2001 IWYMIC Individual

Third World Youth Mathematics Inter-City Competition

Individual Contest
Time allowed : 2 hours
Name ____________________________________________________________________
Team Name _______________________________________________________________

Part I
Write your answer to each of the ten questions on the space provided. Solutions need not be shown. Each of
the questions carries 6 marks.
Question 1
Find all integers n such that 1 + 2 + .. + n is equal to a 3-digit number with identical digits.
Answer:
________________________
D

Question 2

In a convex pentagon ABCDE, ∠A = ∠B = 120°, EA
= AB = BC = 2, CD = DE = 4, Find the area of the
pentagon ABCDE.

E

C

A

B

Answer:
________________________

Question 3
If I place a 6 cm × 6 cm square on a triangle, I can cover up to 60% of the triangle. If I place the triangle on the square,
2
I can cover up to of the square. What is the area of the triangle?
3

Answer:
________________________

Question 4
Find a set of four consecutive positive integers such that the smallest is a multiple of 5, the second is a multiple of 7,
the third is a multiple of 9, and the largest is a multiple of 11.

Answer:
________________________
Question 5
Between 5 and 6 o’clock a lady looked at her watch and mistook the hour hand for the minute hand. She thought the
time was 57 minutes earlier than the correct time. What was the correct time?

Answer:
________________________

Question 6
In a ∆ABC, the incircle touches the sides BC, CA and AB at D, E and F respectively. If the radius of the incircle is 4
units and if BD, CE and AF are consecutive integers, find the length of the three sides of ∆ABC.

Answer:
________________________

Question 7
2
 p + 1 = 2 x
Determine all primes p for which the system  2
has integral solution(s).
2
 p + 1 = 2 y

Answer:
________________________

Question 8
Find all real solutions of the equation:

3x 2 − 18 x + 52 + 2 x 2 − 12 x + 162 = − x 2 + 6 x + 280

Answer:
________________________

Question 9
Simplify the expression into a single numerical value: 12 − 24 + 39 − 104 − 12 + 24 + 39 + 104

Answer:
________________________

Question 10
Let M = 1010101…01 where the digit 1 appears k times. If N = 1001001001001, find the least value of k so that N
divides M?

Answer:
________________________

Part II
Show your organized solution for each of the three questions in the space provided below. Each question carries 20
marks.
Question 1
Given that a and b are unequal positive real numbers, let A =

a+b

and B = ab .
2

(a − b)
B<
< A.
8( A − B)
2

Prove that the following inequality holds:

Question 2
Find the range of p such that the equation 32x – 3x + 1 = p has two different real positive roots.

Question 3
In a scalene ∆ABC with three sides a, b, c where a > b < c, the four vertices of a square lie on the three sides of ∆ABC.
There are three possible ways to come up with the diagram. e.g. one vertex on side a, one vertex on side b and two
vertices on side c. Which of the three possible ways will give you the maximum area? Justify your answer.

Third World Youth Mathematics Inter-City Competition

Team Contest
Team Name _______________________________________________________________
Question 1
Fill in the numbers 1 to 16 in the vertices of the 2 cubes shown below. One number in each vertex and cannot be
repeated, such that the sum of 4 numbers in the 4 vertices of each face is the same.

Third World Youth Mathematics Inter-City Competition

Team Contest
Team Name _______________________________________________________________

Question 2
Arrange the numbers 1-20 in a circular manner such that the sum of 2 adjacent numbers is prime.

Third World Youth Mathematics Inter-City Competition

Team Contest
Team Name _______________________________________________________________

Question 3

In the figure below AB = DE = EF = HA, BC = CD = FG = HG, ∠BCD = ∠FGH = 90°. Divide the given figure into
2 identical regions.
A

B

C

D

H
G

F

E

Third World Youth Mathematics Inter-City Competition

Team Contest
Team Name _______________________________________________________________
Question 4
If a 1 × 1 square is removed from a 8 × 8 square such that the remaining figure can be cut into 21 figures of
and can also be cut into 21 figures of
are removable in the 8 × 8 square?

. This 1 × 1 square is called a removable square. How many squares