problem stringlengths 15 4.7k | solution stringlengths 2 11.9k | answer stringclasses 51
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Each of the numbers $n$, $n+1$, $n+2$, $n+3$ is divisible by its sum of digits in its decimal representation. How many different values can the tens column of $n$ have, if the number in ones column of $n$ is $8$?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ 5... | 1. Define \( S_k \) as the sum of the digits of \( k \) and \( a_k \) as the tens digit of \( k \).
2. Given that the number in the ones column of \( n \) is \( 8 \), we can write \( n \) as \( 10a + 8 \) where \( a \) is the tens digit of \( n \).
3. We need to check the divisibility of \( n \), \( n+1 \), \( n+2 \)... | 1 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
How many pairs of integers $(x,y)$ are there such that $2x+5y=xy-1$?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ 4
\qquad\textbf{(D)}\ 6
\qquad\textbf{(E)}\ 12
$ | 1. Start with the given equation:
\[
2x + 5y = xy - 1
\]
2. Rearrange the equation to group all terms on one side:
\[
xy - 2x - 5y = 1
\]
3. Factor by grouping. Notice that we can factor the left-hand side as follows:
\[
xy - 2x - 5y + 10 - 10 = 1 \implies (x-5)(y-2) = 11
\]
4. Now, we nee... | 4 | Algebra | MCQ | Yes | Yes | aops_forum | false |
What is $o-w$, if $gun^2 = wowgun$ where $g,n,o,u,w \in \{0,1,2,\dots, 9\}$?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 5
\qquad\textbf{(E)}\ \text{None of above}
$ | 1. We are given the equation \( gun^2 = wowgun \) where \( g, n, o, u, w \in \{0,1,2,\dots, 9\} \).
2. We need to find the values of \( g, n, o, u, w \) such that the equation holds true.
3. Let's test the given ideas \( 376 \) and \( 625 \) to see if they satisfy the equation.
- For \( gun = 376 \):
\[
3... | 3 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
How many consequtive numbers are there in the set of positive integers in which powers of all prime factors in their prime factorizations are odd numbers?
$
\textbf{(A)}\ 3
\qquad\textbf{(B)}\ 7
\qquad\textbf{(C)}\ 8
\qquad\textbf{(D)}\ 10
\qquad\textbf{(E)}\ 15
$ | 1. To solve this problem, we need to understand the properties of numbers whose prime factors have only odd powers. Such numbers are called "square-free" numbers, meaning they are not divisible by any perfect square greater than 1.
2. Consider any set of consecutive integers. Among any set of 8 consecutive integers, at... | 7 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
What is the last two digits of base-$3$ representation of $2005^{2003^{2004}+3}$?
$
\textbf{(A)}\ 21
\qquad\textbf{(B)}\ 01
\qquad\textbf{(C)}\ 11
\qquad\textbf{(D)}\ 02
\qquad\textbf{(E)}\ 22
$ | To find the last two digits of the base-3 representation of \(2005^{2003^{2004} + 3}\), we need to follow these steps:
1. **Convert the problem to a simpler form using modular arithmetic:**
We need to find the last two digits in base-3, which is equivalent to finding the number modulo \(3^2 = 9\).
2. **Simplify th... | 11 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
How many primes $p$ are there such that the number of positive divisors of $p^2+23$ is equal to $14$?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 3
\qquad\textbf{(E)}\ \text{None of above}
$ | To solve the problem, we need to find the number of prime numbers \( p \) such that the number of positive divisors of \( p^2 + 23 \) is equal to 14.
1. **Understanding the divisor function**:
The number of positive divisors of a number \( n \) can be determined from its prime factorization. If \( n \) has the pri... | 1 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
How many positive integers which divide $5n^{11}-2n^5-3n$ for all positive integers $n$ are there?
$
\textbf{(A)}\ 2
\qquad\textbf{(B)}\ 5
\qquad\textbf{(C)}\ 6
\qquad\textbf{(D)}\ 12
\qquad\textbf{(E)}\ 18
$ | To determine how many positive integers divide \( f(n) = 5n^{11} - 2n^5 - 3n \) for all positive integers \( n \), we need to find the common divisors of \( f(n) \) for any \( n \).
1. **Check divisibility by 2:**
\[
f(n) = 5n^{11} - 2n^5 - 3n
\]
Since \( 5n^{11} \), \( 2n^5 \), and \( 3n \) are all multip... | 12 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
At least how many weighings of a balanced scale are needed to order four stones with distinct weights from the lightest to the heaviest?
$
\textbf{(A)}\ 4
\qquad\textbf{(B)}\ 5
\qquad\textbf{(C)}\ 6
\qquad\textbf{(D)}\ 7
\qquad\textbf{(E)}\ 8
$ | 1. **Label the stones**: Let the four stones be labeled as \( A, B, C, \) and \( D \).
2. **Initial weighings**:
- Weigh \( A \) against \( B \).
- Weigh \( C \) against \( D \).
3. **Determine the lightest pair**:
- Suppose \( A \) is lighter than \( B \) and \( C \) is lighter than \( D \).
- Weigh the... | 5 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
We have $31$ pieces where $1$ is written on two of them, $2$ is written on eight of them, $3$ is written on twelve of them, $4$ is written on four of them, and $5$ is written on five of them. We place $30$ of them into a $5\times 6$ chessboard such that the sum of numbers on any row is equal to a fixed number and the ... | 1. **Calculate the total sum of all 31 pieces:**
\[
\text{Sum} = 2 \cdot 1 + 8 \cdot 2 + 12 \cdot 3 + 4 \cdot 4 + 5 \cdot 5
\]
\[
= 2 + 16 + 36 + 16 + 25
\]
\[
= 95
\]
2. **Determine the sum of the 30 pieces placed on the chessboard:**
Since the sum of the numbers in each row and each col... | 5 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
For how many different values of integer $n$, one can find $n$ different lines in the plane such that each line intersects with exacly $2004$ of other lines?
$
\textbf{(A)}\ 12
\qquad\textbf{(B)}\ 11
\qquad\textbf{(C)}\ 9
\qquad\textbf{(D)}\ 6
\qquad\textbf{(E)}\ 1
$ | 1. Let \( n \) be the number of lines in the plane, and each line intersects exactly \( 2004 \) other lines. This means each line does not intersect \( n - 2004 - 1 \) lines (since it intersects \( 2004 \) lines out of the total \( n-1 \) other lines).
2. Consider the lines grouped into maximal sets of parallel lines.... | 12 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
For how many triples of positive integers $(x,y,z)$, there exists a positive integer $n$ such that $\dfrac{x}{n} = \dfrac{y}{n+1} = \dfrac{z}{n+2}$ where $x+y+z=90$?
$
\textbf{(A)}\ 4
\qquad\textbf{(B)}\ 5
\qquad\textbf{(C)}\ 6
\qquad\textbf{(D)}\ 7
\qquad\textbf{(E)}\ 9
$ | To solve the problem, we need to find the number of triples \((x, y, z)\) of positive integers such that there exists a positive integer \(n\) satisfying the equation:
\[
\frac{x}{n} = \frac{y}{n+1} = \frac{z}{n+2}
\]
and the condition:
\[
x + y + z = 90
\]
1. Let \(k\) be the common value of the fractions. Then we... | 7 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
What is the sum of real roots of the equation $x^4-4x^3+5x^2-4x+1 = 0$?
$
\textbf{(A)}\ 5
\qquad\textbf{(B)}\ 4
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 2
\qquad\textbf{(E)}\ 1
$ | 1. Given the polynomial equation:
\[
x^4 - 4x^3 + 5x^2 - 4x + 1 = 0
\]
We need to find the sum of the real roots of this equation.
2. Let's try to factorize the polynomial. Notice that the polynomial can be rewritten as:
\[
(x-1)^4 - x^2 = 0
\]
3. Expanding \((x-1)^4\):
\[
(x-1)^4 = x^4 - 4... | 3 | Algebra | MCQ | Yes | Yes | aops_forum | false |
What is the sum of cubes of real roots of the equation $x^3-2x^2-x+1=0$?
$
\textbf{(A)}\ -6
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 8
\qquad\textbf{(D)}\ 11
\qquad\textbf{(E)}\ \text{None of above}
$ | 1. Given the polynomial equation \( f(x) = x^3 - 2x^2 - x + 1 = 0 \), we need to find the sum of the cubes of its real roots.
