2014 Euclid Mathematics Contest Solutions and Problems

2014 Euclid Mathematics Contest Solutions and Problems

The 2014 Euclid Mathematics Contest presents a series of challenging mathematical problems designed for high school students. It includes ten questions covering various topics such as geometry, algebra, and number theory. Each question is structured to test problem-solving skills and mathematical reasoning. This contest is ideal for students preparing for advanced mathematics competitions or seeking to enhance their mathematical abilities. Solutions are provided to help students understand the problem-solving process and improve their skills.

Key Points

  • Includes ten challenging math problems from the 2014 Euclid Contest.
  • Covers topics such as geometry, algebra, and number theory.
  • Designed for high school students preparing for math competitions.
  • Provides detailed solutions to enhance understanding and problem-solving skills.
  • Tyler Xiang
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The CENTRE for EDUCATION
in MATHEMATICS and COMPUTING
cemc.uwaterloo.ca
Euclid Contest
Tuesday, April 15, 2014
(in North America and South America)
Wednesday, April 16, 2014
(outside of North America and South America)
c
2014 University of Waterloo
Do not open this booklet until instructed to do so.
Time: 2
1
2
hours Number of questions: 10
Calculators are permitted, provided Each question is worth 10 marks
they are non-programmable and
without graphic displays.
Parts of each question can be of two types:
1. SHORT ANSWER parts indicated by
• worth 3 marks each
• full marks given for a correct answer which is placed in the box
• part marks awarded only if relevant work is shown in the space provided
2. FULL SOLUTION parts indicated by
• worth the remainder of the 10 marks for the question
• must be written in the appropriate location in the answer booklet
• marks awarded for completeness, clarity, and style of presentation
• a correct solution poorly presented will not earn full marks
WRITE ALL ANSWERS IN THE ANSWER BOOKLET PROVIDED.
• Extra paper for your finished solutions supplied by your supervising teacher must be
inserted into your answer booklet. Write your name, school name, and question number
on any inserted pages.
• Express calculations and answers as exact numbers such as π + 1 and
√
2, etc., rather
than as 4.14 . . . or 1.41 . . ., except where otherwise indicated.
Do not discuss the problems or solutions from this contest online for the next 48 hours.
The name, grade, school and location, and score range of some top-scoring students will be
published on our website, http://www.cemc.uwaterloo.ca. In addition, the name, grade, school
and location, and score of some top-scoring students may be shared with other mathematical
organizations for other recognition opportunities.
TIPS: 1. Please read the instructions on the front cover of this booklet.
2. Write all answers in the answer booklet provided.
3. For questions marked , place your answer in the appropriate box in the
answer booklet and show your work.
4. For questions marked , provide a well-organized solution in the answer
booklet. Use mathematical statements and words to explain all of the steps
of your solution. Work out some details in rough on a separate piece of paper
before writing your finished solution.
5. Diagrams are not drawn to scale. They are intended as aids only.
A Note about Bubbling
Please make sure that you have correctly coded your name, date of birth, grade, and
sex, on the Student Information Form, and that you have answered the question about
eligibility.
1. (a) What is the value of
√
16 +
√
9
√
16 + 9
?
(b) In the diagram, the angles of 4ABC are
shown in terms of x. What is the value of x?
x˚
(x + 10)˚
(x – 10)˚
A
B
C
(c) Lisa earns two times as much per hour as Bart. Lisa works 6 hours and Bart
works 4 hours. They earn $200 in total. How much does Lisa earn per hour?
2. (a) The semi-circular region shown has radius 10.
What is the perimeter of the region?
(b) The parabola with equation y = 10(x + 2)(x − 5) intersects the x-axis at points
P and Q. What is the length of line segment P Q?
(c) The line with equation y = 2x intersects the line segment joining C(0, 60) and
D(30, 0) at the point E. Determine the coordinates of E.
3. (a) Jimmy is baking two large identical triangular
cookies, 4ABC and 4DEF . Each cookie is in
the shape of an isosceles right-angled triangle. The
length of the shorter sides of each of these triangles
is 20 cm. He puts the cookies on a rectangular
baking tray so that A, B, D, and E are at the
vertices of the rectangle, as shown. If the distance
between parallel sides AC and DF is 4 cm, what is
the width BD of the tray?
A
D
B
C
E
F
(b) Determine all values of x for which
x
2
+ x + 4
2x + 1
=
4
x
.
4. (a) Determine the number of positive divisors of 900, including 1 and 900, that are
perfect squares. (A positive divisor of 900 is a positive integer that divides exactly
into 900.)
(b) Points A(k, 3), B(3, 1) and C(6, k) form an isosceles triangle. If ∠ABC = ∠ACB,
determine all possible values of k.
5. (a) A chemist has three bottles, each containing a mixture of acid and water:
• bottle A contains 40 g of which 10% is acid,
• bottle B contains 50 g of which 20% is acid, and
• bottle C contains 50 g of which 30% is acid.
She uses some of the mixture from each of the bottles to create a mixture with
mass 60 g of which 25% is acid. Then she mixes the remaining contents of the
bottles to create a new mixture. What percentage of the new mixture is acid?
(b) Suppose that x and y are real numbers with 3x + 4y = 10. Determine the
minimum possible value of x
2
+ 16y
2
.
6. (a) A bag contains 40 balls, each of which is black or gold. Feridun reaches into the
bag and randomly removes two balls. Each ball in the bag is equally likely to be
removed. If the probability that two gold balls are removed is
5
12
, how many of
the 40 balls are gold?
(b) The geometric sequence with n terms t
1
, t
2
, . . . , t
n−1
, t
n
has t
1
t
n
= 3. Also, the
product of all n terms equals 59 049 (that is, t
1
t
2
···t
n−1
t
n
= 59 049). Determine
the value of n.
(A geometric sequence is a sequence in which each term after the first is obtained
from the previous term by multiplying it by a constant. For example, 3, 6, 12 is
a geometric sequence with three terms.)
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End of Document
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FAQs of 2014 Euclid Mathematics Contest Solutions and Problems

What types of problems are included in the 2014 Euclid Contest?
The 2014 Euclid Contest features a variety of mathematical problems that challenge students in areas such as geometry, algebra, and number theory. Each problem is designed to test not only knowledge but also the ability to apply mathematical concepts in creative ways. The contest encourages critical thinking and problem-solving skills, making it an excellent resource for students preparing for higher-level mathematics.
How can students benefit from the solutions provided in the contest?
The solutions to the 2014 Euclid Contest problems serve as a valuable learning tool for students. By reviewing the solutions, students can gain insights into different problem-solving strategies and methodologies. This helps them understand the reasoning behind each answer and improves their ability to tackle similar problems in the future. Additionally, the solutions can highlight common pitfalls and misconceptions, further enhancing students' mathematical skills.
Who is the target audience for the 2014 Euclid Mathematics Contest?
The 2014 Euclid Mathematics Contest is primarily aimed at high school students, particularly those with an interest in mathematics and problem-solving. It is suitable for students preparing for mathematics competitions or those looking to strengthen their mathematical foundations. Teachers and educators can also use the contest as a resource for enriching their curriculum and providing students with challenging material.
What is the significance of the Euclid Contest in mathematics education?
The Euclid Contest is a prestigious mathematics competition that encourages students to engage deeply with mathematical concepts. It fosters a love for mathematics and promotes critical thinking skills essential for academic and professional success. Participation in such contests can enhance a student's college application and provide recognition for their mathematical abilities. The contest also serves as a platform for students to connect with peers who share similar interests in mathematics.

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