Skoliad - Canadian Mathematical Society

67
SKOLIAD
No. 123
Lily Yen and Mogens Hansen
Please send your solutions to problems in this Skoliad by 1 August, 2010. A
opy of CRUX will be sent to one pre-university reader who sends in solutions
before the deadline. The deision of the editors is nal.
This month's Skoliad ontest is the City Competition of the Croatian
Mathematial Soiety, 2009, Seondary Level, Grade 1. Our thanks go to
Z eljko Hanjs, University of Zagreb, Croatia, for providing us with this ontest
and for permission to publish it.
La redation
souhaite remerier Rolland Gaudet, de College
universitaire de Saint-Bonifae, Winnipeg, MB, d'avoir traduit e onours.
Competition
2009 de la Soiet
e
mathematique
roate
Niveau seondaire, premiere
annee
. Reduire
la fration
1
a4 − 2a3 − 2a2 + 2a + 1
(a + 1)(a + 2)
.
. Si on erit
le hire 3 a gauhe d'un entier a deux positions deimales,
alors on obtient, bien sur,
^ un entier a trois positions deimales.
Si le double
27 fois l'entier a deux positions, quel etait
de l'entier a trois positions egale
l'entier a deux positions, au depart
?
2
. Determiner
le plus grand entier n tel que 3
3
4
. Determiner
le nombre de diviseurs de 288.
. Dans la gure, ABCDEF est un
hexagone regulier
tandis que EF GHI
est un pentagone regulier.
Determiner
l'angle ∠GAF .
5
C
B
n−
5
−2(4n+1) > 6n+5.
3
D
.
I
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.
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A
H
G
. Dans un trapeze
ABCD, l'angle a B est retangle, et la diagonale BD
est perpendiulaire au ot
^ e AD. Le ot
^ e BC est de longueur 5, tandis que la
diagonale BD est de longueur 13. Determiner
la surfae du trapeze
ABCD.
6
68
. Lors de la f^ete de Ther
ese,
le premier oup de sonnette annona la premiere
invitee.
Ensuite, a haque oup de sonnette, le nombre d'invitees
qui se
presentaient
augmentait de deux. Si la sonnette a retentit n fois, ombien
d'invitees
se sont-elles present
ees
a la f^ete ?
7
. Determiner
tous les entiers positifs n tels que n2 − 440 est le arre d'un
entier.
8
City Competition of the Croatian Mathematial
Soiety, 2009
Seondary Level, Grade 1
1
. Redue the fration
a4 − 2a3 − 2a2 + 2a + 1
(a + 1)(a + 2)
.
. If you write the digit 3 on the left side of a two-digit number, you obtain, of ourse, a three-digit number. If twie the three-digit number equals
27 times the two-digit number, what is the original two-digit number?
2
. Find the largest integer n suh that 3
3
4
n−
. Find the number of divisors of 288.
. In the gure, ABCDEF is a regular hexagon while EF GHI is a regular
pentagon. Determine the angle ∠GAF .
5
3
− 2(4n + 1) > 6n + 5.
D
C
5
B
I
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.......................................................................
A
H
G
. In a trapezoid ABCD, the angle at B is a right angle, and the diagonal
is perpendiular to the leg AD. The length of the leg BC is 5, and the
length of the diagonal BD is 13. Find the area of the trapezoid ABCD.
6
BD
. At Tihana's birthday party, the rst guest arrived the rst time the bell
rang. Eah time the bell rang thereafter the number of guests arriving was
two more than the number that had arrived the previous time the bell rang.
If the bell rang n times, how many guests attended the party?
7
. Determine all positive integers n suh that n2 − 440 is the square of an
integer.
8
69
Next follow solutions to the Calgary Mathematial Assoiation Junior
High Shool Mathematis Contest, Part B, 2009, that appeared previously in
Skoliad 117 [2009 : 194{196℄.
. Rihard needs to go from his house to the park by taking a taxi. There
are two taxi ompanies available. The rst taxi ompany harges an initial
ost of $10.00, plus $0.50 for eah kilometre travelled. The seond taxi ompany harges an initial ost of $4.00, plus $0.80 for eah kilometre travelled.
Rihard realises that the ost to go to the park is the same regardless of whih
taxi ompany he hooses. What is the distane in km from his house to the
park?
1
Solution by Jixuan Wang, student, Don Mills Collegiate Institute, Toronto,
.
ON
Let d denote the distane to the park (in kilometres). Then the ost
(in ents) of using the rst taxi ompany is 1000 + 50d while the ost of
using the seond ompany is 400 + 80d. Sine these two osts are equal,
1000 + 50d = 400 + 80d, so 600 = 30d, and, thus, d = 20.
Also solved by GESINE GEUPEL, student, Max Ernst Gymnasium, Bruhl,

NRW, Germany.
2. A radio station runs a ontest in whih eah winner will get to attend
two Calgary Flames playo games and to take one guest to eah game. The
winner does not have to take the same guest to the two games. Lukily,
ve shool friends Alie, Bob, Carol, David, and Eva are all winners of this
ontest. Show how eah winner an hoose two others from this group to be
his or her guests, so that eah pair of the ve friends gets to go to at least
one playo game together.
Solution by Gesine Geupel, student, Max Ernst Gymnasium, Bruhl,

