DIAGRAMS WORK AND ANSWERS: A3
The 2012 Sun Life Financial ĂŶĂĚŝĂŶ KƉĞŶ DĂƚŚĞŵĂƟĐƐ ŚĂůůĞŶŐĞ WůĞĂƐĞ ƉƌŝŶƚ ĐůĞĂƌůLJ ĂŶĚ ĐŽŵƉůĞƚĞ Ăůů ŝŶĨŽƌŵĂƟŽŶ ďĞůŽǁ͘ &ĂŝůƵƌĞ ƚŽ ƉƌŝŶƚ ůĞŐŝďůLJ Žƌ ƉƌŽǀŝĚĞ ĐŽŵƉůĞƚĞ ŝŶĨŽƌŵĂƟŽŶ ŵĂLJ ƌĞƐƵůƚ ŝŶ LJŽƵƌ ĞdžĂŵ ďĞŝŶŐ ĚŝƐƋƵĂůŝĮĞĚ͘ dŚŝƐ ĞdžĂŵ ŝƐ ŶŽƚ ĐŽŶƐŝĚĞƌĞĚ ǀĂůŝĚ ƵŶůĞƐƐ ŝƚ ŝƐ ĂĐĐŽŵƉĂŶŝĞĚ ďLJ LJŽƵƌ ƚĞƐƚ ƐƵƉĞƌǀŝƐŽƌ͛Ɛ ƐŝŐŶĞĚ ĨŽƌŵ͘
First Name: Grade
ϴ ϵ ϭϬ ϭϭ ϭϮ ĠŐĞƉ Last Name:
Other: ;zŽƵƚŚͿ dͲ^Śŝƌƚ ^ŝnjĞ y^ ^ D
ƌĞ LJŽƵ ĐƵƌƌĞŶƚůLJ ƌĞŐŝƐƚĞƌĞĚ ŝŶ ĨƵůůͲƟŵĞ ĂƩĞŶĚĂŶĐĞ Ăƚ ĂŶ ĞůĞŵĞŶƚĂƌLJ͕ ƐĞĐŽŶĚĂƌLJ Žƌ L XL XXL
ĠŐĞƉ ƐĐŚŽŽů͕ Žƌ ŚŽŵĞ ƐĐŚŽŽůĞĚ ĂŶĚ ŚĂǀĞ ďĞĞŶ ƐŝŶĐĞ ^ĞƉƚĞŵďĞƌ ϭϱƚŚ ŽĨ ƚŚŝƐ LJĞĂƌ͍ Date of Birth: Y N y y y y m m d d
A ƌĞ LJŽƵ Ă ĂŶĂĚŝĂŶ ŝƟnjĞŶ Žƌ Ă WĞƌŵĂŶĞŶƚ ZĞƐŝĚĞŶƚ ŽĨ ĂŶĂĚĂ ;ƌĞŐĂƌĚůĞƐƐ ŽĨ ĐƵƌƌĞŶƚ Gender:
;KƉƟŽŶĂůͿ ĂĚĚƌĞƐƐͿ͍ Y N
D & E-mail Address: ^ŝŐŶĂƚƵƌĞ͗ ͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺͺ
INSTRUCTIONS: K EKd KW E d,/^ KK<> d hEd/> /E^dZh d dK K ^K
EXAM: dŚĞƌĞ ĂƌĞ ϯ ƉĂƌƚƐ ƚŽ ƚŚĞ KD ƚŽ ďĞ ĐŽŵƉůĞƚĞĚ ŝŶ Ϯ ŚŽƵƌƐ ĂŶĚ ϯϬ ŵŝŶƵƚĞƐ͘
W Zd ͗ ŽŶƐŝƐƚƐ ŽĨ ϰ ďĂƐŝĐ ƋƵĞƐƟŽŶƐ ǁŽƌƚŚ ϰ ŵĂƌŬƐ ĞĂĐŚ͘ W Zd ͗ ŽŶƐŝƐƚƐ ŽĨ ϰ ŝŶƚĞƌŵĞĚŝĂƚĞ ƋƵĞƐƟŽŶƐ ǁŽƌƚŚ ϲ ŵĂƌŬƐ ĞĂĐŚ͘ DŽďŝůĞ ƉŚŽŶĞƐ ĂŶĚ ĐĂůĐƵůĂƚŽƌƐ
W Zd ͗ ŽŶƐŝƐƚƐ ŽĨ ϰ ĂĚǀĂŶĐĞĚ ƋƵĞƐƟŽŶƐ ǁŽƌƚŚ ϭϬ ŵĂƌŬƐ ĞĂĐŚ͘ ĂƌĞ ŶŽƚ ƉĞƌŵŝƩĞĚ͘
