For this week's blog post, I want everyone to understand how to solve ALL of the problems from the competition you had last Saturday. For those of you who didn't attend the competition, you have an additional homework of solving all the problems on all the tests. This post is due at 3:00 pm by Wednesday, November 23rd. I know this deadline is somewhat early but you have no school so you can devote more time for this. Just for convenience, I am labeling the Sprint round problems 1 - 30, the target round problems 31 - 38, and the team round problems 39 - 48. In addition, each student this time has a "buddy" who will solve nearly the same problems. Pairs of buddies are divided by a space. The first "buddy" to post on the blog will do a regular post while the second "buddy" will have to post and include critiques to his/her partner's solutions (if any). By the way, even if you can't attend the meeting, this blog post is still do and I will check it. For Group 2, there will be 2 "trios."
Minhoo : 1, 13, 19, 25, 31, 37, 43
Gautham : 2, 13, 19, 25, 31, 37, 43
Jeffrey: 2, 13, 19, 25, 31, 37, 43
Aishwarya : 3, 14, 20, 26, 32, 38, 44
Akshara : 4, 14, 20, 26, 32, 38, 44
Lucy: 4, 14, 20, 26, 32, 38, 44
Julia : 5, 15, 21, 27, 33, 39, 45
Mihir : 6, 15, 21, 27, 33, 39, 45
Shivam : 7, 16, 22, 28, 34, 40, 46
Geonil : 8, 16, 22, 28, 34, 40, 46
Adiyan : 9, 17, 23, 29, 35, 41, 47
Vignesh : 10, 17, 23, 29, 35, 41, 47
Sanjana : 11, 18, 24, 30, 36, 42, 48
Melinda : 12, 18, 24, 30, 36, 42, 48
Start Posting! The faster you post to the blog, the better!
-Min Hoo Lee-
ReplyDelete#1: Evaluate 27*27
27*27=729
#13: A circle has area 17. The radius of the circle equals the side length of a square. What is the area of the square?
The radius of the circle = r and the area = 17. So, 17 = pi * r^2. I will call this equation 1.
Since the length of the square is r, the area of the square is r^2. Using equation 1, r^2 = 17/pi. Therefore, the area of the square is 17/pi.
Adiyan Kaul
ReplyDelete#9 The operation is defined by ab=ab+a+bb. Find 9(63).
9*63+9+63*63=
567+9+3969=
4545
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ReplyDeleteAdiyan Kaul
ReplyDelete#35 Adam multiplies a number by the square of its reciprocal. Bob multiplies the square of the same number by the reciprocal of the number. Amazingly, they get the same result. Find Adam’s original number, given that it is a positive integer.
The answer has to be 1 because 1*the square of its reciprocal is 1, so 1= 1*1
Aishwarya Laddha
ReplyDelete3. How many positive 4-digit integers are multiples of 5?
Multiples of five end in 0 or 5 only
From 1000 to 1999 there are 20
From 2000 to 2999 there are 20
From 3000 to 3999 there are 20
From 4000 to 4999 there are 20
From 5000 to 5999 there are 20
From 6000 to 6999 there are 20
From 7000 to 7999 there are 20
From 8000 to 8999 there are 20
From 9000 to 9999 there are 20
20*9= 180
Answer: 180 numbers
Aishwarya Laddha
ReplyDelete14. What is the area of a regular hexagon with a side length of four? Express you answer in simplest radical form.
The formula for finding the area of a regular polygon is
given n is number of sides and s is length of sides
6* (4*4)cot(180/6) simplified it becomes: 6*4cot*30
4 which is 24cot30 = 24√3
Answer is : 24√3
Aishwarya Laddha
ReplyDelete32. In rhombus ABCD, <A=300. If AB = 6, what is the area of ABCD?
A= (1/2)d1d2
A= ½ *6 *6
A= ½ * 36
A= 18
12. 6 / 90 = 20 / x
ReplyDeletex = 300
18. Negative and Positive solutions cancel out.
0
24. 7 * 7 * 143 * 10
= 70070
30. 4[(1 / 2)^4 (1 / 2)^4]
= 1 / 256
36. 2 + 3 + 4 + 5 + 6 + (2 * 3 * 4 * 5 * 6
= 740
42. A = Root 12 + Root 12 ...
A^2 - A - 12 = 0
(A - 4)(A + 3) = 0
A = 4
48. ??
