,
,
,
,
or even expressions of those terms combined as if that would make them defined. I have made this post to answer these questions once and for all, and I politely ask everyone to link this post to threads that are talking about this issue.
,
isn't even rigorously defined in this expression. What exactly do we mean by 
.
,
does not exist in the first place, although you will see why in a calculus course. So the point is that 
?
indeterminate? In analysis (the theory behind calculus and beyond), limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits. However, if the expression obtained after this substitution does not provide sufficient information to determine the original limit, then the expression is called an indeterminate form. For example, we could say that 
,
or ![\[0/0, \infty/\infty, \infty-\infty, \infty \times 0\]](http://latex.artofproblemsolving.com/2/6/0/260f97a1cf1ea8722ff97a29b83bac7bfe83e577.png)
etc etc. So what makes something undefined? In the broader scope, something being undefined refers to an expression which is not assigned an interpretation or a value. A function is said to be undefined for points outside its domain. For example, the function 
given by the mapping 
is undefined for 
.
is undefined because dividing by ![\[\overline{\mathbb{R}}=\mathbb{R}\cup \{-\infty,+\infty\}.\]](http://latex.artofproblemsolving.com/9/9/3/993c78b62ec2c84b5d77daa8e9a8cd6f70fcf904.png)
So instead of treating infinity as an idea, we define infinity (positively and negatively, mind you) as actual numbers in the reals. The advantage of doing this is for two reasons. The first is because we can turn this thing into a totally ordered set. Specifically, we can let 
for each 
which means that via this order topology each subset has an infimum and supremum and 
is therefore compact. While this is nice from an analytic standpoint, extending the reals in this way can allow for interesting arithmetic! In ![\begin{align*}
a + \infty = \infty + a & = \infty, & a & \neq -\infty \\
a - \infty = -\infty + a & = -\infty, & a & \neq \infty \\
a \cdot (\pm\infty) = \pm\infty \cdot a & = \pm\infty, & a & \in (0, +\infty] \\
a \cdot (\pm\infty) = \pm\infty \cdot a & = \mp\infty, & a & \in [-\infty, 0) \\
\frac{a}{\pm\infty} & = 0, & a & \in \mathbb{R} \\
\frac{\pm\infty}{a} & = \pm\infty, & a & \in (0, +\infty) \\
\frac{\pm\infty}{a} & = \mp\infty, & a & \in (-\infty, 0).
\end{align*}](http://latex.artofproblemsolving.com/6/d/8/6d8fc1eca5c791d430a7168cd3b37a02104fa318.png)
So addition, multiplication, and division are all defined nicely. However, notice that we have some indeterminate forms here which are also undefined,![\[\infty-\infty,\frac{\pm\infty}{\pm\infty},\frac{\pm\infty}{0},0\cdot \pm\infty.\]](http://latex.artofproblemsolving.com/a/0/b/a0b67076efcc014c78b2023be1151c84b57c1503.png)
So while we define certain things, we also left others undefined/indeterminate in the process! However, in the context of measure theory it is common to define 
as greenturtle3141 noted below. I encourage to reread what he wrote, it's great stuff! As you may notice, though, dividing by ![\[\mathbb{C}^*=\mathbb{C}\cup\{\tilde{\infty}\}\]](http://latex.artofproblemsolving.com/8/4/e/84ed3d6ab237df970333b87beaef5311caa46185.png)
which we call the Riemann Sphere (it actually forms a sphere, pretty cool right?). As a note, 
means complex infinity, since we are in the complex plane now. Here's the catch: division by ![\[\frac{z}{0}=\tilde{\infty},\frac{z}{\tilde{\infty}}=0.\]](http://latex.artofproblemsolving.com/1/6/0/160cb939707708ff4e0930724df2928af11d7fd1.png)
where 
and 
Furthermore, we actually have some nice properties with multiplication that we didn't have before. In 
it holds that![\[\tilde{\infty}\times \tilde{\infty}=\tilde{\infty}\]](http://latex.artofproblemsolving.com/8/e/0/8e035b4841d20c5ad671c46355cc667ca01e533b.png)
but 
and 
are left as undefined (unless there is an explicit need to change that somehow). One could define the projectively extended reals as we did with ![\[{\widehat {\mathbb {R} }}=\mathbb {R} \cup \{\infty \}.\]](http://latex.artofproblemsolving.com/c/0/4/c044ff37849acad3b8b280b9753539225a1276ca.png)
They behave in a similar way to the Riemann Sphere, with division by 
![$\sqrt[2]{2}$](http://latex.artofproblemsolving.com/1/8/6/1869ac018fb72a26a12ab14499cbf1b1c13b4a10.png)
,![$\sqrt[3]{3}$](http://latex.artofproblemsolving.com/1/7/a/17a6bb5dc11696b8650b6d3a11e770f861f4c326.png)
,![$\sqrt[5]{5}$](http://latex.artofproblemsolving.com/a/b/9/ab90f2d42dc24065e2a64988fe76f22eb9359799.png)
.
such that![\[
\sqrt{x - 2011} + \sqrt{2011 - x} + 10
\]](http://latex.artofproblemsolving.com/4/8/b/48b82d13b733947420a0d763c6af74636770835a.png)
is an integer.
,![\[ \textbf{(A)}\ 24 \qquad
\textbf{(B)}\ 30 \qquad
\textbf{(C)}\ 46 \qquad
\textbf{(D)}\ 66 \qquad
\textbf{(E)}\ 74
\]](http://latex.artofproblemsolving.com/8/2/f/82ffc15072ca99f6cd1abc312781eb4827817356.png)
:o
so 
and since 
is just 
therefore 
so 
Y by
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(This practice test is designed to be slightly harder than the real test. I would recommend you take this like a real test, using a 3 hour time limit and no calculator.)
