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Difference between revisions of "Mock AIME 1 2010 Problems"

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== Problem 1 ==
 
== Problem 1 ==
 
Let <math>a_k = 7k + 4</math>. Find the number of perfect squares among <math>\{a_1, a_2, \ldots, a_{2010}\}</math>.
 
Let <math>a_k = 7k + 4</math>. Find the number of perfect squares among <math>\{a_1, a_2, \ldots, a_{2010}\}</math>.
 +
 +
[[Mock AIME 1 2010 Problems/Problem 1|Solution]]
  
 
== Problem 2 ==
 
== Problem 2 ==
−
Find the last three digits of the number of 7-tuples of positive integers <math>(a_1, a_2, a_3, a_4, a_5, a_6, a_7)</math> such that \linebreak<cmath>0</cmath> <math>a_1 \, | \, a_2 \, | \, a_3 \, | \, a_4 \, | \, a_5 \, | \, a_6 \, | \, a_7 \, | \, 6468</math>, that is, <math>a_1</math> divides <math>a_2</math>, <math>a_2</math> divides <math>a_3</math>, <math>a_3</math> divides <math>a_4</math>, <math>a_4</math> divides <math>a_5</math>, <math>a_5</math> divides <math>a_6</math>, <math>a_6</math> divides <math>a_7</math>, and <math>a_7</math> divides 6468.
+
Find the last three digits of the number of 7-tuples of positive integers <math>(a_1, a_2, a_3, a_4, a_5, a_6, a_7)</math> such that <math>a_1 \, | \, a_2 \, | \, a_3 \, | \, a_4 \, | \, a_5 \, | \, a_6 \, | \, a_7 \, | \, 6468</math>, that is, <math>a_1</math> divides <math>a_2</math>, <math>a_2</math> divides <math>a_3</math>, <math>a_3</math> divides <math>a_4</math>, <math>a_4</math> divides <math>a_5</math>, <math>a_5</math> divides <math>a_6</math>, <math>a_6</math> divides <math>a_7</math>, and <math>a_7</math> divides <math>6468</math>.
 +
 
 +
[[Mock AIME 1 2010 Problems/Problem 2|Solution]]
  
 
== Problem 3 ==
 
== Problem 3 ==
 
Let <math>AB</math> be a line segment of length <math>20 \sqrt{2}</math>, and let <math>S</math> be the set of all points <math>P</math> such that <math>m \angle APB \geq 45^{\circ}</math>. Find the last three digits of the largest integer less than the area of <math>S</math>.
 
Let <math>AB</math> be a line segment of length <math>20 \sqrt{2}</math>, and let <math>S</math> be the set of all points <math>P</math> such that <math>m \angle APB \geq 45^{\circ}</math>. Find the last three digits of the largest integer less than the area of <math>S</math>.
 +
 +
[[Mock AIME 1 2010 Problems/Problem 3|Solution]]
  
 
== Problem 4 ==
 
== 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.
 
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.
 +
 +
[[Mock AIME 1 2010 Problems/Problem 4|Solution]]
  
 
== Problem 5 ==
 
== Problem 5 ==
−
For every integer <math>N</math>, the <math>\emph{balanced ternary}</math> representation of <math>N</math> is defined to be the unique sequence of integers \linebreak<cmath>0</cmath> <math>(b_0, b_1, \ldots, b_m)</math>, with <math>b_i \in \{-1, 0, 1\}</math> and <math>b_m \neq 0</math> such that <math>N = \sum_{i=0}^{m} b_i 3^i</math>. We represent <math>N</math> as <math>c_0 c_1 \ldots c_m</math>, where <math>c_i = b_i</math> if <math>b_i</math> is 0 or 1, and <math>c_i = \underline{1}</math> if <math>b_i = -1</math>. For example, <math>2010 = 3^7 - 3^5 + 3^4 - 3^3 + 3^2 + 3 = 10\underline{1}1\underline{1}110</math>. Find the last three digits of the sum of all integers <math>N</math> with <math>1 \leq N \leq 81</math> such that <math>N</math> has at least one zero when written in balanced ternary form.
+
For every integer <math>N</math>, the <math>\emph{balanced ternary}</math> representation of <math>N</math> is defined to be the unique sequence of integers <math>(b_0, b_1, \ldots, b_m)</math>, with <math>b_i \in \{-1, 0, 1\}</math> and <math>b_m \neq 0</math> such that <math>N = \sum_{i=0}^{m} b_i 3^i</math>. We represent <math>N</math> as <math>c_0 c_1 \ldots c_m</math>, where <math>c_i = b_i</math> if <math>b_i</math> is 0 or 1, and <math>c_i = \underline{1}</math> if <math>b_i = -1</math>. For example, <math>2010 = 3^7 - 3^5 + 3^4 - 3^3 + 3^2 + 3 = 10\underline{1}1\underline{1}110</math>. Find the last three digits of the sum of all integers <math>N</math> with <math>1 \leq N \leq 81</math> such that <math>N</math> has at least one zero when written in balanced ternary form.
 +
 