Adiyan Kaul
ReplyDelete#47 If x^2=x+1, solve x^4-2x^3+2x^2-x+12
x^2(x^2-2x+2)-x+12
(x+1)(x+1-2x+2)-x+12
(x+1)(-x+3)-x+12
-x^2+3x-x+3-x+12
-x-1+x+15
14
Adiyan Kaul
ReplyDelete#9 This is the answer after Ashish changed the problem
The operation /\ is defined by a/\b = (ab+a+b)/b. Find 9/\(6/\3).
6/\3= (18+6+3)/3= 27/3=9
9/\9= (81+9+9)/9= 99/9=11
Adiyan Kaul
ReplyDelete#29 If x+1/y=9, y+1/z=7, and xyz=1, then find the value of z+1/x. Express your answer as a common fraction.
From xyz=1 you can tell y=1/xz Now you substitute that in place of y in x+1/y=9. So it simplifies to x+xz=9. Now you do that to the other equation which is y+1/z=7. So it becomes 1/xz+ 1/z=7. When you multiply the equation with xz you get 1+x=7xz.
Write this as 7xz-x=1
xz+x=9 and add them
you get 8xz=10 so xz= 10/8. Substitute the value into 7xz-x=1
7(10/8)-x=1 So x=62/8.
xz=10/8 From this z=10/62
So z+1/x= 10/62+8/62= 18/62= 9/31
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ReplyDeleteThis comment has been removed by the author.
ReplyDeleteAdiyan Kaul
ReplyDelete#17 A rectangular prism is drawn with edges parallel to the three coordinate axes. If two vertices of the prism are (-2,3,0) and (6,7,9), then what is the volume of the prism?
Since the edges are parallel to the three coordinate axes then you can find the dimensions by subtracting the respective coordinates.
6--2=8, 7-3=4, 9-0=9 When you multiply these together it is 288 cubic units.
Adiyan Kaul
ReplyDelete#41 Let f(x)= 98/12-x. For how many integer values of k is f(k)an integer?
98 has 6 positive and 6 negative factors so there are 12 values that k can be to be an integer.
Aishwarya Laddha
ReplyDelete26. A right cylinder with radius 3 and height 8 is inscribed in a sphere. What is the surface area of the sphere? Express you answer in term of pi.
The height of the cylinder is nothing but the diameter of the sphere.
The formula for finding the surface area of a sphere is S= 4ll (r*r)
diameter= 8
radius= 4
so S= 4ll*4*4
S= 4*4*4ll
_
Answer: S= 64ll
Adiyan Kaul
ReplyDeleteFor integers a and b such that a<b, denote a@b as the sum of the integers from a to b, inclusive. For example, 3@9=3+4+5+6+7+8+9=42. Find the value of (2001@2011)/(1001@1011). Express your answer as a common fraction.
2001@2011=22066
1001@1011=11066
22066/11066= 11033/5533= 1003/503
Aishwarya Laddha
ReplyDelete44. if x2+1/x2 = k where k ≥ 2, then what are the two possible values for x + ½? Express you answer in simplest radical form in terms of k.
(x +1/x)squared – 2=k
(x +1/x)squared= k+2
x + 1/x = ± √(k + 2)
check:
k=7
√9= ± 3
Answer: ± √(k + 2)
-Min Hoo Lee-
ReplyDelete#31: Kevin can crush 100 empty cans in 3 minutes. Zach can crush 100 empty in 6 minutes. If they crush cans at a constant rate, then how many cans can they crush together in 15 minutes?
In 15 minutes, Kevin crushes 500 cans and Zach crushes 250 cans. 750 cans in 15 minutes.
-Min Hoo Lee-
ReplyDelete#37: Compute the number of ordered pairs of integers (x,y) that satisfy x/y + y/z = 173/91.
There are 0 possible pairs because y + z should equal z + y. In this case, they don't equal each other, so there are no cases possible.
Aishwarya Laddha
ReplyDelete20. The inequality, x squared + 6x -12 ≥ is satisfied for all real numbers x. What is the largest possible value for k?
x squared + 6x –(12 +k) ≥ 0
use the quadratic equation formula:
-b ± √(b squared – 4ac)
2a
so b= 6
a=1
c= –(12 +k)
but what the answer is asking for is the part about “b squared – 4ac”
b squared -4ac
(6*6)-4*1*-(12+k)
36- 48+ 4k
place it back into the equation as
36+ 48+ 4k ≥ 0
gives you
4k ≥ -84
k≥ -21
Answer: k ≥ -21
-Min Hoo Lee-
ReplyDelete#43: Evaluate: 9C4 + 10C3 + ... + 15C3.
