December 26-27, 2018 You are given two (2) days to answer these items. Submit all your answers at once via private message (PM). You need to check your answers carefully before you submit them. Good luck!
(2 points for every correct answer) 1. How many factors does 320 + 318 have?
π 20=9 ΔABC is similar to ΔXYZ. If AB = 20 and XY =18, find thethe ratio of their
2. The circumference of a circle is 200 cm. Find the area of a square inscribed in it. 3. Find the solution set of 4.
.
areas.
5. A pair of fair dice is rolled. Find the probability of rolling a sum which is a prime number. 6. A nerdy guy has 2,001 books, but he randomly chooses 1,999 of them for his research. How many ways that can be done? 7. What is the harmonic mean of 10, 100 and 1,000? 8. If today is December, what month is 172,018 months from now? 9. What is the smallest positive co-terminal angle of -2,018°? 10. Evaluate: sin 75° + cos 75°.
(3 points for every correct answer) 11. If 24 jibs is to 36 jabs, 78 jobs is to 65 jubs, and 44 jabs is to 77 jobs, what is the ratio of jibs to jubs? 12. A regular polygon is rotated 11.7° and it is coincident to the original position. Find the minimum number of sides of this polygon. 13. Find the number of ways to arrange the letters of JUSTLOVE such that the vowels are together. 14. Simplify:
+)( +) ( )( ) + + − − − −
∠ =
= = = = 6√ 2 = 16 ∠ = ,, = 20 ,, = 18
15. ABCD is a quadrilateral such that 135°. Find the area of a quadrilateral.
16. Find the smallest integer such that
units,
units,
and
90° and
.
17. A pair of dice is rolled simultaneously. Find the probability that they come out two numbers that differ by at least 2. 18.
In Allan’s point of v
30√ 3
iew, the angle of elevation to the top of a -meter high post is 30°. On the other hand, his friend Ben stands at the other side of the same post and his angle of elevation to
the top of a post is 60°. If Allan, Ben and a post are on a straight line, find the distance between Allan and Ben. 19. Sally has a collection of stamps. If she manages to place 7 stamps per page in her scrapbook, she has 5 stamps left. If she place 10 stamps per page, she has 3 stamps left. What is the least number of stamps does Sally have? 20. If
sincsc=
, what is the value of
sin csc
?
(4 points for every correct answer) 21. How many trailing zeros does 8,102P2,018 have? 22.
The repeating decimal 0.2018888… can be expressed as a terminating numbers are chosen from the set {1, 2, 3, …, 20}. Find the probability that no two numbers 18 202,018 212=0 =0
decimal with minimal base
n. Find n.
23. Five have a difference which is divisible by 5. 24. Find the value of
given that
.
25. A region is bounded by x-axis, y-axis, x = 20 and y = 18. Find the equation of a line (in the form ) passing through (-18, -20) that divides the region into two equal parts. 26. Three congruent circles are tangent to each other. An equilateral triangle is drawn such that each side is tangent to one of the circlesat its midpoint. If the side of an equilateral triangle is 6 cm, what is the radius of one of the circles?
27. Find the largest prime factor of 194+ 64.
28. As the value of increases, the value of what number?
29. Find the least integer N such such that
approaches 1 and the value of
is a perfect cube and
30. A, B and and C are are the angles of a triangle. If
(5 points for every correct answer) 31. Find the value of x in
is a perfect fifth power.
sin= sin= and
, what is
sin2
?
approaches to
14 1 122 1 363 1 ⋯ = 35 1 205 1 807 1 ⋯ tan m∠
32. Two regular hexagons CHRIST and DABEST share a side ST. Find C
H
D
T
R
DRA.
A
S
I
B
E
33. PLAY is a trapezoid such that PY is perpendicular to the bases PL and YA. Point M lies on PY such that PM:MY = 2:1 and point N is a midpoint of LA. If PL = 18 units, YA = 30 units and MN = 25 units, what is the area of a trapezoid? 34. Thirty slips of paper are labeled 2, 4, 8, 16 until 2 30. Three slips are drawn at random. Given that their product is a perfect fourth power, find the probability that their product is not a perfect cube. 35. Find the remainder if 202,018 + 182,018is divided by 49.
= 100 1 00 104 1 04 108⋯ 1 08⋯ 1, 1 , 5 92 92 1, 1 , 5 96 650 6 50 √ √ √ √ 675 √ 700⋯ 7 00⋯ 9,9,975 √ 10,10,000 √ 675 The letters in the string COMMITTEE are arranged so that neither two M’s nor two T’s nor two E’s 2sin3sin : 4sin 2sin 2sin: 3sin4sin 3sin4sin =2:3:4 cos:cos:cos ACNS is a square. B and B’ are inside the square such that AB = BS = CB’ = B’N = BB’ = 5
36. Let
. In terms of , what is the value of
?
37.
are adjacent. Find the number of ways can be done in this way.
38. Given that A, B and C are the angles of a triangle and , what is ? 39.
. Find the
area of a square.
40. Find the smallest positive integer that has exactly 100 positive divisors.
(6 points for every correct answer) 41. How many quadruples of positive integers
, , , , , , =54,000 satisfy
?
42. ABCD is a rectangle with AB = 10 and BC = 5. P and Q are points inside the rectangle such that AP = 4, CR= PAD = 30 QCB = 45°. Find the value of PQ2.
tan5
3√ 2,2, m∠
43. Solve for all values of .
° and m∠ within the range [0, 2π] in the equation 2sin5 √ 3 = 2√ 3cos5 3cos5
44. Three points are located at A(2, 0), B(1, 8) and C(20, 18). What are the coordinates of D such that ABCD is an isosceles trapezoid with AB as one of the bases?
is always the midway betbetween ween two L’s or two I’s or both?
45. How many possible permutations of the letters in the string FAMILYISLOVE such that the letter A 46. A long pathway is constructed such that each row has 3 tiles with numbers written in each tile. The first row has numbers 4, 6 and 9. Each row has numbers which are the cubes of numbers in the previous row. For instance, the second row has numbers 4 3, 6 3 and 93 or 64, 216 and 729 and the third row has 64 3, 2163 and 7293. Suppose a mathematician wants to take the sum of each row in the pathway. What is the product of all of those sums if a pathway has 2,018 rows? Express your answer in exponential form. 47. Four chess pieces are randomly placed on a 8-by-8 chess board. Find the probability that they will form a square whose vertices are those pieces. Not all times that the sides of a square formed are parallel to the sides of a chess board.
tan−
3 25=0, 7,7,777,,
48. If p , q and and r are are the roots of ? 49. What is the remainder if 50. Find
if
what what is the value of
tan tantan− tan−
is divided by 1,235?
= log 2 2√2⋯log 2 √2⋯log 6 6√6⋯log 6 √6⋯log 12 12 12√12⋯⋯log 1 2√12⋯⋯log 9,9,900 9,9,9009,900⋯ =log 2 2√2⋯log 2 √2⋯log 6 6√6⋯log 6 √6⋯log 12 12 12√12⋯⋯log 1 2√12⋯⋯log 9,9,900 9,9,9009,900⋯