 +
[[Mock AIME 1 2010 Problems/Problem 5|Solution]]
  
 
== Problem 6 ==
 
== Problem 6 ==
 
Find the number of Gaussian integers <math>z</math> with magnitude less than 10000 such that there exists a different Gaussian integer <math>w</math> such that <math>z = w^4</math>. (The magnitude of a complex <math>a+bi</math>, where <math>a</math> and <math>b</math> are reals, is defined to be <math>\sqrt{a^2+b^2}</math>. A Gaussian integer is defined to be a complex number whose real and imaginary parts are both integers.)
 
Find the number of Gaussian integers <math>z</math> with magnitude less than 10000 such that there exists a different Gaussian integer <math>w</math> such that <math>z = w^4</math>. (The magnitude of a complex <math>a+bi</math>, where <math>a</math> and <math>b</math> are reals, is defined to be <math>\sqrt{a^2+b^2}</math>. A Gaussian integer is defined to be a complex number whose real and imaginary parts are both integers.)
 +
 +
[[Mock AIME 1 2010 Problems/Problem 6|Solution]]
  
 
== Problem 7 ==
 
== Problem 7 ==
 
Find the number of positive integers <math>n \leq 2010</math> for which there exists a positive integer <math>x</math> such that <math>\frac{n + x}{x}</math> is the square of an integer.
 
Find the number of positive integers <math>n \leq 2010</math> for which there exists a positive integer <math>x</math> such that <math>\frac{n + x}{x}</math> is the square of an integer.
 +
 +
[[Mock AIME 1 2010 Problems/Problem 7|Solution]]
  
 
== Problem 8 ==
 
== Problem 8 ==
−
In the context of this problem, a <math>\emph{square}</math> is a <math>1 \times 1</math> block, a <math>\emph{domino}</math> is a <math>1 \times 2</math> block, and a <math>\emph{triomino}</math> is a <math>1 \times 3</math> block. If <math>N</math> is the number of ways George can place one square, two identical dominoes, and three identical trominoes on a <math>1 \times 20</math> chessboard such that no two overlap, find the remainder when <math>N</math> is divided by 1000.
+
In the context of this problem, a <math>\emph{square}</math> is a <math>1 \times 1</math> block, a <math>\emph{domino}</math> is a <math>1 \times 2</math> block, and a <math>\emph{triomino}</math> is a <math>1 \times 3</math> block. If <math>N</math> is the number of ways George can place one square, two identical dominoes, and three identical triominoes on a <math>1 \times 20</math> chessboard such that no two overlap, find the remainder when <math>N</math> is divided by 1000.
 +
 
 +
[[Mock AIME 1 2010 Problems/Problem 8|Solution]]
  
 
== Problem 9 ==
 
== Problem 9 ==
 
Let <math>\omega_1</math> and <math>\omega_2</math> be circles of radii 5 and 7, respectively, and suppose that the distance between their centers is 10. There exists a circle <math>\omega_3</math> that is internally tangent to both <math>\omega_1</math> and <math>\omega_2</math>, and tangent to the line joining the centers of <math>\omega_1</math> and <math>\omega_2</math>. If the radius of <math>\omega_3</math> can be expressed in the form <math>a \sqrt{b} - c</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are integers, and <math>b</math> is not divisible by the square if any prime, find the value of <math>a + b + c</math>.
 
