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$30$ same balls are put into four boxes $ A$, $ B$, $ C$, $ D$ in such a way that sum of number of balls in $ A$ and $ B$ is greater than sum of in $ C$ and $ D$. How many possible ways are there?
$\textbf{(A)}\ 2472 \qquad\textbf{(B)}\ 2600 \qquad\textbf{(C)}\ 2728 \qquad\textbf{(D)}\ 2856 \qquad\textbf{(E)}\ \t... | 1. **Calculate the total number of ways to distribute 30 balls into 4 boxes:**
The number of ways to distribute \(30\) identical balls into \(4\) distinct boxes can be calculated using the stars and bars theorem. The formula for distributing \(n\) identical items into \(k\) distinct groups is given by:
\[
\bin... | 2600 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
How many ordered integer pairs $(x,y)$ ($0\leq x,y < 31$) are there satisfying $(x^2-18)^2\equiv y^2 (\mod 31)$?
$ \textbf{(A)}\ 59
\qquad\textbf{(B)}\ 60
\qquad\textbf{(C)}\ 61
\qquad\textbf{(D)}\ 62
\qquad\textbf{(E)}\ \text{None}
$ | 1. We start with the given congruence:
\[
(x^2 - 18)^2 \equiv y^2 \pmod{31}
\]
Let \( A = x^2 - 18 \). Then the equation becomes:
\[
A^2 \equiv y^2 \pmod{31}
\]
2. This congruence implies:
\[
(A - y)(A + y) \equiv 0 \pmod{31}
\]
Therefore, \( A \equiv y \pmod{31} \) or \( A \equiv -y \... | 60 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
If every $k-$element subset of $S=\{1,2,\dots , 32\}$ contains three different elements $a,b,c$ such that $a$ divides $b$, and $b$ divides $c$, $k$ must be at least ?
$ \textbf{(A)}\ 17
\qquad\textbf{(B)}\ 24
\qquad\textbf{(C)}\ 25
\qquad\textbf{(D)}\ 29
\qquad\textbf{(E)}\ \text{None}
$ | 1. **Identify the problem**: We need to find the minimum value of \( k \) such that every \( k \)-element subset of \( S = \{1, 2, \dots, 32\} \) contains three different elements \( a, b, c \) such that \( a \) divides \( b \) and \( b \) divides \( c \).
2. **Construct a large subset that does not fulfill the condit... | 25 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
In how many ways can the numbers $0,1,2,\dots , 9$ be arranged in such a way that the odd numbers form an increasing sequence, also the even numbers form an increasing sequence?
$ \textbf{(A)}\ 126
\qquad\textbf{(B)}\ 189
\qquad\textbf{(C)}\ 252
\qquad\textbf{(D)}\ 315
\qquad\textbf{(E)}\ \text{None}
$ | 1. We need to arrange the numbers \(0, 1, 2, \dots, 9\) such that the odd numbers form an increasing sequence and the even numbers form an increasing sequence.
2. The odd numbers in the set are \(1, 3, 5, 7, 9\), and the even numbers are \(0, 2, 4, 6, 8\).
3. Since the odd numbers must form an increasing sequence, thei... | 252 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
What is the largest prime $p$ that makes $\sqrt{17p+625}$ an integer?
$ \textbf{(A)}\ 3
\qquad\textbf{(B)}\ 67
\qquad\textbf{(C)}\ 101
\qquad\textbf{(D)}\ 151
\qquad\textbf{(E)}\ 211
$ | To determine the largest prime \( p \) such that \(\sqrt{17p + 625}\) is an integer, we start by letting \(\sqrt{17p + 625} = q\), where \( q \) is an integer.
1. **Express the equation in terms of \( q \):**
\[
\sqrt{17p + 625} = q \implies 17p + 625 = q^2
\]
Rearrange to solve for \( p \):
\[
17p ... | 67 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Some of $A,B,C,D,$ and $E$ are truth tellers, and the others are liars. Truth tellers always tell the truth. Liars always lie. We know $A$ is a truth teller. According to below conversation,
$B: $ I'm a truth teller.
$C: $ $D$ is a truth teller.
$D: $ $B$ and $E$ are not both truth tellers.
$E: $ $A$ and $B$ are tr... | 1. We know that $A$ is a truth teller.
2. Let's analyze the statements made by $B, C, D,$ and $E$.
### Case 1: Assume $B$ is a truth teller.
- Since $B$ is a truth teller, $B$'s statement "I'm a truth teller" is true.
- $E$'s statement "A and B are truth tellers" must also be true because $E$ is a truth teller.
- Sinc... | 3 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
In how many ways can $7$ red, $7$ white balls be distributed into $7$ boxes such that every box contains exactly $2$ balls?
$ \textbf{(A)}\ 163
\qquad\textbf{(B)}\ 393
\qquad\textbf{(C)}\ 858
\qquad\textbf{(D)}\ 1716
\qquad\textbf{(E)}\ \text{None}
$ | To solve this problem, we need to distribute 7 red and 7 white balls into 7 boxes such that each box contains exactly 2 balls. We will consider different cases based on the number of boxes that contain 2 white balls.
1. **Case 1: 3 boxes with 2 white balls**
- In this case, we will have 3 boxes with 2 white balls, ... | 393 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Let $f(x)=x^3+7x^2+9x+10$. Which value of $p$ satisfies the statement
\[
f(a) \equiv f(b) \ (\text{mod } p) \Rightarrow a \equiv b \ (\text{mod } p)
\]
for every integer $a,b$?
$ \textbf{(A)}\ 5
\qquad\textbf{(B)}\ 7
\qquad\textbf{(C)}\ 11
\qquad\textbf{(D)}\ 13
\qquad\textbf{(E)}\ 17
$ | To solve the problem, we need to find a prime number \( p \) such that if \( f(a) \equiv f(b) \pmod{p} \), then \( a \equiv b \pmod{p} \) for the polynomial \( f(x) = x^3 + 7x^2 + 9x + 10 \).
1. **Evaluate \( f(x) \) at different values:**
\[
f(0) = 0^3 + 7 \cdot 0^2 + 9 \cdot 0 + 10 = 10
\]
\[
f(1) = 1... | 11 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
$x_{n+1}= \left ( 1+\frac2n \right )x_n+\frac4n$, for every positive integer $n$. If $x_1=-1$, what is $x_{2000}$?
$ \textbf{(A)}\ 1999998
\qquad\textbf{(B)}\ 2000998
\qquad\textbf{(C)}\ 2009998
\qquad\textbf{(D)}\ 2000008
\qquad\textbf{(E)}\ 1999999
$ | 1. Given the recurrence relation:
\[
x_{n+1} = \left(1 + \frac{2}{n}\right)x_n + \frac{4}{n}
\]
and the initial condition \( x_1 = -1 \).
2. To simplify the recurrence relation, we multiply both sides by \( n \):
\[
n x_{n+1} = (n + 2)x_n + 4
\]
3. Let \( y_n = x_n + 2 \). Then \( y_1 = x_1 + 2 =... | 2000998 | Algebra | MCQ | Yes | Yes | aops_forum | false |
How many ten digit positive integers with distinct digits are multiples of $11111$?