DIAGRAMS ͗ ŝĂŐƌĂŵƐ ĂƌĞ ŶŽƚ ĚƌĂǁŶ ƚŽ ƐĐĂůĞ͖ ƚŚĞLJ ĂƌĞ ŝŶƚĞŶĚĞĚ ĂƐ ĂŝĚƐ ŽŶůLJ͘
WORK AND ANSWERS:
ůů ƐŽůƵƟŽŶ ǁŽƌŬ ĂŶĚ ĂŶƐǁĞƌƐ ĂƌĞ ƚŽ ďĞ ƉƌĞƐĞŶƚĞĚ ŝŶ ƚŚŝƐ ŬůĞƚ ŝŶ ƚŚĞ ďŽdžĞƐ
ƉƌŽǀŝĚĞĚ͘ DĂƌŬƐ ĂƌĞ ĂǁĂƌĚĞĚ ĨŽƌ ĐŽŵƉůĞƚĞŶĞƐƐ ĂŶĚ ĐůĂƌŝƚLJ͘ &Žƌ ƐĞĐƟŽŶƐ ĂŶĚ ͕ ŝƚ ŝƐ ŶŽƚ ŶĞĐĞƐƐĂƌLJ ƚŽ ƐŚŽǁ
LJŽƵƌ ǁŽƌŬ ŝŶ ŽƌĚĞƌ ƚŽ ƌĞĐĞŝǀĞ ĨƵůů ŵĂƌŬƐ͘ ,ŽǁĞǀĞƌ͕ ŝĨ LJŽƵƌ ĂŶƐǁĞƌ Žƌ ƐŽůƵƟŽŶ ŝƐ ŝŶĐŽƌƌĞĐƚ͕ ĂŶLJ ǁŽƌŬ ƚŚĂƚ LJŽƵ
ĚŽ ĂŶĚ ƉƌĞƐĞŶƚ ŝŶ ƚŚŝƐ ŬůĞƚ ǁŝůů ďĞ ĐŽŶƐŝĚĞƌĞĚ ĨŽƌ ƉĂƌƚ ŵĂƌŬƐ͘ &Žƌ ƐĞĐƟŽŶ ͕ LJŽƵ ŵƵƐƚ ƐŚŽǁ LJŽƵƌ ǁŽƌŬ ĂŶĚ
ƉƌŽǀŝĚĞ ƚŚĞ ĐŽƌƌĞĐƚ ĂŶƐǁĞƌ Žƌ ƐŽůƵƟŽŶ ƚŽ ƌĞĐĞŝǀĞ ĨƵůů ŵĂƌŬƐ͘ /ƚ ŝƐ ĞdžƉĞĐƚĞĚ ƚŚĂƚ Ăůů ĐĂůĐƵůĂƟŽŶƐ ĂŶĚ ĂŶƐǁĞƌƐ ǁŝůů ďĞ ĞdžƉƌĞƐƐĞĚ ĂƐ ĞdžĂĐƚ ŶƵŵďĞƌƐ ƐƵĐŚ ĂƐ ϰʋ͕ Ϯ н яϳ͕ ĞƚĐ͕͘ ƌĂƚŚĞƌ ƚŚĂŶ ĂƐ ϭϮ͘ϱϲϲ͕ ϰ͘ϲϰϲ͕ ĞƚĐ͘ dŚĞ ŶĂŵĞƐ ŽĨ Ăůů ĂǁĂƌĚ ǁŝŶŶĞƌƐ ǁŝůů ďĞ ƉƵďůŝƐŚĞĚ ŽŶ ƚŚĞ ĂŶĂĚŝĂŶ DĂƚŚĞŵĂƟĐĂů ^ŽĐŝĞƚLJ ǁĞď ƐŝƚĞ͘ dŚĞ ĐŽŶƚĞŶƚƐ ŽĨ ƚŚĞ KD ϮϬϭϮ ĂŶĚ LJŽƵƌ ĂŶƐǁĞƌƐ ĂŶĚ ƐŽůƵƟŽŶƐ ŵƵƐƚ ŶŽƚ ďĞ ƉƵďůŝĐĂůůLJ ĚŝƐĐƵƐƐĞĚ͕ ŝŶĐůƵĚŝŶŐ ǁĞď ĐŚĂƚƐ͕ ĨŽƌ Ăƚ ůĞĂƐƚ Ϯϰ ŚŽƵƌƐ͘&Žƌ ŽĸĐŝĂů ƵƐĞ ŽŶůLJ͘ A3 A4 A B3 B4 B C3 C4 C ABC ϭ Ϯ ϭ Ϯ ϭ Ϯ
^hE >/& &/E E / > E / E KW E D d, D d/ ^ , >> E' ϮϬϭϮ WĂŐĞ Ϯ ŽĨ ϭϲ
WĂƌƚ ͗ YƵĞƐƟŽŶ ϭ ;ϰ ŵĂƌŬƐͿ
n
4 A1 Determine the positive integer n such that 8 = 4 .
zŽƵƌ ^ŽůƵƟŽŶ͗ WĂƌƚ ͗ YƵĞƐƟŽŶ Ϯ ;ϰ ŵĂƌŬƐͿ
