Download Getting Started Marathon 3

Getting Started Marathon 3
Compiled by Pulak Mittal
1
Problems
1. (Joml88) How many digits are in the number 417 · 530 ?
2. (chess 64) The product N of three positive integers is 6 times their sum. One of the integers is the
sum of the other two. Find the sum of all possible values of N.
3. (236factorial) I will quit AoPS in 300 days. Now it is August 12. What month will I quit?
4. (aidan) A large watermelon weighs 20 kg, with 98% of its weight being water. It is left to stand in
the sun, and some of the water evaporates, such that now, only 95% of its weight is water. What
does the watermelon now weigh?
5. (236factorial) What is sin 900o ?
6. (chess64) Four positive integers a, b, c, and d have a product of 8! and satisfy
ab + a + b = 524
bc + b + c = 146
cd + c + d = 104
What is a - d?
7. (aidan) The only information that an electronic watch displays is hours as a 2-digit number and
minutes as a 2-digit number. What is the total time in minutes that the digit 2 was visible on the
face of the watch from 15:00 to 16:30 during an afternoon?
8. (chesspro) Mike likes to eat cookies. Every day, he eats 20% of the remaining cookies in the cookie
jar. If 32 cookies were left at the end of the second day that Mike began eating, how many cookies
were originally in the jar?
9. (chess64) Let f be a function satisfying f (xy) =
f (500) = 3, what is the value of f (600)?
f (x)
y
for all positive real numbers x and y. If
10. (PenguinIntegral) Given m, n are positive integers such that mn + m + n = 71, m2 n + mn2 = 880,
find m2 + n2 .
c
a
11. (PenguinIntegral) Given that positive integers a,b,c and d satisfy < < 1, arrange the following
b
d
b d bd (b + d)
in order of increasing magnitude: , , ,
,1
a c ac (a + c)
12. (frt) Solve for x: 4x+1 + 4x+2 + 4x+3 + 4x+4 = 170
13. (chess64) The sum of 49 consecutive integers is 75 . What is their median?
14. (G-Unit) What is the smallest positive integer that can be expressed as the sum of nine consecutive
integers, the sum of ten consecutive integers, and the sum of eleven consecutive integers?
1
15. (PenguinIntegral) Find the smallest positive integer n for which (xy − 3x − 7y − 21)n has at least
1996 terms.
16. (dakyru) How many positive integers less than 10,000 contain the digit ”1” at least once?
17. (G-Unit) A box contains two coins. One coin is heads on both sides and the other is heads on one
side and tails on the other. One coin is selected from the box at random and the face of one side is
observed. If the face is heads what is the probability that the other side is heads?
18. (pkerichang) How many positive integers less than or equal to 1000 can be express as the difference
of two perfect squares??
19. (236factorial) Three distinct digits a, b, and c are used to form 6 distinct three-digit numbers.
What is the greatest positive integer that is a divisor of the sum of those 6 numbers, regardless of
the original numbers chosen?
20. (shyong) There is a rectangle ABCD where E and F lie on line AB and BC respectively. If
4AED = 5cm2 , 4BEF = 4cm2 , 4CDF = 3cm2 . Find 4DEF
21. (chess64) Let
a=
12 22 32
10012
+
+
+ ··· +
1
3
5
2001
b=
12 22 32
10012
+
+
+ ··· +
3
5
7
2003
and
Find the integer closest to a − b.
(A) 500 (B) 501 (C) 999 (D) 1000 (E) 1001
22. (PenguinIntegral) How many 4 digit numbers with first digit 1 have exactly two identical digits
(like 1447, 1005 or 1231)?
23. (236factorial) A triangle has angles of 1328 and 1058 . What is the third angle of the triangle, in
base 8?
24. (236factorial) The surface area of Sphere A is 96% more than that of Sphere B. The volume of
sphere A is r% more than that of Sphere B. What is r? Nearest whole number, please.
25. (chess64) In how many zeroes does
2002!
end?
(1001!)2
(A) 0 (B) 1 (C) 2 (D) 200 (E) 400
√
√
√
26. (236factorial) The numbers 2u − 1, 2u + 1, and 2 u are the side lengths of a triangle. How
many degrees are in the measure of the largest angle?
27. (chess64) What is the difference between the sum of the first 2003 even counting numbers and the
sum of the first 2003 odd counting numbers?
28. (chess64) Find
∞
Y
n=0
n
.
n! + 1
29. (PenguinIntegral) Find the arithmetic mean of (9,99,999.......999999999).
