Mock AIME 1 2010 Problems
Contents
Problem 1
Let . Find the number of perfect squares among
.
Problem 2
Find the last three digits of the number of 7-tuples of positive integers such that \linebreak
, that is,
divides
,
divides
,
divides
,
divides
,
divides
,
divides
, and
divides 6468.
Problem 3
Let be a line segment of length
, and let
be the set of all points
such that
. Find the last three digits of the largest integer less than the area of
.
Problem 4
A round robin tournament is a tournament in which every player plays every other player exactly once. There is a round robin tournament with 2010 people. In each match, the winner scores one point, and the loser scores no points. There are no ties. Find the last three digits of the greatest possible difference between the first and second highest scores appearing among the players.
Problem 5
For every integer , the
representation of
is defined to be the unique sequence of integers \linebreak
, with
and
such that
. We represent
as
, where
if
is 0 or 1, and
if
. For example,
. Find the last three digits of the sum of all integers
with
such that
has at least one zero when written in balanced ternary form.
Problem 6
Find the number of Gaussian integers with magnitude less than 10000 such that there exists a different Gaussian integer
such that
. (The magnitude of a complex
, where
and
are reals, is defined to be
. A Gaussian integer is defined to be a complex number whose real and imaginary parts are both integers.)
Problem 7
Find the number of positive integers for which there exists a positive integer
such that
is the square of an integer.
Problem 8
In the context of this problem, a is a
block, a
is a
block, and a
1 \times 3
N
1 \times 20
N$is divided by 1000.
== Problem 9 ==
Let$ (Error compiling LaTeX. Unknown error_msg)\omega_1\omega_2
\omega_3
\omega_1
\omega_2
\omega_1
\omega_2
\omega_3
a \sqrt{b} - c
a
b
c
b
a + b + c$.
== Problem 10 ==
Find the last three digits of the largest possible value of$ (Error compiling LaTeX. Unknown error_msg)\frac{a^2 b^6}{a^{2 \log_2 a} (a^2 b)^{\log_2 b}},a
b$are positive reals.
== Problem 11 ==
Let$ (Error compiling LaTeX. Unknown error_msg)\triangle ABCAB = 7
BC = 8
CA = 9
D
E
F
DB \perp BA
DC \perp CA
EC \perp CB
EA \perp AB
FA \perp AC
FB \perp BC
AFBDCE
\frac{a \sqrt{b}}{c}
a
c
b
a$.
== Problem 12 ==
Suppose$ (Error compiling LaTeX. Unknown error_msg)a_1 = 32a_2 = 24
a_{n+1} = a_n^{13} a_{n-1}^{37}
n \geq 2
a_{2010}$.
== Problem 13 ==
Suppose$ (Error compiling LaTeX. Unknown error_msg)\triangle ABC\Gamma
B_1
C_1
B
CA
C
AB
D
\overline{B_1 C_1}
\overline{BC}
E
\Gamma
\overline{DA}
A
F
E
BD
BD = 28
EF = \frac{20 \sqrt{159}}{7}
ED^2 + EB^2 = 3050
\tan m \angle ACB
\frac{a \sqrt{b}}{c}
a
c
b
a + b + c$.
== Problem 14 ==
Let$ (Error compiling LaTeX. Unknown error_msg)S_n={1,2,\ldots,n}A=\{a_1,a_2,\ldots,a_k\}
S_n
k > 1
a_1 < a_2 < \cdots < a_k
A
f(A,n)
B=\{b_1,b_2,\ldots,b_k\}
b_1 < b_2 < \cdots < b_k$such that
\begin{enumerate}
\item
<cmath>(i)</cmath>$ (Error compiling LaTeX. Unknown error_msg)B
S_n
A$,
\item
<cmath>(ii)</cmath>$ (Error compiling LaTeX. Unknown error_msg)|a_i-b_i| < |b_i-a_{i+1}|
1 \le i \le k-1$,
\item
<cmath>(iii)</cmath>$ (Error compiling LaTeX. Unknown error_msg)|a_{i+1}-b_{i+1}| < |b_i-a_{i+1}|
1 \le i \le k-1$.
\end{enumerate}
Let$ (Error compiling LaTeX. Unknown error_msg)g(n)=\max_{A\subseteq S_n} f(A,n)
n
g(n)$is over 9000?
== Problem 15 ==
Let$ (Error compiling LaTeX. Unknown error_msg)XX
n \tau(n)
n
X
\tau(n)
n$.