[/quote]
,
be an even prime number and a sequence 
satisfy 
and 
Prove that always exists positive integer 
satisfying for all positive integers 
,
.
is "not congruence"
![$$(\sqrt{x}+1)[2-(x-6)\sqrt{x-3}]=x+8$$](http://latex.artofproblemsolving.com/b/1/d/b1d283f2e7b8e574b6c0ff72236c52a39d3da9e8.png)
is given, 
,
are graphs on the same set of 
connected components respectively. Prove that 
has at least 
connected components.
and 
intersect at points 
and 
,
intersects 
,
intersects 
.
is the circumcenter of 
,
.
and a subset of the numbers written on the board and adds 
be real numbers . Prove that : There exist positive integer 
such that 
Where 
,
,
are pairwise distinct real numbers satisfying 
Prove that
Let 
Prove that
,
denote the sum of the geometric series ![\[12 + 12r + 12r^2 + 12r^3 + \ldots.\]](http://latex.artofproblemsolving.com/1/3/d/13d279ac0064f9e9937642f21a5ebc096f08b69a.png)
Let 
between 
and 
satisfy 
.
.
and 
be positive integers.
and 
.
.
and 
be the two foci of ellipse 
.
is a point on 
.
,
of natural numbers with 
of 
if the product of the elements of 
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 
.