$ \textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1264
\qquad\textbf{(C)}\ 2842
\qquad\textbf{(D)}\ 3456
\qquad\textbf{(E)}\ 11111
$ | 1. **Identify the digits and the divisibility condition:**
- The ten-digit positive integers must have distinct digits, which means they must include all digits from 0 to 9.
- We need to determine how many of these numbers are multiples of 11111.
2. **Check the divisibility by 11111:**
- Since 11111 is coprim... | 3456 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
How many different permutations $(\alpha_1 \alpha_2\alpha_3\alpha_4\alpha_5)$ of the set $\{1,2,3,4,5\}$ are there such that $(\alpha_1\dots \alpha_k)$ is not a permutation of the set $\{1,\dots ,k\}$, for every $1\leq k \leq 4$?
$ \textbf{(A)}\ 13
\qquad\textbf{(B)}\ 65
\qquad\textbf{(C)}\ 71
\qquad\textbf{(D)}\ 461
... | 1. Define \( f(n) \) as the number of permutations of the set \(\{1, 2, \ldots, n\}\) such that for every \(1 \leq k \leq n-1\), the sequence \((\alpha_1, \alpha_2, \ldots, \alpha_k)\) is not a permutation of the set \(\{1, 2, \ldots, k\}\).
2. We need to find \( f(5) \). To do this, we use the recurrence relation:
... | 84 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
A committee with $20$ members votes for the candidates $A,B,C$ by a different election system. Each member writes his ordered prefer list to the ballot (e.g. if he writes $BAC$, he prefers $B$ to $A$ and $C$, and prefers $A$ to $C$). After the ballots are counted, it is recognized that each of the six different permuta... | 1. Define the number of ballots for each permutation of the candidates \(A, B, C\):
\[
\begin{cases}
p = \#(ABC) \\
q = \#(ACB) \\
r = \#(CAB) \\
s = \#(CBA) \\
t = \#(BCA) \\
u = \#(BAC)
\end{cases}
\]
We know that the total number of ballots is 20:
\[
p + q + r + s + t + u = 20
... | 8 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
What is the last two digits of the decimal representation of $9^{8^{7^{\cdot^{\cdot^{\cdot^{2}}}}}}$?
$ \textbf{(A)}\ 81
\qquad\textbf{(B)}\ 61
\qquad\textbf{(C)}\ 41
\qquad\textbf{(D)}\ 21
\qquad\textbf{(E)}\ 01
$ | 1. To find the last two digits of \( 9^{8^{7^{\cdot^{\cdot^{\cdot^{2}}}}}} \), we need to compute \( 9^{8^{7^{\cdot^{\cdot^{\cdot^{2}}}}}} \mod 100 \). Using Euler's theorem, we know that \( a^{\phi(n)} \equiv 1 \mod n \) for \( \gcd(a, n) = 1 \). Here, \( n = 100 \) and \( \phi(100) = 40 \), so it suffices to find \( ... | 21 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
What is the largest possible area of a quadrilateral with sides $1,4,7,8$ ?
$ \textbf{(A)}\ 7\sqrt 2
\qquad\textbf{(B)}\ 10\sqrt 3
\qquad\textbf{(C)}\ 18
\qquad\textbf{(D)}\ 12\sqrt 3
\qquad\textbf{(E)}\ 9\sqrt 5
$ | 1. To maximize the area of a quadrilateral with given side lengths, the quadrilateral should be cyclic. This is because a cyclic quadrilateral has the maximum possible area for given side lengths.
2. We use Brahmagupta's formula to find the area of a cyclic quadrilateral. Brahmagupta's formula states that the area \( A... | 18 | Geometry | MCQ | Yes | Yes | aops_forum | false |
What is the least integer $n\geq 100$ such that $77$ divides $1+2+2^2+2^3+\dots + 2^n$ ?
$ \textbf{(A)}\ 101
\qquad\textbf{(B)}\ 105
\qquad\textbf{(C)}\ 111
\qquad\textbf{(D)}\ 119
\qquad\textbf{(E)}\ \text{None}
$ | 1. We start with the given series sum \(1 + 2 + 2^2 + 2^3 + \dots + 2^n\). This is a geometric series with the first term \(a = 1\) and common ratio \(r = 2\). The sum of the first \(n+1\) terms of a geometric series is given by:
\[
S = \frac{r^{n+1} - 1}{r - 1}
\]
Substituting \(r = 2\), we get:
\[
S... | 119 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Each of the football teams Istanbulspor, Yesildirek, Vefa, Karagumruk, and Adalet, played exactly one match against the other four teams. Istanbulspor defeated all teams except Yesildirek; Yesildirek defeated Istanbulspor but lost to all the other teams. Vefa defeated all except Istanbulspor. The winner of the game Kar... | To solve this problem, we need to determine the number of ways to order the five football teams such that each team, except the last, defeated the next team. Let's denote the teams as follows:
- \( I \) for Istanbulspor
- \( Y \) for Yesildirek
- \( V \) for Vefa
- \( K \) for Karagumruk
- \( A \) for Adalet
From the ... | 9 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
How many primes $p$ are there such that $2p^4-7p^2+1$ is equal to square of an integer?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 4
\qquad\textbf{(D)}\ \text{Infinitely many}
\qquad\textbf{(E)}\ \text{None of the preceding}
$ | 1. We need to find the number of prime numbers \( p \) such that \( 2p^4 - 7p^2 + 1 \) is a perfect square. Let's denote this perfect square by \( k^2 \), where \( k \) is an integer. Therefore, we have:
\[
2p^4 - 7p^2 + 1 = k^2
\]
2. First, let's check small prime numbers to see if they satisfy the equation.... | 1 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
How many real solution does the equation $\dfrac{x^{2000}}{2001} + 2\sqrt 3 x^2 - 2\sqrt 5 x + \sqrt 3$ have?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 11
\qquad\textbf{(D)}\ 12
\qquad\textbf{(E)}\ \text{None of the preceding}
$ | To determine how many real solutions the equation
\[
\frac{x^{2000}}{2001} + 2\sqrt{3} x^2 - 2\sqrt{5} x + \sqrt{3} = 0
\]
has, we need to analyze the behavior of the function
\[
f(x) = \frac{x^{2000}}{2001} + 2\sqrt{3} x^2 - 2\sqrt{5} x + \sqrt{3}.
\]
1. **Analyze the term \(\frac{x^{2000}}{2001}\):**
- The t... | 0 | Inequalities | MCQ | Yes | Yes | aops_forum | false |
How many $5-$digit positive numbers which contain only odd numbers are there such that there is at least one pair of consecutive digits whose sum is $10$?
$
\textbf{(A)}\ 3125
\qquad\textbf{(B)}\ 2500
\qquad\textbf{(C)}\ 1845
\qquad\textbf{(D)}\ 1190
\qquad\textbf{(E)}\ \text{None of the preceding}
$ | 1. We need to find the number of 5-digit positive numbers that contain only odd digits and have at least one pair of consecutive digits whose sum is 10.