Let <math>\omega_1</math> and <math>\omega_2</math> be circles of radii 5 and 7, respectively, and suppose that the distance between their centers is 10. There exists a circle <math>\omega_3</math> that is internally tangent to both <math>\omega_1</math> and <math>\omega_2</math>, and tangent to the line joining the centers of <math>\omega_1</math> and <math>\omega_2</math>. If the radius of <math>\omega_3</math> can be expressed in the form <math>a \sqrt{b} - c</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are integers, and <math>b</math> is not divisible by the square if any prime, find the value of <math>a + b + c</math>.
 +
 +
[[Mock AIME 1 2010 Problems/Problem 9|Solution]]
  
 
== Problem 10 ==
 
== Problem 10 ==
−
Find the last three digits of the largest possible value of  
+
Find the last three digits of the largest possible value of <cmath>\frac{a^2 b^6}{a^{2 \log_2 a} (a^2 b)^{\log_2 b}},</cmath> where <math>a</math> and <math>b</math> are positive reals.
−
<math>\frac{a^2 b^6}{a^{2 \log_2 a} (a^2 b)^{\log_2 b}},</math>
+
 
−
where <math>a</math> and <math>b</math> are positive reals.
+
[[Mock AIME 1 2010 Problems/Problem 10|Solution]]
  
 
== Problem 11 ==
 
== Problem 11 ==
 
Let <math>\triangle ABC</math> be such that <math>AB = 7</math>, <math>BC = 8</math>, and <math>CA = 9</math>. Let <math>D</math>, <math>E</math>, and <math>F</math> be points such that <math>DB \perp BA</math>, <math>DC \perp CA</math>, <math>EC \perp CB</math>, <math>EA \perp AB</math>, <math>FA \perp AC</math>, and <math>FB \perp BC</math>. If the perimeter of hexagon <math>AFBDCE</math> can be expressed in the form <math>\frac{a \sqrt{b}}{c}</math>, where <math>a</math> and <math>c</math> are relatively prime positive integers and <math>b</math> is an integer not divisible by the square of any prime, find <math>a</math>.
 
Let <math>\triangle ABC</math> be such that <math>AB = 7</math>, <math>BC = 8</math>, and <math>CA = 9</math>. Let <math>D</math>, <math>E</math>, and <math>F</math> be points such that <math>DB \perp BA</math>, <math>DC \perp CA</math>, <math>EC \perp CB</math>, <math>EA \perp AB</math>, <math>FA \perp AC</math>, and <math>FB \perp BC</math>. If the perimeter of hexagon <math>AFBDCE</math> can be expressed in the form <math>\frac{a \sqrt{b}}{c}</math>, where <math>a</math> and <math>c</math> are relatively prime positive integers and <math>b</math> is an integer not divisible by the square of any prime, find <math>a</math>.
 +
 +
[[Mock AIME 1 2010 Problems/Problem 11|Solution]]
  
 
== Problem 12 ==
 
== Problem 12 ==
 
Suppose <math>a_1 = 32</math>, <math>a_2 = 24</math>, and <math>a_{n+1} = a_n^{13} a_{n-1}^{37}</math> for all integers <math>n \geq 2</math>. Find the last three digits of <math>a_{2010}</math>.
 
Suppose <math>a_1 = 32</math>, <math>a_2 = 24</math>, and <math>a_{n+1} = a_n^{13} a_{n-1}^{37}</math> for all integers <math>n \geq 2</math>. Find the last three digits of <math>a_{2010}</math>.
 +
 +
[[Mock AIME 1 2010 Problems/Problem 12|Solution]]
  
