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Section A.
In this section, there are 12 questions, each correct answer is worth 5 points. Fill in your answer in the space provided at the end of each question.
1. If both a and b are positive integers greater than 1, find the smallest possible sum
of a and b such that a a =b.
Answer：
2. Find the largest positive integer d, in which there exists at least one integer n
such that d divides both n2+1 and (n+1)2 +1.
Answer： 3. A regular hexagon is inscribed in a circle with radius 12 cm. The hexagon is
divided into 6 congruent equilateral triangles and each of them has a small circle inscribed in it. Another small circle is then drawn touching all the inscribed circles. What is the area of the shaded region? Take
π
=3.14.Answer： cm2 4. Find the smallest positive integer n so that 2n is a perfect square and 7n is a
perfect 7th power.
Answer： 5. Find all possible integer values of x, in which
2 21 21 2 2 2 x 42 x x − − − = + . Answer： 6. If the equation
(
a2 +3b2)
x2 −(
4a+6b x)
+ =7 0has a root of 2017, where a andb are real numbers, find the sum of a and b.
Answer： 7. If x>1, find all possible values of x that satisfies the following equation:
2017 2018 2017 2018 2018 2017 2018 2017 x x x x − − − = − − − . Answer：
8. In the figure below, ABC is an isosceles triangle with base AB. Its orthocentre H
divides its altitude CD into two segments CH and HD, where CH =7cm and
9
HD= cm. Find the perimeter of triangle ABC, in cm.
Answer： cm 9. In quadrilateral ABCD, ∠ABD= ∠ACD= °90 and a point P is on AD so that
APB CPD
∠ = ∠ , as shown in the figure below. If AP=24cm and DP=19cm, find the value of PB PC× .
Answer： 10. A girl tosses a fair coin 100 times and a boy tosses a fair coin 101 times. The
boy wins if he has more heads than the girl has, otherwise, he loses. Find the probability that the boy win this game.
Answer： % 11. A 6 6× chessboard is formed by 36 unit squares. How many different
combinations of 4 unit squares can be selected from the chessboard so that no two unit squares are in the same row or column?
Answer： 12. If abcd is a 4-digit number, where each different letter represents a different
digit such that a<b, c <b and c<d . How many such 4-digit numbers are there? A B C H D A B C P D
Section B.
Answer the following 3 questions, each question is worth 20 points. Partial credits may be awarded. Show your detailed solution in the space provided.
1. Let
α
,β
andγ
be the three roots of the polynomial x3 − +5x 1. If p and q arerelatively prime positive integers such that
(
)(
) (
)(
) (
)(
)
3 3 3 6 1 6 1 6 1 6 1 6 1 6 1 p qα
β
γ
β
γ
α
γ
β
α
− = + + + + + + + + .Find the sum of p and q.
Answer：
2. A 21 21× table contains 21 copies of each of the numbers 1, 2, 3, . . . , 20 and 21. The sum of all the numbers above the main diagonal (diagonal from the top-left cell to the bottom-right cell) is equal to three times the sum of all the numbers below the main diagonal. Find the sum of all the numbers on the main diagonal of the table.
3. In the figure below, ABCD is a square. Points E, Q and P are on sides AB, BC and
CD, respectively, such that PE⊥ AQ and △AQP is an equilateral triangle. Point
F is inside △PQC such that △PFQ and △AEQ are congruent. If EF = 2cm, find
the length of FC, in cm. A B C P Q F E D
