Download ( ) 2016 Math Exploration Day Team Competition

2016 Math Exploration Day
Team Competition
1.
For real numbers x and y , define
Calculate
2.
x ⊕ y = xy − x − y
(((4 ⊕ 3) ⊕ 2) ⊕ 2) .
Two fair dice are tossed. If at least one of the dice shows the number 4, what is the
probability that the sum of the two dice is greater than 7?
3. The product
5
8 × 3 16 can be expressed as 2 n . What is the value of n ?
4.
Detective Teresa receives a piece of chocolate every day plus 2 extra pieces of chocolate
for every mystery that she solves. For October 1 to January 31, inclusive, she received
321 pieces of chocolate. How many mysteries did she solve?
5.
If
6.
What is the digit in the units place of the integer
7.
Farmer Dave has chickens, cows, and horses. Altogether, these animals have 76 feet and
24 heads. If Farmer Dave has 2 more horses than cows, how many cows does he have?
317 + 317 + 317 = 3k , what is k ?
3444 ?
2
?
x
8.
How many ordered pairs of integers (x, y) satisfy the equation y =
9.
A fair die is tossed 6 times. What is the probability that at least 1 six appears?
10.
Find the distance between the centers of two circles whose equations are
x 2 + y 2 − 8 x − 8 y + 16 = 0 and x 2 + y 2 + 6 x + 6 y + 9 = 0 .
11.
To estimate the number of fish in a small lake, researchers caught 200 fish, tagged them,
and returned them to the lake. The next day they caught 100 fish. Of those caught, 25
had a tag. What would be a good guess for the total number of fish in the lake?
12.
The Fibonacci sequence begins with two 1s. Every term after these two is the sum of the
previous two terms. Thus, the sequence begins 1, 1, 2, 3, 5, 8, 13, ....
Let a, b, c, and d be four consecutive terms of the Fibonacci sequence.
If a + b + c = 3194 and b + c + d = 5168, find the value of a.
13.
In one league, soccer teams earn 3 points for a win and 1 point for a draw over the course
of a 38-game season. One team finished the season with 72 points, having 12 more wins
than draws. What was its overall win-loss-draw record for the season?
14.
Express as a fraction in lowest terms:
1 + 2 + 3 + ⋅ ⋅⋅ + 2016
1 + 2 + 3 + ⋅ ⋅⋅ + 2015
15.
The 5 × 5 array of dots represents trees in an orchard.
If you were standing at the central spot marked with an X,
you would not be able to see 8 of the 24 trees (shown as
open circles). If you were standing at the center of a 9 × 9
array of trees, how many of the 80 trees would be hidden?
16.
The point (6, 1) is reflected over the line 2y - x = 6. Find the coordinates of its image.
17.
It takes Megan 3.5 hours to prepare a large flower delivery for sale as small bouquets.
When Ted helps her, the two complete the job in 2 hours. If Ted had to do the entire job
by himself, how much time would he need?
18.
If x +
y = 7 and xy = 28 , what is the sum of the reciprocals of x and y?
19.
Find the perimeter of a right triangle whose area is 210 and whose hypotenuse is 37.
20.
Consider the equilateral triangle centered at the origin. Find the proportion of the
triangle’s area that lies below the x-axis.