Download (°1)+ - Art of Problem Solving

1.
If f (x) = x 2016 , evaluate f (°1) + f (0) + f (1).
2.
Given that the width-to-length ratio of the Romanian Flag is 2 : 3 ,
find the sum of the length and width of a Romanian Flag with area
216 square feet.
3.
What is the remainder when the sum of the squares of the first 100
odd numbers is divided by 10.
4.
Consider a barn with a square base with an area of 288. Brandon and
Barndon are located at opposite corners of the barn. If Brandon walks
at a speed of 3 feet per second , how long does Barndon have to wait
for Brandon to reach him, given that Brandon walks in a straight line
directly to Brandon?
5.
Alice and Bob are picking integers without replacement such that 1 ∑
n ∑ 10. Let the integer that Alice picks be a , and let the integer that
Bob picks be b . What is the probability that ab = 24? Express your
answer as a common fraction.
6.
Consider two sequences of positive integers. Sequence a is defined
as a n = n 2 , and the sequence b is defined as b n = 2n°1 . The first few
terms of each sequence are listed as a 1 = 1.a 2 = 4, a 3 = 9; b 1 = 1, b 2 =
2, b 3 = 4. Find the maximum possible value of a n /b n , and express it
as a common fraction in simplest terms.
2016 NATIONAL MATHCOUNTS MOCK Sprint Round - 04/13
7.
Define a sh0rk-fin to be the shaded region of the circle below, when
6 AOB = 60 degrees, BC ? AO and O is the center of the circle. If the
area ofpthe h0rk-fin of a circle of area 36º can be expressed in the form
aº °
b c
d
find a + b + c + d .
a+b
a°b
= 2016 and a 2 ° b 2 = 1 , find (a + b)2 .
8.
Given that
9.
Find the area of the octagon below, given that each side length shown
p
is 10.
10.
Consider the set S = {A, B,C , D, E }. We define a bad subset of S to be a
subset containing 2 or more of the elements from the set T = {B, A, D}.
Find the number of bad subsets of S.
11.
In the below diagram, three concentric circles are depicted. The areas
of the unshaded region, the blue region, and the red region are all
equal to 4º. If a, b and c are the lengths of the radii of the three circles,
then find a + b + c and express your answer in simplest radical form.
2016 NATIONAL MATHCOUNTS MOCK Sprint Round - 04/13
12.
Given P A = 3, P D = P B and 6 AP D = 60 degrees , find the area of
rectangle ABC D in simplest radical form.
13.
Consider the increasing sequence of positive integers that have exactly 13 factors. Let the remainder when the 13th term of this sequence is divided by 13 be x. Find 13x.
14.
What is the positive difference between the measure of the acute angle formed by the minute and hour hand of a clock at 4:20, and the
measure of the acute angle formed by the minute and hour hand of a
clock at 4:30 ?
15.
An unfair coin has a 2/3 chance of landing heads and a 1/3 chance of
landing tails. Graham flips n of these coins. The chances that n ° 2 of
these coins land heads and 2 of these coins land tails is 8/27. Find n.
16.
Consider the rectangle with vertices (2,3), (2,6), (6,6), (6,3) . Find the
slope of the line through the origin that bisects the area of this rectangle, and express your answer as a fraction in simplest form.
17.
Let ABC DE F be a regular hexagon with side length 6 , and let P be the
intersection of lines BC and F D. Let Q be the intersection of lines C D
and AP . Find the length of BQ , and express your answer in simplest
radical form.
2016 NATIONAL MATHCOUNTS MOCK Sprint Round - 04/13
18.
Two teams, Team Sh0rk and Team Shark, play a best-of-5 series (a
series that ends when one team wins three games). Given that the
probability that Team Sh0rk wins a game is 2/3 when they are ahead
in the series, and 1/2 when they are behind or they both have the
same number of game wins, find the probability that the series lasts 4
games long with Team Shark winning. Express your answer as a common fraction,
19.