2. The odd digits are {1, 3, 5, 7, 9}. We need to count the total number of 5-digit numbers that can be formed using these digits.
3. Each digit in the 5-digit number ... | 1845 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Which of the followings gives the product of the real roots of the equation $x^4+3x^3+5x^2 + 21x -14=0$?
$
\textbf{(A)}\ -2
\qquad\textbf{(B)}\ 7
\qquad\textbf{(C)}\ -14
\qquad\textbf{(D)}\ 21
\qquad\textbf{(E)}\ \text{None of the preceding}
$ | 1. Given the polynomial equation:
\[
x^4 + 3x^3 + 5x^2 + 21x - 14 = 0
\]
We need to find the product of the real roots of this equation.
2. We start by attempting to factorize the polynomial. Notice that we can group terms to facilitate factorization:
\[
x^4 + 3x^3 + 5x^2 + 21x - 14 = (x^4 + 5x^2 - 1... | -2 | Algebra | MCQ | Yes | Yes | aops_forum | false |
What is the largest possible area of an isosceles trapezoid in which the largest side is $13$ and the perimeter is $28$?
$
\textbf{(A)}\ 13
\qquad\textbf{(B)}\ 24
\qquad\textbf{(C)}\ 27
\qquad\textbf{(D)}\ 28
\qquad\textbf{(E)}\ 30
$ | 1. **Define the variables and constraints:**
- Let the lengths of the bases of the isosceles trapezoid be \(a\) and \(b\) with \(a > b\).
- Let the lengths of the legs be \(c\).
- Given that the largest side is 13, we have \(a = 13\).
- The perimeter of the trapezoid is 28, so we have:
\[
a + b + ... | 27 | Geometry | MCQ | Yes | Yes | aops_forum | false |
For how many integers $n$, does the equation system \[\begin{array}{rcl}
2x+3y &=& 7\\
5x + ny &=& n^2
\end{array}\] have a solution over integers?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ 4
\qquad\textbf{(D)}\ 8
\qquad\textbf{(E)}\ \text{None of the preceding}
$ | To determine the number of integer values of \( n \) for which the system of equations
\[
\begin{array}{rcl}
2x + 3y &=& 7 \\
5x + ny &=& n^2
\end{array}
\]
has integer solutions, we will analyze the system step-by-step.
1. **Express \( x \) in terms of \( y \) from the first equation:**
\[
2x + 3y = 7 \implies ... | 8 | Algebra | MCQ | Yes | Yes | aops_forum | false |
How many different solutions does the congruence $x^3+3x^2+x+3 \equiv 0 \pmod{25}$ have?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 4
\qquad\textbf{(D)}\ 5
\qquad\textbf{(E)}\ 6
$ | To determine the number of different solutions to the congruence \(x^3 + 3x^2 + x + 3 \equiv 0 \pmod{25}\), we start by factoring the polynomial.
1. **Factor the polynomial:**
\[
x^3 + 3x^2 + x + 3 = (x+3)(x^2 + 1)
\]
This factorization can be verified by expanding:
\[
(x+3)(x^2 + 1) = x^3 + x + 3x^2... | 6 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle such that midpoints of three altitudes are collinear. If the largest side of triangle is $10$, what is the largest possible area of the triangle?
$
\textbf{(A)}\ 20
\qquad\textbf{(B)}\ 25
\qquad\textbf{(C)}\ 30
\qquad\textbf{(D)}\ 40
\qquad\textbf{(E)}\ 50
$ | 1. **Identify the largest side and midpoints:**
Let \( BC = 10 \) be the largest side of the triangle \( \triangle ABC \). Let \( M_a \), \( M_b \), and \( M_c \) be the midpoints of the sides \( a = BC \), \( b = AC \), and \( c = AB \), respectively. Let \( H_a \), \( H_b \), and \( H_c \) be the midpoints of the ... | 25 | Geometry | MCQ | Yes | Yes | aops_forum | false |
A convex polygon has at least one side with length $1$. If all diagonals of the polygon have integer lengths, at most how many sides does the polygon have?
$
\textbf{(A)}\ 3
\qquad\textbf{(B)}\ 5
\qquad\textbf{(C)}\ 7
\qquad\textbf{(D)}\ 10
\qquad\textbf{(E)}\ \text{None of the preceding}
$ | 1. Let \( A_1A_2\cdots A_n \) be a convex polygon with \( A_1A_2 = 1 \). We need to determine the maximum number of sides \( n \) such that all diagonals have integer lengths.
2. Consider the triangle inequality for \( n \geq 6 \). For any three vertices \( A_1, A_2, A_4 \), the triangle inequality states:
\[
|A... | 5 | Geometry | MCQ | Yes | Yes | aops_forum | false |
If the integers $m,n,k$ hold the equation $221m+247n+323k=2001$, what is the smallest possible value of $k$ greater than $100$?
$
\textbf{(A)}\ 124
\qquad\textbf{(B)}\ 111
\qquad\textbf{(C)}\ 107
\qquad\textbf{(D)}\ 101
\qquad\textbf{(E)}\ \text{None of the preceding}
$ | 1. We start with the given equation:
\[
221m + 247n + 323k = 2001
\]
2. To find the smallest possible value of \( k \) greater than 100, we will consider the equation modulo 13. First, we reduce each coefficient modulo 13:
\[
221 \equiv 0 \pmod{13} \quad \text{(since } 221 = 17 \times 13)
\]
\[
... | 111 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
If the sum of any $10$ of $21$ real numbers is less than the sum of remaining $11$ of them, at least how many of these $21$ numbers are positive?
$
\textbf{(A)}\ 18
\qquad\textbf{(B)}\ 19
\qquad\textbf{(C)}\ 20
\qquad\textbf{(D)}\ 21
\qquad\textbf{(E)}\ \text{None of the preceding}
$ | To solve this problem, we need to determine the minimum number of positive numbers among the 21 real numbers given the condition that the sum of any 10 of them is less than the sum of the remaining 11.
1. **Define the numbers and the condition:**
Let the 21 real numbers be \(a_1, a_2, \ldots, a_{21}\). The conditio... | 21 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
A ladder is formed by removing some consecutive unit squares of a $10\times 10$ chessboard such that for each $k-$th row ($k\in \{1,2,\dots, 10\}$), the leftmost $k-1$ unit squares are removed. How many rectangles formed by composition of unit squares does the ladder have?
$
\textbf{(A)}\ 625
\qquad\textbf{(B)}\ 715... | 1. **Understanding the Problem:**
We need to count the number of rectangles formed by the composition of unit squares in a modified $10 \times 10$ chessboard. The modification is such that for each $k$-th row, the leftmost $k-1$ unit squares are removed. This forms a staircase-like structure.