30. (Iversonfan2005) Circle A is inscribed in a quarter of circle B. If the radius of Circle A=2, then
what is the area of an equilateral triangle inscribed in circle B?
2
31. (chesspro) At my school, 25 of the freshmen and 54 of the sophomores took the AMC 10. If the
number of freshman and sophomore taking the competition are the same, which must be true?
A) There are five times as many sophomores as freshmen.
B) The number of sophomores is twice the freshmen.
C) The number of freshmen is equal to the number of sophomores.
D) The number of freshmen is twice the sophomores.
E) There are five times as many freshman as sophomores.
32. (chess64) The ratio of the volume of a sphere to its surface area is 5. Find the radius.
33. (G-Unit) How many three-digit numbers include at least one seven but have no zeros?
34. (h-s-potter2002) How many 0’s are at the end of (5!)!?
35. (shyong) For every integer from n from 1 to 2005 (inclusive) , how many among them gives us an
integer when
1n + 2n + 3n + 4n + 5n is divided by 5 ?
36. (frt) If
7x − 3y
x−y
= 6, then what is the value of
?
2x + 3y
x+y
37. (236factorial) Point P lies on the line x=-3 and is 10 units from the point (5,2). Find the product
of all possible y-coordinates that satisfy the given conditions.
38. (236factorial) What is the positive difference between the sum of the first 25 even integers and the
sum of the first 20 odd integers?
39. (shyong) if m2 = m + 1 , n2 = n + 1 and m 6= n . Try to compute m5 + n5
40. (chess64) Derive the formula for the sum S = 1 + a + ar + ar2 + · · ·
41. (frt) Find the sum 1 − 4 + 9 − 16 + ... − 1002 .
42. (G-Unit) It takes a cyclist three minutes to ride a kilometre on level ground against the wind, but
only two minutes to return with the wind behind him. How long would it tkae him to ride one
kilometre on a calm day?
43. (dakyru) What are all ordered triples of real numbers (a,b,c) which satisfy ca = b2a , 2c = 2(4a ),
and a + b + c = 10
44. (shyong) There are 2005 people lining in a straight line with the first person labelled as 1 ,second
person as 2 and so on . When the first whistle is blown , those who are at the even number position
are to be left . When the second whistle was blown , the remaining even number position person are
to be left . This process keeps going until there are only two persons left . What are the original
numbers which labelled at those two remaining persons ? ( after the first even number people are
out , the remaining people are labelled again from 1 to 1003 . Then the second whistle to get rid the
even . Then the number is labelled from 1 to 502 and so on .... )
45. (shyong) Factorise this completely a8 + 324b4
46. (frt) The numbers 409, 505, and 745 all leave the same remainder when divided by the positive
integer n. Find the largest possible value of n.
47. (shyong) Given that x2 + x − 1 = 0 , find the value of 5x4 − 2x3 − 4x2 + 15x + 1997
3
48. (aidan) Suppose a and b are two numbers such that
a2 + b2 + 8a − 14b + 65 = 0
Find the value of a2 + ab + b2
49. (shyong) There are 2016 balls which are placed in three different sacks A,B,C . First , we take a
number of balls from A and put into B and C such that the number of balls in B , C are doubled
.Then , we apply to same thing to B and then C . Eventually , three sacks have the same amount of
balls . How many balls are there in sack A , sack B and sack C initially ?
50. (aidan) A train leaves Canberra for Sydney at 12 noon, and another train leaves Sydney for Canberra
forty minutes later. Both trains take the same route and travel at the same uniform speed, taking
3 12 hours to complete the journey. At what time will they pass?
51. (236factorial) What is the largest integral divisor of 214 − 1 that is less than 214 − 1?
52. (frt) If the sum of the interior angles of a regular polygon is 3240, find the number of diagonals.
53. (chess64) Two different prime numbers between 4 and 18 are chosen. When their sum is subtracted
from their product, which of the following numbers could be obtained?
21, 60, 119, 180, or 231
54. (236factorial) Mrs. Walter gave an exam in a math class of 5 students. She entered grades in a
random order, and recalculated the average after each score was entered. She noticed that after each
grade was entered, the average was a whole number. Which was the last grade entered if the grades
were 71, 76, 80, 82, and 91?
55. (aidan) A father in his will left all his money to his children in the following manner: $1000 to the
1
1
first born and 10
of what then remains, then$2000 to the second born and 10
of what then remains,
1
then$3000 to the third born and 10 of what then remains, and so on. When this was done each child
received the same amount of money.
