Download Problem Solving

Problem Solving Group 2
2 35 2
3 4  7
459
, 1  22  32 
, and 12  22  32  42 
. Use
6
6
6
inductive reasoning to make a conjecture about the value of 12  22  32   n2 . Use your
conjecture to determine the value of 12  22  32   10,0002 .
1. Observe that 12  22 
2. Move one segment to make a true statement.
3*. Find the 21-digit number so that when you write the digit 1 in front and behind, the new
number is 99 times the original number.
4*. What are the final two digits of 71997 ?
5*. Given that f 11  11 and f  x  3 
f 14  , f 17  , f  20  ,
f  x 1
for all x , find f  2000  . First find
f  x  1
.
6*. You will throw 6 darts at the given target. Assuming that all six darts hit the target, show
how you can get the following scores:
a) 43
3
7
b) 40
c) 33
d) 29
10
7. Place the numbers 20, 21, 22, 23, 24, and 25 in the circles so that the sum of each side of the
triangle is 67.
8. If 2 x  15 and 15 y  32 , then find the value of xy .
9. For all real numbers x and y , the function f has the property that f  x  y   f  x   f  y  .
If f 1  3 , then find f 10  .
10*. If for all real values of x , 2 f  x   f 1  x   x 2 , then find a formula for the function f .
11. The grades on six tests all range from 0 to 100 inclusive. If the average for the six tests is
93, what is the lowest possible grade on any one test?
12*. What do you get if the sum of the first 8,000,000 positive odd integers is subtracted from
the sum of the first 8,000,000 positive even integers?
13*. How many zeroes are at the end of the number 2300  5600  4400 ?
14*. One electronic device makes a ‘beep’ each 60 seconds. Another electronic device makes a
‘beep’ each 62 seconds. They both ‘beep’ at 10:00 AM. What time will it be when they
will next make a ‘beep’ together?
15. How many digits does the number 836  5110 have?
16. What can you always find in the middle of a cigar?
17. A bottle and a cork together cost $1.10. If the bottle costs $1.00 more than the cork, what
does the cork cost?
18. A person fires a bullet at a tin can, and 2 seconds later hears the sound of the bullet hitting
the can. Assuming that the bullet travels at 3000 feet per second, and that sound travels at
1100 feet per second, how far is the tin can from the person?
1100t2
3000t1
D
19*. Find all pairs of numbers x and y so that
x 3
 x  3.
2y  7
20*. Find all the integer solutions of the equation x 2  5 y  27 or show that there aren’t any.
21. Wilfred stump heard that there is only one number between 2 and 200,000,000,000,000
which is a perfect square, perfect cube, and perfect fifth power. He decided to look for it,
and so far he has checked out every number up to about 100,000 and is beginning to get
discouraged. What is the number he is trying to find?
22*. What is the largest amount of money in U.S. coins(pennies, nickels, dimes, quarters, but no
half-dollars or dollars) you can have and still not have change for a dollar?
23. On a stretch of road 75 miles long, two trucks approach each other. Truck A is traveling at
55 miles per hour, and Truck B is traveling at 80 miles per hour. What is the distance
between the two trucks in miles one minute before their head-on collision?
24. Complete the positioning of the numbers 1-9 into the 9 squares so that the sum of each row,
column, and diagonal is the same. The result is called a magic square.
2
5
8
Can a magic square be built using the numbers 1, 3, 4, 5, 6, 7, 8, 9, and 10? Why? Why
not?
25*. Find the ones digit of 947362 .
26*. a) Decide if the indicated points are inside or outside of the given closed curve. Give a
quick method for determining if a point is inside or outside of a closed curve.
D
B
A
C
b) Now you are only given a portion of the closed curve. If you assume that A is on the
inside, then what about B and C?
B
A
A
C
27. If this pattern continues,
a) Describe the location of
125.
2
1
3
4
6 7 8
9
5
b) Is 125 in a triangle that
points up or down?
c) Describe the location of 457.
d) Is 457 in a triangle that points up or down?
10
28*. A castle in the far away land of Notsuoh was surrounded by four moats. One day the
castle was attacked and captured by a fierce tribe from the north. Guards were stationed at
each bride over the moats. Johann, from the castle, was allowed to take a number of bags
of gold as he went into exile. However, the guard at the first bridge took half of the bags
of gold plus one more bag. The guards at the second third and fourth bridges made
identical demands, all of which Johann met. When Johann finally crossed all the bridges,
he had just one bag of gold left. With how many bags of gold did Johann start?
29*. Peggy is writing the numbers 1 to 9,999. She stops to rest after she has written a total of
1,000 digits. What is the last digit that she wrote?
4
of a cigar and then
5
stops, leaving a cigar butt. If Hal finds 625 cigar butts, how many cigars will he be able to
make and smoke?
30. Hobo Hal makes his cigars by connecting 5 cigar butts. Hal smokes
31*. What is the smallest whole number that when multiplied by 9 gives a number whose digits
are all 5’s?
32. A class of fewer than 40 students took a test. The results were mixed. One-third of the
class received a B, one-fourth received a C, one-sixth received a D, one-ninth of the class
received an F, and the rest of the class received an A. How many students in the class got
an A?
33*. Assume that chicken nuggets come in boxes of 6, 9, and 20. What is the largest number of
chicken nuggets that cannot be ordered exactly?
34*. A man whose end was approaching summoned his sons and said, “Divide my money as I
shall prescribe.” To his eldest son he said, “You are to have $1,000 and a tenth of what is
left.” To his second son he said, “Take $2,000 and a tenth of what remains.” To the third
son he said, “You are to take $3,000 and a tenth of what is left.” Thus he gave each son
$1,000 more than the previous son and a tenth of what remained, and to the last son all that
was left. After following their father’s instructions with care, the sons found that they had
shared their inheritance equally. How many sons were there, and how large was the estate?