2. **Visualizing the S... | 715 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Berk tries to guess the two-digit number that Ayca picks. After each guess, Ayca gives a hint indicating the number of digits which match the number picked. If Berk can guarantee to guess Ayca's number in $n$ guesses, what is the smallest possible value of $n$?
$
\textbf{(A)}\ 9
\qquad\textbf{(B)}\ 10
\qquad\textbf{(... | To solve this problem, we need to determine the smallest number of guesses, \( n \), that Berk needs to guarantee finding Ayca's two-digit number.
1. **Understanding the Problem:**
- Ayca picks a two-digit number, which means there are 90 possible numbers (from 10 to 99).
- After each guess, Ayca provides a hin... | 10 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
If decimal representation of $2^n$ starts with $7$, what is the first digit in decimal representation of $5^n$?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ 5
\qquad\textbf{(D)}\ 7
\qquad\textbf{(E)}\ 9
$ | 1. We start by noting that the decimal representation of \(2^n\) starts with 7. This implies that there exists some integer \(k\) such that:
\[
7 \times 10^k \leq 2^n < 8 \times 10^k
\]
Taking the logarithm base 10 of all parts of the inequality, we get:
\[
\log_{10}(7 \times 10^k) \leq \log_{10}(2^n)... | 1 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Let $ABCD$ be a isosceles trapezoid such that $AB || CD$ and all of its sides are tangent to a circle. $[AD]$ touches this circle at $N$. $NC$ and $NB$ meet the circle again at $K$ and $L$, respectively. What is $\dfrac {|BN|}{|BL|} + \dfrac {|CN|}{|CK|}$?
$
\textbf{(A)}\ 4
\qquad\textbf{(B)}\ 6
\qquad\textbf{(C)}\ ... | 1. **Understanding the Problem:**
We are given an isosceles trapezoid \(ABCD\) with \(AB \parallel CD\) and all sides tangent to a circle. The tangency points imply that the circle is the incircle of the trapezoid. We need to find the value of \(\dfrac{|BN|}{|BL|} + \dfrac{|CN|}{|CK|}\).
2. **Using the Power of a P... | 10 | Geometry | MCQ | Yes | Yes | aops_forum | false |
What is the $33$-rd number after the decimal point of $(\sqrt {10} + 3)^{2001}$?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ 8
$ | 1. We start by considering the expression \((\sqrt{10} + 3)^{2001}\). To find the 33rd digit after the decimal point, we need to understand the behavior of this expression.
2. By the Binomial Theorem, we can write:
\[
(\sqrt{10} + 3)^{2001} = \sum_{k=0}^{2001} \binom{2001}{k} (\sqrt{10})^k \cdot 3^{2001-k}
\]... | 0 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Let $f$ be a real-valued function defined over ordered pairs of integers such that \[f(x+3m-2n, y-4m+5n) = f(x,y)\] for every integers $x,y,m,n$. At most how many elements does the range set of $f$ have?
$
\textbf{(A)}\ 7
\qquad\textbf{(B)}\ 8
\qquad\textbf{(C)}\ 15
\qquad\textbf{(D)}\ 49
\qquad\textbf{(E)}\ \text{In... | 1. Given the functional equation:
\[
f(x+3m-2n, y-4m+5n) = f(x,y)
\]
for all integers \(x, y, m, n\), we need to determine the maximum number of distinct values that the function \(f\) can take.
2. First, consider the case when \(m = 1\) and \(n = 1\):
\[
f(x + 3 \cdot 1 - 2 \cdot 1, y - 4 \cdot 1 + ... | 7 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
How many ordered pairs $(p,n)$ are there such that $(1+p)^n = 1+pn + n^p$ where $p$ is a prime and $n$ is a positive integer?
$
\textbf{(A)}\ 5
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 1
\qquad\textbf{(D)}\ 0
\qquad\textbf{(E)}\ \text{None of the preceding}
$ | To solve the problem, we need to find the ordered pairs \((p, n)\) such that \((1 + p)^n = 1 + pn + n^p\), where \(p\) is a prime number and \(n\) is a positive integer.
1. **Consider the parity of the equation:**
- If \(p\) is an odd prime, then \(1 + p\) is even.
- Therefore, \((1 + p)^n\) is even for any posi... | 1 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
How many integers $0 \leq x < 125$ are there such that $x^3 - 2x + 6 \equiv 0 \pmod {125}$?
$
\textbf{a)}\ 0
\qquad\textbf{b)}\ 1
\qquad\textbf{c)}\ 2
\qquad\textbf{d)}\ 3
\qquad\textbf{e)}\ \text{None of above}
$ | 1. We start by solving the congruence \( x^3 - 2x + 6 \equiv 0 \pmod{125} \). To do this, we first solve the congruence modulo 5.
2. Consider the congruence \( x^3 - 2x + 6 \equiv 0 \pmod{5} \). We test all possible values of \( x \) modulo 5:
- For \( x \equiv 0 \pmod{5} \): \( 0^3 - 2 \cdot 0 + 6 \equiv 6 \not\equ... | 1 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
How many positive roots does polynomial $x^{2002} + a_{2001}x^{2001} + a_{2000}x^{2000} + \cdots + a_1x + a_0$ have such that $a_{2001} = 2002$ and $a_k = -k - 1$ for $0\leq k \leq 2000$?
$
\textbf{a)}\ 0
\qquad\textbf{b)}\ 1
\qquad\textbf{c)}\ 2
\qquad\textbf{d)}\ 1001
\qquad\textbf{e)}\ 2002
$ | To determine the number of positive roots of the polynomial \( P(x) = x^{2002} + 2002x^{2001} - 2001x^{2000} - 2000x^{1999} - \cdots - 2x - 1 \), we can use Descartes' Rule of Signs. Descartes' Rule of Signs states that the number of positive real roots of a polynomial is either equal to the number of sign changes betw... | 1 | Algebra | MCQ | Yes | Yes | aops_forum | false |
How many real roots does the polynomial $x^5 + x^4 - x^3 - x^2 - 2x - 2$ have?
$
\textbf{a)}\ 1
\qquad\textbf{b)}\ 2
\qquad\textbf{c)}\ 3
\qquad\textbf{d)}\ 4
\qquad\textbf{e)}\ \text{None of above}
$ | 1. We start with the polynomial \( P(x) = x^5 + x^4 - x^3 - x^2 - 2x - 2 \).
2. We test for possible rational roots using the Rational Root Theorem, which states that any rational root of the polynomial \( P(x) \) must be a factor of the constant term (-2) divided by a factor of the leading coefficient (1). The possib... | 3 | Algebra | MCQ | Yes | Yes | aops_forum | false |
The lengths of two altitudes of a triangles are $8$ and $12$. Which of the following cannot be the third altitude?
$
\textbf{a)}\ 4
\qquad\textbf{b)}\ 7
\qquad\textbf{c)}\ 8
\qquad\textbf{d)}\ 12
\qquad\textbf{e)}\ 23
$ | 1. Denote the area of the triangle by $\Delta$ and the altitudes from vertices $A, B, C$ to the opposite sides by $h_a, h_b, h_c$, respectively. Given $h_a = 8$ and $h_b = 12$.