(i) How many children were there?
(ii) How much money, altogether, did the father leave for his children?
1
1
√ and b − c =
√
2− 3
2+ 3
Find the value of a2 + b2 + c2 − ab − bc − ac
56. (shyong) Given that a − b =
57. (socrates) compare the numbers 3117 and 1722
58. (aidan) Henry has programmed his computer so that his screen lights up periodically in combinations
of three primary colours. The screen lights up for one second each time.
When all the colours are programmed to appear at the same time the screen colour is white. When
no colours are programmed to appear the screen is black.
Henry programs the primary colours as shown.
Blue : Every 7 seconds
Green : Every 11 seconds
Red : Every 17 seconds
He starts the green program 3 seconds after he starts the blue program.
He starts the red program 12 seconds after he starts the green program.
How many seconds after Henry starts the blue program will the screen appear white?
4
59. (aidan) Two brothers, each aged between 10 and 90, ”combined” their ages by writing them down
one after the other to create a four digit number, and discovered this number to be the square of
an integer. Nine years later they repeated this process (combining their ages in the same order)
and found that the combination was again a square of another integer which was ine larger than the
previous integer. What was the sum of their original ages?
60. (236factorial) What is the sum of the prime factors of the number represented by:
212 − 211 + 210 − 29 + .... + 22 − 21
61. (frt) Find the number of positive integers less than 1000 that are second powers and/or third powers,
and or 4th powers, etc.
62. (bizkits329) In a double elimination tournament of 64 teams, how many games are played?
63. (236 factorial) Find a 5-digit number in which
The number’s square root is a palindrome,
The number’s first two digits is a perfect square,
The number’s middle digit is also a perfect square, and finally
The number’s last two digit express a perfect square.
64. (chess64) Find the number of even positive integers k such that k¡1000 and all digits are different.
65. (236factorial) The smallest case of 5 consecutive odd integers whose sum is a perfect square is 1,
3, 5, 7, and 9. What are the next 5 consecutive odd integers whose sum is a perfect square.
66. (Silverfalcon) The number of real solutions (x, y, z, w) of the simultaneous equations
2y = x +
17
17
17
17
, 2z = y + , 2w = z + , 2x = w +
x
y
z
w
is what?
67. (shyong) A set of integer {1, 2, 3...N } having the property such that the sum of all odd number in
the set minus the sum of all even number gives you exactly 2005 . Find N
68. (deej21) If x = 1 + 2p and y = 1 + 2−p , find y in terms of x
69. (frt) N is the smallest natural number such that
N
N
2
is a perfect square and
N
3
is a perfect cube. Find
70. (aidan) If the tens digit of a perfect square number is 7, how many units digits are possible?
71. (shyong) How many ways you can write down the positive integer from 1 to 2005 in a list such that
the difference between any two consecutive integers in the list is odd ? (of course you can leave your
answer in factorial form)
72. (andreas) ABCD is a trapezium with AD parallel to BC. The diagonals AC and BD intersect at O.
If the area of triangle AOD is x, and the area of triangle BOC is y?
73. (shyong) How many positive integer a satisfying the following inequality ?
√
√
0.04 < a + 1 − a − 1 < 0.05
5
74. (aidan) A circle with center O has diameter AB = 20 , and a point P is located at the midpoint
of one of the arcs AB . Another circle is then constructed such that its center is at P and the
circumference of the second circle contains points A and B . We call the area of the part of circle
O that doesnt lie inside cirlce P as S . And we call the area of the part of circle P that doesnt lie
inside circle O as T . Find the difference between S and T . Leave answer in terms of pi.
75. (andreas) The sum of the first p terms of an arithmetic sequence equals q and the sum of the first q
terms of that sequence is given by p (p 6= q). What is the difference v between two successive terms
of that sequence?
76. (arne) Let a, b, c and d be positive integers such that
a5 = b6
c3 = d4
d − a = 61
Find c − b.
77. (aidan) Griselda needs to mix up a bottle of Love Potion No. 9 for a lovesick client. The recipe calls
for 140 ml of turpentine, 160 ml of Indian ink, and 50 ml of honey as a sweetener for the icky stuff.
Griselda pours 140 ml of turpentine into a 350 ml bottle. Then she adds the Indian ink but her cat
disturbs her and she pours in too much. She only realises her mistake when she finds that the bottle
is already full before she has added all 50 ml of honey.