Let the factor polynomial of an integer k be f k (x) = a 1 x n°1 + a 2 x n°2 +
· · · + a n°1 x + a n . where a 1 , a 2 , . . . , a n are the positive factors of k in
increasing order, and n is the number of positive factors of k. For
example, the factor polynomial of 15 would be f 15 (x) = x 3 +3x 2 +5x +
15. Evaluate
( f 2 (°2) + f 4 (°2) + f 8 (°2) + f 16 (°2) + f 32 (°2) + f 64 (°2) + f 128 (°2)
20.
Given that ABC D is a square with side length 6 , and that the total
area of the net below is 84, find the volume of the pyramid created by
folding the net into a square-base pyramid and express your answer
in simplest radical form.
21.
Wu picks a random one digit positive integer A. Wuu picks a random
two digit integer B . What is the probability that |A ° B | ∑ 50 ? Express
your answer as a common fraction in simplest form.
2016 NATIONAL MATHCOUNTS MOCK Sprint Round - 04/13
22.
Consider the first 9 rows of Pascal’s Triangle, listed below, where each
number is the sum of the two numbers above it to the left and right.
Let a row with n numbers be row n ° 1, and denote by S n the sum of
the numbers in row n. Find 9S 0 + 8S 1 + · · · + 2S 7 + S 8
1
1
1
1
5
7
1
1
8
28
3
10
15
56
1
4
10
20
35
21
1
6
4
6
1
2
3
1
1
1
5
15
35
70
1
1
6
7
21
56
1
28
1
8
1
23.
How many positive integers less than 1000 have at least one 2 among
their digits, but no 3s among their digits?
24.
Consider a triangle ABC with medial triangle DE F , with D on BC , E
on AC , and F on AB (The medial triangle of a triangle is formed by
joining the midpoints of each side of the triangle). Consider the centroids X ,Y , and Z of the medial triangles of triangles AE F , B F D,and
[X Y Z ]
C DE . Find the ratio of
. (Here [X Y Z ] denotes the area of
[ABC ]
4X Y Z .)
25.
We define a trash triple of integers to be a triple (x 1 , x 2 , x 3 ) such that
x 1 + x 2 + x 3 = 0 . Find the number of trash triples (a, b, c) where ab =
2c ° 4a .
26.
How many positive integers exist such that n ∑ 256 and 1/n can be
represented as a terminating decimal when written in base 6?
2016 NATIONAL MATHCOUNTS MOCK Sprint Round - 04/13
27.
Sh0rkWuTer has 10 cards laid out on a table, numbered 1, 2, . . . , 9, 10.
Each minute, he takes two arbitrarypcards numbered a and b , and
replaces them with a card numbered a 2 + b 2 . What is the maximum
possible value of the number of the remaining card he is left with after
9 minutes? Express your answer in simplest radical form.
28.
Consider the 4 by 4 grid of squares depicted below. Four of the 1 by 1
squares are shaded in at random. Find the probability that the centers
of the 4 squares form a parallelogram with exactly one pair of sides
parallel to the sides of the squares.Express your answer as a common
fraction.
29.
Depicted below is a rectangle ABC D , equilateral triangle E FC , and
p
a circle tangent to F E ,AF ,AB , and F E . Let
p F C = 36 3 . If the area
of rectangle ABC D can be expressed as a b + c in simplest radical
form, find a + b + c.
30.
Consider the set of positive integers S = {1, 2, 3, . . . , n}. Consider the
smallest possible value of n such that we can assign 5 integers of set S
to vertices A ,B ,C ,D ,and E and 5 integers of set S to vertices F,G, H , I
and J (that have not already been assigned to vertices A ,B ,C ,D ,and
E ), such that the product of the integers at A ,B ,C ,D ,and E is equal
to the product of the integers at F,G, H , I and J . Find the largest possible product of the integers at vertices A ,B ,C ,D ,and E and for this
value of n.
2016 NATIONAL MATHCOUNTS MOCK Sprint Round - 04/13