Answers 2013

This post is backdated intentionally.


Problem 01: $\frac{163}{60}$
Problem 02: $\frac{9}{32}$
Problem 03: 6
Problem 04: 1620 feet
Problem 05: $\frac{15}{2}$ - NOT 7.5. "Rational number in lowest terms."
Problem 06: Mincle has 22 toes
Problem 07: 5604
Problem 08: 21 minutes
Problem 09: 73
Problem 10: $\frac{5}{6}$
Problem 11: $\sqrt{3}$
Problem 12: 132
Problem 13: $\frac{5}{12}$
Problem 14: 36 square units
Problem 15: $\frac{4\pi\sqrt{3}}{3}$
Problem 16: 15
Problem 17: $\frac{10}{3}$
Problem 18: $\frac{-11}{12}$
Problem 19: 18
Problem 20: $\frac{5}{8}$
Problem 21: $\frac{12}{7}$
Problem 22: 4500
Problem 23: 3375
Problem 24: 162 square units
Problem 25: 19
Problem 26: -8
Problem 27: 122 paths
Problem 28: $\frac{12}{5}$ units
Problem 29: 10 units
Problem 30: 2
Problem 31: (2,1) and (0,2)
Problem 32: 10004
Problem 33: $\frac{1}{66}$
Problem 34: 90
Problem 35: $6\sqrt{10}$
Problem 36: (2, 5/6)
Problem 37: 38
Problem 38: $\frac{3}{8}$
Problem 39: 6 square units
Problem 40: t=8, u=7
Problem 41: $\frac{34}{3}$

Sample box for copying purposes..

Answers to 2007

Problem 01: -14/29 just plug, crank, and pull.
Problem 02: 5/3

Conjugates are the red herring here.
(4√2 + √2)÷(4√2 - √2) = 5√2 ÷ 3√2 =5/3

Problem 03: 2

Draw a vertical line down the middle and the students will quickly see that the two halves are 2x2 and the red portion is 1+1 = 2

Problem 04: 112

Choose three consecutive integers: x-1, x, x+1
(x+1)³ - x³ = 666 + x³ - (x-1)³
x³ + 3x² + 3x +1 - x³ = 666 + x³ - x³ + 3x² - 3x + 1
Stuff cancels all over the place: 6x = 666; x = 111; Answer is 112

Problem 05: 1
Problem 06: 17

(x+2)² = x² + 8²
x² + 4x + 4 = 64 + x²
4x = 60; x=15; hypotenuse = 17. Even if you missed the Pythagorean triple.

Problem 07: 15
Problem 08: 36
Problem 09: 117
Problem 10: 59
Problem 11: 2

9^x + 2(3^x+2) = 243
3^2x + 18(3^x) = 243
complete the square for U = 3^x
u² + 18u + 81 = 243 + 81 = 324 = 18²
u + 9 = +- 18
3^x = 9, -27 .... 2 is only real answer.

Problem 12: 13/14
Problem 13: 930
Problem 14: 140√2

ab = 28; ac = 20; bc=70
a²b²c² = 28*20*70
abc = √(4*7*4*5*7*2*5) = 140√2

Problem 15: 6/31

In these simple dice rolling questions, I like to imagine the full sample space of 36 possibilities. We are told that 6 is not an option and that occurs 5 times, leaving 31 total.
11 12 13 14 15 16
21 22 23 24 25 26
31 32 33 34 35 36
41 42 43 44 45 46
51 52 53 54 55 56
61 62 63 64 65 66
Thus, P(7) = 6/31

Problem 16: 48
Problem 17: 62

Build a rectangle around the figure and subtract the four triangles.
10*10 - ½(4*7 + 3*7 + 3*3 + 6*3) = 62

Problem 18: 1/5
Problem 19: 6/55
Problem 20: 16
Problem 21: 10
Problem 22: 1/13
Problem 23: 8
Problem 24: 4013
Problem 25: 6

log₁₄₄(3^a * 2^b) = c
144^c = 3^a * 2^b
2^4c * 3^2c = 2^b * 3^a
b=4c and a=2c; answer 6

Problem 26: a=3, b=2/13
Problem 27: 1/6
Problem 28: 49
Problem 29: 5π/8
Problem 30: 300
Problem 31: 64/175
Problem 32: 18√3

ABC is a 30-60-90, with B=90^o. AC is 12, split 3,3, and 6. If the altitude is BN, then AN=3 and BN=3√3, making the area = 18√3

Problem 33: 3
Problem 34: 339

4a+4b+4c=124, so a+b+c=31. Square it
a²+ab+ac+ba+b²+bc+ca+cb+c²=961
2ab + 2ac + 2bc = 622
a² + b² + c² = 339, which is also d²

Problem 35: 250

Assume first that p is odd. The list of integers is symmetric:
x-2, x-1, x, x+1, x+2; 5x = 5⁷
In general, then px = 5⁷, making p a power of 5.
p=5, x=5⁶. 5 terms
p=25, x=5⁵. 25 terms
p=125, x=5⁴ = 625 terms.
p=625, x=5³. Impossible because x was the "middle" number and we can't have 312 on either side.