NRW,
.
The table shows a possible hoie of guests.
Winner First guest Seond guest
Alie
Bob
Carol
Bob
Carol
David
Carol
David
Eva
David
Eva
Alie
Eva
Alie
Bob
Germany
Also solved by JIXUAN WANG, student, Don Mills Collegiate Institute, Toronto, ON.
. A lass was given two tests. In eah test, eah student was given a nonnegative integer sore with a maximum possible sore of 10. Adrian notied
that in eah test, only one student sored higher than he did and nobody got
the same sore as he did. But then the teaher posted the averages of the
two sores for eah student, and now there was more than one student with
an average sore higher than Adrian.
3
70
(a) Give an example (using exat sores) to show that this ould happen.
(b) What is the largest possible number of students whose average sore
ould be higher than Adrian's average sore? Explain learly why your
answer is orret.
Solution by Gesine Geupel, student, Max Ernst Gymnasium, Bruhl,

NRW,
.
(a) The table shows a possible set of sores where two students have a
higher average than Adrian.
Germany
Adrian
Bob
Carol
First Test Seond Test Average
7
5
6.0
6
9
7.5
10
4
7.0
(b) To get a better average than Adrian a student must sore better than
Adrian on at least one test. Sine only one student sored better than Adrian
on eah of two tests, at most two students an have a higher average than
Adrian.
Also solved by JIXUAN WANG, student, Don Mills Collegiate Institute, Toronto, ON.
. A retangle with dimensions 6 m by 8 m
is drawn. A irle is drawn irumsribing
this retangle. A square is drawn irumsribing this irle. A seond irle is drawn
that irumsribes this square. What is the
area in m2 of the bigger irle?
4
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.
.
.
.
.
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..... ....... . ......... ....
............................. ...............
............ .............
..........
Solution by Jixuan Wang, student, Don Mills Collegiate Institute, Toronto,
.
ON
The diagonal of the retangle is 62 + 82 = 10 by the Pythagorean
Theorem. But this diagonal is also a diameter of the inner irle, and the
diameter of the inner irle equals the length of the side of the square.
Theorem again,
Thus the square has side length 10. By the
√
√ Pythagorean √
the length of the diagonal of the square is 102 + 102 = 200 = 10 2.
Sine the diagonal of the square
√ is also a diameter of the bigger irle, the
radius
of
the
bigger
irle
is
5 2. It follows that the area of the bigger irle
€ √ Š2
is π 5 2 = 50π.
√
5. If A is a two-digit positive integer that does not ontain zero as a digit, B
is a three-digit positive integer, and A% of B is 400, nd all possible values
of A and B .
71
Solution by Jixuan Wang, student, Don Mills Collegiate Institute, Toronto,
.
ON
Sine (A/100)B = 400, it follows that AB = 40000 = 26 · 54 . Sine
A does not ontain the digit zero, it must have fators of only 2 or only 5.
Therefore, the possible two-digit values for A are 25, 16, 32, and 64.
Solving the equation for B yields that B = 40000/A, so (A, B) is one
of (25, 1600), (16, 2500), (32, 1250), and (64, 625). Sine B has three digits,
the only solution is (A, B) = (64, 625).
6. Find a retangle with the following two properties: (i) its perimeter is an
odd integer; and (ii) none of its sides is an integer.
Next, nd a retangle with the following two properties: (i) its area is
an even integer; and (ii) none of its sides is an integer.
Finally, nd a quadrilateral (not neessarily a retangle) with the following three properties: (i) its perimeter is a positive integer; (ii) its area is
a positive integer; and (iii) none of its sides is an integer.
Solution by Jixuan Wang, student, Don Mills Collegiate Institute, Toronto,
.
ON
part.
The retangle with sides
1
4
and
The retangle with sides 2 +
5
4
has perimeter 3. This solves the rst
√
√
2 and 2 − 2 has perimeter 8 and
€
€√ Š2
√ Š€
√ Š
2 + 2 2 − 2 = 22 −
2
= 4 − 2 = 2.
area
This solves the last two parts.
This issue's prize of one opy of Crux Mathematiorum for the best solutions goes to Jixuan Wang, student, Don Mills Collegiate Institute, Toronto,
ON.
The urrent editors of this setion inherited the name Skoliad, and we
have wondered about its origins. The best we have been able to reonstrut
is this: Olympiad is a derivative of Mount Olympus; the Skoliad setion features problems that are more aessible than Math Olympiad problems and
is therefore named after a less exalted plae, namely the town Skolos (or
Solus) as mentioned in The Iliad, seond song, line 497.
The paragraph above prompted the proofreaders of this issue of Skoliad
to inform us that the name originated with Rihard K. Guy, who found Skolos
as the name of a mountain on a map of Greee at the University of Calgary
library.
We soliit information on the exat loation of Skolos (whih seems to
be an open problem in Greek arhaeology) as well as reader solutions to the
featured ontest.