 35%
(E) 50%
(D) 40%
13
.........1..........................2................
.. ..................................................................................................... ..... 1
. ........................................................................................................................ .
2 .. ..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.. .............................................................................................................................................................................................................................................................. ..
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... .... 2
................................................................................................................................. ...
1 .....
......
................................................
2
1
. (Classes 9-10) Un garon dit toujours la verit
e les jeudis et vendredis,
ment toujours les mardis et, les autres jours de la semaine, soit il dit la verit
e
soit il ment. On lui demanda son nom haque jour de la semaine et les six
premieres
fois, il donna les reponses
suivantes : Jean, Bernard, Jean, Bernard,
Paul, Bernard. Quelle fut sa reponse
le septieme
jour ?
(A) Jean
(B) Bernard (C) Paul
(D) Lu
(E) Pas possible de deider
14
. (Classes 11-12) On insrit un triangle equilat
eral
et un erle M dans un erle K , omme indique dans
la gure. Quel est le rapport de l'aire de K a elle de
M?
(B) 10 : 1
(C) 12 : 1
(A) 8 : 1
(D) 14 : 1
(E) 16 : 1
15
.....................................
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.....K
..... ... ...............
....
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.... .....
...
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..
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.... .M
........ ....
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Next we shall give solutions to the Mathematis Assoiation of Quebe
Contest (Seondary level) February 9, 2006 [2008 : 67-68℄. We apologize to
any readers who sent in solutions to this ontest but whose solutions we have
lost.
1.
A partiular magi square. It is well known that a magi square is obtained
by putting numbers in a square suh that the sum of eah row, olumn, and
diagonal is the same, as for example,
8 1
3 5
4 9
6
7
2
Imagine now that we deide to invent a new form of suh squares by replaing
the sum by a produt. We ask you to nd suh a square by replaing the
asterisks, ∗, by natural numbers, not neessarily distint or onseutive, in
the following square:
∗ 1
4 ∗
∗ ∗
∗
∗
2
7
.
Suppose that the square A below is a magi square. Then the square
B is a magi square for produts. For example, by the Law of Exponents, the
produt along the rst row of B is xa xb xc = xa+b+c and the produt along
the rst olumn of B is xa xd xg = xa+d+g and these are the same beause
a + b + c = a + d + g . The same is true for the other rows, olumns, and
diagonals of B .
Solution by the editor
A =
a b
d e
g h
c
f
i
,
xa
xd
xg
B =
xb
xe
xh
xc
xf
xi
.
Now, if we subtrat 1 from every entry of the rst square given in the question
and if we take x = 2, then the square B below is a solution to the problem.
A =
7 0
2 4
3 8
5
6
1
,
B =
27
22
23
20
24
28
25
26
21
=
128
4
8
1
16
256
32
64
2
.
2.
Clovis' outing. Clovis likes to take an outing in the natural numbers.
Eah day, he starts with a natural number of his hoie, the biggest possible.
Then, during his day, he passes from number to number using the following
rules. Suppose that the sequene of numbers is urrently at n.
(1) If n is divisible by 3 without remainder, then the next number is n/3.
(2) If the remainder after dividing
2n + 1.
n
by
3
is 1, then the next number is
(3) If the remainder after dividing
2n − 1.
n
by
3
is 2, then the next number is
(4) If n = 1, then the sequene stops.
Over the years that he has played this game, he notied that, whatever the
starting number, the sequene always ended up with the number 1. However, he wonders if there is a sequene that inreases indenitely, with larger
and larger numbers on average, or suh that it ends up in a loop of numbers
that does not ontain 1. Determine if suh a sequene is possible and give
an example, or show that suh a sequene does not exist by showing that all
sequenes using the above rules inevitably end up at the number 1.
Here is an example of suh a sequene: Starting with 55, we get 111,
37, 75, 25, 51, 17, 33, 11, 21, 7, 15, 5, 9, 3 and 1, whih ends the sequene.
.