 
== Problem 13 ==
 
== Problem 13 ==
 
Suppose <math>\triangle ABC</math> is inscribed in circle <math>\Gamma</math>. <math>B_1</math> and <math>C_1</math> are the feet of the altitude from <math>B</math> to <math>CA</math> and <math>C</math> to <math>AB</math>, respectively. Let <math>D</math> be the intersection of lines <math>\overline{B_1 C_1}</math> and <math>\overline{BC}</math>, let <math>E</math> be the point of intersection of <math>\Gamma</math> and line <math>\overline{DA}</math> distinct from <math>A</math>, and let <math>F</math> be the foot of the perpendicular from <math>E</math> to <math>BD</math>. Given that <math>BD = 28</math>, <math>EF = \frac{20 \sqrt{159}}{7}</math>, and <math>ED^2 + EB^2 = 3050</math>, and that <math>\tan m \angle ACB</math> can be expressed in the form <math>\frac{a \sqrt{b}}{c}</math>, where <math>a</math> and <math>c</math> are relatively prime positive integers and <math>b</math> is an integer not divisible by the square of any prime, find the last three digits of <math>a + b + c</math>.
 
Suppose <math>\triangle ABC</math> is inscribed in circle <math>\Gamma</math>. <math>B_1</math> and <math>C_1</math> are the feet of the altitude from <math>B</math> to <math>CA</math> and <math>C</math> to <math>AB</math>, respectively. Let <math>D</math> be the intersection of lines <math>\overline{B_1 C_1}</math> and <math>\overline{BC}</math>, let <math>E</math> be the point of intersection of <math>\Gamma</math> and line <math>\overline{DA}</math> distinct from <math>A</math>, and let <math>F</math> be the foot of the perpendicular from <math>E</math> to <math>BD</math>. Given that <math>BD = 28</math>, <math>EF = \frac{20 \sqrt{159}}{7}</math>, and <math>ED^2 + EB^2 = 3050</math>, and that <math>\tan m \angle ACB</math> can be expressed in the form <math>\frac{a \sqrt{b}}{c}</math>, where <math>a</math> and <math>c</math> are relatively prime positive integers and <math>b</math> is an integer not divisible by the square of any prime, find the last three digits of <math>a + b + c</math>.
 +
 +
[[Mock AIME 1 2010 Problems/Problem 13|Solution]]
  
 
== Problem 14 ==
 
== Problem 14 ==
 
Let <math>S_n={1,2,\ldots,n}</math>, and let <math>A=\{a_1,a_2,\ldots,a_k\}</math> be a subset of <math>S_n</math> with <math>k > 1</math> and <math>a_1 < a_2 < \cdots < a_k</math>. For such a set <math>A</math>, let <math>f(A,n)</math> denote the number of sets <math>B=\{b_1,b_2,\ldots,b_k\}</math> with <math>b_1 < b_2 < \cdots < b_k</math> such that  
 
Let <math>S_n={1,2,\ldots,n}</math>, and let <math>A=\{a_1,a_2,\ldots,a_k\}</math> be a subset of <math>S_n</math> with <math>k > 1</math> and <math>a_1 < a_2 < \cdots < a_k</math>. For such a set <math>A</math>, let <math>f(A,n)</math> denote the number of sets <math>B=\{b_1,b_2,\ldots,b_k\}</math> with <math>b_1 < b_2 < \cdots < b_k</math> such that  
−
\begin{enumerate}
+
 
−
\item
+
(i) <math>B</math> is a subset of <math>S_n</math> with the same number of elements as <math>A</math>,
−
<cmath>(i)</cmath> <math>B</math> is a subset of <math>S_n</math> with the same number of elements as <math>A</math>,
+
 
−
\item
+
(ii) <math>|a_i-b_i| < |b_i-a_{i+1}|</math> for <math>1 \le i \le k-1</math>,
−
<cmath>(ii)</cmath> <math>|a_i-b_i| < |b_i-a_{i+1}|</math> for <math>1 \le i \le k-1</math>,
+
 
−
\item
+
(iii) <math>|a_{i+1}-b_{i+1}| < |b_i-a_{i+1}|</math> for <math>1 \le i \le k-1</math>.
−
<cmath>(iii)</cmath> <math>|a_{i+1}-b_{i+1}| < |b_i-a_{i+1}|</math> for <math>1 \le i \le k-1</math>.
+
 
−
\end{enumerate}
 
 
Let <math>g(n)=\max_{A\subseteq S_n} f(A,n)</math>. What is the smallest positive integer <math>n</math> such that <math>g(n)</math> is over 9000?
 
