year: 2017
contest: Mock AMC10 Test 1
number of questions: 25
answers: CEDABDBCBADECBCEACDEDEBCA
Q1
How many different triangles are there in the picture shown?
(A)65 (B) 75 (C) 85 (D) 90 (E)105
Q2
What is the value of \(\frac { 4 }{ 3 } -\frac { 7 }{ 12 } +\frac { 9 }{ 20 } -\frac { 11 }{ 30 } +\frac { 13 }{ 42 } -\frac { 15 }{ 56 } +\frac { 17 }{ 72 } -\frac { 19 }{ 90 } \)?
(A) \(\frac { 1 }{ 3 } \)
(B) \(\frac { 2 }{ 3 } \)
(C) \(\frac { 4}{ 5 } \)
(D) \(\frac { 7 }{ 8 } \)
(E) \(\frac { 9 }{ 10 } \)
Q3
N is a positive integer. The minimum sum of two different positive divisors of N is 4. The maximum sum of two different positive divisors of N is 2940. What is the value of N?
(A) 2103
(B) 2133
(C) 2145
(D) 2205
(E) 2409
Q4
The picture is showing an isosceles right triangle and a semicircle. What’s the area of the shaded regions?
(A) 9
(B) 12
(C) \(12+\pi\)
(D) 14
(E) \(12-\pi\)
Q5
When a rectangle’s width is increased by a factor of \(\frac{2}{7}\) and its length is decreased by a factor of \(\frac{1}{3}\). The rectangle’s perimeter doesn’t change. What’s the rectangle’s original area?
(A) 950
(B) 1050
(C) 1100
(D) 1150
(E) 1200
Q6
How many ways are there to color the regions A, B, C, D, E with 4 colors: red, blue, green and yellow such that each region is colored with one color and adjacent regions are not colored with the same color?
(A) 48
(B) 72
(C) 80
(D) 96
(E) 120
Q7
In the game of archery, the score for each round of shooting is an integer between 0 and 10. Two people A and B both shoot 5 rounds. The total score is the sum of the scores of each round. If we calculate the product of their individual round’s scores, they are tied at 1764. A’s total score is 4 more than B’s. What’s the sum of their total scores?
(A) 48
(B) 52
(C) 55
(D) 64
(E) 72
Q8
Rotate semicircle with AB as diameter about point A 60 degree anticlockwise so that B becomes C. What’s the area that arc AB swipe through if AB=6?
(A) \(4\pi+\sqrt{3}\)
(B) \(6\pi-\sqrt{3}\)
(C) \(6\pi\)
(D) \(6\pi+\sqrt{3}\)
(E) \(8\pi\)
Q9
10 couples attended a party. Every husband shakes hands with everyone else except his own wife. The wives don’t shake hands with each other. What’s the total amount of handshakes?
(A) 95
(B) 135
(C) 145
(D) 190
(E) 200
Q10
A job can be finished by team A and B working together in 12 days. After team A worked alone for 14 days, team B spent 9 days working alone to finish the job. In how many days can team A finish the job alone?
(A) 20
(B) 25
(C) 30
(D) 32
(E) 35
Q11
Three distinct prime numbers \(a, b, c\) satisfy \(a{b^b}c+a=2000\). What’s the value of \(a+b+c\)?
(A) 32
(B) 36
(C) 37
(D) 42
(E) 44
Q12
A boy and a girl are walking up on an upwards moving escalator. The boy climbs 1 step per second. The girl climbs 2 steps every 3 seconds. They both start at the bottom at the same time. The boy reached the top in 50 seconds, 10 second less than the girl. How many steps does the escalator have?
(A) 80
(B) 85
(C) 90
(D) 95
(E) 100
Q13
There are 150 light bulbs lined up in a row in a long room. Each bulb has its own switch and is currently on. Each bulb is numbered consecutively from 1 to 150. Now we flip the switch on each bulb if its number is multiple of 3. Then we flip the switch on each bulb if its number is multiple of 5. How many bulbs are still on at the end?
