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1. Let ( ) 9 9 3 x x f x = + . Calculate f
( ) ( )
20171 + f 20172 + +... f( ) ( )
20172015 + f 20162017 .Answer:
2. In △ABC, point M is between A and B such that AM : MB = 1 : 2. Points N and P are between C and M such that CN : NM = 3 : 2 , CP : PM = 1 : 5. Segments AN and BC intersect at point Q. Segments PQ and AC intersect at point L. Find the ratio CL : LA.
Answer: CL : LA ＝ ：
A B C N L M P Q3. Find the largest integer p such that 142017 +22017 is divisible by 2p.
4. In pentagon ABCDE, points M, P, N and Q are midpoints of AB, BC, CD and DE respectively. While points K and L are midpoints of QP and MN, respectively, as shown in the figure below. If KL=25cm, find the length of EA, in cm.
A B C N L D K E M P Q
Answer:
cm
5. Let x and y be positive integers, where 0< < <x y 2018. How many ordered pairs (x, y) are there such that x2 +20182 = y2 +2017 ?2
6. Points A, B, C, D and E are on the circumference. Chord AC is a diameter of the circle, as shown in the figure below. If ∠ABE = ∠EBD= ∠DBC, BE =16 cm and BD=12 3 cm, find the area of pentagon ABCDE.
A E D C B
Answer:
cm
7. A 10 10× chessboard is dissected into thirty-three 1 3× or 3 1× rectangles and one unit square. In how many different positions can this unit square be, if the chessboard may not be reflected or rotated?
8. Prove the inequality: 2 100 99 101 98 102 ... 1 199 4 π × + × + + × < .
9. A computer randomly chooses three different points on the given grid below (all points have the equal chance of being chosen).
Let p
q be the probability to form a triangle with these points (this fraction is
written in its irreducible form). Find the sum of p and q.
10. Jane has 12 marbles, where in one is fake. She are not certain if the fake marble is heavier or lighter than the real marble. What is the minimum number of weightings needed to find the fake marble and determine whether the fake marble is heavier or lighter than the real marble? Explain your answer.