Let <math>g(n)=\max_{A\subseteq S_n} f(A,n)</math>. What is the smallest positive integer <math>n</math> such that <math>g(n)</math> is over 9000?
 +
 +
[[Mock AIME 1 2010 Problems/Problem 14|Solution]]
  
 
== Problem 15 ==
 
== Problem 15 ==
 
Let <math>X</math> be the set of all integers less than or equal to 2010 such that when its divisors are listed in increasing order, they are alternatingly odd and even. For example, 6 belongs to <math>X</math>, since the divisors of 6 are 1, 2, 3, and 6, which are odd, even, odd, and even in that order. Find the last three digits of the largest possible value of <math>n \tau(n)</math>, where <math>n</math> lies in <math>X</math> and <math>\tau(n)</math> denotes the number of divisors of <math>n</math>.
 
Let <math>X</math> be the set of all integers less than or equal to 2010 such that when its divisors are listed in increasing order, they are alternatingly odd and even. For example, 6 belongs to <math>X</math>, since the divisors of 6 are 1, 2, 3, and 6, which are odd, even, odd, and even in that order. Find the last three digits of the largest possible value of <math>n \tau(n)</math>, where <math>n</math> lies in <math>X</math> and <math>\tau(n)</math> denotes the number of divisors of <math>n</math>.
 +
 +
[[Mock AIME 1 2010 Problems/Problem 15|Solution]]
 +
 +
== See Also ==
 +
*[[Mock AIME 1 2010]]
 +
*[[Mock AIME 1 2010 Answer Key]]

Latest revision as of 12:20, 2 August 2024

Problem 1

Let $a_k = 7k + 4$. Find the number of perfect squares among $\{a_1, a_2, \ldots, a_{2010}\}$.

Solution

Problem 2

Find the last three digits of the number of 7-tuples of positive integers $(a_1, a_2, a_3, a_4, a_5, a_6, a_7)$ such that $a_1 \, | \, a_2 \, | \, a_3 \, | \, a_4 \, | \, a_5 \, | \, a_6 \, | \, a_7 \, | \, 6468$, that is, $a_1$ divides $a_2$, $a_2$ divides $a_3$, $a_3$ divides $a_4$, $a_4$ divides $a_5$, $a_5$ divides $a_6$, $a_6$ divides $a_7$, and $a_7$ divides $6468$.

Solution

Problem 3

Let $AB$ be a line segment of length $20 \sqrt{2}$, and let $S$ be the set of all points $P$ such that $m \angle APB \geq 45^{\circ}$. Find the last three digits of the largest integer less than the area of $S$.

Solution

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.

Solution

Problem 5

For every integer $N$, the $\emph{balanced ternary}$ representation of $N$ is defined to be the unique sequence of integers $(b_0, b_1, \ldots, b_m)$, with $b_i \in \{-1, 0, 1\}$ and $b_m \neq 0$ such that $N = \sum_{i=0}^{m} b_i 3^i$. We represent $N$ as $c_0 c_1 \ldots c_m$, where $c_i = b_i$ if $b_i$ is 0 or 1, and $c_i = \underline{1}$ if $b_i = -1$. For example, $2010 = 3^7 - 3^5 + 3^4 - 3^3 + 3^2 + 3 = 10\underline{1}1\underline{1}110$. Find the last three digits of the sum of all integers $N$ with $1 \leq N \leq 81$ such that $N$ has at least one zero when written in balanced ternary form.

Solution

Problem 6

Find the number of Gaussian integers $z$ with magnitude less than 10000 such that there exists a different Gaussian integer $w$ such that $z = w^4$. (The magnitude of a complex $a+bi$, where $a$ and $b$ are reals, is defined to be $\sqrt{a^2+b^2}$. A Gaussian integer is defined to be a complex number whose real and imaginary parts are both integers.)

Solution

Problem 7

Find the number of positive integers $n \leq 2010$ for which there exists a positive integer $x$ such that $\frac{n + x}{x}$ is the square of an integer.

