Download Problems for the test

The first five years – questions 1,2 and 3,4 (for the junior division)
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*
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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?
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.
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?

3n + 5n
Determine the value of the infinite series

8n
n 0
Find the largest integer n for which 12n evenly divides 20!
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?
If we set C = 19!, then express 21! –20! In terms of C.
Find the hypotenuse of a right triangle whose legs are 20,806 and 408
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
We define a * b to equal
. Evaluate
.
ab
5*7
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?
A square with side length 1 is rotated about one vertex by an angle  , where 0o <  < 90o and
4
cos  = . Find the area of the shaded region that is common to both squares. ***
5
What is the greatest possible number of points of intersection among four lines and a circle
in the plane?
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?
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}?
Find the maximum possible value for x + y given that 3x +2y  7 and 2x + 4y  8
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).
Between which two consecutive integers is 999 ?
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?
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?
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
27
(-1,1,1)
290
64/15
8
6:08 pm
400C
20,810
2
7/3
33
1/2
14
3
7
11/4
Andy and
Clair
31 and 32
16
91/216
(c 5 )/2
terms of c?
jr
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. ***
jr
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
jr
Solve for x in the equation 2(16 ) = 16(2 )
Jr/sr 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?
jr
Define an operation * by declaring that a * b = (a + b)/(a – b). Find a number x such that 3 *
x = 3.
jr
Find the smallest positive integer greater than 1 which leaves a remainder of 1 when divided
by 2, 3, …, 8, and 9.
jr
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.)
Jr/sr 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?
sr
Find the measure, in degrees, of the smallest positive angle  for which sin 3  = cos 7  .
jr
Find the average of all distinct four-digit numbers formed by permuting the digits of 1993.
jr
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?
jr
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?
jr
A point is chosen at random inside a square of side length 2 cm. What is the probability that
1
the point is within cm of at least one of the sides?
4
Jr/cd Find the smallest positive integer which can be written as a product of two factors (each
greater than 1) in exactly three different ways. For example, 70 is one such integer since 70
= 2*35 = 5*14 = 7*10. Note that the order of the factors does not matter.
jr
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
3
two-year period. What was Kalispell’s population at the beginning of 2004?
jr
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?
jr
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?
jr
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. ****
Jr/sr 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).
52 (units
squared)
30 miles
per hour
2/3
21/40
3/2
2521
29 and 41
600
9o
6110.5
2/3
24
7/16
24
45,000
4
2
10
201
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.
jr
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Jr/sr
Which number is larger, 3 6 2 or 6 2 3 ?
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. ***
Suppose that a1, a2,…, a9, are positive integers, each greater than 1, which satisfy
1/3
6 2 3
2 3
512
9
a1  a  3 a  …  a . Find the smallest possible value for a9. ***subscripts**
Jr/sr 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 ? ***
Jr/cd Suzy drives around the equator of a small planet. If her car’s tires have a diameter of 100cm
and the planet is spherical with radius 1000km, then how many rotations do her wheels
make during the trip?
Jr/cd Find the maximum value that the expression (a – b)/ c attains if a, b, and c are distinct
integers greater than 100 but less than 200.
sr
If the parabola defined by y = ax2 + 6 is tangent to the line y = x, then calculate the constant
a.
Jr/sr Marija is really bored, so she writes all the integers from 1 to 1,000,000. How many times
does she write the digit 1?
Jr/cd The amount of current which flows through a wire is directly proportional to the wire’s
conductance G. Two wires have G = 1 and G = 10 and together carry a current of 10
amperes. How much current (in amperes) does the wire with the lower conductance carry?
Jr/sr Two teams compete in a relay race; they are composed of four members, each of whom runs
100 meters. All the members of team B run at b meters per second (mps). The first three
members of team A can only run 5b/6 mps, while the fourth member runs at a mps. Find a/b
if the race is a tie.
3 /3
2,000,000
49/51
1/24
600,001
10/11
5/2
Proof round
1. Prove that p2 – 1 is divisible by 24 if p is a prime not less than 5.
Solution
If p is prime not less than 5 than p is odd.
We can therefore represent p = 2a + 1, so
p2 -1 = (2a + 1)2 – 1
= 4a2 +4a
= 4a(a +1)
Which is divisible by 8 as a(a + 1) is even ( the product of two consecutive integers is even)
It is true that exactly one of p -1, p, p + 1 is divisible by 3. But p is prime, and not less than 5, so p can’t be
divisible by 3.
Thus, one of p – 1, p + 1
is divisible by 3
Therefore (p – 1)(p + 1)
is divisible by 3
. ..
p2 – 1
is divisible by 3
. . . p2 – 1
is divisible by 8*3 = 24.
2. Show that the equation
x - 11 = y has an infinite number of solutions where x and y are integers.
x
Find the solution where the ratio
has the greatest possible value.
y
Solution
If
x - 11 =
y then
( x - 11 )2 = y
. ..
x - 2 11x + 11 = y
.
. .
x + 11 - y = 2 11x
Thus 11x must be a perfect square.
Hence x = 11n2 for integer n, and when this is so,
x + 11 – y = 22n
.
. .
11n2 + 11 – y = 22n
. . . y = 11n2 – 22n + 11
. . . y = 11( n – 1)2
Thus, as n  1 can take infinitely many values, there are infinitely many solutions of the form
x = 11n2 , y = 11(n – 1)2 .
2
1 
n 2 
n2
x
Now
=
=(
) = 1 
 4
2
n 1
y
(n  1)
 n 1
3. green book Competition Mathematics (part 2)
solution is on page 124 problem 4.10
ABCD is a cyclic quadrilateral. The edges AB, DC produced intersect in R and DA, CB produced in S.
Prove that the bisectors of angle BRC and ASB are perpendicular.
**** scan the picture on p18****
4. green book Competition Mathematics (part 2)
solution is on page 133 problem 4.40
ABCD is a parallelogram. BC is produced to E such that BC = CE. F is the midpoint of DC. AF and DB
1
intersect at G. Show that the area of  DFG is exactly
of the area of the entire parallelogram.
12
5. green book Competition Mathematics (part 2)
solution is on page 224 problem 16.16
Let x, y, and z be positive real numbers with sum 1. Prove that
1
1
1
+
+
 9.
x
z
y