Mathematical Society of the Philippines- NCR Chapter

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MATHEMATICS WIZARD 2009

February 28, 2009Bulwagang Manileño, Pamantasan ng Lungsod ng Maynila

Final Round

LEVEL I

Question No. 1

The 5-digit number 5A55B is divisible by 72. What does A represent?

Question No. 2

How many pairs of integers x,y are there which satisfy the equation

?2

111

yx

Question No. 3

Find the sum of all prime numbers p such that p 1(mod3) and p < 40.

Question No. 4

Donald Duck can eat 2 pizzas in 3 minutes, while Goofy can eat 3 pizzas in 2 minutes. At these rates, how many pizzas can they eat together in an hour?

Question No. 5

A Dog and Cat club found it could achieve a membership ratio of 2 dogs to one cat either by inducting 24 dogs or by expelling x cats. What is x?

Question No. 6

The nth triangular number is defined as the sum of the first n positive integers. Find the 35th triangular number.

Question No. 7

How many triples of real numbers (x,y,z) are there such that xy = z, xz = y and yz = x?

Question No. 8

A right isosceles triangle is drawn with legs of 12 cm. A triangle is then inscribed by connecting the midpoints of all the sides of the first triangle. If triangles are continuously inscribed in the same fashion, what is the sum of the areas of every triangle drawn?

Question No. 9

Let r, s, t be the roots of x3 – 7x2 + 8x + 2 = 0. What is the value of

?111

tsr

Question No. 10

During a recent period of time, eleven days had some rain was always followed by a clear afternoon. An afternoon rain was always preceded by a clear morning. A total of nine mornings and twelve afternoons were clear. How many days had no rain at all?

LEVEL II

Question No. 1

What is the ratio of the area of a square inscribed in a semi-circle to the area of a square inscribed in the entire circle?

Question No. 2

What is the smallest positive integer a for which there is an integer c and a right triangle with side lengths a and 17 and a hypotenuse of length c?

Question No. 3

Given the relation

with a1= 1, evaluate

81 nn aa

nn

a

lim

Question No. 4

How many positive integers less than 10,000 are of the form x8 + y8 for some integers x > 0 and y > 0?

Question No. 5

Ajay, Bjay, Cjay, Djay and Ejay put their hats in a pile. When they pick up their hats later, each one gets someone else’s hat. How many ways can this be done?

Question No. 6

Twelve people are equally spaced around a large circle. What is the largest number of wires that can be stretched between pairs of people so that no two wires intersect at any point inside the circle?

Question No. 7

The lower two vertices of a square lie on the y-axis and the upper two vertices of the square lie on the parabola y = 15 – x2. What is the area of the square?

Question No. 8

Three problems were given to participants of a math contest. Each participant got 0, 1, 2, or 3 points for each problem. After the papers graded it turned out that no pair of participants received matching scores for more than one problem. What is the largest possible number of participants?

Question No. 9

On the television station MYX, A% of the airtime is devoted to the airing of quality music, B% of the airing of other music, C% of the airing of advertisement, and D% of the airing of other programming, A, B, C, and D are positive integers and add up to 100. A<B, C is twice B, and D is four times C. If D is a multiple of 9, find A.

Question No. 10

For how many values of b, 0 < b < 24, could p2 b (mod 24) for some prime number p?

LEVEL III

Question No. 1

The following inequalities hold for all positive integers n:

What is the greatest integer which is less than

114

11

nn

nnn

24

1 14

1

n n

Question No. 2

Someone rolls three dice where you cannot see them, and states (truthfully) that there is at least one four. What is the probability the three dice sum to 12?

Question No. 3

Evaluate

)5(

132

k

k

Question No. 4

Let an and bn be two arithmetic progressions with n > 0, the sum of the first n terms of which are Sa(n) and Sb(n), respectively. Given that

and a2 = 4, determine b4.

82

95

)(

)(

n

n

nS

nS

b

a

Question No. 5

Evaluate

324 45

1

n nn

Question No. 6

Four distinct numbers are chosen from the first nine natural numbers. What is the probability that 5 is the second largest of those chosen?

Question No. 7

Consider the relation an=3an-1+(1/2)an

for a1 = 1 and a2 = 2.

Find the 2009th term.

Question No. 8

Let f be a function defined for the positive integers such that for every positive integer n, (i) f(n) is a positive integer;(ii) f(n+1)>f(n); and(iii) f(f(n)) = f(n)How many such functions are there?

Question No. 9

A train that is one mile long is moving at a constant speed. A rabbit, who runs faster than the train, starts at the back of the train and runs alongside until it reaches the front of the train. At that instant, it immediately turns around and runs back (at the same rate) until it again reaches the back of the train. At that instant, the back of the train is now precisely where the front of the train was when the rabbit started running. In total, how far did the rabbit run?

Question No. 10

Find the last digit of777777

Congratulations to the MSP-NCR

Math Wizard 2009winners!