2. First, we determine the number of real roots by evaluating the polynomial at several points:
\[
f(-1) = (-1)^3 - 2(-1)^2 - (-1) + 1 = -1 - 2 + 1 + 1 = -1
\]
\[
f(0) = 0^3 - 2(0... | 11 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Let $h_1$ and $h_2$ be the altitudes of a triangle drawn to the sides with length $5$ and $2\sqrt 6$, respectively. If $5+h_1 \leq 2\sqrt 6 + h_2$, what is the third side of the triangle?
$
\textbf{(A)}\ 5
\qquad\textbf{(B)}\ 7
\qquad\textbf{(C)}\ 2\sqrt 6
\qquad\textbf{(D)}\ 3\sqrt 6
\qquad\textbf{(E)}\ 5\sqrt 3
$ | 1. Let $\triangle ABC$ be a triangle with sides $a = 5$, $b = 2\sqrt{6}$, and $c$ being the third side. Let $h_1$ and $h_2$ be the altitudes to the sides $a$ and $b$, respectively.
2. The given inequality is $5 + h_1 \leq 2\sqrt{6} + h_2$.
3. Recall that the area of the triangle can be expressed using the base and the ... | 7 | Geometry | MCQ | Yes | Yes | aops_forum | false |
Which of the following divides $3^{3n+1} + 5^{3n+2}+7^{3n+3}$ for every positive integer $n$?
$
\textbf{(A)}\ 3
\qquad\textbf{(B)}\ 5
\qquad\textbf{(C)}\ 7
\qquad\textbf{(D)}\ 11
\qquad\textbf{(E)}\ 53
$ | To determine which of the given numbers divides \(3^{3n+1} + 5^{3n+2} + 7^{3n+3}\) for every positive integer \(n\), we will check each option one by one.
1. **Check divisibility by 3:**
\[
3^{3n+1} + 5^{3n+2} + 7^{3n+3} \pmod{3}
\]
- \(3^{3n+1} \equiv 0 \pmod{3}\) because any power of 3 is divisible by 3.... | 7 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Which of the following does not divide $n^{2225}-n^{2005}$ for every integer value of $n$?
$
\textbf{(A)}\ 3
\qquad\textbf{(B)}\ 5
\qquad\textbf{(C)}\ 7
\qquad\textbf{(D)}\ 11
\qquad\textbf{(E)}\ 23
$ | 1. We start with the expression \( n^{2225} - n^{2005} \). We can factor this as:
\[
n^{2225} - n^{2005} = n^{2005}(n^{220} - 1)
\]
This factorization helps us analyze the divisibility properties of the expression.
2. To determine which of the given numbers does not divide \( n^{2225} - n^{2005} \) for eve... | 7 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
We call a number $10^3 < n < 10^6$ a [i]balanced [/i]number if the sum of its last three digits is equal to the sum of its other digits. What is the sum of all balanced numbers in $\bmod {13}$?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 5
\qquad\textbf{(C)}\ 7
\qquad\textbf{(D)}\ 11
\qquad\textbf{(E)}\ 12
$ | 1. We need to find the sum of all balanced numbers in the range \(10^3 < n < 10^6\) modulo 13. A balanced number is defined as a number where the sum of its last three digits is equal to the sum of its other digits.
2. Let's denote a balanced number as \(n = 1000a + b\), where \(a\) is the integer part of the number w... | 0 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
For the real pairs $(x,y)$ satisfying the equation $x^2 + y^2 + 2x - 6y = 6$, which of the following cannot be equal to $(x-1)^2 + (y-2)^2$?
$
\textbf{(A)}\ 2
\qquad\textbf{(B)}\ 9
\qquad\textbf{(C)}\ 16
\qquad\textbf{(D)}\ 23
\qquad\textbf{(E)}\ 30
$ | 1. Start with the given equation:
\[
x^2 + y^2 + 2x - 6y = 6
\]
We can complete the square for both \(x\) and \(y\).
2. For \(x\):
\[
x^2 + 2x = (x+1)^2 - 1
\]
3. For \(y\):
\[
y^2 - 6y = (y-3)^2 - 9
\]
4. Substitute these into the original equation:
\[
(x+1)^2 - 1 + (y-3)^2 - 9 =... | 2 | Geometry | MCQ | Yes | Yes | aops_forum | false |
What is the maximum value of the difference between the largest real root and the smallest real root of the equation system \[\begin{array}{rcl}
ax^2 + bx+ c &=& 0 \\
bx^2 + cx+ a &=& 0 \\
cx^2 + ax+ b &=& 0
\end{array}\], where at least one of the reals $a,b,c$ is non-zero?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\q... | 1. We start with the given system of quadratic equations:
\[
\begin{array}{rcl}
ax^2 + bx + c &=& 0 \\
bx^2 + cx + a &=& 0 \\
cx^2 + ax + b &=& 0
\end{array}
\]
We need to find the maximum value of the difference between the largest real root and the smallest real root of this system.
2. First,... | 0 | Algebra | MCQ | Yes | Yes | aops_forum | false |
$100$ stones, each weighs $1$ kg or $10$ kgs or $50$ kgs, weighs $500$ kgs in total. How many values can the number of stones weighing $10$ kgs take?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 3
\qquad\textbf{(E)}\ 4
$ | To solve this problem, we need to determine how many different values the number of stones weighing 10 kg can take, given the constraints. Let's denote:
- \( x \) as the number of stones weighing 1 kg,
- \( y \) as the number of stones weighing 10 kg,
- \( z \) as the number of stones weighing 50 kg.
We are given the ... | 1 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Ali chooses one of the stones from a group of $2005$ stones, marks this stone in a way that Betül cannot see the mark, and shuffles the stones. At each move, Betül divides stones into three non-empty groups. Ali removes the group with more stones from the two groups that do not contain the marked stone (if these two gr... | 1. **Understanding the Problem:**
- Ali marks one stone out of 2005 stones.
- Betül divides the stones into three non-empty groups.
- Ali removes the group with more stones from the two groups that do not contain the marked stone.
- The game continues until two stones remain, and Ali reveals the marked ston... | 11 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
If $p$ and $p^2+2$ are prime numbers, at most how many prime divisors can $p^3+3$ have?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ 5
$ | 1. **Identify the given conditions:**
- \( p \) is a prime number.
- \( p^2 + 2 \) is also a prime number.
2. **Analyze the possible values of \( p \):**
- If \( p = 2 \):
\[
p^2 + 2 = 2^2 + 2 = 4 + 2 = 6
\]
Since 6 is not a prime number, \( p \) cannot be 2.
- If \( p = 3 \):
\[
... | 3 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
$a_1=-1$, $a_2=2$, and $a_n=\frac {a_{n-1}}{a_{n-2}}$ for $n\geq 3$. What is $a_{2006}$?
$
\textbf{(A)}\ -2
\qquad\textbf{(B)}\ -1
\qquad\textbf{(C)}\ -\frac 12
\qquad\textbf{(D)}\ \frac 12
\qquad\textbf{(E)}\ 2
$ | 1. First, we list out the initial terms given in the problem:
\[
a_1 = -1, \quad a_2 = 2
\]
2. Using the recurrence relation \(a_n = \frac{a_{n-1}}{a_{n-2}}\) for \(n \geq 3\), we calculate the next few terms:
\[
a_3 = \frac{a_2}{a_1} = \frac{2}{-1} = -2
\]
\[
a_4 = \frac{a_3}{a_2} = \frac{-2}{2... | 2 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
What is the sum of $3+3^2+3^{2^2} + 3^{2^3} + \dots + 3^{2^{2006}}$ in $\mod 11$?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 5
\qquad\textbf{(E)}\ 10
$ | 1. **Identify the sequence pattern:**
We need to find the sum of the series \(3 + 3^2 + 3^{2^2} + 3^{2^3} + \dots + 3^{2^{2006}}\) modulo 11. First, we observe the behavior of powers of 3 modulo 11.
2. **Calculate the first few powers of 3 modulo 11:**
\[
3^1 \equiv 3 \mod 11
\]
\[
3^2 \equiv 9 \mod ... | 6 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
What is the sum of the real roots of the equation $4x^4-3x^2+7x-3=0$?
$
\textbf{(A)}\ -1
\qquad\textbf{(B)}\ -2
\qquad\textbf{(C)}\ -3
\qquad\textbf{(D)}\ -4
\qquad\textbf{(E)}\ \text {None of above}
$ | 1. Given the polynomial equation:
\[
4x^4 - 3x^2 + 7x - 3 = 0
\]
we need to find the sum of the real roots.
2. We start by using synthetic division to check for possible rational roots. Let's test \( x = \frac{1}{2} \):
\[
\text{Synthetic division of } 4x^4 - 3x^2 + 7x - 3 \text{ by } (2x - 1):
\]... | -1 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Let $P(x)=x^3+ax^2+bx+c$ where $a,b,c$ are positive real numbers. If $P(1)\geq 2$ and $P(3)\leq 31$, how many of integers can $P(4)$ take?