A2 Let x be the average of the following six numbers: {12, 412, 812, 1212, 1612, 2012}. Determine the value of x.
zŽƵƌ ^ŽůƵƟŽŶ͗
^hE >/& &/E E / > E / E KW E D d, D d/ ^ , >> E' ϮϬϭϮ WĂŐĞ ϯ ŽĨ ϭϲ ^hE >/& &/E E / > E / E KW E D d, D d/ ^ , >> E' ϮϬϭϮ WĂŐĞ Ϯ ŽĨ ϭϲ
WĂƌƚ ͗ YƵĞƐƟŽŶ ϯ ;ϰ ŵĂƌŬƐͿ
Determine the value of r.
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zŽƵƌ ^ŽůƵƟŽŶ͗ zŽƵƌ ^ŽůƵƟŽŶ͗
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F A B C D E A4 Twelve different lines are drawn on the coordinate plane so that each line is parallel to exactly two other lines. Furthermore, no three lines intersect at a point. Determine the total number of intersection points among the twelve lines.
A3 Let ABCDEF be a hexagon all of whose sides are equal in length and all of whose angles are equal. The area of hexagon ABCDEF is exactly r times the area of triangle ACD.
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^hE >/& &/E E / > E / E KW E D d, D d/ ^ , >> E' ϮϬϭϮ WĂŐĞ ϰ ŽĨ ϭϲ
WĂƌƚ ͗ YƵĞƐƟŽŶ ϭ ;ϲ ŵĂƌŬƐͿ
1 Alice and Bob run in the clockwise direction around a circular track, each running at a constant speed. Alice can complete a lap in t seconds, and Bob can complete a lap in 60 seconds. They start at diametrically-opposite points.