Griselda does a quick calculation, mixesthe potion thoroughly and pours some of the mixture in the
bottle down the sink. She replaces this amount with a honey and turpentine mix which results in
the final mixture in the bottle matching the amounts required by the recipe.
1. In what ratio must Griselda have mixed the honey and turpentine together for the replacement?
2. If Griselda poured 70 ml of fluid down the sink, how much of each ingredient was wasted?
78. (aidan) From four numbers, three are chosen, averaged and the fourth one added to it. This can be
done four ways, leaving out a different number each time. The four results are 17, 21, 23, and 29.
What is the largest of the original numbers?
79. (andreas) David finds the sum of the digits for each 8-digit number. What is/are the sum that
occurs/occur most often?
1 1
1
80. (andreas) On a blackboard are written 100 numbers: 1, , , ...,
. It is possible to delete two
2 3
100
numbers (chosen randomly) a and b and rewrite only one number, that is a + b + ab instead of them.
After 99 operation of this kind on the blackboard there is a number. What is this number?
81. (aidan) A four digit number N leaves remainder 10 when divided by 21, remainder 11 when divided
by 23 and remainder 12 when divided by 25. What is the sum of the digits of N?
√
82. (andreas) It is x = 99 − 70 2 Then:
1
(A) x ≤ −
100
1
(B) −
<x<0
100
(C) x = 0
1
(D) 0 < x <
100
1
(E) x ≥
100
6
83. (arne) If x2 + y 2 = 2005xy find
x−y
x+y
84. (236factorial) Prove that none of the numbers 11, 111, 1111, 11111, 111111, ... are perfect squares.
85. (shyong) How many pairs of positive integer a, b ≤ 2005 having the property such that 12a is a
divisor of 18b
86. (Gabriel-fr) For b > 1 and x > 0 solve the equation: (2x)logb 2 − (3x)logb 3 = 0
87. (frt) Given that x + 2y = 6, x > 1, and y ≥ 0, what is the maximum value of xy?
88. (G-Unit) I have 4 pairs of socks to be hung out side by side on a straight clothes line. The socks
in each pair are identical but the pairs themselves have different colours. How many different colour
patterns can be made if no sock is to be next to its mate?
89. (Melissa) Find the natural number n such that
n
50
<
1
2·5
+
1
5·8
+
1
8·11
+ ... +
1
200·203
<
n+1
50 .
90. (G-Unit) Is it possible to cut a 39 x 55 rectangle into 5 x 11 rectangles?
91. (Melissa) In the sequence of fractions
1 2 1 3 2 1 4 3 2 1 5
, , , , , , , , , , , ..., what is the 2005th fraction?
1 1 2 1 2 3 1 2 3 4 1
92. (G-Unit) How many nine-digit numbers have an even sum of their digits?
93. (236factorial) What is
1
1
1
+
+ ... +
?
1·2·3·4 2·3·4·5
28 · 29 · 30 · 31
94. (236factorial) Find all the complex solutions of the equation x5 − 2x2 = 9x + 6
95. (chess64) Let P (n) and S(n) denote the product and sum, respectively, of the digits of the integer
n. Suppose N is a two-digit number such that N = P (N ) + S(N ). What is the units digit of N ?
96. (riddler) find the nth term of this cubic sequence, 7, 23, 55, 108
97. (236factorial) It is given that the system of equations
x − by = 1 y − ax = 1 bx + ay = 1
has solutions. Prove that a2 + b2 + ab + a + b = 1.
98. (Riddler) Riddler has no problems to post. He searches ”geometry problem” on google and gets
about 83,000 results. He does all of these problems. (Assume that each result had exactly one math
problem on it.) 5,000 of them take him 1 hour to do (total), 10,000 of them take him 2 hours to do,
15,000 of them take him 3 hours to do, 20,000 of them take him 4 hours to do, 25,000 of them take
him 5 hours to do, and 8,000 of them take him 6 hours to do. What is the expected value of the time
in hours it takes for him to do a problem?
99. (236factorial) Prove that sin x cos3 x − cos x sin3 x =
1
4
sin 4x
100. (K.W.M.A.N.) I have a square with side z and a circle, lying in that square with radius z/3. I
randomly pick out points lying in the square. After having picked out 10000 points, I counted that
3498 of them were in the circle too. Now estimate the value of Pi.
