[/quote]
,
with 
prime and ![\[a^p+b^p=p!.\]](http://latex.artofproblemsolving.com/f/e/b/feb27894b1f6497559eeb094cfe92256bc26688e.png)
such that 
.
such that 
.
.
be a cyclic quadrilateral with circumcenter 
,
is not a diameter of its circumcircle. The lines 
and 
intersect at point 
,
lies between 
and 
lies between 
and 
is acute and let 
be its orthocenter. The points 
and 
on the lines 
and 
.
,
.
which satisfy the condition![\[ f(m + n) \geq f(m) + f(f(n)) - 1
\]](http://latex.artofproblemsolving.com/4/b/d/4bddce124cf0da82cb8bb6c6ae2325239354f27c.png)
Find all possible values of 
.
be a positive integer and 
be a positive integer coprime to 
.
,
,
Find, in terms of 
,
for which there exists an index 
such that 
is divisible by 
.
table are initially white. Alice and Bob play a game. First Alice paints 
rows and 
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 
approaches infinity.
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 
.
