Download Junior Individual Test

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Junior Individual test
Multiple choice section
LIST ONLY THE LETTER OF YOUR ANSWER ON THE ANSWER SHEET
1.
2.
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
square
d)
Shaq scores an average of 18.6 points per game for five basketball 27
games. How many points must he score in the next game to raise
his average to 20 points per game?
A
a.
b.
c.
d.
e.
D
a.
b.
c.
d.
e.
3.
4.
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?
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?
2/3
B
a.
b.
c.
d.
e.
16
D
a.
b.
c.
d.
e.
5.
6.
7.
8.
9.
o
The point (1,1,1) is rotated 180 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.
(-1,1,1)
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?
A point is chosen at random inside a square of side length 2 cm.
1
What is the probability that the point is within cm of at least one
4
of the sides?
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.)
2
The increasing sequence of positive integers a1, a2, a3,… has the
property that an + 2 = an + an+1, for all n>=1. Suppose that a7 =
120. What is a8 ?
10. Nine chairs in a row are to be occupied by six students and
Professors Alpha, Beta, and Gamma. The three professors arrive
before the students and decide to choose their chairs so that each
professor will be between two students. In how many ways can
professors Alpha, Beta, and Gamma choose their chairs?
A
C
7/16
C
a.
b.
c.
d.
e.
a.
b.
c.
d.
e.
a.
b.
c.
d.
e.
29 and
41
A
194
D
60
C
a.
b.
c.
d.
e.
a.
b.
c.
d.
e.
a.
b.
c.
d.
e.
52
44
28
26
20
25
25.6
26
27
28
1
2/3
1/2
1/3
0
12
12.5
15
16
25
(-1,1,1)
(1,-1,1)
( 1,1,-1)
(-1,-1,1)
(-1,1,-1)
0
1
2
3
4
5/8
9/16
7/16
3/8
5/16
29 and 41
85 and 41
63 and 41
63 and 29
63 and 07
128
144
168
194
210
12
36
60
84
630
Page 2 of 2
Junior Individual Test
Short Answer Section
ALL ANSWERS MUST BE IN SIMPLIFIED AND REDUCED FORMS.
3
ALL EQUIVALENT ANSWERS WILL BE ACCEPTED (E.G.
AND 0.75).
4
ONLY ANSWERS ON THE ANSWER SHEET WILL BE MARKED.
1. 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?
2. Which number is larger,
6 2 3 ?
3 6 2 or
x
x
3. Solve for x in the equation 2(16 ) = 16(2 )
4. 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?
5. Find the largest integer n for which 12n evenly divides 20!
6. 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?
7. Define an operation # by declaring that a # b = (a + b)/(a – b). Find a number x such that 3 # x =
3.
8. Find the average of all distinct four-digit numbers formed by permuting the digits of 1993.
9. 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?
10. 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.
290
6 2 3
2/3
11/30
8
6:08 pm
3/2
6110.5
24
2 3