101. (G-Unit) Consider the following fraction:
1
n
What must be true of n such that,
1
n
is a terminating decimal?
7
102. (Snowstorm) Triangle ABC is a right triangle with ∠ACB as its right angle, m∠ABC = 60, and
AB = 10. Let P be a randomly√chosen point inside 4ABC, and extend BP to meet AC at D. What
is the probability that BD > 5 2?
√
103. (deej21) The medians from the acute angles of a right triangle have length 5 and 40. Find the
length of the hypotenuse.
104. (SnowStorm) A triangle with sides of 5, 12, and 13 has both an inscribed and a circumscribed
circle. What is the distance between the centers of those circles?
n
X
k 9
105. (deej21) Evaluate
2
3
k=0
106. (Melissa) Evaluate
1 1
1
1
1
1
+ +
+
+
+ ... +
2 6 12 20 30
22350
107. (Paranoid Android) For a set of five whole numbers, the mean is 4, the mode is 1, and the median
is 5. What are the five numbers?
108. (frt) Solve for x:
|x| + 2|x − 3| = x + 2
109. (Melissa) Between 5am and 6am the hands of a clock are at right angles exactly twice. How many
minutes elapse between these times?
110. (chess64) In a rectangular solid, the area of the top face is 135, and the area of the front face is 30,
and the area of the right face is 50. Find the volume of the solid.
111. (Iversonfan2005) How many digits are there in 47 510 ?
112. (Iversonfan2005) How many times does 24 go into 100! evenly?
113. (frt) What are the values of p and q such that x2 + 2x + 5 is a factor of x4 + px2 + q?
114. (cbong) How many digits are there in 22005 ? Give your answer to the nearest hundred.
√
√
115. (shyong) Given that x = 3 4 + 2 2 + 1 , find the value of (x − x1 − 2)3
116. (arne) the sum of 3 consecutive positive integers divides the sum of the cubes of these integers.
117. (SnowStorm) How many 15-letter arrangements of 5 A’s, 5 B’s, and 5 C’s have no A’s in the first
5 letters, no B’s in the next 5 letters, and no C’s in the last 5 letters?
118. (Melissa) If x + y + z + u = 5 y + z + u + v = 1 z + u + v + x = 2 u + v + x + y = 0 v + x + y + z = 4,
what is the value of v + x?
119. (goyoungha) Given that a, b, and c are all positive, and that ab = 21, ac = 18, and bc = 42, find
a - b - c.
120. (kunny) Let x be a positive number satisfying x2 + 4x − 1 = 0. Set A = 1 + (1 + x) + (1 + x)2 +
A
(1 + x)3 , B = (1 + x)3 , find the value of .
B
p
121. (SilverFalcon) If y = 1 + 31 36 − (x − 2)2 , then the maximum for y is
(A) 2 (B) 3 (C) 5 (D) -1 (E) 4
122. (SilverFalcon) Find the last (i.e. rightmost) three digits in the expansion of 32001 .
8
123. (shyong) compute the value of
14
1
2
100
+ 4
+ ... +
2
2
4
+1 +1 2 +2 +1
100 + 1002 + 1
124. (frt) A triangle has lengths of sides 3, 4 and 5. Find the volume of the figure formed if the triangle
is rotated about its longest side.
125. (236factorial) How many prime numbers p exist such that p2 + 21p − 1 is also prime?
126. (aidan) The ratio of the speed of two trains is equal to the ratio of the time they take to pass each
other going in the same direction to the time they take to pass each other going in opposite directions.
What is the ratio of the speeds of the two trains?
127. (shyong) Sam is having his 17th birthday on 16 November 2005 . How many years later will he have
his birthday falls on the same weekday again ?
128. (invisal) A + B = 2 Show that A4 + B 4 ≥ A3 + B 3
129. (shyong) Find the largest positive integer k such that 4x+7y=k has exactly 71 non-negative integer
solution for x,y .
130. (shyong) Given four box A,B,C,D , each with a ball inside . Now all ball is taken out and we put
them into each box at random (one box may only contain one ball) . What is the number of way to
put the balls so that none of them are placed back into the same box ?
131. (Melissa) Four vehicles travelled on a road with constant velocities. The car overtook the scooter
at 12h00, then met the bike at 14h00 and the motorcycle at 16h00. The motorcycle met the scooter
at 17h00 and overtook the bike at 18h00. At what time did the bike and the scooter meet?
