Individual Contest
Part I
Write your answer to each of the twelve questions on the answer sheets. Working need not be shown. Each of the questions carries 5 marks.
Question 1
Find the unit digit of 172000.
Question 2
Two fractions are removed from the six fractions 1 3,
1 6,
1 9,
1 12,
1 15,
1
18 so that the sum of the
remaining four fractions is 2
3. Find the product of the two ramoved fractions.
Question 3
Let A be a multiple of 11 and which lies between 100 and 1000. The hundreds digit of A is greater
than its units digit. Find the smallest A satisfying all the above conditions.
Question 4
Find the sum of all integers, between 150 and 650, such that when each is divided by 10, the remainder is 4.
Question 5
Let N=111…1222…2, in which there are two thousand 1’s and two thousand 2’s. If N is divided by
the 2000-digit number 666…6, find the quotient.
Question 6
The famous Goldbach conjecture says: any even number greater than 7 can be expressed as the sum
of two unequal prime numbers; for example, 10=3+7. Find two unequal prime numbers p and q so
that p+q=192 and 2p-q is as large as possible.
Question 7
D is a point on the side BC of a triangle ABC such that
AC=CD and angle CAB=angle ABC + 45°
Find angle BAD.
Question 8
If 100000035811ab12=1000000cde2247482444265735361, where a, b, c, d, e are non-negative
integers less than 10, find the value of a+b+c-d-e.
Question 9
P is a point inside a rectangle ABCD. If PA=4, PB=6 and PD=9, find the length of PC.
Question 10
and 160 degrees at the boiling point of water. Find the temperature in this new scale when it is 215 degrees Celsius. For your reference, water freezes at 0 degree Celsius and boils at 100 degrees Celsius.
Question 11
A square ABCD is inscribed in a circle. The circle is folded to form a semi-sircle, and another square EFGH is inscribed in this semi-cirlcle. Find the ratio of the area of the square EFGH to the area of the square ABCD.
Question 12
A positive integer is written on each of a pile of ten cards. It is found that the numbers on any three consective cards in the pile add up to 20. if the number on the first card in the pile is 2 and that on the ninth card is 8, find the number on the fifth card in the pile.
Part II
Show your steps for each of the three questions in the spaces provided on the answer sheets. Each question carries 20 marks.
Question 1
The figure shows a square ABCD which is folded along EF so that the vertex A coincides with a point A’ on the side BC and the vertex D is mapped onto a point D’. Let G be the intersection of A’D’ and FC. Prove that A’E+FG=A’G.
Question 2
Twenty distinct positive integers are written on the front and back of ten cards, with one integer on each face of every card. The sum of the two integers on each card is the same for all the ten cards. Also, the sum of the ten integers on the front of the cards is equal to the sum of the the ten integers at the back of these cards. If nine of the ten integers on the front of the cards are
2, 5, 17, 21, 24, 31, 35, 36, 42, find the integer on the front of the tenth card.
Question 3
Let a, b, c be positive integers such that a and b are 3-digit numbers, and c is a 4-digit number. If the sum of the digits of the numbers a+b, b+c and c+a are all equal to 2, find the largest possible sum of the digits of the number a+b+c.
F
G E
D
C B
A
D’