This equals 16C4 - (3C3 + 4C3 + 5C3 + 6C3 + 7C3 + 8C3)
Another way to write this is 16C4 - 9C4. This equals 1694.
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ReplyDelete2. Evaluate: 1+3+5+7+9+11+13+15+17+19.
ReplyDelete=20+20+20+20+20
=100
Answer: 100
13. A circle has area 17. The radius of the circle equals the side length of a square. What is the area of the square?
Area of square = r^2
17=πr^2
17/ π=r^2
Answer: 17/ π
19. The radius of a cone is equal to the diameter of a cylinder. If the volume of the cone equals the volume of the cylinder, then the height of the cone is k times the height of the cylinder. Express k as a common fraction.
V of Cone= πr^2h
V of Cylinder=1/3 πr^2h
If radius of cylinder is 1, then radius of cone is 2
Cylinder = Cone
πr^2h = 1/3 πr^2h
πh = 1/3 π4h
h = 1/3 (4) h
h = 4/3 h
3/4h = h
Answer: k= 3/4
25. In triangle ABC, point D lies on AB such that AD=4 and DB=9. Points E, F lie on AC such that AE=2, EF=5, FC=5, and AF>AE. Find[ADE][ABC], where [X] denotes the area of region X.
31. Kevin can crush 100 empty cans in 3 minutes. Zach can crush 100 empty cans in 6 minutes. If they crush cans at a constant rate, then how many cans can they crush together in 15 minutes?
Kevin=100cans/3min
Zach=100cans/6min OR 50cans/3min
So, 150 cans can crushed in 3 minutes.
Answer: In 15 minutes, 750 cans can crushed.
37. Compute the number of ordered pairs of integers (x,y) that satisfy x/y+y/x=173/91.
Answer: 0
43. Evaluate: (9C3)+(10C3)+…+(15C3)
(9C3)+ (10C3)+ (11C3)+ (12C3)+ (13C3)+ (14C3)+ (15C3)
84+120+165+220+286+364+455
Answer: 1694
-Geonil Kim
ReplyDelete8. For an integer n, F(n)=n!(n-1)!(n-2)!1. What is the value of F(6) / (F(5)F(2))?
= 6!5!4!/5!4!3!2!
=[720 x 120 x 24] / [120 x 24 x 6 x 2]
=720/ [6 x 2]
=60
Answer: 60
16. When arranged in groups of three, Mr. Kim's class has 2 students left over. When arranged in groups of 7, Mr. Kim's class has 4 students left over. Find the number of students in Mr. Kim's classroom, given that there are between 20 and 40 students in the class.
Possible for groups of 3 and leftovers of 2: 23, 26, 29, 32, 35, 38
Possible for groups of 7 and leftovers of 4: 25, 32, 39
Answer: 32 students
22. In tetrahedron ABCD, points E,F, and G are the midpoints of AB, AC, and AD, respectively. If tetrahedron ABCD has volume 72, then what is the volume of tetrahedron AEFG?
I know that the volume made by midpoints of original sides is 1/8 of the original volume. For example, cube with 2cm x 2cm x 2cm is decreased to 1cm x 1cm x 1cm. The decreased volume of cube is one-eighth of the original cube. Due to that,
=72 x 1/8
=9
Answer: 9
28. In isosceles trapezoid ABCD, AB=12, BC=5, CD=18, and DA=5. The feet of the altitudes from A and B to CD are E and F, respectively. If the intersection of AE and BD is G, then what is the area of AGF?
For this problem, I used coordinate graph. If point D is the origin (0, 0), EF is 12 and CD is 18. DE and FC are 3, now. We can find AE and BF by using hypotenuse theorem. Since AD is 5, and ED is 3, and AE is an altitude, we can find that AE is 4. The intersection point, G, have the x-coordinate as 3, so we put 3 in x of the formula: y= (4/15)x. y is 12/15. If we subtract 12/15 from 4, AG comes out as 3 and 1/5. Now since we have the base and height, we can use the formula: a= (1/2)bh.
