[/quote]
,
.
is in the interior of ![\[\angle PAD:\angle PBA:\angle DPA=1:2:3=\angle CBP:\angle BAP:\angle BPC\]](http://latex.artofproblemsolving.com/9/3/8/9388a91325842f1ec67a4693f0f10af46ed66c6d.png)
Prove that the following three lines meet in a point: the internal bisectors of angles 
and 
and the perpendicular bisector of segment 
.
with 
.
of 
.
and the length of 
is 
intersects 
and 
.
.
midpoint of side 
in an isosceles triangle 
with 
.
on a smaller arc 
of circumcircle of triangle 
.
point inside of angle 
such that 
and 
.
does not depend on choice of 
)
.
is chosen on arc 
,
,
intersect 
,
.
respectively, such that 
is a parallelogram. Point 
,
.
intersects 
at 
,
intersects 
at 
.
intersects 
at points 
,
and 
respectively.
.
,
and 
are relatively prime positive integers. Find 
.
be the incircle of the triangle 
.
be the circumcircle of the triangle 
.
intersect at the point 
be the intersection of the common tangents of the circles 
is tangent to the circumcircle of the triangle 
is the foot of the altitude from 
.
and 
lie on the circles 
and 
,
and 
.
and 
be vertices of regular pentagon inscribed in a circle whose radius is 
and center is at 
.
respectively. Let 
is a parallelogram. Suppose that 
.
a convex quadrilateral such that each of the segments 
and 
is tangent to 
be the circumcircle of the triangle 
.
beyond 
beyond 
meets 
and 
beyond ![\[A D+D T+T X+X A=C D+D Y+Y Z+Z C.\]](http://latex.artofproblemsolving.com/8/2/6/8261276053071cc6cc06dda824a883426bf1d4ac.png)
.
are 
is the center of the square. Prove that points 
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 
.