Alice Bob When they meet for the first time, Alice has completed exactly 30 laps. Determine all possible values of t. zŽƵƌ ^ŽůƵƟŽŶ͗
^hE >/& &/E E / > E / E KW E D d, D d/ ^ , >> E' ϮϬϭϮ WĂŐĞ ϰ ŽĨ ϭϲ ^hE >/& &/E E / > E / E KW E D d, D d/ ^ , >> E' ϮϬϭϮ WĂŐĞ ϱ ŽĨ ϭϲ
WĂƌƚ ͗ YƵĞƐƟŽŶ Ϯ ;ϲ ŵĂƌŬƐͿ
2 For each positive integer n, define ϕ(n) to be the number of positive divisors of n. For exam- ple, ϕ(10) = 4, since 10 has 4 positive divisors, namely {1, 2, 5, 10}.
Suppose n is a positive integer such that ϕ(2n) = 6. Determine the minimum possible value of ϕ(6n).
zŽƵƌ ^ŽůƵƟŽŶ͗
^hE >/& &/E E / > E / E KW E D d, D d/ ^ , >> E' ϮϬϭϮ WĂŐĞ ϲ ŽĨ ϭϲ
WĂƌƚ ͗ YƵĞƐƟŽŶ ϯ ;ϲ ŵĂƌŬƐͿ
3 Given the following 4 by 4 square grid of points, determine the number of ways we can label ten different points A, B, C, D, E, F, G, H, I, J such that the lengths of the nine segments AB, BC, CD, DE, EF, F G, GH, HI, IJ are in strictly increasing order.
zŽƵƌ ^ŽůƵƟŽŶ͗
^hE >/& &/E E / > E / E KW E D d, D d/ ^ , >> E' ϮϬϭϮ WĂŐĞ ϲ ŽĨ ϭϲ ^hE >/& &/E E / > E / E KW E D d, D d/ ^ , >> E' ϮϬϭϮ WĂŐĞ ϳ ŽĨ ϭϲ
WĂƌƚ ͗ YƵĞƐƟŽŶ ϰ ;ϲ ŵĂƌŬƐͿ
4 In the following diagram, two lines that meet at a point A are tangent to a circle at points B and C. The line parallel to AC passing through B meets the circle again at D. Join the segments CD and AD. Suppose AB = 49 and CD = 28. The length of AD is a positive integer n. Determine n.
B D A C
zŽƵƌ ^ŽůƵƟŽŶ͗
^hE >/& &/E E / > E / E KW E D d, D d/ ^ , >> E' ϮϬϭϮ WĂŐĞ ϴ ŽĨ ϭϲ
WĂƌƚ ͗ YƵĞƐƟŽŶ ϭ ;ϭϬ ŵĂƌŬƐͿ
2 C1 Let f (x) = x and g(x) = 3x − 8.
(a) (2 marks) Determine the values of f (2) and g(f (2)). (b) (4 marks) Determine all values of x such that f (g(x)) = g(f (x)).
(c) (4 marks) Let h(x) = 3x − r. Determine all values of r such that f(h(2)) = h(f(2)).
zŽƵƌ ^ŽůƵƟŽŶ͗
^hE >/& &/E E / > E / E KW E D d, D d/ ^ , >> E' ϮϬϭϮ WĂŐĞ ϵ ŽĨ ϭϲ ^hE >/& &/E E / > E / E KW E D d, D d/ ^ , >> E' ϮϬϭϮ WĂŐĞ ϴ ŽĨ ϭϲ
^hE >/& &/E E / > E / E KW E D d, D d/ ^ , >> E' ϮϬϭϮ WĂŐĞ ϭϬ ŽĨ ϭϲ
WĂƌƚ ͗ YƵĞƐƟŽŶ Ϯ ;ϭϬ ŵĂƌŬƐͿ
C2 We fill a 3 × 3 grid with 0s and 1s. We score one point for each row, column, and diagonal whose sum is odd.
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1 For example, the grid on the left has 0 points and the grid on the right has 3 points.
(a) (2 marks) Fill in the following grid so that the grid has exactly 1 point. No additional work is required. Many answers are possible. You only need to provide one.
