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Team: Score:
For Juries Use Only
No. 1 2 3 4 5 6 Total
Score Score
You are allowed 60 minutes for this paper, consisting of 6 questions printed on separate sheets. For questions 1, 3 and 5, only numerical answers are required. For questions 2, 4 and 6, full solutions are required.
Each question is worth 40 points. For odd-numbered questions, no partial credits are given. There are no penalties for incorrect answers, but you must not give more than the number of answers asked for. For questions asking for several answers, full credit will only be given if all correct answers are found. For even-numbered questions, partial credits may be awarded.
Diagrams shown may not be drawn to scale.
Instructions:
Write down your team’s name on the spaces provided on every question sheet. Enter your answers in the spaces provided after individual questions on the question paper.
During the first 10 minutes, the three team members examine the questions together, with discussion. Then they distribute the questions among themselves, with each team member allotted at least 1 question.
During the next 50 minutes, the three team members write down the solutions to their allotted problems on the respective question sheets, with no further
communication among themselves.
You may not use instruments such as protractors, calculators and electronic devices. At the end of the contest, you must hand in the envelope containing all question sheets and all scrap papers.
International Young Mathematicians’ Convention
Senior Level
TEAM CONTEST
Team
：
Score
：
1. If x and y are positive real numbers, find the smallest value of
2 2 2 2
225 15 2− x+x + 200−20y+ y + x − 2xy+ y .
2. The sum of 2016 real numbers is 2017 and each of them is less than 2017 2015.
Prove that the sum of any two of the numbers is greater than or equal to 2017
International Young Mathematicians’ Convention
Senior Level
TEAM CONTEST
Team
：
Score
：
3. There are 2016 unit cubes, each of which can be painted black or white. How
many values of n is it possible to construct an n n n× × cube with n3 unit cubes such that each cube shares a common face with exactly three cubes of the opposite colour?
4. P is the midpoint of the arc AC of the circumcircle of an equilateral triangle ABC. M is another point on this arc and N is the midpoint of BM. K is the projection of P on the line MC, as shown in the diagram below. If the length of NA is 19 cm,
find the length of NK in cm.
Answer:
cm
B A P C N M KInternational Young Mathematicians’ Convention
Senior Level
TEAM CONTEST
Team
：
Score
：
5. Let Sn denote the n-th sequence so that every word in a sequence consists only of
the letters A and B. The first word has only one letter A. For k ≥2, the k-th word
is obtained from the (k −1)-th by simultaneously replacing every A by AAB and every B by A. Then every word is an initial part of the next word. For example,
1
S = A, S2 = AAB, S3 = AABAABA and S4 = AABAABAAABAABAAAB.
Find the number of As in S . 10
6. D, E and F are points on the sides BC, CA and AB, respectively, of triangle ABC
such that AD, BE and CF are concurrent. The area of triangle ABC is 2016 cm2. If
there exists a point P such that both BDPE and AFCP are parallelograms, as shown in the diagram below. Find the area of triangle of DEF, in cm2.