(A) 70
(B) 80
(C) 90
(D) 100
(E) 110
Q14
In a math contest, Alice missed \(\frac{1}{9}\) of the questions. Bob answered 7 questions correctly. \(\frac{1}{6}\) of the questions were answered correctly by both Alice and Bob. How many questions did Alice answer correctly?
(A) 16
(B) 32
(C) 42
(D) 48
(E) 50
Q15
Equilateral triangle ABC is inscribed in circle O. D is middle point of BC. Chord EF passes through D and is parallel to AB. DE is shorter than DF. If AB=2, what is the length of DE?
(A) \(\frac{1}{2}\)
(B) 1
(C) \(\frac{\sqrt{5}-1}{2}\)
(D) \(\frac{\sqrt{5}+1}{2}\)
(E) \(\frac{\sqrt{3}}{2}\)
Q16
In a bag, there are 100 balls. 28 of them are red. 20 of them are green. 12 of them are yellow. 20 of them are blue. 10 of them are white. 10 of them are black. How many balls do you need to take out from the bag to guarantee you can have at least 15 balls that are the same color?
(A) 16
(B) 31
(C) 41
(D) 61
(E) 75
Q17
\(x\) is a real number, \(x^3+x^2+x+1=0\), what’s the value of \(x^{97}+x^{98}+x^{99}+…+x^{103}\)?
(A) -1
(B) 0
(C) 1
(D) \(\sqrt{2}\)
(E) 2
Q18
There are 8 bacteria on a plate. When one bacteria becomes mature, it splits into 8 new bacteria. Which following number could be the total number of bacteria on the place?
(A) 2015
(B) 2016
(C) 2017
(D) 2018
(E) 2019
Q19
Positive integers \(m, n\) satisfy \(8m+9n=mn+6\). What’s the maximum value of \(m\)?
(A) 63
(B) 68
(C) 72
(D) 75
(E) 80
Q20
\(x, y\) are real numbers. \(xy+x+y=17\), \(x^2 y+xy^2=66\). What is the value of \(x^4+x^3 y+x^2 y^2+xy^3+y^4\)?
(A) 9640
(B) 10800
(C) 11280
(D) 12368
(E) 12499
Q21
Expand \((x^2−x+1)^6\) get \(a_{12} x^{12}+a_{11} x^{11}+…+a_1 x+a_0\). What’s the value of \(a_{12}+a_{10}+a_8+a_6+a_4+a_2+a_0\)?
(A) 250
(B) 300
(C) 344
(D) 365
(E) 400
Q22
Positive integer ordered triplets \(\{a, b, c\}\) satisfy \(\frac{1}{a}+\frac{1}{b}=\frac{1}{7}\) and \(c=a+b\). Define a triplet’s value as \(abc\)? What’s the sum of all the different triplets’ values?
(A) 5488
(B) 28672
(C) 34160
(D) 46382
(E) 62832
Q23
Quadrilateral ABCD is inscribed in a circle. AC is the diameter of the circle. AC=3. AB=BD. AC and BD intersect at point P. PC=0.6. What’s the perimeter of quadrilateral ABCD.
(A) \(\frac{1}{2}+3\sqrt{2}+\sqrt{3}+\sqrt{6}\)
(B) \(1+2\sqrt{2}+\sqrt{3}+\sqrt{6}\)
(C) \(\frac{1}{2}+\sqrt{2}+2\sqrt{3}+\sqrt{6}\)
(D) \(2+\sqrt{2}+2\sqrt{3}+\sqrt{6}\)
(E) \(1+\sqrt{2}+2\sqrt{3}+2\sqrt{6}\)
Q24
How many days in year 2000 have the following property: the product of the year number, the month number and the day number can be expressed as product of three consecutive multiple of 5, for example 5, 10, 15?
(A) 3
(B) 6
(C) 8
(D) 9
(E) 12
Q25
Point P is inside parallelogram ABCD. Area of triangle PAB is 7. Area of triangle PAD is 3. What is the area of triangle PAC?
(A) 4
(B) 5
(C) 6
(D) 7
(E) 8