Question 1
What is the value of ?
Solution
Question 2
For what value does ?
Solution
Question 3
For every dollar Ben spent on bagels, David spent cents less. Ben paid more than David. How much did they spend in the bagel store together?
Solution
Question 4
The remainder can be defined for all real numbers and with by where denotes the greatest integer less than or equal to . What is the value of ?
Solution
Question 5
A rectangular box has integer side lengths in the ratio . Which of the following could be the volume of the box?
Solution
Question 6
Ximena lists the whole numbers through once. Emilio copies Ximena's numbers, replacing each occurrence of the digit by the digit . Ximena adds her numbers and Emilio adds his numbers. How much larger is Ximena's sum than Emilio's?
Solution
Question 7
The mean, median, and mode of the data values are all equal to . What is the value of ?
Solution
Question 8
Trickster Rabbit agrees with Foolish Fox to double Fox's money every time Fox crosses the bridge by Rabbit's house, as long as Fox pays coins in toll to Rabbit after each crossing. The payment is made after the doubling, Fox is excited about his good fortune until he discovers that all his money is gone after crossing the bridge three times. How many coins did Fox have at the beginning?
Solution
Question 9
A triangular array of coins has coin in the first row, coins in the second row, coins in the third row, and so on up to coins in the th row. What is the sum of the digits of ?
Solution
Question 10
A rug is made with three different colors as shown. The areas of the three differently colored regions form an arithmetic progression. The inner rectangle is one foot wide, and each of the two shaded regions is foot wide on all four sides. What is the length in feet of the inner rectangle?
Solution
Question 11
Find the area of the shaded region.jj
Solution
Question 12
Three distinct integers are selected at random between and , inclusive. Which of the following is a correct statement about the probability that the product of the three integers is odd?
Solution
Question 13
Five friends sat in a movie theater in a row containing seats, numbered to from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?
Solution
The seats are numbered 1 through 5, and let each letter (A,B,C,D,E) correspond to a number. Let a move to the left be subtraction and a move to the right be addition.
We know that 1+2+3+4+5=A+B+C+D+E=15. After everyone moves around, however, our equation looks like (A+x)+B+2+C-1+D+E=15 because D and E switched seats, B moved two to the right, and C moved 1 to the left.
For this equation to be true, x has to be -1, meaning A moves 1 left from her original seat. Since A is now sitting in a corner seat, the only possible option for the original placement of A is in seat 2.
Question 14
How many ways are there to write as the sum of twos and threes, ignoring order? (For example, and are two such ways.)
Solution
Question 15
Seven cookies of radius inch are cut from a circle of cookie dough, as shown. Neighboring cookies are tangent, and all except the center cookie are tangent to the edge of the dough. The leftover scrap is reshaped to form another cookie of the same thickness. What is the radius in inches of the scrap cookie?
Solution
Question 16
A triangle with vertices , , and is reflected about the -axis, then the image is rotated counterclockwise about the origin by to produce . Which of the following transformations will return to ?
counterclockwise rotation about the origin by .
clockwise rotation about the origin by .
reflection about the -axis
reflection about the line
reflection about the -axis.
Solution
Question 17
Let be a positive multiple of . One red ball and green balls are arranged in a line in random order. Let be the probability that at least of the green balls are on the same side of the red ball. Observe that and that approaches as grows large. What is the sum of the digits of the least value of such that ?
Solution
N=5:
we have 6 spaces in a line, and 5 green balls + 1 red ball to arrange.
No matter where you place the red ball, it is certain that on 1 side of the red ball, there are 3/5 of green balls. So P(5) = 1
N=10:
we have 11 spaces in a line, and 5*2=10 green balls + 1 red ball to arrange.
Unless you put red ball in the middle space - 6th space - it is certain that on 1 side of the red ball, there are 3/5 of green balls. So P(10) = 10/11
N=15:
we have 16 spaces in a line, and 5*3=15 green balls + 1 red ball to arrange.
Unless you put red ball in the middle spaces - 8th and 9th space - it is certain that on 1 side of the red ball, there are 3/5 of green balls. So P(15) = 14/16
The trend of P(N) as N goes from 5 to a large number is P(N) will get smaller from 1 to approach 4/5.
So for what N, we will have P(N)=321/400? If we can find this N, then the next number, N+1, will make P(N)<321/400.
You can do a few tries as above (N=5, 10, 15, etc.), and you will see that the ball "works" in places
from 1 to 2/5 * N + 1, and places 3/5 * N +1 to N+1. This is a total of 4/5 * N + 2 spaces, over a total of N+1 spaces:
(4/5 * N + 2)/(N + 1)
Let the above = 321/400:
(4/5 * N + 2)/(N + 1) = 321/400
We have N=479.
So the next N, N+1=479+480, will make P(480)
Question 18
Each vertex of a cube is to be labeled with an integer through , with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How many different arrangements are possible?