132. (Melissa) One day Fred of Bikeville and Sam of Cycletown set off on their bicycles at the same
time, each riding to the other’s town. Each rode at a constant speed, but Fred rode faster than Sam.
They met at Mount Saddle. Fred had ridden 2456 metres. They continued their rides. Fred took 81
minutes to get to Cycletown from Mount Saddle, and Sam took 100 minutes to get to Bikevill from
Mount Saddle. How far is it from Bikeville to Cycletown?
133. (eminem) prove 2x4 + 1402 = y 4 has no integral solution
134. (shyong) 288 workers , with 11 working hours per day , use up 5 days to dig a ground with length
132 , width 6 and depth 2 .Now there are another 112 workers , with 9 working hours per day , dig
a ground with length 210 , width 8 and depth 3. If the effeciencies of the former and latter workers
have the ratio 4: 5 , how many days would the latter workers use to complete their work ?
135. (Melissa) If p3 − p − 1 = 0 then what is p5 − p4 − 1 equal to?
136. (shyong) A person is rowing a boat upstream for 60m where he drops his empty bottle into river .
He then rows ahead for another 15 minutes and turn back to row down stream . When he reaches
his original place where he started to row the boat , his bottle also reaches there at the same time .
What is the speed of the current (in meter/min)?
137. (Fermat2) I spend exactly 30 buying some 19p stamps and some 25p stamps so that the difference
in number between the two kinds of stamps is as small as possible. How many stamps did I buy?
138. (goyoungha) A fair standard six-sided die is tossed three times. Given that the sum of the first two
tosses equals the third, what is the probability that at least one 2 is tossed?
139. (Kurt Godel) Given that m, n ∈ N0 : m2 + 2n = n2 + 2m + 5. Find n.
9
140. (SnowStorm) In triangle ABC, side AC and the perpendicular bisector of BC meet in point D,
and BD bisects ∠ABC. If AD = 9 and DC = 7, what is the area of triangle ABD?
141. (frt) In the sequence 6, x, y, 16, the first three terms are in arithmetic progression and the last three
terms are in geometric progression. Find all possible ordered pairs (x, y).
142. (SnowStorm) Consider the sequence of numbers: 4, 7, 1, 8, 9, 7, 6, . . . For n > 2, the nth term of the
sequence is the units digit of the sum of the two previous terms. Let Sn denote the sum of the first
n terms of this sequence. What is the smallest value of n for which Sn > 10, 000?
143. (agolsme) How many nonempty subsets of {1, 2, 3, . . . , 12} have the property that the sum of the
largest element and the smallest element is 13?
144. (frt) How many minutes before 1 o’clock will the minute hand and the hour hand be the same
”distance” from the 12?
145. (catcurio) How many ordered pairs (x, y) of positive integers x and y satisfy the equation 2x + 3y =
137?
x log7 11 x log7 9
146. (JavaMan)
−
= 0 for which positive integral value of x?
9
11
147. (FMako) If there are 15 teams to play in a tournament, 2 teams per game, in how many ways
can the tournament be organized if each team is to participate in exactly 5 games against different
opponents?
148. (Hokkage) Line segment PQ is tangent to circle O at Q. Line segment PR is tangent to O at R. R
is not Q. A third tangent is introduced and intersects segment PQ at C and segment PR at D. If PQ
is 20, what is the perimeter of 4P CD?
149. (dogseatcheese)
1 1 1
1
+ + + ... +
2 4 8
1024
150. (dogseatcheese)
1
1
1 1
+ +
+ ... +
2 6 12
8190
151. (t0rajir0u)
1
1
1
1
1
+
+
+
+ ... +
4 10 18 28
8008
152. (t0rajir0u) Find the length of a line from the centroid of a regular tetrahedron of side length 1 to
any of its vertices.
153. (t0rajir0u) Two positive real numbers x, y are random chosen between 0 and 1, inclusive. What is
the probability that x2 + y 2 < 1?
154. (t0rajir0u) Find the circumradius of a triangle with side lengths 17, 25, 26.
155. (catcurio) What is the value of
19993 − 7203 − 12793
?
1999 · 720 · 1279
156. (t0rajir0u) How many positive integers less than 2006 are multiples of 5 or multiples of 7, but not
multiples of 35?
157. (catcurio) Find the length of the diagonal of an isosceles trapezoid of bases 16 and 20 and legs 10.
158. (t0rajir0u) Two circles of radii 4 and 7 have their centers 15 apart. A ”belt” is looped around them
such that it contains arcs of both circles and two shared internal tangents (that is, the belt crosses
itself between the circles). What is the total length of the belt?