Assume p is even.
x and x+1 are the middle two terms.
x-3, x-2, x-1, x, x+1, X+2, x+3, x+4 : 8 terms => 8x + 4
In general, p terms = px + .5p = p(x+1/2).
Since x is integer, we need a 2: q(2x+1) = 5⁷.
5(2x+1)=5⁷: x=5⁶, means ten terms.
25(2x+1)= 5⁷: x=1562, 50 terms
125(2x+1)=5⁷: x=310, 250 terms
625(2x+1)=5⁷: x=62, impossible for x to be the middle.

Problem 36: 46

log_ab + 10log_ba = 7
log_ab + 10/log_ab = 7
U + 10/U = 7
U² + 10 = 7U
U² - 7U + 10 = 0 ... (U-5)(U-2) = 0 .... log_ab = 2, 5
a⁵ = b has three candidates in range: 2,32; 3,243; 4;1024
since 45² = 2025, a² = b has 43 candidates in range. 2² = 4 through 44² = 1836
46 candidates.

Problem 37: 19
Problem 38: a=9 and b=11

Two equations:
386_a = 272_b
146_a=102_b
Starting with the second: 1a² + 4a + 6 = 1b² + 0b + 2
a² + 4a + 4 = b²
(a+2)² = b² gives us a+2 = b
Substitute into equation 1
3a² + 8a + 6 = 2b² + 7b + 2
3a² + 8a + 6 = 2(a+2)² + 7(a+2) + 2
expand
3a² + 8a + 6 = 2a² + 8a + 8 + 7a + 14 + 2
3a² + 8a + 6 = 2a² + 15a + 24
a2 - 7a - 18 = 0
(a+2)(a-9) gives us a = -2, 9
a = -2 is impossible so we're left with a=9 and thus b=11
back to number 38.

Problem 39: 224x³

Simplify by canceling an x³ and then by substituting u = x³.  Now for some long division. Yay!

      u666 +2u663 +3u660 + u657 + ...

-----------------------

u3-1) u669 + u666 + u663 + u660 + ...
      u669 - u666
      -----------

            2u666 + u663
            2u666 -2u663
            ------------

                  3u663 + u660etc.

So each column adds one and projecting down the line gets you to a remainder of 224x³

Hang on. I messed that up. Must revisit.

      u668 + u667 + u666 + u657 + ...
.   -----------------------
u-1) u669 +  0    + 0 +   u666 + 0 + 0 + u663 + u660 + ...
      u669 - u668
      -----------
            u668 + 0
            u668 - u667
            ------------
                  u667 + u666
                  u667 - u666
                 ------------
                       2u666 + 0

When you are finished, you will have 224/u-1 but we can't leave it there. We removed a x³ from numerator and denominator so the last bit is really 224x³ / (x⁶-x³) so the remainder is really 224x³.

Good thing I wasn't taking the test for real.

Problem 40: 3/2

d² = x² +(2x)² +(5-x)² + (2x)² + (5-2x)² + (5-x)² + (5-2x)² + x²
Expand all that and cancel/clean up
sum of d² = 20x² - 30x + 50
min when d(sum)=0
40x - 60 = 0
x = 3/2
Kids get in trouble here because they want to differentiate the square root or get off the idea that the sum of the distances must be at a minimum.

