Fall 2010 McNabb GDCTM Contest
Geometry
NO Calculators Allowed
1. An automobile goes y/9 yards in d seconds. How many feet does it travel
in two minutes time?
40y 40d 3y 120y
(A) (B) (C) (D) 120yd (E)
d 3y 40d d
2. The well-known formula f = (9/5)c + 32 relates the temperature f in
Fahrenheit to the temperature c in Celcius. For how many values of f
satisfying 32 ≤ f ≤ 212, will the temperature be an integer in both of these
scales?
(A) 9 (B) 10 (C) 19 (D) 20 (E) 21
3. Suppose the converse of the following statement is true:
If Zerb is from Xanlor, then Zerb is blue.
Which of the following statements must be true?
I. If Zerb is not from Xanlor, then Zerb
is not blue.
II. If Zerb is from Xanlor, then Zerb is
blue.
III. If Zerb is blue, then Zerb is not from
Xanlor.
(A) I only (B) II only (C) III only
(D) I and II only (E) I and III only
4. At Zeke’s Zucchini Stand, 3 zucchini’s and 2 squash cost $4.75, while 2
zucchini’s and 3 squash cost $5.25. How much would 3 zucchini’s and 3
squash cost?
(A) $5.50 (B) $5.75 (C) $6 (D) $6.25 (E) $6.50
Fall 2010 Geometry 1
5. A square is inscribed in a right triangle with sides of length 3, 4, and 5, so
that one of the sides of the square is contained in the hypotenuse of the
right triangle. What is the side length of the square?
60 12
(A) (B) 2 (C) (D) 3 (E) cannot be determined
37 5
6. In how many ways can the the letters in the string ABECEDA be arranged
so that the consonants are in alphabetical order?
(A) 90 (B) 105 (C) 120 (D) 180 (E) 210
7. Points A, B, C, and D are collinear and occur in the same order as given.
If the ratio AB : BC equals 3 and the ratio BD : AB equals 8/3, then
determine the ratio CD : AC.
5 3 7 8
(A) (B) (C) 2 (D) (E)
3 2 4 3
8. A baseball team has won 50 games out of 75 so far played. If there are 45
games yet to be played, how many of these must be won in order for the
team to finish its season having won exactly 60% of its games?
(A) 20 (B) 21 (C) 22 (D) 23 (E) 72
9. In 4 ABC, ∠ A = 60◦ , ∠C = 40◦ , BD ⊥
−→
AC and BE bisects ∠ ABC. Find the
measure of ∠ DBE in degrees. A C
D E
(A) 8 (B) 10 (C) 12 (D) 14 (E) 20
10. A bag contains 4 quarters and 2 dimes. If 3 coins are randomly removed
from the bag, what is the expected total value in cents of these three coins?
(A) 50 (B) 55 (C) 60 (D) 65 (E) 75
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11. The midpoints of the sides of a triangle are (7, 4), (1, 2), and (1, 6). What
is the area of this triangle?
(A) 12 (B) 24 (C) 30 (D) 36 (E) 48
12. On co-planar lines l and m we choose points P1 , P2 , P3 , P4 , and P5 on the
former; and points Q1 , Q2 , Q3 , Q4 on the latter. Draw all possible segments
with one endpoint one of the P’s and the other one of the Q’s. What is
the maximum total number of points that can be formed by intersection of
pairs of these segments?
(A) 75 (B) 60 (C) 45 (D) 30 (E) 20
13. The set S contains seven numbers whose mean is 202. The mean of the
four smallest numbers in S equals 100, while the mean of the four largest
numbers in S equals 300. What is the median of all the numbers in S?
(A) 184 (B) 186 (C) 192 (D) 196 (E) 200
A C
14. Quadrilaterals ABCD and BEGF are
rhombi and are situated as in the dia- E F
D
gram. If ∠EBF = 20◦ and ∠ A = 50◦ ,
what is ∠ DEG?
G
(A) 40◦ (B) 45◦ (C) 50◦ (D) 55◦ (E) 60◦
15. While Xerxes marched on Greece his army streched out for 50 miles. A
dispatch rider had to ride from the rear to the head of the army, then
instantly turn about and return to the rear. While he did this, the army
advanced 50 miles. How many miles did the rider ride?
√ √ √
(A) 100 (B) 50 + 50 2 (C) 100 2 (D) 150 (E) 50 + 100 2
Fall 2010 Geometry 3
16. How many non-congruent scalene triangles with integer side lengths exist
with two sides of lengths 13 and 7 respectively?
(A) 10 (B) 11 (C) 12 (D) 13 (E) 14
17. An isosceles trapezoid has bases of 11 and 21 units and legs of 13 units.
What is the area of the trapezoid?
(A) 144 (B) 160 (C) 176 (D) 192 (E) 208
18. Hezy and Zeke were employed at different daily wages. At the end of a
certain number of days Hezy received $300, while Zeke, who had been
absent from work two of those days, received only $192. However, had
it been the other way around, had Zeke worked all those days and Hezy
been absent twice, then both would have received the same amount. What
was Hezy’s daily wage?
(A) 30 (B) 40 (C) 50 (D) 60 (E) 70
19. Suppose that the two legs of a certain right triangle are in the ratio 3 : 4.
What is the greatest possible area of such a right triangle, if one of its
altitudes measures 24?
(A) 216 (B) 384 (C) 486 (D) 600 (E) 726
20. A five-digit integer, with all distinct digits which in this problem must
be 1,2,3,4, and 5 in some order, is called alternating if the digits alternate
between increasing and decreasing in size as read from left to right. They
may start on an increasing or decreasing foot. For instance, both 34152 and
53412 are alternating while 12354 is not, for example. How many of this
kind of 5 digit integer are alternating?
(A) 32 (B) 28 (C) 24 (D) 20 (E) 16
Fall 2010 Geometry 4
