Download Solutions to Cal Math League Contest 2 of 28 Nov 2006

ACHS Math Team
Solutions to Math League Contest 2 of 28 Nov 2006
Peter S. Simon
Problem 2-1
What is the only number less than
√
2006 whose square is 2006?
Problem 2-1
What is the only number less than
√
2006 whose square is 2006?
There
is equal to 2006. They
√ numbers whose
√ are only two
√ square √
are
2006
and
−
2006.
Since
−
2006
<
2006, the answer is
√
− 2006.
Problem 2-2
A rectangle is divided into two
congruent trapezoids, as shown. The
lengths of the legs of both trapezoids
are 4 and 5. The length of the shorter
base of both trapezoids is 3. What is
the area of the rectangle?
3
5
4
3
4
Problem 2-2
A rectangle is divided into two
congruent trapezoids, as shown. The
lengths of the legs of both trapezoids
are 4 and 5. The length of the shorter
base of both trapezoids is 3. What is
the area of the rectangle?
3
3
5
4
3
Draw perpendiculars as shown. Each is of length 4.
4
3
Problem 2-2
A rectangle is divided into two
congruent trapezoids, as shown. The
lengths of the legs of both trapezoids
are 4 and 5. The length of the shorter
base of both trapezoids is 3. What is
the area of the rectangle?
3
3
3
5
4
3
4
3
3
Draw perpendiculars as shown. Each is of length 4.
Middle line segments must be of length 3 by Pythagorean theorem.
The area of rectangle is then ((3 + 3 + 3)(4) = 9 × 4 = 36
Problem 2-3
One 4-tuple that satisfies a2 + b2 + c 2 + d 2 = abcd is (2, 2, 2, 2).
What is the largest number x for which (2, 2, 2, x) satisfies
a2 + b2 + c 2 + d 2 = abcd?
Problem 2-3
One 4-tuple that satisfies a2 + b2 + c 2 + d 2 = abcd is (2, 2, 2, 2).
What is the largest number x for which (2, 2, 2, x) satisfies
a2 + b2 + c 2 + d 2 = abcd?
22 + 22 + 22 + x 2 = 8x
Problem 2-3
One 4-tuple that satisfies a2 + b2 + c 2 + d 2 = abcd is (2, 2, 2, 2).
What is the largest number x for which (2, 2, 2, x) satisfies
a2 + b2 + c 2 + d 2 = abcd?
22 + 22 + 22 + x 2 = 8x
12 + x 2 = 8x
Problem 2-3
One 4-tuple that satisfies a2 + b2 + c 2 + d 2 = abcd is (2, 2, 2, 2).
What is the largest number x for which (2, 2, 2, x) satisfies
a2 + b2 + c 2 + d 2 = abcd?
22 + 22 + 22 + x 2 = 8x
12 + x 2 = 8x
x 2 − 8x + 12 = 0
Problem 2-3
One 4-tuple that satisfies a2 + b2 + c 2 + d 2 = abcd is (2, 2, 2, 2).
What is the largest number x for which (2, 2, 2, x) satisfies
a2 + b2 + c 2 + d 2 = abcd?
22 + 22 + 22 + x 2 = 8x
12 + x 2 = 8x
x 2 − 8x + 12 = 0
(x − 2)(x − 6) = 0
Problem 2-3
One 4-tuple that satisfies a2 + b2 + c 2 + d 2 = abcd is (2, 2, 2, 2).
What is the largest number x for which (2, 2, 2, x) satisfies
a2 + b2 + c 2 + d 2 = abcd?
22 + 22 + 22 + x 2 = 8x
12 + x 2 = 8x
x 2 − 8x + 12 = 0
(x − 2)(x − 6) = 0
so x = 2 or x = 6
Problem 2-4
I filled 49 packages with big and small pens. No package was left
empty, and all the packages were filled differently. For example, a
package could have been filled with 3 pens in only 4 ways: 3 big, 2
big and 1 small, 1 big and 2 small, 3 small. What is the least total
number of pens I could have put in these packages?
5
Problem 2-4
I filled 49 packages with big and small pens. No package was left
empty, and all the packages were filled differently. For example, a
package could have been filled with 3 pens in only 4 ways: 3 big, 2
big and 1 small, 1 big and 2 small, 3 small. What is the least total
number of pens I could have put in these packages?
There are 2 ways to package 1 pen (1 big, 1 small).