Problem 41: 150

Cleaning up Jonathan's comment a bit:
The semiperimeter, S = (5+6+7)/2 = 9.
Heron's:
A = √(9*4*3*2) = √216 = 6√6.
To get Altitude (height):
Area = ½*5*h = 6√6 so h = 12/5√6
Use CA = 6 and h to find that CP_h = 6/5.
Using h and Pythagoras:
q₁ = (1/5)² + h²
q₂ = (4/5)² + h²
q₃ = (9/5)² + h²
q₄ = (13/5)² + h²
Sum:
4(12/5*√6)² + (1+16+81+196)/25.
(4*144*6 + 1+16+81+196)/25 = 3750/25
= 150

Eric P said:
First find cos(C) by law of cosines
Cos(C)=[6²-7²-5²]/(-2*7*5)=19/35
q₁² = 49 +1 -2(7)(1)(19/35)
q₂² = 49 +4 -2(7)(2)(19/35)
q₃² = 49 +9 -2(7)(3)(19/35)
q₄² = 49 +16-2(7)(4)(19/35)
Lining it up this way and using some quick mental math tricks makes the addition simpler: SUM=4(49)+30-2(7)(10)(19/35)=150

I think I have to side with Eric on this one.

Answers to 2004

Problem 01:

Just an extended fraction: $\dfrac{53}{22}$

Problem 02: 3
Problem 03: 2
Problem 04: 1/25
Problem 05: 4pi
Problem 06: 8/3
Problem 07: 30°
Problem 08: 125
Problem 09: 5

Problem 10: 10
Problem 11: 25% decrease
Problem 12: 1/2
Problem 13: 630
Problem 14: 6
Problem 15: 25/4
Problem 16: 6
Problem 17: 35/36
Problem 18: 225
Problem 19: 4008

Problem 20: 5
Problem 21: 6√3-3π
Problem 22: -4/5
Problem 23: 33/4
Problem 24: 0 or 4/3
Problem 25: 46
Problem 26: $\sqrt[3]{2}$
Problem 27: 139
Problem 28: 240
Problem 29: 35

Problem 30: (x,y) = (66, 55)
Problem 31: 3/4
Problem 32: 704
Problem 33: 1003
Problem 34: 1/6
Problem 35: 41
Problem 36: 9/8
Problem 37: 8√3
Problem 38: 42
Problem 39: 76

Problem 40: 1/20
Problem 41:

$(x,y)=\dfrac{3}{2},\dfrac{1}{2}$

Answers to 2002.

Problem 01: 4/5
Problem 02: 8 pounds

3/4 of the gas weighs 18 pounds. Gas = 24 pounds. Thus tank is 8

Problem 03: 30

$\frac{1}{(\frac{1}{5}-\frac{1}{6})} = \frac{1}{\frac{1}{30}} = 30$

Problem 04: 99%
Problem 05: 20°

BAD = 20° arcAC = 140° arcACE = 180°,
thus arcCE=40° and CAE = 20°
BAC = 60° thus DAE = 60° - 20° - 20° = 20°

Problem 06: 30°
Problem 07: -2+4i
Problem 08: 2008

$f(2002)=f(3*\dfrac{2002}{3})=\dfrac{1}{1+\dfrac{2002}{3}}=\dfrac{3}{2005}$

Problem 09: 36

4 years * 3 days * 3 months

Problem 10: 25%

$\dfrac{1}{\dfrac{4}{3}a}+\dfrac{1}{\dfrac{4}{3}b}=\dfrac{3}{4}(\dfrac{1}{a}+\dfrac{1}{b})$

Problem 11: -2 or 17

Cancel x+2, then multiply out both sides and simplify. Remember the x+2 when done.

Problem 12: $\frac{11}{21}$

The number of 3-element subsets with 3 evens: C(4,3) = 4
The number of 3-element subsets with 2 odd, 1 evens: C(5,2)*4 = 40

Problem 13: -1

The terms pattern out as 1, 2, -1, 2

Problem 14: $\frac{25}{6}$

(10-x)(0.4)+x(1.0)=0.65(10)

Problem 15: $\sqrt{6}$

Square the second expression.

Problem 16: $\frac{2}{3}$

Half-angle formula for sin divided by the half-angle formula for cos.

Problem 17: $\sqrt{3}$ or $\frac{1}{9}$

OMG, this was a pain. I'll post if requested.

Problem 18: 44
Problem 19: 12
Problem 20: -1
Problem 21: 26
Problem 22: $\frac{9}{2}$; 4.5
Problem 23: 11
Problem 24: 100°; length = $\dfrac{10\pi}{9}$
Problem 25: 9
Problem 26: 34
Problem 27: $6\sqrt{3}$
Problem 28: $\frac{29}{2}$; 14.5
Problem 29: $\frac{4}{9}$
Problem 30: $(\frac{3}{2}, \frac{5}{2})$
Problem 31: 264
Problem 32: 6π
Problem 33: d = 9/5 miles
Problem 34: n = 1, 2
Problem 35: 105 ways
Problem 36: 1001/334
Problem 37: 7/2