10
159. (chinesechess4all) If the geometric mean of A and B is 10 and the arithmetic mean of A and B is
the number of this problem, what is (a4 + b4 )(a3 + b3 )?
160. (catcurio) Find the smallest possible sum A + B + C such that A, B, and C are pairwise relatively
prime positive integers and A log56 7 + B log56 2 = C
161. (g1sk8erz) A mathematician said, ”Three fourths of my age two years ago is greater than or equal
to two thirds of my age three years from now.” He is now at least x years old, where x = ?
A) 39 B) 40 C) 41 D) 42 E) none of the above
162. (catcurio) 3 runners from each of 2 teams race against each other. An nth place finish earns n points
for one’s team, and the team with the fewest total points wins. How many different winning scores
are there?
163. (catcurio) Tom and Jimmy are running toward each other. They’re 50 meters apart initially. Tom
m
runs 2 sec
faster than Jimmy, and they meet in 10 seconds. How fast was Jimmy running?
164. (shyong) Riddler,robinhe and shyong walk at the speed of 5m/s , 7m/s ,10m/s respectively . If
riddler starts walking for 2 hours before robinhe starts to walk , when should shyong start to walk(time
is counted right after robinhe starts his journey) so that robinhe and shyong can chase up riddler at
the same time .
165. (robinhe) A Martian day, called a sol, is 24 hours, 39 minutes long in Earth time. If it is noon at
a particular location on Earth and noon at the same moment at a particular location on Mars, after
how many sols will it first be noon at these same locations at the same moment in the time again?
166. (chinesechess4all) If a person runs 5 mph and then a car leaves 20 minutes later, how fast must
the car drive at a constant speed to catch up to the person an hour 20 minutes after the car leaves?
167. (robinhe) Three friends, Ralph, Emerson and Waldo, each select a number form the set
{1, 2, 3, 11, 12, 13, 21, 22, 23} and remove it. Then they add their three numbers together. If they put
their numbers back and repeat this process for all possible combinations, how many different sums
can they get?
168. (catcurio) If you have a coin and a weighted die where there is a 14 chance of getting a one, 15 chance
of getting a two , and an equal chance of getting any other number, what is the probability that
when you roll the die twice, the product of the rolls exceeds 8 and then you flip a heads on the coin?
169. (catcurio) Joan and Jerry receive monthly allowances equal to their ages. Each age is a whole
number of years. Joan is older than Jerry. If Joan’s allowance is doubled, their combined allowance
would be less than 36 dollars, but if Jerry’s allowance is doubled, their combined allowance would be
more than 30 dollars. If the sum of their ages is S, how many different values of S are possible?
170. (robinhe) Reid took seven tests. On the first five tests that he took, his mean score is 76. His mean
score for all seven tests is 78 points. Each of his scores is an integer and no score is less than 70
points. If the mode is unique and has a frequency of 3, what is the greatest possible value of the
mode for his seven test scores?
171. (catcurio) Alex’s telepathy is 99% accurate, and Brad’s is 85%. They are to detect whether suspect
George is lying or not. Brad says he is, while Alex says he isn’t. What is the probability that George
is lying?
172. (catcurio) 5 boys and 5 girls sit around a round table. If all seats are identical and the 5 girls sit
together, how many seating arrangements are possible?
11
173. (frt) Solve for x:
32+x + 32−x = 82
174. (catcurio) Two concentric circles are such that the smaller divides the larger into two regions of
equal area. Given point P on the larger circle and point T on the smaller circle, P T is tangent to
the smaller circle. If the radius of the smaller circle is r, what is the length of segment P T ?
175. (Hokkage) Given the unit cube x, y, z ≤ 1, cut it with planes x = y, y = z, x = z. How many pieces
are there?
176. (chinesechess4all) A square ABCD has side length of 12 inches. Let P be the center of a circle
passing by C and D and tangent to AB. What is the area in square inches of the triangle PDC?
177. (Hokkage) Find 99 consecutive composite numbers.
178. (Farenhajt) Solve the equation bxc =
largest integer not greater than x.
2x−1
3 ,
where bxc denotes the floor function of x, i.e. the
179. (shyong) Three groups of people join a meeting (Lets call it A,B,C) . During the meeting , the
chairman wants to know how many people attends so he asks the leader of each group ( provided
that the chairman belongs to neither groups) .Here is how they answer ....