5
Problem 2-4
I filled 49 packages with big and small pens. No package was left
empty, and all the packages were filled differently. For example, a
package could have been filled with 3 pens in only 4 ways: 3 big, 2
big and 1 small, 1 big and 2 small, 3 small. What is the least total
number of pens I could have put in these packages?
There are 2 ways to package 1 pen (1 big, 1 small). There are 3
ways to package 2 pens (3 big, 2 big and 1 small, 1 big and 2
small, 3 small).
5
Problem 2-4
I filled 49 packages with big and small pens. No package was left
empty, and all the packages were filled differently. For example, a
package could have been filled with 3 pens in only 4 ways: 3 big, 2
big and 1 small, 1 big and 2 small, 3 small. What is the least total
number of pens I could have put in these packages?
There are 2 ways to package 1 pen (1 big, 1 small). There are 3
ways to package 2 pens (3 big, 2 big and 1 small, 1 big and 2
small, 3 small). In general, if a package is to contain n pens, we
can place 0, 1, 2, . . . , n − 1, or n big pens in it (with the remaining
pens small). So in general there are n + 1 ways to fill a bag with n
pens.
5
Problem 2-4 (Continued)
Let’s make a table showing the number of pens used to fill
packages with increasing amounts of pens:
# Pens/Pkg
1
# Pkgs Used
2
# Pens Used
2
Total # Pens
2
Total # Pkgs
2
Problem 2-4 (Continued)
Let’s make a table showing the number of pens used to fill
packages with increasing amounts of pens:
# Pens/Pkg
1
2
# Pkgs Used
2
3
# Pens Used
2
6
Total # Pens
2
8
Total # Pkgs
2
5
Problem 2-4 (Continued)
Let’s make a table showing the number of pens used to fill
packages with increasing amounts of pens:
# Pens/Pkg
1
2
3
# Pkgs Used
2
3
4
# Pens Used
2
6
12
Total # Pens
2
8
20
Total # Pkgs
2
5
9
Problem 2-4 (Continued)
Let’s make a table showing the number of pens used to fill
packages with increasing amounts of pens:
# Pens/Pkg
1
2
3
4
# Pkgs Used
2
3
4
5
# Pens Used
2
6
12
20
Total # Pens
2
8
20
40
Total # Pkgs
2
5
9
14
Problem 2-4 (Continued)
Let’s make a table showing the number of pens used to fill
packages with increasing amounts of pens:
# Pens/Pkg
1
2
3
4
5
# Pkgs Used
2
3
4
5
6
# Pens Used
2
6
12
20
30
Total # Pens
2
8
20
40
70
Total # Pkgs
2
5
9
14
20
Problem 2-4 (Continued)
Let’s make a table showing the number of pens used to fill
packages with increasing amounts of pens:
# Pens/Pkg
1
2
3
4
5
6
# Pkgs Used
2
3
4
5
6
7
# Pens Used
2
6
12
20
30
42
Total # Pens
2
8
20
40
70
112
Total # Pkgs
2
5
9
14
20
27
Problem 2-4 (Continued)
Let’s make a table showing the number of pens used to fill
packages with increasing amounts of pens:
# Pens/Pkg
1
2
3
4
5
6
7
# Pkgs Used
2
3
4
5
6
7
8
# Pens Used
2
6
12
20
30
42
56
Total # Pens
2
8
20
40
70
112
168
Total # Pkgs
2
5
9
14
20
27
35
Problem 2-4 (Continued)
Let’s make a table showing the number of pens used to fill
packages with increasing amounts of pens:
# Pens/Pkg
1
2
3
4
5
6
7
8
# Pkgs Used
2
3
4
5
6
7
8
9
# Pens Used
2
6
12
20
30
42
56
72
Total # Pens
2
8
20
40
70
112
168
240
Total # Pkgs
2
5
9
14
20
27
35
44
Problem 2-4 (Continued)
Let’s make a table showing the number of pens used to fill
packages with increasing amounts of pens:
# Pens/Pkg
1
2
3
4
5
6
7
8
# Pkgs Used
2
3
4
5
6
7
8
9
# Pens Used
2
6
12
20
30
42
56
72
Total # Pens
2
8
20
40
70
112
168
240
Total # Pkgs
2
5
9
14
20
27
35
44
In order to use all 49 packages, we will fill the remaining 5
packages with 9 pens each, using up 45 more pens. The total
number of pens uses is therefore at least
Problem 2-4 (Continued)
Let’s make a table showing the number of pens used to fill
packages with increasing amounts of pens:
# Pens/Pkg
1
2
3
4
5
6
7
8
# Pkgs Used
2
3
4
5
6
7
8
9
# Pens Used
2
6
12
20
30
42
56
72
Total # Pens
2
8
20
40
70
112
168
240
Total # Pkgs
2
5
9
14
20
27
35
44
In order to use all 49 packages, we will fill the remaining 5
packages with 9 pens each, using up 45 more pens. The total
number of pens uses is therefore at least
240 + 45 = 285.