$
\textbf{(A)}\ 3
\qquad\textbf{(B)}\ 4
\qquad\textbf{(C)}\ 5
\qquad\textbf{(D)}\ 6
\qquad\textbf{(E)}\ \text{None of above}
$ | 1. Given the polynomial \( P(x) = x^3 + ax^2 + bx + c \) where \( a, b, c \) are positive real numbers, we need to find the possible integer values of \( P(4) \) given the constraints \( P(1) \geq 2 \) and \( P(3) \leq 31 \).
2. First, evaluate \( P(1) \):
\[
P(1) = 1^3 + a \cdot 1^2 + b \cdot 1 + c = 1 + a + b ... | 4 | Inequalities | MCQ | Yes | Yes | aops_forum | false |
What is the greatest integer $k$ which makes the statement "When we take any $6$ subsets with $5$ elements of the set $\{1,2,\dots, 9\}$, there exist $k$ of them having at least one common element." true?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ 5
$ | To solve this problem, we need to determine the greatest integer \( k \) such that in any selection of 6 subsets of 5 elements each from the set \(\{1, 2, \dots, 9\}\), there exist \( k \) subsets that have at least one common element.
1. **Calculate the total number of subsets:**
The set \(\{1, 2, \dots, 9\}\) has... | 4 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
$P(x)=ax^2+bx+c$ has exactly $1$ different real root where $a,b,c$ are real numbers. If $P(P(P(x)))$ has exactly $3$ different real roots, what is the minimum possible value of $abc$?
$
\textbf{(A)}\ -3
\qquad\textbf{(B)}\ -2
\qquad\textbf{(C)}\ 2\sqrt 3
\qquad\textbf{(D)}\ 3\sqrt 3
\qquad\textbf{(E)}\ \text{None of ... | 1. Given that \( P(x) = ax^2 + bx + c \) has exactly one real root, we know that the discriminant of \( P(x) \) must be zero. The discriminant of a quadratic equation \( ax^2 + bx + c \) is given by:
\[
\Delta = b^2 - 4ac
\]
For \( P(x) \) to have exactly one real root, we must have:
\[
\Delta = 0 \im... | -2 | Algebra | MCQ | Yes | Yes | aops_forum | false |
How many positive integers $n$ are there such that $n!(2n+1)$ and $221$ are relatively prime?
$
\textbf{(A)}\ 10
\qquad\textbf{(B)}\ 11
\qquad\textbf{(C)}\ 12
\qquad\textbf{(D)}\ 13
\qquad\textbf{(E)}\ \text{None of the above}
$ | 1. First, we need to determine the prime factorization of 221:
\[
221 = 13 \times 17
\]
Therefore, \( n!(2n+1) \) and 221 are relatively prime if and only if \( n!(2n+1) \) is not divisible by either 13 or 17.
2. Consider the factorial part \( n! \). For \( n! \) to be relatively prime to 221, \( n \) must... | 10 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Starting from the number $123456789$, at each step, we are swaping two adjacent numbers which are different from zero, and then decreasing the two numbers by $1$. What is the sum of digits of the least number that can be get after finite steps?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 3
\qquad\text... | 1. We start with the number \(123456789\).
2. At each step, we swap two adjacent numbers which are different from zero and then decrease both numbers by 1.
3. We need to determine the sum of the digits of the least number that can be obtained after a finite number of steps.
To understand the process, let's consider th... | 5 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
How many positive integers $n<10^6$ are there such that $n$ is equal to twice of square of an integer and is equal to three times of cube of an integer?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 3
\qquad\textbf{(E)}\ \text{None of the above}
$ | To solve the problem, we need to find the positive integers \( n < 10^6 \) such that \( n \) is both twice the square of an integer and three times the cube of an integer. Let's denote these integers as follows:
1. Let \( n = 2a^2 \) for some integer \( a \).
2. Let \( n = 3b^3 \) for some integer \( b \).
Equating t... | 2 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Let $K$ be the point of intersection of $AB$ and the line touching the circumcircle of $\triangle ABC$ at $C$ where $m(\widehat {A}) > m(\widehat {B})$. Let $L$ be a point on $[BC]$ such that $m(\widehat{ALB})=m(\widehat{CAK})$, $5|LC|=4|BL|$, and $|KC|=12$. What is $|AK|$?
$
\textbf{(A)}\ 4\sqrt 2
\qquad\textbf{(B)}... | 1. Given that $K$ is the point of intersection of $AB$ and the line touching the circumcircle of $\triangle ABC$ at $C$, and $L$ is a point on $[BC]$ such that $m(\widehat{ALB}) = m(\widehat{CAK})$, $5|LC| = 4|BL|$, and $|KC| = 12$.
2. We need to find $|AK|$.
3. Since $\angle{ABL} = \angle{ABC} = \angle{KCA}$, we have... | 8 | Geometry | MCQ | Yes | Yes | aops_forum | false |
How many integers $n$ are there such that $n^3+8$ has at most $3$ positive divisors?
$
\textbf{(A)}\ 4
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 1
\qquad\textbf{(E)}\ \text{None of the above}
$ | To determine how many integers \( n \) exist such that \( n^3 + 8 \) has at most 3 positive divisors, we need to consider the structure of numbers with at most 3 divisors.
A number can have at most 3 positive divisors in the following cases:
1. The number is of the form \( p^2 \) where \( p \) is a prime (total numbe... | 2 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Let $ABCD$ be a quadrilateral such that $m(\widehat{A}) = m(\widehat{D}) = 90^\circ$. Let $M$ be the midpoint of $[DC]$. If $AC\perp BM$, $|DC|=12$, and $|AB|=9$, then what is $|AD|$?
$
\textbf{(A)}\ 4
\qquad\textbf{(B)}\ 6
\qquad\textbf{(C)}\ 9
\qquad\textbf{(D)}\ 12
\qquad\textbf{(E)}\ \text{None of the above}
$ | 1. Given that \(ABCD\) is a quadrilateral with \(m(\widehat{A}) = m(\widehat{D}) = 90^\circ\), we can infer that \(AB\) and \(AD\) are perpendicular to each other, and \(AD\) and \(DC\) are perpendicular to each other.
2. Let \(M\) be the midpoint of \(DC\). Since \(M\) is the midpoint, we have \(|DM| = |MC| = \frac{|D... | 6 | Geometry | MCQ | Yes | Yes | aops_forum | false |
Let $n$ and $m$ be integers such that $n\leq 2007 \leq m$ and $n^n \equiv -1 \equiv m^m \pmod 5$. What is the least possible value of $m-n$?
$
\textbf{(A)}\ 4
\qquad\textbf{(B)}\ 5
\qquad\textbf{(C)}\ 6
\qquad\textbf{(D)}\ 7
\qquad\textbf{(E)}\ 8
$ | To solve the problem, we need to find integers \( n \) and \( m \) such that \( n \leq 2007 \leq m \) and both \( n^n \equiv -1 \pmod{5} \) and \( m^m \equiv -1 \pmod{5} \). We also need to find the least possible value of \( m - n \).
1. **Identify the residues modulo 5:**
We start by examining the residues of pow... | 7 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
$n$ integers are arranged along a circle in such a way that each number is equal to the absolute value of the difference of the two numbers following that number in clockwise direction. If the sum of all numbers is $278$, how many different values can $n$ take?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C... | 1. Given that $n$ integers are arranged along a circle such that each number is equal to the absolute value of the difference of the two numbers following it in a clockwise direction. We need to determine how many different values $n$ can take if the sum of all numbers is $278$.
2. Let the integers be $a_1, a_2, \ldot... | 2 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
For how many primes $p$ less than $15$, there exists integer triples $(m,n,k)$ such that
\[
\begin{array}{rcl}
m+n+k &\equiv& 0 \pmod p \\
mn+mk+nk &\equiv& 1 \pmod p \\
mnk &\equiv& 2 \pmod p.
\end{array}
\]
$
\textbf{(A)}\ 2
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ 4
\qquad\textbf{(D)}\ 5
\qquad\textbf{(E)}\ 6
$ | To solve the problem, we need to find the number of primes \( p \) less than 15 for which there exist integer triples \((m, n, k)\) satisfying the given congruences:
\[
\begin{array}{rcl}
m+n+k &\equiv& 0 \pmod{p} \\
mn+mk+nk &\equiv& 1 \pmod{p} \\
mnk &\equiv& 2 \pmod{p}.