Solution
Now, consider the oppsite sides (the two sides which are parallel but not in same face of the cube, eg, ab vs hg). They must have same sum value too: for example, since a+b+c+d=c+d+h+g=18, so a+b=h+g.
Now consider 1 and 8. if they are not sharing the same side, they must be on opposite sides. And if they are on oppsite sides, we will have 1+X=8+Y. And this is not possible for any X that is between 2 to 7. So we conclude 1 and 8 must be sharing the same side.
If 1 and 8 are on same side, this side adds up to 9. So will be its opposite side and the sides sharing the same faces with side 1-8. This means we will have 4 parallel sides: 1-8, 2-7, 3-6, and 4-5.
Look from another dimension, we have to make sure the end points of the 4 parallel lines add up to 18 as well. This will work with 1,7,6,4 on one face and 8,2,3,5 on the other face.
So our job is just to arrange 4 different points on 1 face. And there are 4!/4 = 6 possibilities.
Question 19
In rectangle and . Point between and , and point between and are such that . Segments and intersect at and , respectively. The ratio can be written as where the greatest common factor of and is 1. What is ?
Solution 1
Solution 2
Since DP/PB = 3/1 and DP+PB=BD, we know PB = 1/4 * BD
(DQ+QP)/PB = 3/1 = DQ/PB + QP/PB = (3/5 * BD)/(1/4 * BD) + QP/(1/4 * BD)
That is:
(3/5 * BD)/(1/4 * BD) + QP/(1/4 * BD) = 3/1
Rearrange to get:
QP = 3/20 * BD
So PB:QP:QD = 1/4 : 3/20 : 3/5 = 5:3:12
Question 20
For some particular value of , when is expanded and like terms are combined, the resulting expression contains exactly terms that include all four variables and , each to some positive power. What is ?
Solution
All the desired terms are in the form a^x * b^y * c^z * d^w * 1^t, where x + y + z + w + t = N (the 1^t part is necessary to make stars and bars work better.) Since x, y, z, and w must be at least 1 (t can be 0), let x' = x - 1, y' = y - 1, z' = z - 1, and w' = w - 1, so x' + y' + z' + w' + t = N - 4. Now, the quesiton becomes distributing N-4 among x' + y' + z' + w' + t. Using the stars and bars rule, the number of ways is C(N, 4). To make C(N, 4) = 1001, we have N=14.
Question 21
Circles with centers and , having radii and , respectively, lie on the same side of line and are tangent to at and , respectively, with between and . The circle with center is externally tangent to each of the other two circles. What is the area of triangle ?
Solution
Question 22
For some positive integer , the number has positive integer divisors, including and the number . How many positive integer divisors does the number have?
Solution
Question 23
A binary operation has the properties that and that for all nonzero real numbers and . (Here represents multiplication). The solution to the equation can be written as , where and are relatively prime positive integers. What is
Solution
Question 24
A quadrilateral is inscribed in a circle of radius . Three of the sides of this quadrilateral have length . What is the length of the fourth side?
Solution
And remember to use cosine for angle FCD, which is the same as angle BOC(You can prove it yourself). Refer to the diagram below. You will find this problem quite easy.
Question 25
How many ordered triples of positive integers satisfy and ?
Solution
If LCM(x, y) = 2^3 * 3^2, then when x = 2^3 * 3^1, the exponent of factor 3 in y must be 2, ie y must be 2^3 * 3^2, or 2^2 * 3^2, or 2^1 * 3^2, or 2^0 * 3^2, etc. This is because 2^3 * 3^2 must be both a common multiple of x and y, and must be the smallest common multiple.
Now we prime factorize 72,600, and 900. The prime factorizations are
2^3 * 3^2
2^3 * 3 * 5^2
2^2 * 3^2 * 5^2
Let
x=2^a * 3^b * 5^c,
y=2^d * 3^e * 5^f and
z=2^g * 3^h * 5^i.
From the question, we know that
max(a,d)=3
max(b,e)=2
max(a,g)=3
max(b,h)=1
max(c,i)=2
max(d,g)=2
max(e,h)=2
We also know c=f=0, since lcm}(x,y) isn't a multiple of 5. Since max(d,g)=2 we know that a=3. We also know that since max(b,h)=1, then e=2. With this kind of exercise, we will see that
max(b,h)=1
max(d,g)=2
Are the only two important ones left. We do casework on each now. If max(b,h)=1 then (b,h)=(1,0),(0,1) or (1,1). Similarly if max(d,g)=2 then (d,g)=(2,0),(2,1),(2,2),(1,2),(0,2). Thus our answer is 5 * 3 = 15.
Answer Keys
Question 1: B
Question 2: C
Question 3: C
Question 4: B
Question 5: D
Question 6: D
Question 7: D
Question 8: C
Question 9: D
Question 10: B
Question 11: D
Question 12: A
Question 13: B
Question 14: C
Question 15: A
Question 16: D
Question 17: A
Question 18: C
Question 19: E
Question 20: B
Question 21: D
Question 22: D
Question 23: A
Question 24: E
Question 25: A