Note that Clovis' sequene starting with 55 has a dereasing subsequene that goes to 1, given by the underlined numbers: 55, 111, 37, 75, 25,
Solution by the editor
8
51, 17, 33, 11, 21, 7, 15, 5, 9, 3, 1.
We will show that for any number a > 1
in one of Clovis' sequenes, there is always a number b oming after a in the
sequene suh that a > b. Thus, if Clovis starts with n > 1, then there will
be a subsequene n, m, p, . . . with n > m > p > · · · and this subsequene
must eventually hit the number 1 (beause all of the terms in it are positive,
it annot derease forever).
If a > 1 and a = 3k, k > 1, then by rule (1) the number b = k omes
right after a and a > b.
If a > 1 and a = 3k + 1, k > 0, then by rule (2) the number 2a + 1 =
2(3k+1)+1 = 6k+3 omes right after the number a, and then by rule (1) the
number (6k + 3)/3 = 2k + 1 omes after 2a + 1. Sine a = 3k + 1 > 2k + 1,
we see that the number b = 2k + 1 omes after a and a > b.
If a > 1 and a = 3k + 2, k ≥ 0, then by rule (3) the number 2a − 1 =
2(3k+2)+1 = 6k+3 omes right after the number a, and then by rule (1) the
number (6k + 3)/3 = 2k + 1 omes after 2a − 1. Sine a = 3k + 2 > 2k + 1,
we see that the number b = 2k + 1 omes after a and a > b.
Thus, in all ases where a > 1, there is a number b oming after a in
the sequene suh that a > b, and we are done.
3.
Eight balls in two urns. We give you two similar urns, four white balls,
and four blak balls. You must separate the balls amongst the two urns
(not neessarily the same number in eah urn), after whih both urns will
be made indistinguishable. How should the balls be distributed to maximize
the hanes that, if you draw a ball randomly from a randomly hosen urn,
you will obtain a white ball?
.
Solution by the editor
Put 1 white ball in one urn and all the other balls in the other urn. The
probability of hoosing the urn with 1 white ball and then drawing that white
ball from it is 12 ·1 = 12 and the probability of hoosing the other urn and then
3
3
= 14
drawing a white ball from it is 12 · 3+4
. Thus, with this distribution, the
overall probability of ultimately obtaining a white ball is p = 12 ·1+ 12 · 37 = 57 .
Now let p1 and p2 be the probabilities of drawing white balls from the
two urns (p1 = 1 and p2 = 37 above). The overall probability of ultimately
p2
obtaining a white ball is then p = 12 p1 + 12 p2 = p1 +
, whih is the average
2
5
of the probabilities p1 and p2 . Therefore, p > 7 implies that p1 > 57 or
p2 > 57 . To make an urn with p1 > 57 (say) we must have (w, b) = (1, 0),
(2, 0), (3, 0), (3, 1), (4, 0), or (4, 1), where the urn ontains w white balls
and b blak balls. These are the only distributions that ould yield p > 75 .
In the ase of (w, b) = (2, 0), (3, 0), or (4, 0) we have p < 57 , as moving all
but one white ball to the other urn inreases p2 but leaves p1 = 1. Finally,
(w, b) = (3, 1) or (4, 1) yields p = 12 or p = 25 , eah less than 57 .
Our rst distribution maximizes our hane of obtaining a white ball.
9
4.
. Three big ylindrial
barrels, lying parallel to the earth, are attahed by
a steel able at their ontat points, A and B , suh
that they stay xed in plae. Knowing that the two
smaller ones eah have a radius of 4 metres and the
Bq
qA
biggest one has a radius of 9 metres, what is the
length of the steel able?
Solution by the editor.
Join the entres of the smaller barrels
and drop a perpendiular to this segment from
the entre of the larger barrel, as in the diaC
gram at right. Sine the barrels rest on the
B
A
earth, the length of CZ is the dierene of
q
q
their radii, that is, |CZ| = 9 − 4 = 5. Also X
Y
Z
the length of CY is the sum of the radii, that
is, |CY | = 9 + 4 √
= 13. By the Pythagorean
Theorem |ZY | = 132 − 52 = 12. Thus, |XY | = 24. Finally, sine triangle
9
216
CAB is similar to triangle CXY , we have |AB| =
|XY | =
.