\end{array}
\]
1. **Formulating the Polynomia... | 3 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
What is the third digit after the decimal point of the decimal representation of $\sqrt[3]{2+\sqrt 5} + \sqrt[3]{2-\sqrt 5}$?
$
\textbf{(A)}\ 8
\qquad\textbf{(B)}\ 5
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 1
\qquad\textbf{(E)}\ \text{None of the above}
$ | 1. Let \( x = \sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}} \).
2. To find \( x \), we first cube both sides:
\[
x^3 = \left( \sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}} \right)^3
\]
3. Using the binomial theorem, we expand the right-hand side:
\[
x^3 = \left( \sqrt[3]{2 + \sqrt{5}} \right)^3 + \... | 0 | Algebra | MCQ | Yes | Yes | aops_forum | false |
A triangle with sides $a,b,c$ is called a good triangle if $a^2,b^2,c^2$ can form a triangle. How many of below triangles are good?
(i) $40^{\circ}, 60^{\circ}, 80^{\circ}$
(ii) $10^{\circ}, 10^{\circ}, 160^{\circ}$
(iii) $110^{\circ}, 35^{\circ}, 35^{\circ}$
(iv) $50^{\circ}, 30^{\circ}, 100^{\circ}$
(v) $90^{\c... | To determine if a triangle with sides \(a, b, c\) is a good triangle, we need to check if \(a^2, b^2, c^2\) can form a triangle. This is equivalent to checking if the triangle is acute, as the squares of the sides of an acute triangle will also form a triangle.
We will analyze each given triangle to see if it is acute... | 2 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
A positive integer $n$ is called a good number if every integer multiple of $n$ is divisible by $n$ however its digits are rearranged. How many good numbers are there?
$
\textbf{(A)}\ 3
\qquad\textbf{(B)}\ 4
\qquad\textbf{(C)}\ 6
\qquad\textbf{(D)}\ 12
\qquad\textbf{(E)}\ \text{Infinitely many}
$ | To determine how many good numbers there are, we need to understand the properties of such numbers. A good number \( n \) is defined such that every integer multiple of \( n \) remains divisible by \( n \) even if its digits are rearranged.
1. **Understanding the properties of good numbers:**
- A number \( n \) is ... | 3 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
How many pairs of positive integers $(x,y)$ are there such that $\sqrt{xy}-71\sqrt x + 30 = 0$?
$
\textbf{(A)}\ 8
\qquad\textbf{(B)}\ 18
\qquad\textbf{(C)}\ 72
\qquad\textbf{(D)}\ 2130
\qquad\textbf{(E)}\ \text{Infinitely many}
$ | 1. Given the equation:
\[
\sqrt{xy} - 71\sqrt{x} + 30 = 0
\]
Let's set \( \sqrt{x} = a \) and \( \sqrt{y} = b \). Then the equation becomes:
\[
ab - 71a + 30 = 0
\]
2. We can factor out \( a \) from the equation:
\[
a(b - 71) + 30 = 0
\]
Rearranging, we get:
\[
a(b - 71) = -30
... | 8 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Let the vertices $A$ and $C$ of a right triangle $ABC$ be on the arc with center $B$ and radius $20$. A semicircle with diameter $[AB]$ is drawn to the inner region of the arc. The tangent from $C$ to the semicircle touches the semicircle at $D$ other than $B$. Let $CD$ intersect the arc at $F$. What is $|FD|$?
$
\te... | 1. **Identify the given elements and their relationships:**
- \( A \) and \( C \) are on an arc with center \( B \) and radius \( 20 \).
- A semicircle with diameter \( AB \) is drawn inside the arc.
- \( C \) is tangent to the semicircle at \( D \).
- \( CD \) intersects the arc at \( F \).
2. **Define th... | 4 | Geometry | MCQ | Yes | Yes | aops_forum | false |
How many positive integers $n$ are there such that $\sqrt{n+\sqrt{n+\sqrt{n+\sqrt{n}}}}$ is an integer?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ \text{Infinitely many}
\qquad\textbf{(E)}\ \text{None of the above}
$ | 1. Let \( x = \sqrt{n + \sqrt{n + \sqrt{n + \sqrt{n}}}} \). We need \( x \) to be an integer.
2. Squaring both sides, we get:
\[
x^2 = n + \sqrt{n + \sqrt{n + \sqrt{n}}}
\]
3. Let \( y = \sqrt{n + \sqrt{n + \sqrt{n}}} \). Then:
\[
x^2 = n + y
\]
4. Squaring both sides again, we get:
\[
y^2 = n +... | 1 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
Let $ABC$ be a right triangle with $m(\widehat{A})=90^\circ$. Let $APQR$ be a square with area $9$ such that $P\in [AC]$, $Q\in [BC]$, $R\in [AB]$. Let $KLMN$ be a square with area $8$ such that $N,K\in [BC]$, $M\in [AB]$, and $L\in [AC]$. What is $|AB|+|AC|$?
$
\textbf{(A)}\ 8
\qquad\textbf{(B)}\ 10
\qquad\textbf{(C... | 1. Given that \( \triangle ABC \) is a right triangle with \( \angle A = 90^\circ \), and \( APQR \) is a square with area 9 such that \( P \in [AC] \), \( Q \in [BC] \), \( R \in [AB] \).
2. Since the area of square \( APQR \) is 9, the side length of the square is \( \sqrt{9} = 3 \).
3. Let \( AQ \) be the angle bise... | 12 | Geometry | MCQ | Yes | Yes | aops_forum | false |
Let $A=\frac{2^2+3\cdot 2 + 1}{3! \cdot 4!} + \frac{3^2+3\cdot 3 + 1}{4! \cdot 5!} + \frac{4^2+3\cdot 4 + 1}{5! \cdot 6!} + \dots + \frac{10^2+3\cdot 10 + 1}{11! \cdot 12!}$. What is the remainder when $11!\cdot 12! \cdot A$ is divided by $11$?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 5
\qquad\text... | 1. We start with the given expression for \( A \):
\[
A = \sum_{n=2}^{10} \frac{n^2 + 3n + 1}{(n+1)! \cdot (n+2)!}
\]
2. We need to find the remainder when \( 11! \cdot 12! \cdot A \) is divided by 11. First, let's simplify the expression for \( A \):
\[
A = \sum_{n=2}^{10} \frac{n^2 + 3n + 1}{(n+1)! \c... | 10 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
The angles $\alpha, \beta, \gamma$ of a triangle are in arithmetic progression. If $\sin 20\alpha$, $\sin 20\beta$, and $\sin 20\gamma$ are in arithmetic progression, how many different values can $\alpha$ take?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ \t... | 1. Given that the angles $\alpha, \beta, \gamma$ of a triangle are in arithmetic progression, we can write:
\[
\beta = \alpha + d \quad \text{and} \quad \gamma = \alpha + 2d
\]
Since the sum of the angles in a triangle is $\pi$, we have:
\[
\alpha + \beta + \gamma = \pi \implies \alpha + (\alpha + d) ... | 3 | Geometry | MCQ | Yes | Yes | aops_forum | false |
In a sequence with the first term is positive integer, the next term is generated by adding the previous term and its largest digit. At most how many consequtive terms of this sequence are odd?
$
\textbf{(A)}\ 2
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ 4
\qquad\textbf{(D)}\ 5
\qquad\textbf{(E)}\ 6
$ | 1. Let \( a_1 \) be the first term of the sequence, which is a positive integer.
2. The next term \( a_{n+1} \) is generated by adding the previous term \( a_n \) and its largest digit. We need to determine the maximum number of consecutive odd terms in this sequence.
3. Consider the last digit of \( a_n \). If \( a_n ... | 5 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
Find the least positive integer $n$ such that $15$ divides the product
\[a_1a_2\dots a_{15}\left (a_1^n+a_2^n+\dots+a_{15}^n \right )\]
, for every positive integers $a_1, a_2, \dots, a_{15}$. | 1. **Define the product and sum:**
Let \( S_n = a_1a_2 \dots a_{15} \left( a_1^n + a_2^n + \dots + a_{15}^n \right) \).
2. **Check divisibility by 3:**
- If any \( a_i \equiv 0 \pmod{3} \), then \( 3 \mid S_n \) because \( a_1a_2 \dots a_{15} \) will be divisible by 3.
- If all \( a_i \not\equiv 0 \pmod{3} \)... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $N>1$ be an integer. We are adding all remainders when we divide $N$ by all positive integers less than $N$. If this sum is less than $N$, find all possible values of $N$. | 1. **Define the problem and notation:**
Let \( N > 1 \) be an integer. We need to find all possible values of \( N \) such that the sum of the remainders when \( N \) is divided by all positive integers less than \( N \) is less than \( N \). Let \( r(N, q) \) be the remainder when \( N \) is divided by \( q \).
