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Invitational World Youth Mathematics Intercity Competition

Individual Contest

Time limit: 120 minutes

Information:

         You are allowed 120 minutes for this paper, consisting of 12 questions in Section A to which only numerical answers are required, and 3 questions in Section B to which full solutions are required.

         Each question in Section A is worth 5 points. No partial credits are given. There are no penalties for incorrect answers, but you must not give more than the number of answers being asked for. For questions asking for several answers, full credit will only be given if all correct answers are found. Each question in Section B is worth 20 points. Partial credits may be awarded.

         Diagrams shown may not be drawn to scale.

Instructions:

         Write down your name, your contestant number and your team’s name in the space provided on the first page of the question paper.

         For Section A, enter your answers in the space provided after the individual questions on the question paper. For Section B, write down your solutions on spaces provided after individual questions.

         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 and all scratch papers.

English Version

Team:           Name:                No.:           Score:        

For Juries Use Only

Section A.

In this section, there are 12 questions. Fill in the correct answer in the space provided at the end of each question. Each correct answer is worth 5 points.

1.         The diagram shows five collinear towns connected by semicircular roads. A journey is defined as travelling between two towns along a semicircle. In how many possible ways can we start and end at Town 5 after four journeys if the journeys may be repeated?

1 5

1.         Let m and n be positive integers such that all possible values of m  n .

Answer：       ways

m(n  m)  11n  8 . Find the sum of

Answer：           

1.         Ann tosses a fair coin twice while Bob tosses the same coin three times. The probability that they obtain the same number of heads at the end of the game is expressed as an irreducible fraction. What is the sum of its numerator and its denominator?

1.         Let p and q be prime numbers such that Find the largest possible value of p  q .

p2  3 pq  q2

Answer：            

is the square of an integer.

Answer：            

1.         Two circles

k1 and

k2 with the same radius intersect at points B and C. The

center O1 of

k1 lies on

k2 and the center O2

of k2 lies on

k1 . AB is a diameter

of k1 , and

AO2

intersects

k2 at points M and N, with M between A and

O2 .

The extensions of CM and NB intersect at point P. Find CP：CN. P

B

N

O2

1

M Answer：     ：     

A C

1.         Given is the product 1!2!3!..... 99!100! How many consecutive 0s are there at the end of this product?

1.         Let

Answer：            

P(x)  x4  ax3  bx2  cx  d , where a, b, c and d are real constants.

Suppose

P(1)  7 ,

P(2)  52

and

P(3)  97 . Find the value of

P(9)  P(5) .

4

Answer：            

1.         In quadrilateral ABCD, AD is parallel to BC and AB  AC . F is a point on BC

such that DF is perpendicular to BC. AC intersects DF at E. If

BE  2DF

and

BE bisects

ABC , find the measure, in degrees, of

BAD .

A D

B F C



Answer：            

1.         Arrange the numbers 1, 2, 3, 4, 5, 6 and 7 in a row such that none of the first number, the sum of the first two numbers, the sum of the first three numbers, and so on, up to the sum of all seven numbers, is divisible by 3. In how many ways can this be done?

Answer：       ways

2.         An equilateral triangle and a regular 7-sided polygon are inscribed in the same circle of circumference 84 cm, divided by the vertices into ten arcs. What is the maximum possible length, in cm, of the shortest arc?

Answer：         cm

 a  b  8 3