2016 IYMC 高中組個人賽試題

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International Young Mathematicians’ Convention

Senior Level

Individual Contest

Time limit: 90 minutes

Team: Name: No.: Score:

Information:

You are allowed 90 minutes for this paper, consisting of 8 questions to

which only numerical answers are required.

Each question is worth 10 points. 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.

Diagrams shown may not be drawn to scale.

Instructions:

Write down your name, your contestant number and your team’s name

on the answer sheet.

Enter your answers in the spaces provided on the answer sheet.

You must use either a pencil or a ball-point pen which is either black or

blue.

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 the

question paper, your answer sheet and all scratch papers.

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1. Find the number of all real solutions of the system of equations 5 3 4 5 1 5 4 5 1 2 5 5 1 2 3 5 1 2 3 4 5 2 3 4 5 ( ) 3888 ( ) 3888 ( ) 3888 ( ) 3888 ( ) 3888 x x x x x x x x x x x x x x x x x x x x + + = + + = + + = + + = + + =

2. What is the simplified value of 5+ 52+ 54+ 58 +_{⋯ ?}

3. If a is a positive integer so that a2 +20162 is divisible by 2016a, find the number of the possible values of a.

4. Let f x( ) x 20 x +

= and f x_n( )= f f( (_⋯( ( ))f x _⋯)) be the n-fold composite off. For example, ₂ 20 20 21 20 ( ) 20 ₂₀ x x x f x x _x x + ₊ + = ₊ = + and 3 21 20 20 41 420 20 ( ) 21 20 ₂₁ ₂₀ 20 x x x f x x _x x + ₊ + + = ₊ = + + .

Let S be the complete set of real solutions of f x_n( )= x. What is the maximal number of the elements in S ?

5. Given D and E are points on the sides BC and CA, respectively, of triangle ABC. If ∠ADC =130°, ∠BEA= °25 and BE bisects ∠ABC, as shown in the

diagram below. Find the measure of ∠EDC, in degrees.

6. The sum of ten numbers on a circle is 2016. The sum of any three numbers in a row is at least 585. Determine the minimal number n such that for any such set of ten, none of them is greater than n.

7. Anna tosses 2016 coins and Boris tosses 2017 coins. Whoever has more heads wins. If they have the same number of heads, then Anna wins. What is the probability of Anna winning?

8. In triangle ABC, AC = BC. D is a point on AB such that the inradius of triangle CAD is equal to the exradius of triangle BCD opposite

C, as shown in the right diagram.

If the length of the altitude AH is 36 cm, find the length of this common radius. A E C B D B A C D H