Leader A: half of people in group B and one-third from group C is the number of people in my group.
Leader B: one-fourth of people in group C and one-fifth people from group A is the number of people
in my group.
Leader C: Well , as far as I know , there are not more than 400 people attend this meeting.
The question is,what is the number of people in each group?
180. (catcurio) How many ways are there to roll a 9 with three 6-sided dice? (Order matters)
181. (Farenhajt) Find the smallest positive integer which has exactly 2006 divisors (including 1 and the
number itself).
182. (Robinhe) Zan tosses one fair 6-sided die with faces labeled 1 through 6 and one fair 4-sided die
with faces labeled 1 through 4. If at least one die shows a 3 what is the probability that the sum is
5? Express your answer as a common fraction.
183. (frt) If
x+y
y+z
z+x
xy + yz + zx
=
=
(6= 0), compute 2
3
4
5
x + y2 + z2
184. (Farenhajt) ABCD is a square with side s. Let M, N, P, Q be the midpoints of the sides. Circles
s
s
s
s
(M, ), (N, ), (P, ), (Q, ) are drawn. Find the area of the ”flower” they form inside the square.
2
2
2
2
((T, a) stands for ”center T and radius a”.)
185. (riddler) Find n such that log10 sin x + log10 cos x = −1 , log10 (sin x + cos x) = (log10 n − 1)/2
186. (Farenhajt) A regular trilateral prism, whose base side is s and height is H, is cut by a plane
passing through one side of the lower base and the opposite vertex of the upper base. Find the area
of the section if s = 4, H = 2
187. (frt) Bob starts from the east end and Jane from the west end of a swimming pool, and both swim
two lengths of the pool at constant rates. They pass each other twice, each time going in opposite
directions. The first time they pass they are 20 feet from the east end, and the second time they are
18 feet from the west end. Assuming that each made an instantaneous turn when they reached an
end of the pool, how long is the pool, in feet?
12
188. (Farenhajt) Solve the equation ||x2 − 5| − 2| = 1
189. (frt) In 4ABC, D is a point on AB such that AD : DB = 4 : 3. Also, E is a point on AC such
that AE : EC = 5 : 3. If the area of quadrilateral BCED is 18, find the area of 4ADE
190. (shyong) Find
1 1
3
5
2
+ + 2 + 3 + 4 + ···
1 2 2
2
2
Where the numerator is Fibonanci series while the denominator is geometric series with common
ratio 2 .
191. (Farenhajt) Solve the equation |x − 3|3x
2 −10x+3
=1
192. (shyong) Find time of maximum velocity of a person doing a bungee jump if the graph v-t is given
by the equation- v = 100 cos(30 − t/2)
193. (shadysaysurspammed) iF x3 + x + 1 = 0 and roots are a,b,c,find sum of cubes of the reciprocals
of the roots...
194. (Farenhajt) Solve the following system in the simplest manner:
x + 7y + 3v + 5u = 16
8x + 4y + 6v + 2u = −16
2x + 6y + 4v + 8u = 16
5x + 3y + 7v + u = −16
195. (shyong) A 2006-digit integer n has q and r as its quotient and remainder respectively upon divided
by 100 . If q + r is divisible by 11 , how many n satisfies it ?
196. (chess64) A triangle is formed by connecting the points (-5,0), (0,6), and (5,0). The resulting triangle
is then rotated about the x-axis to form a solid 3D figure. The original triangle is then rotated about
the y-axis to form a different solid. What is the positive difference between the number of cubic units
in the volumes of the two resulting solids?
197. (Snowstorm) Pat is to select six cookies from a tray containing only chocolate chip, oatmeal, and
peanut butter cookies. There are at least six of each of these three kinds of cookies on the tray. How
many different assortments of six cookies can be selected?
198. (chinesechess4all) The base 10 representation of 55 is 3125 . What is the base-5 representation of
3124 ?
p
p
√
√
3
3
199. (shadysaysurspammed) Find the sum...
5 + 2 13 + 5 − 2 13
200. (frt) After the Christmas shopping season was over, a department store inventory showed several
toys of the same kind with a total value of $160. Four of these were broken and a total loss; the
remainder were each solde for 75 cents less than they cost the store, and the store lost a total of $65
on the lot. How many of this kind of toy were left after Christmas? Note: assume that inventories
show only whole number of items.
201. (catcurio) Each face of a regular tetrahedron is to be painted either blue or red. How many distinct
paintings are possible?
13