13
13
The three attahed barrels
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... ...............................
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5.
The magi words. An illusionist is searhing for
magi words to aompany his many magi triks.
R
He deides to onstrut his magi words starting D
with the diagram on the right. He takes a path
through the diagram and jots downs the letters he
A
nds on it. Eah magi word must have exatly 11
letters and must start and end with the letter A. C
B
Two onseutive letters must never be idential.
How many magi words are there?
Note: Here are two possible magi words: ABRACADABRA and
ARADCABARBA.
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Let Mk be the number of k-letter words that start with A and end with
A and that an be formed by travelling through the bowtie. Let Nk be the
number of k-letter words starting with A but not ending with the letter A
that an be similarly formed.
If k ≥ 2, then we see that Mk = Nk−1 , beause removing the letter A
from a word ending with A leaves a word not ending in A (but still starting
with A) and the proess an be reversed. Similarly, by deleting the last letter
of a word of length k that starts with A but does not end in A, we see that
Nk = Nk−1 + 4Mk−1 , beause any of the four letters dierent from A an
be added to a word not ending in A or else there is only way to extend a
(k − 1)-letter word not ending in A to one that still does not end in A. Sine
Mk−1 = Nk−2 the last equation beomes Nk = Nk−1 + 4Nk−2 , where
k ≥ 1.
Solution by the editor
10
We now have N1 = 0, N2 = 4, N3 = N2 + 4N1 = 4 + 4 · 0 = 4, and so
forth. The results of alulating the Ni are summarized in the table below:
N1
0
N2
4
N3
4
N4
20
N5
36
N6
116
N7
260
N8
724
N9
1764
N10
4660
Finally M11 = N10 , so there are 4660 magi words altogether.
6.
All ten digits. Find the smallest positive natural number N suh that, in
the deimal notation, N and 2N together use all ten digits: 0, 1, 2, . . . , 9.
.
Solution by the editor
We have 2(13485) = 26970, and we will prove that if N1 and 2N1
together use all ten digits and N1 ≤ N = 13485, then N1 = N .
As N1 has ve digits and N1 ≤ N , then N1 = 1 . . . and 2N1 = 2 . . ..
Digits 1 and 2 are now used and 2N1 uses 0 (otherwise N1 uses 0 and 2N1
then uses 0 or 1, a ontradition). Thus, N1 = 13 . . .. The smallest available
digit for N1 is now a 4 and N1 ≤ N , hene N1 = 134 . . . and 2N1 = 26 . . ..
The number N1 uses 5, beause 2N1 uses 0. If N1 = 1345x, then x is a digit
greater than 5 and 2N1 = 2691y, a ontradition. Thus, N1 = 134x5 ≤ N .
Finally, x 6= 7, hene x = 8.
Therefore, N1 = N and N is the smallest positive integer with the
given property.
[Ed.: Rolland Gaudet oers the solution N = 6792 if initial zeroes are
allowed, for then 2(6792) = 013584.℄
7. The pizza toppings. At the Julio pizzeria, all the pizzas have heese and
tomato saue on them. The hoie of toppings is limited to blak olives,
anhovies, and sausage. Of the 200 lients Julio had yesterday, 40 took anhovies, 80 took blak olives, 120 took sausage, 60 took at the same time
blak olives and sausage, but none took at the same time anhovies and
blak olives or anhovies and sausage. How many lients took none of the
three toppings?
.
Let t be the number of ustomers who took at least one topping. Any
ustomer who took anhovies took no other topping, so t = 40 + x where x
is the number of ustomers who took blak olives or sausage (or both). There
were 60 ustomers who took both blak olives and sausage, so 20 = 80 − 60
took just blak olives and nothing else. Similarly, 60 = 120 − 60 ustomers
took sausage and nothing else. Thus, t = 40+x = 40+(20+60+60) = 180,
and the number of ustomers who took no toppings is 200 − t = 20.
Solution by the editor