Problem 2-5
What is the only pair of real numbers (a, b) that satisfies both
a3 + ab2 = 30
and b3 + ba2 = 90?
Problem 2-5
What is the only pair of real numbers (a, b) that satisfies both
a3 + ab2 = 30
a(a2 + b2 ) = 30,
and b3 + ba2 = 90?
b(a2 + b2 ) = 90,
Problem 2-5
What is the only pair of real numbers (a, b) that satisfies both
a3 + ab2 = 30
a(a2 + b2 ) = 30,
and b3 + ba2 = 90?
b(a2 + b2 ) = 90, =⇒ b/a = 3
Problem 2-5
What is the only pair of real numbers (a, b) that satisfies both
a3 + ab2 = 30
a(a2 + b2 ) = 30,
a3 + a(3a)2 = 30
and b3 + ba2 = 90?
b(a2 + b2 ) = 90, =⇒ b/a = 3
Problem 2-5
What is the only pair of real numbers (a, b) that satisfies both
a3 + ab2 = 30
a(a2 + b2 ) = 30,
and b3 + ba2 = 90?
b(a2 + b2 ) = 90, =⇒ b/a = 3
a3 + a(3a)2 = 30 ⇐⇒ a3 (1 + 9) = 30
Problem 2-5
What is the only pair of real numbers (a, b) that satisfies both
a3 + ab2 = 30
a(a2 + b2 ) = 30,
and b3 + ba2 = 90?
b(a2 + b2 ) = 90, =⇒ b/a = 3
a3 + a(3a)2 = 30 ⇐⇒ a3 (1 + 9) = 30 ⇐⇒ a3 = 3
Problem 2-5
What is the only pair of real numbers (a, b) that satisfies both
a3 + ab2 = 30
a(a2 + b2 ) = 30,
and b3 + ba2 = 90?
b(a2 + b2 ) = 90, =⇒ b/a = 3
a3 + a(3a)2 = 30 ⇐⇒ a3 (1 + 9) = 30 ⇐⇒ a3 = 3 ⇐⇒ a =
p
3
3
Problem 2-5
What is the only pair of real numbers (a, b) that satisfies both
a3 + ab2 = 30
a(a2 + b2 ) = 30,
and b3 + ba2 = 90?
b(a2 + b2 ) = 90, =⇒ b/a = 3
a3 + a(3a)2 = 30 ⇐⇒ a3 (1 + 9) = 30 ⇐⇒ a3 = 3 ⇐⇒ a =
(a, b) =
p
p
3
3
3, 3 3
p
3
3
Problem 2-6
What is the area of a semicircular region tangent to two sides of a
unit square, with endpoints of its diameter on the other two sides?
Problem 2-6
What is the area of a semicircular region tangent to two sides of a
unit square, with endpoints of its diameter on the other two sides?
The hard part is drawing the
figure. . .
D
r
O
r
B
b
Problem 2-6
What is the area of a semicircular region tangent to two sides of a
unit square, with endpoints of its diameter on the other two sides?
Draw perpendiculars from O
(the center of the circle) to the
sides of the square.
C
D
r
O
F
b
r
B
A
E
Problem 2-6
What is the area of a semicircular region tangent to two sides of a
unit square, with endpoints of its diameter on the other two sides?
∠OBA ≡ ∠DOE =⇒ △OAB ∼ △DEO
C
OB ≡ OD
△OAB ≡ △DEO
D
AB = OE = 1 − r = OA
△OAB is right isosceles!
p
1 − r = r/ 2
p
p
2−r 2=r
√
√
p
1− 2
2
r=
√ ·
√ =2− 2
1+ 2 1− 2
p
Area = πr 2 /2 = π(3 − 2 2)
r
O
F
b
r
B
A
E