Download Junior Individual Test - answers no mult choice

Junior Individual Test
10 multiple choice and 10 short answer
These questions can also be used for the countdown round and team rounds
1. The point (1,1,1) is rotated 180o about the y-axis, then reflected through the y-z plane, reflected
through the x-z plane, rotated about the y-axis, and reflected through the x-z plane. Find the
coordinates of the point now.
2. Shaq scores an average of 18.6 points per game for five basketball games. How many points
must he score in the next game to raise his average to 20 points per game?
3. Bilal arranges the counting numbers in a triangle by writing 1 at the apex, then writing 2 and 3
on the second row, then, 4, 5, and 6 on the third row, and so on. What is the sum of the first and
last integers on the seventeenth row?
4. Find the largest integer n for which 12n evenly divides 20!
5. A man is climbing a 75 foot cliff. He climbs up 12 feet every 8 minutes, but then tires and slides
down 9 feet in the next 2 minutes before repeating this process. If he starts climbing at 2:30pm,
at what time does he reach the top of the cliff?
6. There are five musicians in the band. Sabrina plays banjo and bagpipes, Celia plays keyboard
and drums, Lukas lays castanets and bagpipes, Zoe plays banjo and keyboard, and Sam plays
drums and castanets. In how many different ways can the musicians choose their instruments so
that all five instruments are played?
1/ a  1/ b
3*5
7. We define a * b to equal
. Evaluate
.
ab
5*7
8. How many different triples of numbers (a, b, c) satisfy the equation a2 + bc = b2 + ac, if a, b,
and c are integers from 1 to 5, inclusive?
9. What is the greatest possible number of points of intersection among four lines and a circle in
the plane?
10. Let a and b be positive integers. If a!/b! is a multiple of 4 but not a multiple of 8, then what is
the largest possible value for a – b?
11. How many three-term arithmetic sequences contain one term from each of the following sets:
{1, 3, 9, 27}, {1, 5, 25, 125}, and {1, 7, 49, 343}?
12. Find the maximum possible value for x + y given that 3x +2y  7 and 2x + 4y  8
13. Andy says, “Exactly three of us are liars.” Bill says, “Andy is a liar.” Clair says, “Bill is a liar.”
Daisy says, “My favorite movie is Dukes of Hazzard.” Each person is either lying or telling the
truth. Name the liar(s).
14. Between which two consecutive integers is 999 ?
15. Two circles with centers (0, 0) and (24, 7) and radii of length 3 and 4, respectively, are drawn in
the coordinate plane. What is the radius of the smallest circle which contains both of them?
16. Rocky and Bullwinkle are playing Risk. Rocky rolls one six-sided die, while Bullwinkle rolls
two of them. What is the probability that Rocky’s roll is as high as Bullwinkle’s largest
number?
17. Triangle ABC has a right angle at C. Let D and E be the midpoints of the sides
BC and AC respectively. We also write AD = s, BE = t, and AB = c. If a right triangle were
constructed with legs of lengths s and t, then what would be the length of the hypotenuse, in
terms of c?
(-1,1,1)
27
290
8
6:08 pm
2
7/3
33
14
3
7
11/4
Andy and
Clair
31 and 32
16
91/216
(c 5 )/2
18. In the diagram, a square is built on hypotenuse AC of right triangle ABC. If
AB = 4 and BC = 6, then compute the area of the square.
52 (units
squared)
19. Two bikers are seven-eighths of the way through a mile-long tunnel when they hear a train
approaching the closer end at 40 miles per hour. The riders take off at the same speed but in
opposite directions and each escapes the tunnel just as the train passes them. How fast did they
ride?
x
x
20. Solve for x in the equation 2(16 ) = 16(2 )
21. Twenty concentric circles are drawn with radii 1, 2, 3, …, and 20. The regions between the
circles are painted, alternating between red and black, beginning with the interior of the smallest
circle, which is painted red. If a point is chosen at random inside the largest circle, what is the
probability that it lies in a black region?
22. Define an operation * by declaring that a * b = (a + b)/(a – b). Find a number x such that 3 * x =
3.
23. Find the smallest positive integer greater than 1 which leaves a remainder of 1 when divided by
2, 3, …, 8, and 9.
24. The square of an integer may end with which of the following two digit pairs: 07, 29, 41, 63, or
85? (Your answer may include several of these.)
25. In how many ways can three squares be chosen from a five by five grid of squares so that no
two chosen squares lie in the same row or column?
26. Find the measure, in degrees, of the smallest positive angle  for which sin 3  = cos 7  .
27. Find the average of all distinct four-digit numbers formed by permuting the digits of 1993.
28. The sum of the charges of the quarks in a particle gives the overall charge of the particle. Two
up quarks and a down quark makes a proton, which has a charge 1. On the other hand, two
down quarks and an up quark make a neutron, which has charge 0. What is the charge of an up
quark?
