2018年欧几里得数学竞赛(EMC)真题加详解
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The CENTRE for EDUCATION in MATHEMATICS and COMPUTING
cemc.uwaterloo.ca
2018 Euclid Contest
Wednesday, April 11, 2018
(in North America and South America)
Also, OC = 6.
Since the equation of the line is y = 2x, then its slope is 2.
OC Since the slope of the line is 2, then = 2.
√ CD2 = BC2 − BD2 = (12 2)2 − 122 = 122(2) − 122 = 122
Since CD > 0, then CD = 12. Therefore, AC = AD + DC = 5 + 12 = 17.
2018 Euclid Contest Solutions
Page 4
(c) Solution 1
The area of the shaded region equals the area of square OABC minus the area of
Since square OABC has side length 6, then its area is 62 or 36.
(b) By the Pythagorean Theorem in ADB,
AD2 = AB2 − BD2 = 132 − 122 = 169 − 144 = 25 √
Since AD > 0, then AD = 25 = 5. By the Pythagorean Theorem in CDB,
Thursday, April 12, 2018
(outside of North America and South America)
Solutions
©2018 University of Waterloo
2018 Euclid Contest Solutions
Page 3
1. (a) When x = 11,
x + (x + 1) + (x + 2) + (x + 3) = 4x + 6 = 4(11) + 6 = 50
Alternatively,
x + (x + 1) + (x + 2) + (x + 3) = 11 + 12 + 13 + 14 = 50
a6 (b) We multiply the equation + = 1 by 18 to obtain 3a + 6 = 18.
6 18 Solving, we get 3a = 12 and so a = 4.
(c) Solution 1
Since the cost of one chocolate bar is $1.00 more than that of a pack of gum, then if we
replace a pack of gum with a chocolate bar, then the price increases by $1.00.
2. (a) Suppose that the five-digit integer has digits abcde. The digits a, b, c, d, e are 1, 3, 5, 7, 9 in some order. Since abcde is greater than 80 000, then a ≥ 8, which means that a = 9. Since 9bcde is less than 92 000, then b < 2, which means that b = 1. Since 91cde has units (ones) digit 3, then e = 3. So far, the integer is 91cd3, which means that c and d are 5 and 7 in some order. Since the two-digit integer cd is divisible by 5, then it must be 75. This means that the the five-digit integer is 91753.
Starting with one chocolate bar and two packs of gum, we replace the two packs of gum
with two chocolate bars.
This increases the price by $2.00 from $4.15 to $6.15.
In
other
words,
three
chocolate
bars
cost
$6.15,
and
so
one
chocolate
bar
costs
1 3($6.15)or $2.05.Solution 2 Let the cost of one chocolate bar be $x. Let the cost of one pack of gum be $y. Since the cost of one chocolate bar and two packs of gum is $4.15, then x + 2y = 4.15. Since one chocolate bar costs $1.00 more than one pack of gum, then x = y + 1. Since x = y + 1, then y = x − 1.
Since x + 2y = 4.15, then x + 2(x − 1) = 4.15.
Solving, we obtain x + 2x − 2 = 4.15 or 3x = 6.15 and so x = 2.05. In other words, the cost of one chocolate bar is $2.05.
cemc.uwaterloo.ca
2018 Euclid Contest
Wednesday, April 11, 2018
(in North America and South America)
Also, OC = 6.
Since the equation of the line is y = 2x, then its slope is 2.
OC Since the slope of the line is 2, then = 2.
√ CD2 = BC2 − BD2 = (12 2)2 − 122 = 122(2) − 122 = 122
Since CD > 0, then CD = 12. Therefore, AC = AD + DC = 5 + 12 = 17.
2018 Euclid Contest Solutions
Page 4
(c) Solution 1
The area of the shaded region equals the area of square OABC minus the area of
Since square OABC has side length 6, then its area is 62 or 36.
(b) By the Pythagorean Theorem in ADB,
AD2 = AB2 − BD2 = 132 − 122 = 169 − 144 = 25 √
Since AD > 0, then AD = 25 = 5. By the Pythagorean Theorem in CDB,
Thursday, April 12, 2018
(outside of North America and South America)
Solutions
©2018 University of Waterloo
2018 Euclid Contest Solutions
Page 3
1. (a) When x = 11,
x + (x + 1) + (x + 2) + (x + 3) = 4x + 6 = 4(11) + 6 = 50
Alternatively,
x + (x + 1) + (x + 2) + (x + 3) = 11 + 12 + 13 + 14 = 50
a6 (b) We multiply the equation + = 1 by 18 to obtain 3a + 6 = 18.
6 18 Solving, we get 3a = 12 and so a = 4.
(c) Solution 1
Since the cost of one chocolate bar is $1.00 more than that of a pack of gum, then if we
replace a pack of gum with a chocolate bar, then the price increases by $1.00.
2. (a) Suppose that the five-digit integer has digits abcde. The digits a, b, c, d, e are 1, 3, 5, 7, 9 in some order. Since abcde is greater than 80 000, then a ≥ 8, which means that a = 9. Since 9bcde is less than 92 000, then b < 2, which means that b = 1. Since 91cde has units (ones) digit 3, then e = 3. So far, the integer is 91cd3, which means that c and d are 5 and 7 in some order. Since the two-digit integer cd is divisible by 5, then it must be 75. This means that the the five-digit integer is 91753.
Starting with one chocolate bar and two packs of gum, we replace the two packs of gum
with two chocolate bars.
This increases the price by $2.00 from $4.15 to $6.15.
In
other
words,
three
chocolate
bars
cost
$6.15,
and
so
one
chocolate
bar
costs
1 3($6.15)or $2.05.Solution 2 Let the cost of one chocolate bar be $x. Let the cost of one pack of gum be $y. Since the cost of one chocolate bar and two packs of gum is $4.15, then x + 2y = 4.15. Since one chocolate bar costs $1.00 more than one pack of gum, then x = y + 1. Since x = y + 1, then y = x − 1.
Since x + 2y = 4.15, then x + 2(x − 1) = 4.15.
Solving, we obtain x + 2x − 2 = 4.15 or 3x = 6.15 and so x = 2.05. In other words, the cost of one chocolate bar is $2.05.