2.... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In a qualification group with $15$ volleyball teams, each team plays with all the other teams exactly once. Since there is no tie in volleyball, there is a winner in every match. After all matches played, a team would be qualified if its total number of losses is not exceeding $N$. If there are at least $7$ teams quali... | 1. **Determine the total number of matches:**
Each team plays with every other team exactly once. The total number of matches can be calculated using the combination formula for choosing 2 teams out of 15:
\[
\binom{15}{2} = \frac{15 \cdot 14}{2} = 105
\]
2. **Establish the condition for qualification:**
... | 3 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $m, n, a, k$ be positive integers and $k>1$ such that the equality $$5^m+63n+49=a^k$$
holds. Find the minimum value of $k$. | To find the minimum value of \( k \) such that \( 5^m + 63n + 49 = a^k \) holds for positive integers \( m, n, a, k \) with \( k > 1 \), we will analyze the cases for \( k = 2, 3, 4 \) and show that they do not have solutions. Then, we will verify that \( k = 5 \) is a valid solution.
1. **Case \( k = 2 \):**
\[
... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that a sequence $(a_n)_{n=1}^{\infty}$ of integers has the following property: For all $n$ large enough (i.e. $n \ge N$ for some $N$ ), $a_n$ equals the number of indices $i$, $1 \le i < n$, such that $a_i + i \ge n$. Find the maximum possible number of integers which occur infinitely many times in the sequence... | 1. **Define the Set \( S_n \)**:
We start by defining the set \( S_n \) as follows:
\[
S_n = \{ i : 1 \leq i < n, a_i + i \geq n \}
\]
By the problem's condition, we have:
\[
a_n = |S_n|
\]
2. **Boundedness of the Sequence**:
We claim that the sequence \( (a_n) \) is bounded. To prove this, ... | 2 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_1,a_2,a_3,a_4$ be positive integers, with the property that it is impossible to assign them around a circle where all the neighbors are coprime. Let $i,j,k\in\{1,2,3,4\}$ with $i \neq j$, $j\neq k$, and $k\neq i $. Determine the maximum number of triples $(i,j,k)$ for which
$$
({\rm gcd}(a_i,a_j))^2|a_k.
$$ | To solve this problem, we need to determine the maximum number of triples \((i, j, k)\) for which \((\gcd(a_i, a_j))^2 \mid a_k\). We are given that it is impossible to arrange \(a_1, a_2, a_3, a_4\) around a circle such that all neighbors are coprime.
1. **Initial Assumptions and Setup**:
- Assume \(a_1, a_2, a_3,... | 8 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$a, b, c$ are positive real numbers such that $$(\sqrt {ab}-1)(\sqrt {bc}-1)(\sqrt {ca}-1)=1$$
At most, how many of the numbers: $$a-\frac {b}{c}, a-\frac {c}{b}, b-\frac {a}{c}, b-\frac {c}{a}, c-\frac {a}{b}, c-\frac {b}{a}$$ can be bigger than $1$? | 1. **Given Condition:**
We start with the given condition:
\[
(\sqrt{ab} - 1)(\sqrt{bc} - 1)(\sqrt{ca} - 1) = 1
\]
where \(a, b, c\) are positive real numbers.
2. **Claim:**
We need to determine how many of the numbers:
\[
a - \frac{b}{c}, \quad a - \frac{c}{b}, \quad b - \frac{a}{c}, \quad b -... | 4 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $p$ be a prime number such that $\frac{28^p-1}{2p^2+2p+1}$ is an integer. Find all possible values of number of divisors of $2p^2+2p+1$. | To solve the problem, we need to determine the number of divisors of \(2p^2 + 2p + 1\) given that \(\frac{28^p - 1}{2p^2 + 2p + 1}\) is an integer for a prime number \(p\).
1. **Identify the divisors of \(2p^2 + 2p + 1\):**
Let \(q\) be a prime divisor of \(2p^2 + 2p + 1\). Since \(q\) divides \(2p^2 + 2p + 1\), it... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There is a stone on each square of $n\times n$ chessboard. We gather $n^2$ stones and distribute them to the squares (again each square contains one stone) such that any two adjacent stones are again adjacent. Find all distributions such that at least one stone at the corners remains at its initial square. (Two square... | 1. **Understanding the Problem:**
We start with an \( n \times n \) chessboard where each square initially contains one stone. We need to redistribute these stones such that each square again contains one stone, and any two stones that were adjacent in the initial configuration remain adjacent in the new configurati... | 3 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ with $S_{ABC} = S_{ADC}$ intersect at $E$. The lines through $E$ parallel to $AD$, $DC$, $CB$, $BA$
meet $AB$, $BC$, $CD$, $DA$ at $K$, $L$, $M$, $N$, respectively. Compute the ratio $\frac{S_{KLMN}}{S_{ABC}}$ | 1. Given a convex quadrilateral \(ABCD\) with diagonals \(AC\) and \(BD\) intersecting at \(E\), and the areas of triangles \(ABC\) and \(ADC\) are equal, i.e., \(S_{ABC} = S_{ADC}\).
2. Lines through \(E\) parallel to \(AD\), \(DC\), \(CB\), and \(BA\) meet \(AB\), \(BC\), \(CD\), and \(DA\) at points \(K\), \(L\), \(... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the maximum number of pairwise disjoint sets of the form
$S_{a,b} = \{n^{2}+an+b | n \in \mathbb{Z}\}$, $a, b \in \mathbb{Z}$. | 1. **Observation on the form of sets $S_{a,b}$:**
- We start by noting that the sets $S_{a,b} = \{n^2 + an + b \mid n \in \mathbb{Z}\}$ can be simplified based on the parity of $a$.
- If $a$ is even, let $a = 2k$. Then:
\[
S_{a,b} = \{n^2 + 2kn + b \mid n \in \mathbb{Z}\}
\]
We can rewrite thi... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In a football league, whenever a player is transferred from a team $X$ with $x$ players to a team $Y$ with $y$ players, the federation is paid $y-x$ billions liras by $Y$ if $y \geq x$, while the federation pays $x-y$ billions liras to $X$ if $x > y$. A player is allowed to change as many teams as he wishes during a se... | 1. **Initial Setup:**
- There are 18 teams, each starting with 20 players.
- Total initial number of players: \( 18 \times 20 = 360 \).
2. **Final Distribution:**
- 12 teams have 20 players each.
- Remaining 6 teams have 16, 16, 21, 22, 22, and 23 players.
3. **Total Number of Players at the End:**
- T... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Each of $A$, $B$, $C$, $D$, $E$, and $F$ knows a piece of gossip. They communicate by telephone via a central switchboard, which can connect only two of them at a time. During a conversation, each side tells the other everything he or she knows at that point. Determine the minimum number of calls for everyone to know a... | To determine the minimum number of calls required for everyone to know all six pieces of gossip, we can break down the problem into two parts: proving that 9 calls are sufficient and proving that 9 calls are necessary.
1. **Proving 9 calls are sufficient:**
- Let's denote the six people as \( A, B, C, D, E, \) and ... | 9 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
We are given 5040 balls in k different colors, where the number of balls of each color is the same. The balls are put into 2520 bags so that each bag contains two balls of different colors. Find the smallest k such that, however the balls are distributed into the bags, we can arrange the bags around a circle so that no... | 1. **Claim**: The smallest number of colors \( k \) such that we can always arrange the bags around a circle so that no two balls of the same color are in two neighboring bags is \( \boxed{6} \).
2. **Proof that \( k = 5 \) fails**:
- Label the colors \( A, B, C, D, E \).
- Each color has \( \frac{5040}{5} = 1... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the integer which is closest to the value of the following expression:
$$((7 + \sqrt{48})^{2023} + (7 - \sqrt{48})^{2023})^2 - ((7 + \sqrt{48})^{2023} - (7 - \sqrt{48})^{2023})^2$$ | 1. Consider the given expression:
\[
\left( (7 + \sqrt{48})^{2023} + (7 - \sqrt{48})^{2023} \right)^2 - \left( (7 + \sqrt{48})^{2023} - (7 - \sqrt{48})^{2023} \right)^2
\]
2. Let \( a = (7 + \sqrt{48})^{2023} \) and \( b = (7 - \sqrt{48})^{2023} \). The expression can be rewritten as:
\[
(a + b)^2 - (a ... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The equation $ ax^3\plus{}bx^2\plus{}cx\plus{}d\equal{}0$ has three distinct solutions. How many distinct solutions does the following equation have:
$ 4(ax^3\plus{}bx^2\plus{}cx\plus{}d)(3ax\plus{}b)\equal{}(3ax^2\plus{}2bx\plus{}c)^2?$ | 1. Let \( f(x) = ax^3 + bx^2 + cx + d \). Given that \( f(x) = 0 \) has three distinct solutions, denote these solutions by \( \alpha, \beta, \gamma \) where \( \alpha < \beta < \gamma \).