29. Jayne writes the integers from 1 to 2000 on a piece of paper. She erases all the multiples of 3,
then all the multiples of 5, and so on, erasing all the multiples of each odd prime. How many
numbers are left when she finishes?
30. A point is chosen at random inside a square of side length 2 cm. What is the probability that the
1
point is within cm of at least one of the sides?
4
31. The population of Kalispell, Montana decreased 20% during 2004, but then increased by
2
16 % during 2005. Overall, the population dropped by a total of 3000 people during this two3
year period. What was Kalispell’s population at the beginning of 2004?
32. In a round robin tournament, each player competes against every other player exactly once. If
every player wins an even number of games, what is the smallest number of players there could
be in the tournament?
33. Goldbach’s Conjecture asserts that every integer n  4 can be written as the sum of two primes.
How many ways are there to do this when n = 32?
34. Quadrilateral ABCD has side lengths AB = 6, BC = 7, CD = 8, and AD =
9. A circle is drawn that is tangent to AD , AB , and BC , as shown. If X
and Y are the points of tangency to AD and BC , then compute CY +
DX.
30 miles
per hour
35. Let C be the sum of the first 100 positive even numbers and let D be the sum of the first 100
positive odd numbers. Calculate (C + D)/(C – D).
201
2/3
21/40
3/2
2521
29 and 41
600
9o
6110.5
2/3
24
7/16
45,000
4
2
10
36. A real number b is chosen at random from the interval –3  b  3. Find the probability that x2 +
bx + 1 has two distinct roots.
37. Which number is larger,
3 6 2 or
1/3
6 2 3 ?
6 2 3
38. The two circles shown are concentric with radii of length 1 and 2. Find the
length of a chord of the larger circle which is tangent to the smaller circle.
2 3
39. Suppose that a1, a2,…, a9, are positive integers, each greater than 1, which satisfy
a1  a 2  3 a 3  …  9 a9 . Find the smallest possible value for a9.
512
40. Triangle ABC has a right angle C, m  B = 30o, and AB = 12. Let P be chosen at
random inside  ABC and draw cervian AD through P as shown, with D on
side BC . What is the probability that AD  6 2 ?
41. A quarter weighs the same as two pennies. If a pound of quarters is worth $25, then how much
is pound of pennies worth? (One quarter = $0.25; one penny = $0.01)
42. A flag stands at each corner of a 100 meter by 60 meter rectangular field. Jessica stands at a
point 24 meters from one of the short sides and 7 meters from one of the long sides. What is the
least distance she could run to grab all four flags?
43. Alessandra draws 300 circles with diameter one, Ben draws 400 squares with side length one,
and Cyril draws 500 equilateral triangles with side length one. If all use the same type of pen,
who uses the most ink?
44. At a certain time, Janice notices that her digital watch reads a minutes after two o’clock.
Fifteen minutes later, it reads b minutes after three o’clock. She is amused to note that a is six
times greater than b. What time was it when she looked at her watch for the second time?
45. How many of the following numbers get smaller when squared:
1
1
6
3
- ,
0,
,
,
,
5
2
3
7
2
46. Tessa takes a square piece of paper, ABCD, and cuts it along a line running from A to the
midpoint of BC. What is the area of the larger of the two resulting pieces if AB = 4?
47. How many natural numbers n less than 100 have exactly one divisor other than 1 and n?
48. A quarter ($0.25), a dime ($0.10), and a nickel ($0.05) lie heads up on a table. Raven chooses
any two coins and flips them over. She again flips over any two coins, then repeats this process
one more time. Finally, she pockets the coins which are now heads up. What is the most
money she could have taken?
49. What is the positive difference between the two roots of x2 – 5x + 5?
50. Ben, Fang, and Venetia are playing a game in which a card numbered 2, 3, 4, 5, 6, 7, or 8 is
stuck to each of their foreheads, so that each player can see the other two numbers but not their
own. Joey walks in and observes loudly, "The three numbers are not all different, and the
product of the three numbers is a perfect square!" How many of the three players can now
deduce the numbers on their foreheads?
51. One afternoon Marija notices that the current time is 10% of the way from 3:00 to 4:00. What
fraction (in lowest terms) of the time has elapsed from 2:00 to 5:00?
52. The Lakers and the Heat each need to win two of the remaining three games to take the
Championship Series. The Lakers have probabilities of 2/5, 2/5, and 1/2 to win the fifth, sixth,
and seventh games, respectively. What is the probability that the Lakers win the Series?
3 /3
$2
245
meters
Ben
3:09
2
12
4
40 cents
($.40)
5
3 (all)
11/30
2/5
53. Each month in the country of Aibres, residents wait until the first Thursday, then have festivals
on every subsequent Tuesday until the end of the month. If a given month has 31 days, what is
the probability that there are four festival days during that month?
54. Dr. Diggler has three dials: the leftmost contains the digits 1 and 2, the middle shows the digits
0, 4 and 8, and the rightmost has the digits 3, 5, 6, and 7. How many three-digit prime numbers
can he create using the dials?
5/7
3