(b) (4 marks) Determine all grids with exactly 8 points.
(c) (4 marks) Let E be the number of grids with an even number of points, and O be the number of grids with an odd number of points. Prove that E = O.
zŽƵƌ ^ŽůƵƟŽŶ͗
^hE >/& &/E E / > E / E KW E D d, D d/ ^ , >> E' ϮϬϭϮ WĂŐĞ ϭϭ ŽĨ ϭϲ ^hE >/& &/E E / > E / E KW E D d, D d/ ^ , >> E' ϮϬϭϮ WĂŐĞ ϭϬ ŽĨ ϭϲ
^hE >/& &/E E / > E / E KW E D d, D d/ ^ , >> E' ϮϬϭϮ WĂŐĞ ϭϮ ŽĨ ϭϲ
WĂƌƚ ͗ YƵĞƐƟŽŶ ϯ ;ϭϬ ŵĂƌŬƐͿ
C3 Let ABCD be a parallelogram. We draw in the diagonal AC. A circle is drawn inside ∆ABC tangent to all three sides and touches side AC at a point P .
A B P D C (a) (2 marks) Prove that DA + AP = DC + CP .
(b) (4 marks) Draw in the line DP . A circle of radius r 1 2 is drawn inside ∆DAP tangent to all three sides. A circle of radius r is drawn inside ∆DCP tangent to all three sides.
Prove that 1 r AP 2 = . r P C
(c) (4 marks) Suppose DA + DC = 3AC and DA = DP . Let r 1 2 1 , r 2 be the two radii defined in (b). Determine the ratio r /r .
zŽƵƌ ^ŽůƵƟŽŶ͗
^hE >/& &/E E / > E / E KW E D d, D d/ ^ , >> E' ϮϬϭϮ WĂŐĞ ϭϯ ŽĨ ϭϲ ^hE >/& &/E E / > E / E KW E D d, D d/ ^ , >> E' ϮϬϭϮ WĂŐĞ ϭϮ ŽĨ ϭϲ
^hE >/& &/E E / > E / E KW E D d, D d/ ^ , >> E' ϮϬϭϮ WĂŐĞ ϭϰ ŽĨ ϭϲ
WĂƌƚ ͗ YƵĞƐƟŽŶ ϰ ;ϭϬ ŵĂƌŬƐͿ
C4 For any positive integer n, an n-tuple of positive integers (x 1 , x 2 n ) is said to be super- , · · · , x squared if it satisfies both of the following properties:
(1) x 1 > x 2 > x 3 n . 2 > · · · > x 2 2 (2) The sum x + x is a perfect square for each 1 ≤ k ≤ n. 1 2 + · · · + x k 2 2 2 For example, (12, 9, 8) is super-squared, since 12 > 9 > 8, and each of 12 , 12 + 9 , and 2 2 2 12 + 9 + 8 are perfect squares.
(a) (2 marks) Determine all values of t such that (32, t, 9) is super-squared. (b) (2 marks) Determine a super-squared 4-tuple (x 1 , x 2 , x 3 , x 4 ) with x 1 < 200.
(c) (6 marks) Determine whether there exists a super-squared 2012-tuple.
zŽƵƌ ^ŽůƵƟŽŶ͗
^hE >/& &/E E / > E / E KW E D d, D d/ ^ , >> E' ϮϬϭϮ WĂŐĞ ϭϱ ŽĨ ϭϲ ^hE >/& &/E E / > E / E KW E D d, D d/ ^ , >> E' ϮϬϭϮ WĂŐĞ ϭϰ ŽĨ ϭϲ
^hE >/& &/E E / > E / E KW E D d, D d/ ^ , >> E' ϮϬϭϮ WĂŐĞ ϭϲ ŽĨ ϭϲ
Canadian Mathematical Society Société mathématique du Canada ĂŶĂĚŝĂŶ KƉĞŶ DĂƚŚĞŵĂƟĐƐ ŚĂůůĞŶŐĞ