InhumanCalculator

We know that the tens digit cannot be 0, so we can count the total number of integers between 10 and 99 with 0 as a unit's digit:

10

20

30

40

50

60

70

80

90

Hope this helped. Let me know if I did anything wrong!

If the two lines are parallel, then they have the same slope. We can calculate the slope of the first equation to be -1/3 (remember, the slope in standard form is -A/B for Ax+By=C). Now that we know the slope, we can calculate the x and y intercepts using the point and extending it:

x-intercept = (-1,0)

y-intercept = (0,-1/3)

You can now plug this back into the equation and solve for A and B.

I hope this helped. Let me know if I did anything wrong!

Well, this depends. Are you asking -(2)^2 or (-2)^2? This matters because, as per the order of operations, exponents are calculated first. So, in the first situation:

-(2)^2 = -4

In the second situation, however, we are squaring the entire expression of -2. A negative squared is a positive, so (-2)^2 = 4.

Does the question look like -2^2 or (-2)^2? Because that will change the answer.

Let me know if I did anything wrong! Hope this helps.

Since we know that f(4) = 1, we can plug that information into the second equation and get:

f(2*4) = 3f(4)

f(8) = 3*1

f(8) = 3

f(2*8) = 3f(8)

f(16) = 3*3

f(16) = 9

f(2*16) = 3f(16)

f(32) = 3*9

f(32) = 27

f(2*32) = 3f(32)

f(64) = 3*27

f(64) = 81

Hope this helped. Let me know if I did anything wrong!

We can use the distance formula, or s = d/t.

s = d/12 (since the distance remains constant we can use it for the second equation too)

s - 35 = d/27

s = d/27 + 35

d/12 = d/27 + 35

I believe you can solve for d, and then plug that back into one of the original equations to get the value of s. The speed in light traffic is "s", and the speed in heavy traffic is "s-35".

Hope this helps. Let me know if I did anything wrong!

If Angle A is 27 degrees, and A, B, and C form an arithmatic sequence in that order, then we can say that:

A + (A + d) + (A + 2d) = 180, where d is the common difference.

= 27 + (27 + d) + (27 + 2d) = 180

You can then solve for d, and thus solve for C.

Hope this helped! Let me know if I did anything wrong! If you don't understand my method then you can reply and I will do my best to explain more thoroughly!

I am not smart enough yet to make AIME, but when I saw this problem I definitely got a little suspicious, as I knew one of my friends was taking the test and it was happening at that time. Later confirmed it was a real AIME problem. I wish good luck to everyone who attempted AIME this year PROPERLY, cheaters get nowhere in life. Hope to make it next year myself though!

I don't believe this has an actual computable answer... even if you enter it into a calcuator the value is too high to display. By hand is a whole other matter. 

To find the power of the full expression, we can find the sum of the numbers from 1-100. So, using the Gauss Formula:

n(n+1)/2

100(101)/2

10100/2

5050

We can then find patterns, since we are trying to find the tens and ones digits.

7^1 = 07

7^2 = 49

7^3 = 343

7^4 = 2401

7^5 = 16807

7^6 = 11649

7^7 = 823543

Use the pattern to solve the answer. Hope this helps!

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Plug in all of the points and create 3 seperate equations:

#1: 3 = a(-3)^2 + b(-3) + c

#2: 3 = a(1)^2 + b(1) + c

#3: 0 = a(0)^2 + b(0) + c

You can use the last equation to get the value of c, which is 0. You can then use substitution or elimination, whichever you choose, to simplify and solve for a and b. Hope this helps!

Let me know if I did anything wrong! 

First we need to find a list of what numbers from both sets {1,2,3,4,5,6,7,8,9,10,11,12} and {1,2,3,4,5,6,7,8,9,10,11,12} will add up to 19 or greater. These are:

12+7=19

12+8=20

12+9=21

12+10=22

12+11=23

12+12=24

11+8=19

11+9=20

11+10=21

11+11=22

10+9=19

10+10=20

This amounts to 12 combinations. Since this can happen on opposite dice, we multiply by 2 to get 24 outcomes. The total number of outcomes are 12*12=144. So, the answer is 24/144 = 1/6.

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Since P(G(a)) has to be defined, the term under the square root has to be greater than or equal to 0. This means that (4-8x)+3 >= 0. This simplifies to 7 >= 8x, which further simplifies to x <= 7/8. This means that the largest possible value of x is 7/8.

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We know that since the angle of a square is 90 degrees, and an equilateral triangle has an angle of 60. So, we know that BEC = 90-60 = 30. 

CE = CD (because it is an equilateral triangle)

CD = CB (because it is a square)

CE = CB

Thus, CBE = CEB

Since a triangle's angles add up to 180, we know:

CBE + CEB + 30 = 180

CBE + CEB = 150

Since CBE and CEB are equal, they both equal 150/2 = 75

This means that Angle CBE = 75 degrees.

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I may be wrong here, as Probability is not my strongest subject, but it seems to me that the solution should be quite simple. Let's say you pick any random point. You will notice that only 2 other points are 1 unit away... all the others are farther. So, assuming that you can't pick the same point twice, you have 7 options for picking another point, and as 2 are not going to work, the answer is 5/7.

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You want to have a quadratic one one side that is equal to zero. You can do this by adding 2-x to both sides, leaving you with:

x^2 + 2x + 2 >= 0

You will notice that this is not factorable but it does not need to be factored. You can see that the discrimant, which is the term under the square root of the Quadratic Formula (b^2 - 4ac) is negative. When the Discrimant is negative, it means that the quadratic has no real solutions, or no real x-intercepts. Since the a value is positive, thus meaning that the parabola opens upwards, we know that all values of x result in positive y values. So, the answer to the question is:

(-inf,inf), or, in other words, all values of x will suffice.

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I believe that you simply need to find the LCM of the 3 numbers. Since they are all primes, the LCM(13,17,19)=13*17*19=4199.

You then need to find 4199 divided by each of the numbers, giving you the values of x, y, and z. These should be the lowest values. Add them up and you should have your answer.

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I believe this requires the use of Partial Fraction Decomposition:

As the denominators of the two fractions on the RHS are factors of the denominator on the LHS, you can multiply both sides by x^2-7x+10. You are left with A(x-5) + B(x-2) = 5x+16. You can plug in 5 for x to cancel out A and solve for B. Similarly, you can plug in 2 for x to cancel out B and solve for A. 

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If f(0)=0 then 0=a(0)^2+b(0)+c, which means that c=0
If f(1)=2 then 2=a(1)^2+b(1), which means that a+b=2

If f(2)=-5 then -5=a(-5)^2+b(-5), which means that 25a-5b=-5, which means that b+5a=1

Solve the system of equations.

b+5a=1

b+a=2

4a=-1

a=-1/4

Solve for b and then plug in 3 for x. You should get your answer. Let me know if I did anything wrong!