2. The area $\Delta$ can be expressed in terms of the altitudes and the corresponding sides:
\[
\Delta = \frac{1}{2} a h_a = \frac{1}{2} ... | 4 | Geometry | MCQ | Yes | Yes | aops_forum | false |
The thousands digit of a five-digit number which is divisible by $37$ and $173$ is $3$. What is the hundreds digit of this number?
$
\textbf{a)}\ 0
\qquad\textbf{b)}\ 2
\qquad\textbf{c)}\ 4
\qquad\textbf{d)}\ 6
\qquad\textbf{e)}\ 8
$ | 1. **Determine the least common multiple (LCM) of 37 and 173:**
\[
\text{LCM}(37, 173) = 37 \times 173 = 6401
\]
Since 37 and 173 are both prime numbers, their LCM is simply their product.
2. **Identify the five-digit multiples of 6401:**
\[
6401 \times 2 = 12802
\]
\[
6401 \times 3 = 19203
... | 2 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
What is the least number of weighings needed to determine the sum of weights of $13$ watermelons such that exactly two watermelons should be weighed in each weigh?
$
\textbf{a)}\ 7
\qquad\textbf{b)}\ 8
\qquad\textbf{c)}\ 9
\qquad\textbf{d)}\ 10
\qquad\textbf{e)}\ 11
$ | 1. **Understanding the problem**: We need to determine the sum of the weights of 13 watermelons using the least number of weighings, where each weighing involves exactly two watermelons.
2. **Initial analysis**: Let's denote the weights of the 13 watermelons as \( x_1, x_2, \ldots, x_{13} \). We need to find the sum \... | 8 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Let $ABC$ be triangle such that $|AB| = 5$, $|BC| = 9$ and $|AC| = 8$. The angle bisector of $\widehat{BCA}$ meets $BA$ at $X$ and the angle bisector of $\widehat{CAB}$ meets $BC$ at $Y$. Let $Z$ be the intersection of lines $XY$ and $AC$. What is $|AZ|$?
$
\textbf{a)}\ \sqrt{104}
\qquad\textbf{b)}\ \sqrt{145}
\qquad... | 1. **Apply Menelaus' Theorem**: Menelaus' Theorem states that for a triangle \( \triangle ABC \) with a transversal line intersecting \( BC \), \( CA \), and \( AB \) at points \( D \), \( E \), and \( F \) respectively, the following relation holds:
\[
\frac{BD}{DC} \cdot \frac{CE}{EA} \cdot \frac{AF}{FB} = 1
... | 10 | Geometry | MCQ | Yes | Yes | aops_forum | false |
What is the coefficient of $x^5$ in the expansion of $(1 + x + x^2)^9$?
$
\textbf{a)}\ 1680
\qquad\textbf{b)}\ 882
\qquad\textbf{c)}\ 729
\qquad\textbf{d)}\ 450
\qquad\textbf{e)}\ 246
$ | To find the coefficient of \(x^5\) in the expansion of \((1 + x + x^2)^9\), we can use the multinomial theorem. The multinomial theorem states that:
\[
(a_1 + a_2 + \cdots + a_m)^n = \sum_{k_1+k_2+\cdots+k_m=n} \frac{n!}{k_1! k_2! \cdots k_m!} a_1^{k_1} a_2^{k_2} \cdots a_m^{k_m}
\]
In our case, \(a_1 = 1\), \(a_2 = ... | 882 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
How many primes $p$ are there such that $39p + 1$ is a perfect square?
$
\textbf{a)}\ 0
\qquad\textbf{b)}\ 1
\qquad\textbf{c)}\ 2
\qquad\textbf{d)}\ 3
\qquad\textbf{e)}\ \text{None of above}
$ | 1. Let \( 39p + 1 = n^2 \) for some integer \( n \). This implies:
\[
39p = n^2 - 1
\]
Since \( n^2 - 1 \) can be factored as \( (n+1)(n-1) \), we have:
\[
39p = (n+1)(n-1)
\]
2. The factors of \( 39p \) are \( 1, 3, 13, 39, p, 3p, 13p, \) and \( 39p \). We need to test each of these factors to se... | 3 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Let $ABCD$ be a trapezoid and a tangential quadrilateral such that $AD || BC$ and $|AB|=|CD|$. The incircle touches $[CD]$ at $N$. $[AN]$ and $[BN]$ meet the incircle again at $K$ and $L$, respectively. What is $\dfrac {|AN|}{|AK|} + \dfrac {|BN|}{|BL|}$?
$
\textbf{(A)}\ 8
\qquad\textbf{(B)}\ 9
\qquad\textbf{(C)}\ 1... | 1. **Understanding the Problem:**
We are given a trapezoid \(ABCD\) with \(AD \parallel BC\) and \(|AB| = |CD|\). The quadrilateral is tangential, meaning it has an incircle that touches all four sides. The incircle touches \(CD\) at point \(N\). We need to find the value of \(\dfrac{|AN|}{|AK|} + \dfrac{|BN|}{|BL|}... | 10 | Geometry | MCQ | Yes | Yes | aops_forum | false |
For how many integers $x$ is $|15x^2-32x-28|$ a prime number?
$
\textbf{a)}\ 0
\qquad\textbf{b)}\ 1
\qquad\textbf{c)}\ 2
\qquad\textbf{d)}\ 4
\qquad\textbf{e)}\ \text{None of above}
$ | 1. We start with the given expression \( |15x^2 - 32x - 28| \) and need to determine for how many integer values of \( x \) this expression is a prime number.
2. First, we factorize the quadratic expression inside the absolute value:
\[
15x^2 - 32x - 28 = (3x + 2)(5x - 14)
\]
3. We need \( |(3x + 2)(5x - 14)| ... | 2 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
How many positive integers $A$ are there such that if we append $3$ digits to the rightmost of decimal representation of $A$, we will get a number equal to $1+2+\cdots + A$?
$
\textbf{a)}\ 0
\qquad\textbf{b)}\ 1
\qquad\textbf{c)}\ 2
\qquad\textbf{d)}\ 2002
\qquad\textbf{e)}\ \text{None of above}
$ | 1. Let the three digits appended be \( B = \overline{abc} \), where \( a, b, c \) are digits (possibly all zeros). The new number formed by appending these digits to \( A \) is \( 1000A + B \).
2. According to the problem, this new number equals the sum of the first \( A \) positive integers:
\[
1000A + B = 1 + ... | 1 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
If $2^n$ divides $5^{256} - 1$, what is the largest possible value of $n$?
$
\textbf{a)}\ 8
\qquad\textbf{b)}\ 10
\qquad\textbf{c)}\ 11
\qquad\textbf{d)}\ 12
\qquad\textbf{e)}\ \text{None of above}
$ | To determine the largest possible value of \( n \) such that \( 2^n \) divides \( 5^{256} - 1 \), we need to analyze the factorization of \( 5^{256} - 1 \) and count the powers of 2 in the factorization.
