Download (1) A particular convex polygon with seven sides has exactly one

(1)
(2)
A particular convex polygon with seven sides has exactly one right angle.
How many diagonals does this seven-sided polygon have?
What integer is closest to the value of 10π +
√
99?
(3)
In a group of cows and chickens, the number of legs is 14 more than twice
the number of heads. How many cows are there in the group?
(4)
On Sunday, Jan. 7, 2007, Rob has $5 he saved from his birthday, and he
receives his first allowance payment of $35. He will continue to receive a $35 allowance
payment every Sunday. However, starting Jan. 7, 2007, Rob is also responsible for paying
for his five lunches each week at $3 per lunch and his weekly Saturday movie that costs
$9.50. If Rob always adds the remainder of his allowances each week to his current savings,
after how many total allowance payments will he have enough money to pay for the $50
CD player he wants and still be able to cover his expenses for the rest of that week?
(5)
Zan has created this rule for generating sequences of whole numbers:
If a number is 15 or less, triple the number.
Therefore, if Zan starts with 10, she
If a number is more than 15, subtract 13 from it.
gets the sequence 10, 30, 17, 4, 12, . . . If the first number in Zan’s sequence is 34, what
is the 8th number in the sequence?
(6)
Two boys (Jake and Miles) and two girls (Betty and Abby) are elected to
the four student council offices (president, vice-president, treasurer and secretary). If a girl
is elected president and Jake is elected vice-president, in how many ways can the four
students fill the four offices?
(7)
Each of four test scores in Connie’s class is to be weighted equally. On the
first three tests Connie scored 79%, 87%, and 98%. What percent must she score on her
fourth tests to have an overall average of exactly 90%?
(8)
It costs 2.5 cents to copy a page. How many pages can you copy for $20?
(9)
Quentin spent $480 to purchase 30 books. Using the same average price per
book, how many dollars will 45 books cost?
(10)
For what value of k does the line represented by the equation 1 − kx = −3y
contain the point (4, −3)?
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Answer Sheet
Number
1
2
3
4
5
6
7
8
9
10
Answer
14 diagonals
41
7 cows
5 payments
7
4 ways
96 percent
800 pages
720 dollars
-2
Problem ID
C4D
D121
D3CB
5D33
D221
23
0BB
0231
13D
C331
Copyright MATHCOUNTS Inc. All rights reserved
Solutions
(1) 14 diagonals
ID: [C4D]
For each vertex, we can create a diagonal by connecting it to any non-adjacent vertex. If
there are n vertices, there are n(n − 3) diagonals we draw. But we are over-counting by a
factor of 2 since each diagonal can be created from 2 vertices. So there are n(n − 3)/2
diagonals. In this problem, since n = 7, there are 7 · 4/2 = 14 diagonals.
(2) 41
ID: [D121]
No solution is available at this time.
(3) 7 cows
ID: [D3CB]
Let the number of cows be a. Each cow has one head and four legs. Let the number of
chickens be b. Each chicken has one head and two legs. The total number of legs is
4a + 2b. The total number of heads is a + b. Since the number of legs is 14 more than
twice the number of heads, we have the equation 4a + 2b = 14 + 2(a + b). Simplifying this
equation yields a = 7. Thus, there are 7 cows in the group.
(4) 5 payments
ID: [5D33]
Each week, he receives 35 dollars and must spend 24.5 dollars, so his net earnings are
35 − 24.5 = 10.5 dollars. When his net earnings are added to his savings, he must have at
least 50 dollars (he will still have enough money to cover his other expenses because we
factored that into his net earnings). Let the number of allowance payments be x. Rob has
5 + 10.5x dollars. In order for this to be at least 50, x must be at least 5, so Rob needs 5
allowance payments.
(5) 7
ID: [D221]
No solution is available at this time.
(6) 4 ways
ID: [23]
We can choose the president in 2 ways. Since Jake must be in the role of vice-president,
that leaves 2 choices for the treasurer, at which point the secretary is fixed. Thus there are
2 · 2 = 4 ways to fill the offices.
(7) 96 percent
ID: [0BB]
No solution is available at this time.
(8) 800 pages
ID: [0231]
Twenty dollars is 2000 cents. Since every page costs 2.5 cents, you can copy
2000/2.5 = 800 pages.
(9) 720 dollars
ID: [13D]
45 books cost 45/30 = 1.5 times as much as 30 books, or 1.5 · 480 = 720 dollars.
(10) -2
ID: [C331]
Since (4, −3) lies on the line, we plug x = 4 and y = −3 into the equation to get
1 − 4k = −3 · −3 ⇒ k = −2 .
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