Answer= 19 and 1/5
34. How many integers x satisfy the inequality x^2 A 3x + 41 O 100?
I have no idea.
40. ∆ABC is similar to ∆XYZ. If m <BAC = 90, XY = 12, and XZ =5, then what is YZ?
Sine ∆XYZ also is a right triangle, We can use hypotenuse formula on this. a^2 + b^2 = c^2
144+ 25 = c^2
169= c^2
c= 13
Answer: YZ is 13
46. At Unhappy University, all of the 350 students must take a foreign language class. 180 students take Spanish, 201 students take French, and 157 students take German. 104 students take both Spanish and French, 71 students take both French and German, and 84 students take both German and Spanish. How many students take Spanish, French, and German?
350=201+180+157-104-71-84+x
x=71
Answer: 71
-Min Hoo Lee-
ReplyDeleteProblems with Jeffrey's Solution to #19
#19
It seems that though you got the answer correct, you switched the equation for the volume of the cone with the volume of the cylinder.
Instead of solving it that way, you should solve it like this:
Volume of the Cone: 1/3πR^2H
Volume of the Cylinder: πr^2h
According to the question,
R = 2r
H = kh
Then by substituting you get:
1/3π(2r)^2(kh) = πr^2h
1/3π * 4r^2 * kh = πr^2h
By reducing we get, k = 3/4
Answer: k = 3/4
Aishwarya,
ReplyDeleteProblems with your #26:
The height of the cylinder is not equal to the diameter of the sphere because a sphere is rounded while a cylinder is flat. The cylinder doesn't reach all the way to the edge of the sphere so that the surfaces don't completely touch. However, the sphere and the cylinder do have points of intersection such as around the circular base of the cylinder (these touch the edges of the sphere). We know the height and radius of the cylinder. Using this, we can cut the height of the cylinder in half and use this and the radius to form an L shape in the cylinder. The radius reaches all the way from the center of the cylinder to one of the points of intersection (we are using the radius on the base). From this point, we draw a line back to the point where half the cylinder's height is represented. This is also the center of the circle. With these lines, we formed a right triangle. The base is the radius of the cylinder(3), the height is half the height of the cylinder(4), and the hypotenuse is the radius of the circle. Using the Pythagorean Theorem, we can do height^+base^=hypotenuse^, which is the radius. 4^+3^=25. The square root of 25 is 5 so the radius of the sphere is 5. The formula for surface area of the sphere is 4*pi*r^. Plug in 5 for r and you get: 4*25*pi.
The surface area of the sphere is 100pi.
14. What is the area of a regular hexagon with side length 4? Express your answer in simplest radical form.
ReplyDeleteA regular hexagon is made up of 6 equilateral triangles with one of their sides being one of the sides of the hexagon. Using this, all the side lengths of all the small triangles is 4. Focus on one triangle. You can cut the triangle in half to get you a right triangle. Considering the Pythagorean Theorem, we can now find the height of the triangle. b^+h^= hy^ so hy^_b^=h^. We know the hypotenuse of the triangle is 4 and the base is 2. 16-4=12. To find the height of the triangle, we need to find the square root of 12 in simplest form. √12= √4+√3= 2√3. So the height of the triangle is 2 √3. To find the area of the triangle we multiply this by the base(4) and then divide by two: 4*2√3/2. The twos cancel out giving you 4√3 as the area of one triangle. To find the entire hexagon's area you multiply this by 6 for the 6 triangles in the hexagon. 4√3*6=24√3.
The area of a regular hexagon with side length 4 is 24√3.
I don't see the inequality sign in 20.
ReplyDelete4. Express 36/14 as a common fraction.
ReplyDelete36/14=18/7=2& 4/7
32. In rhombus ABCD, angle A is 30 degrees. If AB=6, what is the area of ABCD?
ReplyDeleteTo find the area of a rhombus, you multiply base times height. We know that the base of the rhombus is 6 because all 4 sides of a rhombus are equal. To find the height of a rhombus, we use a special formula. We label the angle that we know is 30 degrees sine theta. Sin theta=h/AD (or any line connected to sine theta) so h=AD*sin theta. We know AD is equal to to 6 and sin theta is equal to sine 30. 6*sine(30) is the height. You multiply this by the base 6 to get 6*6*sin(30)=36*sin(30). If you have a sine calculator you can find sine(30)(approx. -.998) and multiply it by 36.
akshara there is the inequality sign in number 20
ReplyDeleteMIHIR KHAMBETE
ReplyDelete#6
Evaluate 111+221+331-11-21-31
notice that 11, 21, and 31 are subtracted from(111+221+331)
=100+200+300
=600
#15
How many ordered triplets satisfy a+b+c=10?
