2002-03COMCContest
The Canadian Mathematical Society in MATHEMATICS and COMPUTING
Canadian Open Mathematics Challenge NOTE:1.
Please read the instructions on the front cover of this booklet.2.
Write solutions in the answer booklet provided.3.
It is expected that all calculations and answers will be expressed as exact numbers such as 427π,+, etc.4.Calculators are not allowed.
PART A
1.In triangle PQR , F is the point on QR so that PF is perpendicular
to QR . If PR =13, RF =5, and FQ =9, what is the perimeter
of ?PQR ?2.
If x y +=4 and xy =?12, what is the value of x xy y 225++?3. A regular pentagon is a five-sided figure which has all of its
angles equal and all of its side lengths equal. In the diagram,
TREND is a regular pentagon , PEA is an equilateral triangle, and
OPEN is a square. Determine the size of ∠EAR .4.In a sequence of numbers, the sum of the first n terms is equal to 562n n +. What is the sum of the
3rd, 4th and 5th terms in the original sequence?
5.If m and n are non-negative integers with m n <, we define m n ? to be the sum of the integers from m to n , including m and n . For example, 58567826?=+++=.For every positive integer a , the numerical value of
212111a a a a ?()?+()[]?()?+()[]
is the same. Determine this value.6.Two mirrors meet at an angle of 30o at the point
V . A beam of light, from a source S , travels parallel to one mirror and strikes the other mirror at point A, as shown. After a number of reflections, the beam comes back to S . If SA and AV are both
1 metre in length, determine the total distance travelled by the beam.
7.N is a five-digit positive integer. A six-digit integer P is constructed by placing a 1 at the right-hand
end of N . A second six-digit integer Q is constructed by placing a 1 at the left-hand end of N . If P is three times Q , determine the value of N .
95A
S V 30°
8.Suppose that M is an integer with the property that if x is randomly chosen from the set
1239991000,,,,,K {}, the probability that x is a divisor of M is 1100. If M ≤1000, determine the maximum possible value of M .
PART B
1.Square ABCD has vertices A 00,(), B 08,(), C 88,(), and D 80,(). The points P 05,() and Q 03,() are on
side AB , and the point F 81,() is on side CD .
(a)What is the equation of the line through Q parallel to the line through P and F ?
(b)If the line from part (a) intersects AD at the point G , what is the equation of the line through F
and G ?
(c)The centre of the square is the point H 44,(). Determine the equation of the line through H
perpendicular to FG .
(d) A circle is drawn with centre H that is tangent to the four sides of the square. Does this circle
intersect the line through F and G ? Justify your answer. (A sketch is not sufficient justification.)2.(a)Let A and B be digits (that is, A and B are integers between 0 and 9 inclusive). If the product of
the three-digit integers 25A and 13B is divisible by 36, determine with justification the four possible ordered pairs A B ,().
(b)
An integer n is said to be a multiple of 7 if n k =7 for some integer k .
(i)If a and b are integers and 107a b m += for some integer m , prove that a b ?2 is a multiple
of 7.
(ii)If c and d are integers and 54c d + is a multiple of 7, prove that 4c d ? is also a multiple
of 7.3.There are some marbles in a bowl. Alphonse, Beryl and Colleen each take turns removing one or two
marbles from the bowl, with Alphonse going first, then Beryl, then Colleen, then Alphonse again, and so on. The player who takes the last marble from the bowl is the loser, and the other two players are the winners.
(a)If the game starts with 5 marbles in the bowl, can Beryl and Colleen work together and force
Alphonse to lose?(b)The game is played again, this time starting with N marbles in the bowl. For what values of N
can Beryl and Colleen work together and force Alphonse to lose?
4.Triangle DEF is acute. Circle C 1 is drawn with DF as its
diameter and circle C 2 is drawn with DE as its diameter.Points Y and Z are on DF and DE respectively so that EY and
FZ are altitudes of ?DEF . EY intersects C 1 at P , and FZ
intersects C 2 at Q . EY extended intersects C 1 at R , and FZ
extended intersects C 2 at S . Prove that P , Q , R, and S are
concyclic points.D E F C 2C 1