1. **Factorization using difference of powers:**
\[
5^{256} - 1 = (5 - 1)(5 + 1)(5^2 + 1)(5^4 + 1)(5^8 + 1)(5... | 10 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
How many positive integers $n$ are there such that the equation $\left \lfloor \sqrt[3] {7n + 2} \right \rfloor = \left \lfloor \sqrt[3] {7n + 3} \right \rfloor $ does not hold?
$
\textbf{a)}\ 0
\qquad\textbf{b)}\ 1
\qquad\textbf{c)}\ 7
\qquad\textbf{d)}\ \text{Infinitely many}
\qquad\textbf{e)}\ \text{None of above}... | To determine how many positive integers \( n \) exist such that the equation
\[
\left\lfloor \sqrt[3]{7n + 2} \right\rfloor = \left\lfloor \sqrt[3]{7n + 3} \right\rfloor
\]
does not hold, we need to analyze the conditions under which the floor functions of the cube roots of \( 7n + 2 \) and \( 7n + 3 \) are equal.
1.... | 0 | Inequalities | MCQ | Yes | Yes | aops_forum | false |
The keys of a safe with five locks are cloned and distributed among eight people such that any of five of eight people can open the safe. What is the least total number of keys?
$
\textbf{a)}\ 18
\qquad\textbf{b)}\ 20
\qquad\textbf{c)}\ 22
\qquad\textbf{d)}\ 24
\qquad\textbf{e)}\ 25
$ | 1. **Understanding the problem**: We need to distribute the keys of a safe with five locks among eight people such that any group of five people can open the safe. We need to find the minimum total number of keys required.
2. **Key distribution requirement**: Each key must be distributed among at least 4 people. If a ... | 20 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
The numbers $1, 2, \dots ,N$ are arranged in a circle where $N \geq 2$. If each number shares a common digit with each of its neighbours in decimal representation, what is the least possible value of $N$?
$
\textbf{a)}\ 18
\qquad\textbf{b)}\ 19
\qquad\textbf{c)}\ 28
\qquad\textbf{d)}\ 29
\qquad\textbf{e)}\ \text{No... | To solve this problem, we need to find the smallest \( N \) such that the numbers \( 1, 2, \ldots, N \) can be arranged in a circle where each number shares a common digit with each of its neighbors.
1. **Understanding the Problem:**
- We need to arrange the numbers \( 1, 2, \ldots, N \) in a circle.
- Each num... | 29 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
How many positive integers $n$ are there such that $3n^2 + 3n + 7$ is a perfect cube?
$
\textbf{a)}\ 0
\qquad\textbf{b)}\ 1
\qquad\textbf{c)}\ 3
\qquad\textbf{d)}\ 7
\qquad\textbf{e)}\ \text{Infinitely many}
$ | To determine how many positive integers \( n \) exist such that \( 3n^2 + 3n + 7 \) is a perfect cube, we start by setting up the equation:
\[ 3n^2 + 3n + 7 = y^3 \]
We need to analyze this equation modulo 3. First, we consider the expression \( 3n^2 + 3n + 7 \) modulo 3:
\[ 3n^2 + 3n + 7 \equiv 0n^2 + 0n + 7 \equiv... | 0 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
For each integer $i=0,1,2, \dots$, there are eight balls each weighing $2^i$ grams. We may place balls as much as we desire into given $n$ boxes. If the total weight of balls in each box is same, what is the largest possible value of $n$?
$
\textbf{a)}\ 8
\qquad\textbf{b)}\ 10
\qquad\textbf{c)}\ 12
\qquad\textbf{d)}\... | To solve this problem, we need to determine the largest possible number of boxes, \( n \), such that the total weight of balls in each box is the same. Each integer \( i = 0, 1, 2, \ldots \) corresponds to eight balls each weighing \( 2^i \) grams.
1. **Determine the total weight of all balls:**
Each integer \( i \... | 15 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
If $a^5 +5a^4 +10a^3 +3a^2 -9a-6 = 0$ where $a$ is a real number other than $-1$, what is $(a + 1)^3$?
$
\textbf{a)}\ 1
\qquad\textbf{b)}\ 3\sqrt 3
\qquad\textbf{c)}\ 7
\qquad\textbf{d)}\ 8
\qquad\textbf{e)}\ 27
$ | 1. Given the polynomial equation:
\[
a^5 + 5a^4 + 10a^3 + 3a^2 - 9a - 6 = 0
\]
we need to find the value of \((a + 1)^3\).
2. Notice that the polynomial can be rewritten in terms of \((a + 1)\). Let \(b = a + 1\). Then \(a = b - 1\).
3. Substitute \(a = b - 1\) into the polynomial:
\[
(b-1)^5 + 5(b-... | 7 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Let $P$ be a polynomial such that $(x-4)P(2x) = 4(x-1)P(x)$, for every real $x$. If $P(0) \neq 0$, what is the degree of $P$?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 3
\qquad\textbf{(E)}\ \text{None of the preceding}
$ | 1. Given the polynomial equation:
\[
(x-4)P(2x) = 4(x-1)P(x)
\]
for every real \( x \), we need to determine the degree of \( P \) given that \( P(0) \neq 0 \).
2. First, substitute \( x = 4 \) into the equation:
\[
(4-4)P(8) = 4(4-1)P(4) \implies 0 = 12P(4)
\]
This implies \( P(4) = 0 \). Ther... | 2 | Algebra | MCQ | Yes | Yes | aops_forum | false |
How many primes $p$ are there such that $5p(2^{p+1}-1)$ is a perfect square?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 3
\qquad\textbf{(E)}\ \text{None of the preceding}
$ | 1. Let \( N = 5p(2^{p+1} - 1) \). For \( N \) to be a perfect square, the product \( 5p(2^{p+1} - 1) \) must be a perfect square.
2. Since \( N \) includes a factor of \( p \), for \( N \) to be a perfect square, one of \( 5 \) and \( 2^{p+1} - 1 \) must be divisible by \( p \).
3. First, consider the case where \( p ... | 1 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
For which of the following integers $n$, there is at least one integer $x$ such that $x^2 \equiv -1 \pmod{n}$?
$
\textbf{(A)}\ 97
\qquad\textbf{(B)}\ 98
\qquad\textbf{(C)}\ 99
\qquad\textbf{(D)}\ 100
\qquad\textbf{(E)}\ \text{None of the preceding}
$ | To determine for which of the given integers \( n \) there exists an integer \( x \) such that \( x^2 \equiv -1 \pmod{n} \), we need to analyze each option.
1. **Option (A): \( n = 97 \)**
- \( 97 \) is a prime number.
- We need to check if there exists an integer \( x \) such that \( x^2 \equiv -1 \pmod{97} \).... | 97 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
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 |
If the sum of digits in decimal representaion of positive integer $n$ is $111$ and the sum of digits in decimal representation of $7002n$ is $990$, what is the sum of digits in decimal representation of $2003n$?