2. The derivative of \( f(x) \) is:
\[
f'(x) = 3ax^2 + 2bx + c
\]
3. The second derivative of \( f(x) \) is:
\[
f'... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $ 0 \le \alpha, \beta, \gamma \le \frac{\pi}{2}$ satisfy the conditions
$ \sin \alpha\plus{}\sin \beta\plus{}\sin \gamma\equal{}1, \sin \alpha \cos 2\alpha\plus{}\sin \beta\cos 2\beta\plus{}\sin \gamma \cos 2\gamma\equal{}\minus{}1.$
Find all possible values of $ \sin^2 \alpha\plus{}\sin^2 \beta\plus{}\sin^2 ... | 1. Given the conditions:
\[
0 \le \alpha, \beta, \gamma \le \frac{\pi}{2}
\]
\[
\sin \alpha + \sin \beta + \sin \gamma = 1
\]
\[
\sin \alpha \cos 2\alpha + \sin \beta \cos 2\beta + \sin \gamma \cos 2\gamma = -1
\]
2. For any \( x \in \left[0, \frac{\pi}{2} \right] \), we have:
\[
\cos ... | 1 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
A polygon on a coordinate grid is built of $ 2005$ dominoes $ 1 \times 2$. What is the smallest number of sides of an even length such a polygon can have? | 1. **Understanding the Problem:**
We are given a polygon built from 2005 dominoes, each of size \(1 \times 2\). We need to determine the smallest number of sides of even length that such a polygon can have.
2. **Initial Considerations:**
Each domino covers two unit squares. Therefore, the total area covered by t... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The squadron of $10$ powerful destroyers and $20$ small boats is about to attack the island. All ships are positioned on the straight line, and are equally spaced.
Two torpedo boats with $10$ torpedoes each want to protect the island. However, the first torpedo boat can shoot only $10$ successive boats, whereas the s... | To solve this problem, we need to determine the maximum number of destroyers that can avoid being hit by the torpedoes, regardless of the torpedo boats' targeting strategy.
1. **Label the Ships**:
We label the positions of the ships from 1 to 30. We divide these positions into three groups:
\[
A_1 = \{1, 2, ... | 3 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find all primes $ p$ for that there is an integer $ n$ such that there are no integers $ x,y$ with $ x^3 \plus{} y^3 \equiv n \mod p$ (so not all residues are the sum of two cubes).
E.g. for $ p \equal{} 7$, one could set $ n \equal{} \pm 3$ since $ x^3,y^3 \equiv 0 , \pm 1 \mod 7$, thus $ x^3 \plus{} y^3 \equiv 0 ,... | 1. **Identify the problem and given example:**
We need to find all prime numbers \( p \) such that there exists an integer \( n \) for which there are no integers \( x \) and \( y \) satisfying \( x^3 + y^3 \equiv n \pmod{p} \). For example, for \( p = 7 \), we can set \( n = \pm 3 \) since \( x^3, y^3 \equiv 0, \pm... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $x_1,x_2,\cdots,x_n$ be postive real numbers such that $x_1x_2\cdots x_n=1$ ,$S=x^3_1+x^3_2+\cdots+x^3_n$.Find the maximum of $\frac{x_1}{S-x^3_1+x^2_1}+\frac{x_2}{S-x^3_2+x^2_2}+\cdots+\frac{x_n}{S-x^3_n+x^2_n}$ | Given the problem, we need to find the maximum value of the expression:
\[
\frac{x_1}{S - x_1^3 + x_1^2} + \frac{x_2}{S - x_2^3 + x_2^2} + \cdots + \frac{x_n}{S - x_n^3 + x_n^2}
\]
where \( x_1, x_2, \ldots, x_n \) are positive real numbers such that \( x_1 x_2 \cdots x_n = 1 \) and \( S = x_1^3 + x_2^3 + \cdots + x_... | 1 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Find all positive integers $n$ for which the largest prime divisor of $n^2+3$ is equal to the least prime divisor of $n^4+6.$ | 1. Let \( p \) be the largest prime divisor of \( n^2 + 3 \) and also the smallest prime divisor of \( n^4 + 6 \). Since \( n^4 + 6 = (n^2 + 3)(n^2 - 3) + 15 \), we get that \( p \mid 15 \).
2. The prime divisors of 15 are 3 and 5. We will consider each case separately.
3. **Case 1: \( p = 3 \)**
- Since \( p \) i... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$ABCD$ is a rectangle. The segment $MA$ is perpendicular to plane $ABC$ . $MB= 15$ , $MC=24$ , $MD=20$. Find the length of $MA$ . | 1. **Identify the given information and the goal:**
- We have a rectangle \(ABCD\).
- The segment \(MA\) is perpendicular to the plane \(ABC\).
- The distances from \(M\) to the vertices of the rectangle are given:
\[
MB = 15, \quad MC = 24, \quad MD = 20
\]
- We need to find the length of \(... | 7 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In a circle, let $AB$ and $BC$ be chords , with $AB =\sqrt3, BC =3\sqrt3, \angle ABC =60^o$. Find the length of the circle chord that divides angle $ \angle ABC$ in half. | 1. **Identify the given elements and apply the angle bisector theorem:**
- Given: \(AB = \sqrt{3}\), \(BC = 3\sqrt{3}\), \(\angle ABC = 60^\circ\).
- Let the angle bisector of \(\angle ABC\) intersect \(AC\) at \(D\).
- By the angle bisector theorem, \(\frac{AD}{DC} = \frac{AB}{BC} = \frac{\sqrt{3}}{3\sqrt{3}}... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Diagonals of trapezium $ABCD$ are mutually perpendicular and the midline of the trapezium is $5$. Find the length of the segment that connects the midpoints of the bases of the trapezium. | 1. Let $ABCD$ be a trapezium with $AB \parallel CD$ and diagonals $AC$ and $BD$ intersecting at point $O$ such that $AC \perp BD$.
2. Let $E$ and $G$ be the midpoints of $AB$ and $CD$ respectively. The line segment $EG$ is the midline of the trapezium, and it is given that $EG = 5$.
3. Let $F$ and $H$ be the midpoints ... | 5 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The kid cut out of grid paper with the side of the cell $1$ rectangle along the grid lines and calculated its area and perimeter. Carlson snatched his scissors and cut out of this rectangle along the lines of the grid a square adjacent to the boundary of the rectangle.
- My rectangle ... - kid sobbed. - There is somet... | Let's denote the sides of the original rectangle as \(a\) and \(b\), and the side length of the square that Carlson cut out as \(x\).
1. **Calculate the area and perimeter of the original rectangle:**
- The area of the original rectangle is \(ab\).
- The perimeter of the original rectangle is \(2(a + b)\).
2. *... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
For positive integer $n$, define $S_n$ to be the minimum value of the sum \[ \sum_{k=1}^n \sqrt{(2k-1)^2+a_k^2}, \] where $a_1,a_2,\ldots,a_n$ are positive real numbers whose sum is 17. There is a unique positive integer $n$ for which $S_n$ is also an integer. Find this $n$. | 1. **Geometric Interpretation**: Consider \( n \) right triangles joined at their vertices, with bases \( a_1, a_2, \ldots, a_n \) and heights \( 1, 3, \ldots, 2n-1 \). The sum of their hypotenuses is the value of \( S_n \). The minimum value of \( S_n \) is the length of the straight line connecting the bottom vertex ... | 12 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
In triangle $ABC$, $A'$, $B'$, and $C'$ are on the sides $BC$, $AC$, and $AB$, respectively. Given that $AA'$, $BB'$, and $CC'$ are concurrent at the point $O$, and that \[\frac{AO}{OA'}+\frac{BO}{OB'}+\frac{CO}{OC'}=92,\] find \[\frac{AO}{OA'}\cdot \frac{BO}{OB'}\cdot \frac{CO}{OC'}.\] | 1. Let \( \frac{AO}{OA'} = k_1 \), \( \frac{BO}{OB'} = k_2 \), and \( \frac{CO}{OC'} = k_3 \). We are given that:
\[
k_1 + k_2 + k_3 = 92
\]
2. By the properties of cevians in a triangle, we know that:
\[
\frac{1}{k_1 + 1} + \frac{1}{k_2 + 1} + \frac{1}{k_3 + 1} = 1
\]
3. To find \( k_1 k_2 k_3 \), we... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The number $r$ can be expressed as a four-place decimal $0.abcd,$ where $a, b, c,$ and $d$ represent digits, any of which could be zero. It is desired to approximate $r$ by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The closest such fraction to $r$ is $\frac 27.$ What is the number of p... | To solve this problem, we need to find the number of possible values for \( r \) such that \( r \) is closest to \(\frac{2}{7}\) when expressed as a four-place decimal \(0.abcd\).