Let me know any suggestions for improvement on test quality, difficulty, problem selection, problem placement, test topics, etc. for the next tests that I make!
Practice AIME
1.
Positive integers a, b, and c satisfy a + b + c = 49 and ab + bc + ca = 471. Find the value of the product abc.
2.
Find the integer closest to the value of (69^(1/2) + 420^(1/2))^2.
3.
Let G and A be two points that are 243 units apart. Suppose A_1 is at G, and for n > 1, A_n is the point on line GA such that A_nA_(n-1) = 243, and A_n is farther from A than G. Let L be the locus of points T such that GT + A_6T = 2025. Find the maximum possible distance from T to line GA as T varies across L.
4.
Find the value of (69 + 12 * 33^(1/2))^(1/2) + (69 - 12 * 33^(1/2))^(1/2).
5.
Find the sum of the numerator and denominator of the probability that two (not necessarily distinct) randomly chosen positive integer divisors of 900 are relatively prime, when expressed as a fraction in lowest terms.
6.
Find the limit of (1x^2 + 345x^6)/(5x^6 + 78x + 90) as x approaches infinity.
7.
Find the slope of the line tangent to the graph of y = 6x^2 + 9x + 420 at the point where y = 615 and x is positive.
8.
Find the smallest positive integer n such that the sum of the positive integer divisors of n is 1344.
9.
Find the first 3 digits after the decimal point in the decimal expansion of the square root of 911.
10.
Let n be the smallest positive integer in base 10 such that the base 2 expression of 60n contains an odd number of 1’s. Find the sum of the squares of the digits of n.
11.
Find the sum of the 7 smallest positive integers n such that n is a multiple of 7, and the repeating decimal expansion of 1/n does not have a period of 6.
12.
Let n be an integer from 1 to 999, inclusive. How many different numerators are possible when n/1000 is written as a common fraction in lowest terms?
13.
How many ways are there to divide a pile of 15 indistinguishable bricks?
14.
Let n be the unique 3-digit positive integer such that the value of the product 100n can be expressed in bases b, b + 1, b + 2, and b + 3 using only 0’s and 1’s, for some integer b > 1. Find n.
15.
For positive integers n, let f(n) be the sum of the positive integer divisors of n. Suppose a positive integer k is untouchable if there does not exist a positive integer a such that f(a) = k + a. For example, the integers 2 and 5 are untouchable, by the above definition. Find the next smallest integer after 2 and 5 that is untouchable.
Answer key:
Let me know any suggestions for improvement on test quality, difficulty, problem selection, problem placement, test topics, etc. for the next tests that I make!
Practice AIME
1.
Positive integers a, b, and c satisfy a + b + c = 49 and ab + bc + ca = 471. Find the value of the product abc.
2.
Find the integer closest to the value of (69^(1/2) + 420^(1/2))^2.
3.
Let G and A be two points that are 243 units apart. Suppose A_1 is at G, and for n > 1, A_n is the point on line GA such that A_nA_(n-1) = 243, and A_n is farther from A than G. Let L be the locus of points T such that GT + A_6T = 2025. Find the maximum possible distance from T to line GA as T varies across L.
4.
Find the value of (69 + 12 * 33^(1/2))^(1/2) + (69 - 12 * 33^(1/2))^(1/2).
5.
Find the sum of the numerator and denominator of the probability that two (not necessarily distinct) randomly chosen positive integer divisors of 900 are relatively prime, when expressed as a fraction in lowest terms.
6.
Find the limit of (1x^2 + 345x^6)/(5x^6 + 78x + 90) as x approaches infinity.
7.
Find the slope of the line tangent to the graph of y = 6x^2 + 9x + 420 at the point where y = 615 and x is positive.
8.
Find the smallest positive integer n such that the sum of the positive integer divisors of n is 1344.
9.
Find the first 3 digits after the decimal point in the decimal expansion of the square root of 911.
10.
Let n be the smallest positive integer in base 10 such that the base 2 expression of 60n contains an odd number of 1’s. Find the sum of the squares of the digits of n.
11.
Find the sum of the 7 smallest positive integers n such that n is a multiple of 7, and the repeating decimal expansion of 1/n does not have a period of 6.
12.
Let n be an integer from 1 to 999, inclusive. How many different numerators are possible when n/1000 is written as a common fraction in lowest terms?
13.
How many ways are there to divide a pile of 15 indistinguishable bricks?
14.
Let n be the unique 3-digit positive integer such that the value of the product 100n can be expressed in bases b, b + 1, b + 2, and b + 3 using only 0’s and 1’s, for some integer b > 1. Find n.
15.
For positive integers n, let f(n) be the sum of the positive integer divisors of n. Suppose a positive integer k is untouchable if there does not exist a positive integer a such that f(a) = k + a. For example, the integers 2 and 5 are untouchable, by the above definition. Find the next smallest integer after 2 and 5 that is untouchable.
Answer key:
,
,
satisfy 
and 
.
.
.
and 
be two points that are 243 units apart. Suppose 
is at 
,
is the point on line 
such that 
,
be the locus of points 
such that 
.
.
as 
at the point where 
and 
such that the sum of the positive integer divisors of 
contains an odd number of 1’s. Find the sum of the squares of the digits of 
does not have a period of 6.
is written as a common fraction in lowest terms?
indistinguishable bricks?
can be expressed in bases 
,
,
using only 0’s and 1’s, for some integer 
.
be the sum of the positive integer divisors of 
is untouchable if there does not exist a positive integer 
.