$
\textbf{(A)}\ 309
\qquad\textbf{(B)}\ 330
\qquad\textbf{(C)}\ 550
\qquad\textbf{(D)}\ 555
\qquad\textbf{... | Let \( S(x) \) represent the sum of the digits of the number \( x \).
Given:
1. \( S(n) = 111 \)
2. \( S(7002n) = 990 \)
We need to find \( S(2003n) \).
First, let's analyze the given information using the properties of digit sums and carries.
### Step 1: Understanding the Sum of Digits
The sum of the digits of a n... | 555 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Let the circles $S_1$ and $S_2$ meet at the points $A$ and $B$. A line through $B$ meets $S_1$ at a point $D$ other than $B$ and meets $S_2$ at a point $C$ other than $B$. The tangent to $S_1$ through $D$ and the tangent to $S_2$ through $C$ meet at $E$. If $|AD|=15$, $|AC|=16$, $|AB|=10$, what is $|AE|$?
$
\textbf{(... | 1. **Identify the given information and the relationships between the points:**
- Circles \( S_1 \) and \( S_2 \) intersect at points \( A \) and \( B \).
- A line through \( B \) intersects \( S_1 \) at \( D \) and \( S_2 \) at \( C \).
- Tangents to \( S_1 \) at \( D \) and to \( S_2 \) at \( C \) meet at \(... | 24 | Geometry | 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 |
Let $a_1 = \sqrt 7$ and $b_i = \lfloor a_i \rfloor$, $a_{i+1} = \dfrac{1}{b_i - \lfloor b_i \rfloor}$ for each $i\geq i$. What is the smallest integer $n$ greater than $2004$ such that $b_n$ is divisible by $4$? ($\lfloor x \rfloor$ denotes the largest integer less than or equal to $x$)
$
\textbf{(A)}\ 2005
\qquad\t... | 1. We start with \( a_1 = \sqrt{7} \).
2. Calculate \( b_1 = \lfloor a_1 \rfloor = \lfloor \sqrt{7} \rfloor = 2 \).
3. Next, we compute \( a_2 \):
\[
a_2 = \frac{1}{a_1 - b_1} = \frac{1}{\sqrt{7} - 2}
\]
Rationalize the denominator:
\[
a_2 = \frac{1}{\sqrt{7} - 2} \cdot \frac{\sqrt{7} + 2}{\sqrt{7} + ... | 2005 | Logic and Puzzles | 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 write one of the numbers $0$ and $1$ into each unit square of a chessboard with $40$ rows and $7$ columns. If any two rows have different sequences, at most how many $1$s can be written into the unit squares?
$
\textbf{(A)}\ 198
\qquad\textbf{(B)}\ 128
\qquad\textbf{(C)}\ 82
\qquad\textbf{(D)}\ 40
\qquad\textbf{(E... | To solve this problem, we need to maximize the number of 1's on a 40-row by 7-column chessboard, ensuring that each row has a unique sequence of 0's and 1's. We will use combinatorial methods to determine the maximum number of 1's.
1. **Determine the number of distinct sequences:**
Each row must have a unique seque... | 198 | Combinatorics | 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 |
What is the difference between the maximum value and the minimum value of the sum $a_1 + 2a_2 + 3a_3 + 4a_4 + 5a_5$ where $\{a_1,a_2,a_3,a_4,a_5\} = \{1,2,3,4,5\}$?
$
\textbf{(A)}\ 20
\qquad\textbf{(B)}\ 15
\qquad\textbf{(C)}\ 10
\qquad\textbf{(D)}\ 5
\qquad\textbf{(E)}\ 0
$ | 1. We are given the sum \( S = a_1 + 2a_2 + 3a_3 + 4a_4 + 5a_5 \) where \(\{a_1, a_2, a_3, a_4, a_5\} = \{1, 2, 3, 4, 5\}\). We need to find the difference between the maximum and minimum values of \( S \).
2. To find the maximum value of \( S \), we should assign the largest coefficients to the largest values of \(a_... | 20 | 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 |
What is the largest possible value of $8x^2+9xy+18y^2+2x+3y$ such that $4x^2 + 9y^2 = 8$ where $x,y$ are real numbers?
$
\textbf{(A)}\ 23
\qquad\textbf{(B)}\ 26
\qquad\textbf{(C)}\ 29
\qquad\textbf{(D)}\ 31
\qquad\textbf{(E)}\ 35
$ | To find the largest possible value of \(8x^2 + 9xy + 18y^2 + 2x + 3y\) subject to the constraint \(4x^2 + 9y^2 = 8\), we can use the method of Lagrange multipliers.
1. **Define the objective function and the constraint:**
\[
f(x, y) = 8x^2 + 9xy + 18y^2 + 2x + 3y
\]
\[
g(x, y) = 4x^2 + 9y^2 - 8 = 0
\... | 26 | Calculus | MCQ | Yes | Yes | aops_forum | false |
If $a$ and $b$ are the roots of the equation $x^2-2cx-5d = 0$, $c$ and $d$ are the roots of the equation $x^2-2ax-5b=0$, where $a,b,c,d$ are distinct real numbers, what is $a+b+c+d$?
$
\textbf{(A)}\ 10
\qquad\textbf{(B)}\ 15
\qquad\textbf{(C)}\ 20
\qquad\textbf{(D)}\ 25
\qquad\textbf{(E)}\ 30
$ | Given the equations:
1. \( x^2 - 2cx - 5d = 0 \) with roots \( a \) and \( b \)
2. \( x^2 - 2ax - 5b = 0 \) with roots \( c \) and \( d \)
We need to find \( a + b + c + d \).
1. **Sum and Product of Roots for the First Equation:**
- By Vieta's formulas, for the equation \( x^2 - 2cx - 5d = 0 \):
\[
a + ... | 30 | 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 which $k$, there is no integer pair $(x,y)$ such that $x^2 - y^2 = k$?
$
\textbf{(A)}\ 2005
\qquad\textbf{(B)}\ 2006
\qquad\textbf{(C)}\ 2007
\qquad\textbf{(D)}\ 2008
\qquad\textbf{(E)}\ 2009
$ | 1. We start with the given equation \(x^2 - y^2 = k\). This can be factored as:
\[
x^2 - y^2 = (x-y)(x+y) = k
\]
2. We need to determine for which values of \(k\) there are no integer pairs \((x, y)\) that satisfy this equation.
3. Consider the factorization \(k = (x-y)(x+y)\). For \(k\) to be expressed as a ... | 2006 | 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 |
How many natural number triples $(x,y,z)$ are there such that $xyz = 10^6$?
$
\textbf{(A)}\ 568
\qquad\textbf{(B)}\ 784
\qquad\textbf{(C)}\ 812
\qquad\textbf{(D)}\ 816
\qquad\textbf{(E)}\ 824
$ | 1. We start with the equation \(xyz = 10^6\). We can factorize \(10^6\) as follows:
\[
10^6 = (2 \cdot 5)^6 = 2^6 \cdot 5^6
\]
Therefore, we need to find the number of natural number triples \((x, y, z)\) such that \(x, y, z\) are products of the factors \(2^a \cdot 5^b\) where \(a + b = 6\).