1. **Convert \(\frac{2}{7}\) to a decimal:**
\[
\frac{2}{7} \approx 0.285714285714\ldots
\]
The decimal representation of \(\fr... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ S$ be the set of points in the Cartesian plane that satisfy
\[ \Big|\big|{|x| \minus{} 2}\big| \minus{} 1\Big| \plus{} \Big|\big|{|y| \minus{} 2}\big| \minus{} 1\Big| \equal{} 1.
\]
If a model of $ S$ were built from wire of negligible thickness, then the total length of wire required would be $ a\sqrt {b},$ w... | 1. **Understanding the given equation:**
\[
\Big|\big|{|x| - 2}\big| - 1\Big| + \Big|\big|{|y| - 2}\big| - 1\Big| = 1
\]
We need to analyze the behavior of the function \( f(a) = \Big|\big||a| - 2\big| - 1\Big| \).
2. **Analyzing \( f(a) \):**
- For \( 0 \leq a \leq 1 \):
\[
f(a) = \Big|\big|a... | 10 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that $x,$ $y,$ and $z$ are three positive numbers that satisfy the equations $xyz=1,$ $x+\frac{1}{z}=5,$ and $y+\frac{1}{x}=29.$ Then $z+\frac{1}{y}=\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ | 1. Given the equations:
\[
xyz = 1, \quad x + \frac{1}{z} = 5, \quad y + \frac{1}{x} = 29
\]
2. From the second equation, solve for \(x\):
\[
x + \frac{1}{z} = 5 \implies x = 5 - \frac{1}{z}
\]
3. From the third equation, solve for \(y\):
\[
y + \frac{1}{x} = 29 \implies y = 29 - \frac{1}{x}
... | 5 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Given that $z$ is a complex number such that $z+\frac 1z=2\cos 3^\circ,$ find the least integer that is greater than $z^{2000}+\frac 1{z^{2000}}.$ | 1. **Express \( z \) in exponential form:**
Given \( z + \frac{1}{z} = 2 \cos 3^\circ \), we can write \( z \) as \( z = e^{i\theta} \). Therefore, \( \frac{1}{z} = e^{-i\theta} \).
2. **Simplify the given equation:**
\[
e^{i\theta} + e^{-i\theta} = 2 \cos 3^\circ
\]
Using Euler's formula, \( e^{i\theta... | 0 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the least positive integer $n$ such that \[ \frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}. \] | 1. We start with the given equation:
\[
\frac{1}{\sin 45^\circ \sin 46^\circ} + \frac{1}{\sin 47^\circ \sin 48^\circ} + \cdots + \frac{1}{\sin 133^\circ \sin 134^\circ} = \frac{1}{\sin n^\circ}
\]
2. Notice that \(\sin(180^\circ - x) = \sin x\). Therefore, we can rewrite the sines of angles greater than \(90^... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
An equilateral triangle is inscribed in the ellipse whose equation is $x^2+4y^2=4.$ One vertex of the triangle is $(0,1),$ one altitude is contained in the $y$-axis, and the length of each side is $\sqrt{\frac mn},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ | 1. **Transform the ellipse to a circle:**
The given ellipse is \( x^2 + 4y^2 = 4 \). We can rewrite it as:
\[
\frac{x^2}{4} + y^2 = 1
\]
This represents an ellipse with semi-major axis \( a = 2 \) and semi-minor axis \( b = 1 \).
To transform this ellipse into a circle, we perform a scaling transform... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Given that
\begin{eqnarray*}&(1)& \text{x and y are both integers between 100 and 999, inclusive;}\qquad \qquad \qquad \qquad \qquad \\ &(2)& \text{y is the number formed by reversing the digits of x; and}\\ &(3)& z=|x-y|. \end{eqnarray*}How many distinct values of $z$ are possible? | 1. **Express \( x \) and \( y \) in terms of their digits:**
Let \( x = \overline{abc} \), where \( a, b, \) and \( c \) are the digits of \( x \). Therefore, we can write:
\[
x = 100a + 10b + c
\]
Similarly, since \( y \) is the number formed by reversing the digits of \( x \), we have:
\[
y = 100... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Circles $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ intersect at two points, one of which is $(9,6),$ and the product of the radii is $68.$ The x-axis and the line $y=mx$, where $m>0,$ are tangent to both circles. It is given that $m$ can be written in the form $a\sqrt{b}/c,$ where $a,$ $b,$ and $c$ are positive integers, ... | 1. **Identify the properties of the circles:**
- The circles \(\mathcal{C}_1\) and \(\mathcal{C}_2\) intersect at \((9,6)\).
- The product of the radii of the circles is \(68\).
- Both circles are tangent to the x-axis and the line \(y = mx\).
2. **Consider a general circle tangent to the x-axis and the line ... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Given that $\log_{10} \sin x + \log_{10} \cos x = -1$ and that $\log_{10} (\sin x + \cos x) = \textstyle \frac{1}{2} (\log_{10} n - 1)$, find $n$. | 1. Given the equation:
\[
\log_{10} \sin x + \log_{10} \cos x = -1
\]
Using the properties of logarithms, we can combine the logs:
\[
\log_{10} (\sin x \cos x) = -1
\]
This implies:
\[
\sin x \cos x = 10^{-1} = \frac{1}{10}
\]
2. Using the double-angle identity for sine, we know:
\[... | 12 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
For positive integers $n$, let $\tau (n)$ denote the number of positive integer divisors of $n$, including $1$ and $n$. For example, $\tau (1)=1$ and $\tau(6) =4$. Define $S(n)$ by \[S(n)=\tau(1)+ \tau(2) + ... + \tau(n).\] Let $a$ denote the number of positive integers $n \leq 2005$ with $S(n)$ odd, and let $b$ denote... | 1. **Understanding the function $\tau(n)$:**
- $\tau(n)$ denotes the number of positive integer divisors of $n$. For example, $\tau(1) = 1$ and $\tau(6) = 4$.
- $\tau(n)$ is odd if and only if $n$ is a perfect square. This is because the divisors of a non-square number come in pairs, while a square number has a m... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given that $O$ is a regular octahedron, that $C$ is the cube whose vertices are the centers of the faces of $O$, and that the ratio of the volume of $O$ to that of $C$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m+n$. | 1. **Set the side length of the octahedron to 1 for simplicity.**
- The volume of a regular octahedron with side length \( a \) is given by:
\[
V_O = \frac{\sqrt{2}}{3} a^3
\]
- For \( a = 1 \):
\[
V_O = \frac{\sqrt{2}}{3}
\]
2. **Determine the height of the octahedron.**
- An o... | 11 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ S_i$ be the set of all integers $ n$ such that $ 100i\leq n < 100(i \plus{} 1)$. For example, $ S_4$ is the set $ {400,401,402,\ldots,499}$. How many of the sets $ S_0, S_1, S_2, \ldots, S_{999}$ do not contain a perfect square? | 1. **Identify the range of each set \( S_i \):**
Each set \( S_i \) contains integers \( n \) such that \( 100i \leq n < 100(i + 1) \). This means each set \( S_i \) contains exactly 100 integers.
2. **Determine the range of perfect squares:**
We need to find the perfect squares within the range of each set \( S... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let
\[ p(x,y) \equal{} a_0 \plus{} a_1x \plus{} a_2y \plus{} a_3x^2 \plus{} a_4xy \plus{} a_5y^2 \plus{} a_6x^3 \plus{} a_7x^2y \plus{} a_8xy^2 \plus{} a_9y^3.