7+2+1, 7, 1,2
6+3+1, 6,1,3, 6,2,2,
5, 4,1, 5,1,4, 5,3,2 5,2,3
...........
Do you realize a pattern?
2+3+4+5+6+7+8
=35
#21
Find the sum of vertices, edges, and faces of an octahedron.
An octahedron is like two pyramids together. One"pyramid" points up and the other points down. Be mindful that the "bases" are common.
Count:
Edges: 12
Vertices: 6
Faces :8
12+6+8=26.
#27
x=1 is a solution to 2xcubed+6xsquared-x-7=0
What is the sum of the other two solutions?
since 1 is a solution x-1 is a solution.
the final product is of 3 parts-the factors
You need to factorize
2xcubed+3xsquared-x-7=0
(x-1)[2xsquared+8x+7]=0
(x-1)[2xsquared+8x+8-1]=0
(x-1)[(2)(xsquared+4x+4)-1]=0
(x-1)[(2)(x+2)squared -1] =0
(x-1)[(x+2)(square root of 2)+1][(x+2)(root 2)-1)]=0
(x+2)root 2 =1
(x+2+root 2 =-1
x+2=1/root 2
or
x+2=-1/root 2
x=1/root2 -2
or
x = -1/root2 -2
-2-2
=-4
#33.
The sum of five consecutive integers is 265. What is the value of the largest integer?
find the mean.
265/5=53.
then it is 51,52,53,54,55.
So the largest integer is 55.
#39
a two digit number is randomly chosen. What is the probability of it being a multiple of 11?
how many 2 digit numbers?
(90)
how many multiples of 11?
(9)
9/90 =1/10/
#45.the perimeter of an equilateral triangle is equal to the perimeter of a square.What is the ratio of their areas?
equilateral triangle perimeter-3s
square perimeter-4s
assume perimeter =12
area of equilateral triangle=1/4(s squared)root 3
1/4(16)root 3
=[4 (root 3)]/9
Where is the sign? Maybe I accidentally deleted it before printing.
ReplyDelete10)The amount of bacteria in a petri dish triples every 4 hours. At 6:15 PM, there were 9,000,000 bacteria in the dish. At what time were there exactly 1,000,000 bacteria in the dish?
ReplyDeleteI divided twice by 3 to get 1,000,000 and that means 8 hours passed. THe question asks for time so 8 hours before 6:15 pm is 10:15 am.
17)A rectangular prism is drawn with edges parallel to the three coordinate axes. If two vertices of the prism are (-2,3,0) and (6,7,9), then what is the volume of the prism?
Since the axes are parallel to the edges, you can find the area by subtracting the vertices value from each other and multiplying the answers together. (6- -2)(7-3)(9-0)
8x4x9
288.
23)For integers a and b such that a<b, denote a@b as the sume of the integers from a to b, inclusive. For example, 3@9=3+4+5+6+7+8+9=42. Find the value of (2001@2011)/(1001@1011). Express your answer as a common fraction.
First I solved for 2001@2011=22066 and 1001@1011=11066
Then I simplified 22066/11066 by 22/22 to get 1003/503.
29) If x+1/y=9, y+1/z=7, and xyz=1, then find the value of z+1/x. Express your answer as a common fraction.
I didn't get how to do this one even after looking at my buddies post
35) Adam multiplies a number by the square of its reciprocal. Bob multiplies the square of the same number by the reciprocal of the number. Amazingly, they get the same result. Find Adam’s original number, given that it is a positive integer.
I got 1 because the reciprocal of 1 is 1 and the square of 1 is one so the answer is always 1.
41) I found the number of integer factors that 98 has and that was my answer. Since it is integer negative number count. I found 6 positive integers 1, 98 2, 49 7,14 and 6 negative integers -1,-98 -2, -49 -7,-14. Therefore i got 14 integer values for k.
47) I didn't get this problem either
Oh, I see the inequality sign now that I got my Mathleague test back. I did delete the sign! :]
ReplyDelete