2. We treat t... | 784 | Number Theory | 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 |
There are $20$ people in a certain community. $10$ of them speak English, $10$ of them speak German, and $10$ of them speak French. We call a [i]committee[/i] to a $3$-subset of this community if there is at least one who speaks English, at least one who speaks German, and at least one who speaks French in this subset.... | 1. **Calculate the total number of 3-subsets from 20 people:**
\[
\binom{20}{3} = \frac{20 \times 19 \times 18}{3 \times 2 \times 1} = 1140
\]
This represents all possible committees of 3 people from the 20-person community.
2. **Calculate the number of 3-subsets that do not include at least one English sp... | 1020 | Combinatorics | 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 |
There are $27$ unit cubes. We are marking one point on each of the two opposing faces, two points on each of the other two opposing faces, and three points on each of the remaining two opposing faces of each cube. We are constructing a $3\times 3 \times 3$ cube with these $27$ cubes. What is the least number of marked ... | 1. **Understanding the Problem:**
We have 27 unit cubes, each with points marked on their faces. We need to construct a \(3 \times 3 \times 3\) cube using these unit cubes and determine the least number of marked points visible on the faces of the new cube.
2. **Classifying the Unit Cubes:**
- **Corner Cubes:** ... | 90 | Combinatorics | 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 |
How many positive integers are there such that $\left \lfloor \frac m{11} \right \rfloor = \left \lfloor \frac m{10} \right \rfloor$? ($\left \lfloor x \right \rfloor$ denotes the greatest integer not exceeding $x$.)
$
\textbf{(A)}\ 44
\qquad\textbf{(B)}\ 48
\qquad\textbf{(C)}\ 52
\qquad\textbf{(D)}\ 54
\qquad\textb... | To solve the problem, we need to find the number of positive integers \( m \) such that \(\left \lfloor \frac{m}{11} \right \rfloor = \left \lfloor \frac{m}{10} \right \rfloor\).
1. **Define the floor functions:**
Let \( k = \left \lfloor \frac{m}{11} \right \rfloor = \left \lfloor \frac{m}{10} \right \rfloor \). T... | 55 | Inequalities | MCQ | Yes | Yes | aops_forum | false |
Let $d_1$ and $d_2$ be parallel lines in the plane. We are marking $11$ black points on $d_1$, and $16$ white points on $d_2$. We are drawig the segments connecting black points with white points. What is the maximum number of points of intersection of these segments that lies on between the parallel lines (excluding t... | 1. **Understanding the Problem:**
We need to find the maximum number of intersection points of segments connecting black points on line \(d_1\) with white points on line \(d_2\). The intersection points must lie between the parallel lines \(d_1\) and \(d_2\).
2. **Counting the Segments:**
- There are 11 black po... | 6600 | Combinatorics | 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 |
The integer $k$ is a [i]good number[/i], if we can divide a square into $k$ squares. How many good numbers not greater than $2006$ are there?
$
\textbf{(A)}\ 1003
\qquad\textbf{(B)}\ 1026
\qquad\textbf{(C)}\ 2000
\qquad\textbf{(D)}\ 2003
\qquad\textbf{(E)}\ 2004
$ | 1. **Understanding the problem**: We need to determine how many integers \( k \) (not greater than 2006) can be the number of squares into which a larger square can be divided. These integers are termed as "good numbers."
2. **Initial observations**:
- We cannot divide a square into exactly 2 smaller squares.
- ... | 2003 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
How many integer pairs $(x,y)$ are there such that \[0\leq x < 165, \quad 0\leq y < 165 \text{ and } y^2\equiv x^3+x \pmod {165}?\]
$
\textbf{(A)}\ 80
\qquad\textbf{(B)}\ 99
\qquad\textbf{(C)}\ 120
\qquad\textbf{(D)}\ 315
\qquad\textbf{(E)}\ \text{None of above}
$ | We are asked to find the number of integer pairs \((x, y)\) such that
\[0 \leq x < 165, \quad 0 \leq y < 165 \text{ and } y^2 \equiv x^3 + x \pmod{165}.\]
First, note that \(165 = 3 \cdot 5 \cdot 11\). By the Chinese Remainder Theorem, we can solve the congruence \(y^2 \equiv x^3 + x \pmod{165}\) by solving it modulo... | 99 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
What is the greatest integer not exceeding the number $\left( 1 + \frac{\sqrt 2 + \sqrt 3 + \sqrt 4}{\sqrt 2 + \sqrt 3 + \sqrt 6 + \sqrt 8 + 4}\right)^{10}$?
$
\textbf{(A)}\ 2
\qquad\textbf{(B)}\ 10
\qquad\textbf{(C)}\ 21
\qquad\textbf{(D)}\ 32
\qquad\textbf{(E)}\ 36
$ | 1. We start with the given expression:
\[
\left( 1 + \frac{\sqrt{2} + \sqrt{3} + \sqrt{4}}{\sqrt{2} + \sqrt{3} + \sqrt{6} + \sqrt{8} + 4} \right)^{10}
\]
2. First, simplify the denominator:
\[
\sqrt{2} + \sqrt{3} + \sqrt{6} + \sqrt{8} + 4
\]
3. Notice that:
\[
\sqrt{8} = 2\sqrt{2} \quad \text{... | 32 | Calculus | MCQ | Yes | Yes | aops_forum | false |
Ali who has $10$ candies eats at least one candy a day. In how many different ways can he eat all candies (according to distribution among days)?
$
\textbf{(A)}\ 64
\qquad\textbf{(B)}\ 126
\qquad\textbf{(C)}\ 243
\qquad\textbf{(D)}\ 512
\qquad\textbf{(E)}\ 1025
$ | 1. **Understanding the problem**: Ali has 10 candies and eats at least one candy each day. We need to find the number of different ways he can distribute the candies over several days.
2. **Formulating the problem**: Since Ali eats at least one candy each day, the problem can be seen as distributing 10 candies over a ... | 512 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
How many integer triples $(x,y,z)$ are there such that \[\begin{array}{rcl} x - yz^2&\equiv & 1 \pmod {13} \\ xz+y&\equiv& 4 \pmod {13} \end{array}\] where $0\leq x < 13$, $0\leq y <13$, and $0\leq z< 13$?
$
\textbf{(A)}\ 10
\qquad\textbf{(B)}\ 23
\qquad\textbf{(C)}\ 36
\qquad\textbf{(D)}\ 49
\qquad\textbf{(E)}\ \te... | To solve the problem, we need to find the number of integer triples \((x, y, z)\) that satisfy the following system of congruences modulo 13:
\[
\begin{array}{rcl}
x - yz^2 &\equiv & 1 \pmod{13} \\
xz + y &\equiv & 4 \pmod{13}
\end{array}
\]
where \(0 \leq x < 13\), \(0 \leq y < 13\), and \(0 \leq z < 13\).
1. **Expre... | 13 | Number Theory | 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 |
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