Third Invitational World Youth Mathematics Inter-City Competition
Tagaytay City, Philippines
Team Contest
Instructions :
• The team contest consists of 8 problems to be solved in 150 minutes
• Answers and all solutions must be written on the spaces provided below each question. The use of calculator is not allowed
• Five minutes will be allotted for the four contestants to distribute the first six problems among themselves.
• Each contestant should solve the problems individually and submit their answers at the end of 60 minutes.
• The last two problems are to be solved by the team members together for 90 minutes.
Question 1
Fill in the numbers 1 to 16 in the vertices of the 2 cubes shown below. One number in each vetex and can not be repeated, such that the sum of 4 numbers in the 4 vertices of each face is the same.
Question 2
Arrange the numbers 1 – 20 in a circular manner such that the sum of 2 adjacent numbers is prime.
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.
Question 4
If a 1 × 1 square is removed from an 8 × 8 square such that the remaining figure can be cut into
21 figures of and can also be cut into 21 figures of . this 1 × 1 square is
called a removable square. How many squares are removable in the 8 × 8 square?
Question 5
A 2n-digits natural number K is called a Kabulek number if it satisfies the following conditions: 1. It can be divided into 2 parts. The first part consists of n digits and the second part
consists of another n digits.
2. The square of the sum of the first part and the second part equals to that 2n-digits numbers.
For example, 3025 is a Kabulek number since (30 + 25)2 = 3025. Find all 4-digit Kabulek
numbers other than 3025.
Question 6
In the equilateral ∆ABC, point P is an interior point such that PA=4, PB=4 3 and PC = 8. Find the area of ∆ABC.
P A
B C
8 4
Question 7
The fraction
4 1 66664 16666
= has an interesting characteristic such that if we add the digit 6 after the
first digit 1 of the numerator n times and add the digit 6 before the unit’s digit 4 of the denominator n times also, the fraction has the same value. List all fractions having the same characteristics and present the solution.
Note: All the other fractions with the same characteristics as above do not necessarily add the digit 6. It can also be other digits as long as the number times it is added in the numerator is the same as that of the denominator.
Question 8
There are seven shapes formed of three a four equilateral triangles connected edge-to-edge, as shown in the 2×5 chart below. For each of the numbered spaces in the chart, find a figure which can be formed from copies of the shape at the head of the row, and from copies of the shape at the head of the column. The problem in the firs space has been solved as an illustration (Rotations and reflections are allowed.)