\]Suppose that
\begin{align*}p(0,0) &\equal{} p(1,0) \equal{} p( \minus{} 1,0) \equal{} p(0,1) \equal{} p(0, \minus{} 1) \\&\equal{} p(1,1) \equal{} p(1, \minu... | 1. Given the polynomial:
\[
p(x,y) = a_0 + a_1x + a_2y + a_3x^2 + a_4xy + a_5y^2 + a_6x^3 + a_7x^2y + a_8xy^2 + a_9y^3
\]
and the conditions:
\[
p(0,0) = p(1,0) = p(-1,0) = p(0,1) = p(0,-1) = p(1,1) = p(1,-1) = p(2,2) = 0
\]
2. From \( p(0,0) = 0 \), we get:
\[
a_0 = 0
\]
3. From \( p(1,... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $ a$ and $ b$ be positive real numbers with $ a\ge b$. Let $ \rho$ be the maximum possible value of $ \frac{a}{b}$ for which the system of equations
\[ a^2\plus{}y^2\equal{}b^2\plus{}x^2\equal{}(a\minus{}x)^2\plus{}(b\minus{}y)^2\]has a solution in $ (x,y)$ satisfying $ 0\le x<a$ and $ 0\le y<b$. Then $ \rho^2$ ca... | 1. Consider the given system of equations:
\[
a^2 + y^2 = b^2 + x^2 = (a - x)^2 + (b - y)^2
\]
We need to find the maximum possible value of \(\frac{a}{b}\) for which this system has a solution \((x, y)\) satisfying \(0 \le x < a\) and \(0 \le y < b\).
2. Let's analyze the geometric interpretation of the e... | 7 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For real number $x$ let $\lfloor x\rfloor$ be the greatest integer less than or equal to $x$, and define $\{x\}=x-\lfloor x\rfloor$ to be the fractional part of $x$. For example, $\{3\}=0$ and $\{4.56\}=0.56$. Define $f(x)=x\{x\}$, and let $N$ be the number of real-valued solutions to the equation $f(f(f(x)))=17$ for $... | 1. **Understanding the function \( f(x) \):**
\[
f(x) = x \{ x \} = x (x - \lfloor x \rfloor)
\]
Here, \( \{ x \} = x - \lfloor x \rfloor \) is the fractional part of \( x \).
2. **Analyzing the equation \( f(f(f(x))) = 17 \):**
We need to find the number of real-valued solutions to this equation for \(... | 10 | Other | math-word-problem | Yes | Yes | aops_forum | false |
A street has parallel curbs $ 40$ feet apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The length of the curb between the stripes is $ 15$ feet and each stripe is $ 50$ feet long. Find the distance, in feet, between the stripes.
$ \textbf{(A)}\ 9 \qquad \textbf{(B)}\ 10 \qquad \te... | 1. **Identify the given information and the goal:**
- The distance between the parallel curbs is \(40\) feet.
- The length of the curb between the stripes is \(15\) feet.
- Each stripe is \(50\) feet long.
- We need to find the distance between the stripes.
2. **Visualize the problem:**
- Draw a diagram... | 12 | Geometry | MCQ | Yes | Yes | aops_forum | false |
Both roots of the quadratic equation $ x^2 \minus{} 63x \plus{} k \equal{} 0$ are prime numbers. The number of possible values of $ k$ is
$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ \textbf{more than four}$ | 1. **Identify the roots using Vieta's formulas:**
By Vieta's formulas, for the quadratic equation \(x^2 - 63x + k = 0\), the sum of the roots \(a\) and \(b\) is given by:
\[
a + b = 63
\]
and the product of the roots is:
\[
ab = k
\]
2. **Determine the nature of the roots:**
Since both \(a\)... | 1 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
How many ordered triples of positive integers $(x,y,z)$ satisfy $(x^y)^z=64$?
$\textbf{(A) }5\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$ | To solve the problem, we need to find all ordered triples \((x, y, z)\) of positive integers such that \((x^y)^z = 64\). We start by expressing 64 as a power of a prime number:
\[ 64 = 2^6 \]
This means we need to find all combinations of \(x\), \(y\), and \(z\) such that:
\[ (x^y)^z = 2^6 \]
This can be rewritten ... | 9 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Let $P(x)=kx^3+2k^2x^2+k^3$. Find the sum of all real numbers $k$ for which $x-2$ is a factor of $P(x)$.
$\textbf{(A) }-8\qquad\textbf{(B) }-4\qquad\textbf{(C) }0\qquad\textbf{(D) }5\qquad\textbf{(E) }8$ | 1. Given the polynomial \( P(x) = kx^3 + 2k^2x^2 + k^3 \), we need to find the sum of all real numbers \( k \) for which \( x-2 \) is a factor of \( P(x) \).
2. If \( x-2 \) is a factor of \( P(x) \), then \( P(2) = 0 \). We substitute \( x = 2 \) into \( P(x) \):
\[
P(2) = k(2)^3 + 2k^2(2)^2 + k^3 = 8k + 8k^2 +... | -8 | Algebra | MCQ | Yes | Yes | aops_forum | false |
What is the smallest integer $n$ for which any subset of $\{1,2,3,\ldots,20\}$ of size $n$ must contain two numbers that differ by $8$?
$\textbf{(A) }2\qquad\textbf{(B) }8\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }15$ | 1. **Identify the problem constraints**: We need to find the smallest integer \( n \) such that any subset of \(\{1, 2, 3, \ldots, 20\}\) of size \( n \) must contain two numbers that differ by 8.
2. **Consider the Pigeonhole Principle**: The Pigeonhole Principle states that if \( n \) items are put into \( m \) conta... | 9 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
A grocer makes a display of cans in which the top row has one can and each lower row has two more cans than the row above it. If the display contains $ 100$ cans, how many rows does it contain?
$ \textbf{(A)}\ 5\qquad
\textbf{(B)}\ 8\qquad
\textbf{(C)}\ 9\qquad
\textbf{(D)}\ 10\qquad
\textbf{(E)}\ 11$ | 1. Let's denote the number of rows by \( n \). The number of cans in the \( k \)-th row is given by \( 2k - 1 \). Therefore, the total number of cans in the display is the sum of the first \( n \) odd positive integers.
2. The sum of the first \( n \) odd positive integers is given by the formula:
\[
1 + 3 + 5 + ... | 10 | Algebra | MCQ | Yes | Yes | aops_forum | false |
The equations $ 2x \plus{} 7 \equal{} 3$ and $ bx\minus{}10 \equal{} \minus{}\!2$ have the same solution for $ x$. What is the value of $ b$?
$ \textbf{(A)}\minus{}\!8 \qquad
\textbf{(B)}\minus{}\!4 \qquad
\textbf{(C)}\minus{}\!2 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 8$ | 1. First, solve the equation \(2x + 7 = 3\) for \(x\):
\[
2x + 7 = 3
\]
Subtract 7 from both sides:
\[
2x = 3 - 7
\]
Simplify the right-hand side:
\[
2x = -4
\]
Divide both sides by 2:
\[
x = \frac{-4}{2} = -2
\]
2. Now, substitute \(x = -2\) into the second equation \(bx -... | -4 | Algebra | MCQ | Yes | Yes | aops_forum | false |
A sphere is inscribed in a cube that has a surface area of $ 24$ square meters. A second cube is then inscribed within the sphere. What is the surface area in square meters of the inner cube?
$ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 12$ | 1. **Determine the side length of the outer cube:**
- The surface area of the outer cube is given as \(24\) square meters.
- A cube has 6 faces, so the area of each face is:
\[
\frac{24}{6} = 4 \text{ square meters}
\]
- The side length \(s\) of the outer cube can be found by taking the square r... | 8 | Geometry | MCQ | Yes | Yes | aops_forum | false |
How many positive integers $n$ satisfy$$\dfrac{n+1000}{70} = \lfloor \sqrt{n} \rfloor?$$(Recall that $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.)
$\textbf{(A) } 2 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 30 \qquad\textbf{(E) } 32$ | 1. Let $\lfloor \sqrt{n} \rfloor = k$. This implies that $k \leq \sqrt{n} < k+1$, which translates to:
\[
k^2 \leq n < (k+1)^2
\]
2. Given the equation:
\[
\frac{n + 1000}{70} = k
\]
we can solve for $n$:
\[
n = 70k - 1000
\]
3. Substituting $n = 70k - 1000$ into the inequality $k^2 \leq n... | 6 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
The base-nine representation of the number $N$ is $27{,}006{,}000{,}052_{\rm nine}$. What is the remainder when $N$ is divided by $5?$
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4$ | 1. **Convert the base-nine number to its decimal (base-10) equivalent:**
The given number in base-nine is \(27{,}006{,}000{,}052_{\text{nine}}\). To convert this to base-10, we use the formula for base conversion:
\[
N = 2 \cdot 9^{10} + 7 \cdot 9^9 + 0 \cdot 9^8 + 0 \cdot 9^7 + 6 \cdot 9^6 + 0 \cdot 9^5 + 0 \... | 3 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
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