Pretty hefty polynomial expansion. I'll post it if requested.
Update: I'll post the "trick" (which I got from the test creators; my method was hairy):
Make the substitution $x=3^t-9$ and $y=9^t-3$
Then the student should notice that the equation becomes $x^3 + y^3 = (x+y)^3$

This is only possible if the middle terms of the expansion cancel each other.
So, $3x^2y-3xy^2 = 0$, and factoring gives us $3xy(x+y)=0$ and three corresponding results.
$x = 3^t-9 = 0$ which means that t = 2
$y = 9^t-3 = 0$ which means that t = 0.5
and $9^t+3^t-12 = 0$ and t = 1
Final answer is the sum of the three values is $\dfrac{7}{2}$

Problem 38: 22/3

Law of Cosines, solve for cos A: $cosA = \frac{11}{21}$, then plug back in for side AD.

Problem 39: -2

f(2002) = f(1001) + 1 = f(500) = f(250) + 1 = f(125) + 2 = f(62) + 1
= f(31) + 2 = f(15) + 1 = f(7) = f(3) - 1 = f(1) - 2 = -2

Problem 40: 4/3

FH is the altitude.
By similar triangles, AH/FH = 4/1 and HE/FH = AE/1.
Add the equations: AH/FH + HE/FH = 4 + AE; AE/FH = 4 +AE
Swap: FH = AE /(4 + AE)
1/6 = 1/2*AE*(FH) = 1/2 AE*[AE/(4+AE)]
Algebraic manipulation and x = AE: 3x² - x - 4 = (3x - 4)(x + 1)
AE is either 4/3 or -1.

Problem 41: 74

3 squares = 50. Original triangle + other easy triangle = 12.
Angle α in the upper left so the obtuse angle is 180°-α. Sin(α) = 4/5 = sin(180°-α), making the area of the upper obtuse triangle A = ½*3*5*4/5 = 6. Similar logic gives us the area of the lower triangle A = ½*4*5*3/5 = 6.
Total 62 + 6 + 6 = 74.

Answers to 1990

This post is backdated intentionally.


Problem 01: 54
Problem 02: $\frac{13}{60}$
Problem 03: 9
Problem 04: 119. Thank you Jonathan. Looking back at my notes, I did the exact same reasoning but, for some reason, put 720 - 1 = 719. I really have to pay closer attention. Maybe bad handwriting? Somehow I got an extra 6.
Problem 05: 150
Problem 06: Have not looked at this yet.
Problem 07: 4
Problem 08: $\pi$
Problem 09: 38
Problem 10: 6
Problem 11: 11, 19, 43
Problem 12: 100 or 1
Problem 13: $\frac{19}{90}$
Problem 14: 18 square units
Problem 15: a = -2, b = 4, c = 1
Problem 16: 126/17
Problem 17: 6 square units
Problem 18: 7*6*5*4 = 840; Thanks to Marc and Eric !
Problem 19: 8/95

PeggyU: I assumed there were 20 X 19 or 380 ways of picking two different numbers. There were 20 pairs where one number was the double of the other, and 12 pairs where one was the triple of the other, and 20 + 12 = 32, so 32/380 = 8/95 ...?

Problem 20: 5, 6, & 7
Problem 21: $3^3 + 4^3 + 5^3 = 6^3$
Problem 22: I got $\frac{\pi}{6}$ and $\frac{-\pi}{2}$
Problem 23: 24 ~ each face will have four edges, making a new cube. I think.
Problem 24: 13,440
Problem 25: 27 -- look at the equivalent lengths. PM is PQ + QM ...
Problem 26: 1
Problem 27: 2310
Problem 28: LCM of 1414 and 1000 = 707000. But this feels wrong.
Problem 29:  √3
Problem 30: 40
Problem 31: 1990.  Found it! There is always a problem that contains the year.
Problem 32:
Problem 33: I got 27$\pi$. A commenter got 3$\pi$. Confirmation?
Problem 34: 44.

Consider the function $x^{17} - 45x^{16}$.
This function is "Overall S-Shaped" and $\lim_{x\rightarrow -\infty}f(x)=-\infty$. It has 16 roots at 0, making a very flat broad shelf, and one root at 45, through which the function is passing at a positive slope (m>20). If the function is translated vertically 27 units, the roots will shift ever so slightly to the left, 44<r<45.

Problem 35:
Problem 36:
Problem 37:
Problem 38: 2, 3, 4
Problem 39: 9/4
Problem 40:
Problem 41:

Any thoughts on any of these answers?.