Solution

Problem 8

In the context of this problem, a $\emph{square}$ is a $1 \times 1$ block, a $\emph{domino}$ is a $1 \times 2$ block, and a $\emph{triomino}$ is a $1 \times 3$ block. If $N$ is the number of ways George can place one square, two identical dominoes, and three identical triominoes on a $1 \times 20$ chessboard such that no two overlap, find the remainder when $N$ is divided by 1000.

Solution

Problem 9

Let $\omega_1$ and $\omega_2$ be circles of radii 5 and 7, respectively, and suppose that the distance between their centers is 10. There exists a circle $\omega_3$ that is internally tangent to both $\omega_1$ and $\omega_2$, and tangent to the line joining the centers of $\omega_1$ and $\omega_2$. If the radius of $\omega_3$ can be expressed in the form $a \sqrt{b} - c$, where $a$, $b$, and $c$ are integers, and $b$ is not divisible by the square if any prime, find the value of $a + b + c$.

Solution

Problem 10

Find the last three digits of the largest possible value of \[\frac{a^2 b^6}{a^{2 \log_2 a} (a^2 b)^{\log_2 b}},\] where $a$ and $b$ are positive reals.

Solution

Problem 11

Let $\triangle ABC$ be such that $AB = 7$, $BC = 8$, and $CA = 9$. Let $D$, $E$, and $F$ be points such that $DB \perp BA$, $DC \perp CA$, $EC \perp CB$, $EA \perp AB$, $FA \perp AC$, and $FB \perp BC$. If the perimeter of hexagon $AFBDCE$ can be expressed in the form $\frac{a \sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers and $b$ is an integer not divisible by the square of any prime, find $a$.

Solution

Problem 12

Suppose $a_1 = 32$, $a_2 = 24$, and $a_{n+1} = a_n^{13} a_{n-1}^{37}$ for all integers $n \geq 2$. Find the last three digits of $a_{2010}$.

Solution

Problem 13

Suppose $\triangle ABC$ is inscribed in circle $\Gamma$. $B_1$ and $C_1$ are the feet of the altitude from $B$ to $CA$ and $C$ to $AB$, respectively. Let $D$ be the intersection of lines $\overline{B_1 C_1}$ and $\overline{BC}$, let $E$ be the point of intersection of $\Gamma$ and line $\overline{DA}$ distinct from $A$, and let $F$ be the foot of the perpendicular from $E$ to $BD$. Given that $BD = 28$, $EF = \frac{20 \sqrt{159}}{7}$, and $ED^2 + EB^2 = 3050$, and that $\tan m \angle ACB$ can be expressed in the form $\frac{a \sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers and $b$ is an integer not divisible by the square of any prime, find the last three digits of $a + b + c$.

Solution

Problem 14

Let $S_n={1,2,\ldots,n}$, and let $A=\{a_1,a_2,\ldots,a_k\}$ be a subset of $S_n$ with $k > 1$ and $a_1 < a_2 < \cdots < a_k$. For such a set $A$, let $f(A,n)$ denote the number of sets $B=\{b_1,b_2,\ldots,b_k\}$ with $b_1 < b_2 < \cdots < b_k$ such that

(i) $B$ is a subset of $S_n$ with the same number of elements as $A$,

(ii) $|a_i-b_i| < |b_i-a_{i+1}|$ for $1 \le i \le k-1$,

(iii) $|a_{i+1}-b_{i+1}| < |b_i-a_{i+1}|$ for $1 \le i \le k-1$.

Let $g(n)=\max_{A\subseteq S_n} f(A,n)$. What is the smallest positive integer $n$ such that $g(n)$ is over 9000?

Solution

Problem 15

Let $X$ be the set of all integers less than or equal to 2010 such that when its divisors are listed in increasing order, they are alternatingly odd and even. For example, 6 belongs to $X$, since the divisors of 6 are 1, 2, 3, and 6, which are odd, even, odd, and even in that order. Find the last three digits of the largest possible value of $n \tau(n)$, where $n$ lies in $X$ and $\tau(n)$ denotes the number of divisors of $n$.

Solution

See Also

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