Question 1
One ticket to a show costs at full price. Susan buys 4 tickets using a coupon that gives her a 25% discount. Pam buys 5 tickets using a coupon that gives her a 30% discount. How many more dollars does Pam pay than Susan?
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Question 2
An aquarium has a rectangular base that measures 100 cm by 40 cm and has a height of 50 cm. It is filled with water to a height of 40 cm. A brick with a rectangular base that measures 40 cm by 20 cm and a height of 10 cm is placed in the aquarium. By how many centimeters does the water rise?
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Question 3
The larger of two consecutive odd integers is three times the smaller. What is their sum?
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Question 4
Kate rode her bicycle for 30 minutes at a speed of 16 mph, then walked for 90 minutes at a speed of 4 mph. What was her overall average speed in miles per hour?
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Question 5
Last year Mr. Jon Q. Public received an inheritance. He paid in federal taxes on the inheritance, and paid of what he had left in state taxes. He paid a total of for both taxes. How many dollars was his inheritance?
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Question 6
Triangles and are isosceles with and . Point is inside triangle , angle measures 40 degrees, and angle measures 140 degrees. What is the degree measure of angle ?
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Question 7
Let , and be five consecutive terms in an arithmetic sequence, and suppose that . Which of or can be found?
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Question 8
A star-polygon is drawn on a clock face by drawing a chord from each number to the fifth number counted clockwise from that number. That is, chords are drawn from 12 to 5, from 5 to 10, from 10 to 3, and so on, ending back at 12. What is the degree measure of the angle at each vertex in the star polygon?
Solution
Question 9
Yan is somewhere between his home and the stadium. To get to the stadium he can walk directly to the stadium, or else he can walk home and then ride his bicycle to the stadium. He rides 7 times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his home to his distance from the stadium?
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Question 10
A triangle with side lengths in the ratio is inscribed in a circle with radius 3. What is the area of the triangle?
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Question 11
A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might begin with the terms 247, 475, and 756 and end with the term 824. Let be the sum of all the terms in the sequence. What is the largest prime factor that always divides ?
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Question 12
Integers and , not necessarily distinct, are chosen independently and at random from 0 to 2007, inclusive. What is the probability that is even?
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Question 13
A piece of cheese is located at in a coordinate plane. A mouse is at and is running up the line . At the point the mouse starts getting farther from the cheese rather than closer to it. What is ?
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Question 14
Let a, b, c, d, and e be distinct integers such that
What is ?
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Question 15
The set is augmented by a fifth element , not equal to any of the other four. The median of the resulting set is equal to its mean. What is the sum of all possible values of ?
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Question 16
How many three-digit numbers are composed of three distinct digits such that one digit is the average of the other two?
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Question 17
Suppose that and . What is ?
Solution
https://www.homesweetlearning.com/resources/math/math910/trigonometry/trigonometric_identity.html
Question 18
The polynomial has real coefficients, and What is
Solution
https://www.homesweetlearning.com/resources/math/grade_11_and_12_math/complex_number.html
Then have a review of the complex conjugate root theorem:
In mathematics, the complex conjugate root theorem states that if P is a polynomial in one variable with real coefficients, and a+bi is a root of P with a and b being real numbers, then its complex conjugate a-bi, is also a root of P.
And the complex conjugate root property:
If we multiply a complex number by its conjugate we get the sum of the squares of its real and imaginary parts:
(a+bi)(a-bi) = a^2 + b^2
The polynomial we have is:
f(x) = x^4 + ax^3 + bx^2 + cx +d
When x = 1:
f(1) = 1 + a + b + c + d, ie: a+b+c+d = f(1)-1
Since the polynomial at question has 2 roots 2i and 2+i, then it must have their complex conjugate -2i and 2-i as roots:
f(x) = (x-(2+i))(x-(2-i))(x-2i)(x+2i) = (x-2-i)(x-2+i)(x-2i)(x+2i)
When x=1:
f(1) = (1-2-i)(1-2+i)(1-2i)(1+2i)
Applying the rule (a+bi)(a-bi) = a^2 + b^2, we have:
f(1) = (1^2+1^2)(1^2+2^2) = 10
Therefore:
f(1)-1 = a+b+c+d = 9
Question 19
Triangles and have areas and respectively, with and What is the sum of all possible x-coordinates of ?
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Question 20
Corners are sliced off a unit cube so that the six faces each become regular octagons. What is the total volume of the removed tetrahedra?
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Question 21
The sum of the zeros, the product of the zeros, and the sum of the coefficients of the function are equal. Their common value must also be which of the following?
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Question 22
For each positive integer , let denote the sum of the digits of For how many values of is
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Question 23
Square has area and is parallel to the x-axis. Vertices , and are on the graphs of and respectively. What is
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Question 24
For each integer , let be the number of solutions to the equation on the interval . What is ?
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Question 25
Call a set of integers spacy if it contains no more than one out of any three consecutive integers. How many subsets of including the empty set, are spacy?
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Answer Keys
Question 1: C
Question 2: D
Question 3: A
Question 4: A
Question 5: D
Question 6: D
Question 7: C
Question 8: C
Question 9: B
Question 10: A
Question 11: D
Question 12: E
Question 13: B
Question 14: C
Question 15: E
Question 16: C
Question 17: B
Question 18: D
Question 19: E
Question 20: B
Question 21: A
Question 22: D
Question 23: A
Question 24: D
Question 25: E