problem stringlengths 2 5.64k | solution stringlengths 2 13.5k | answer stringlengths 1 43 | problem_type stringclasses 8
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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 ways are there to partition $7$ students into the groups of $2$ or $3$?
$
\textbf{(A)}\ 70
\qquad\textbf{(B)}\ 105
\qquad\textbf{(C)}\ 210
\qquad\textbf{(D)}\ 280
\qquad\textbf{(E)}\ 630
$ | To solve the problem of partitioning 7 students into groups of 2 or 3, we need to consider the possible groupings and count the number of ways to form these groups.
1. **Identify the possible groupings:**
- One possible way to partition 7 students is to have one group of 3 students and two groups of 2 students.
2.... | 105 | Combinatorics | 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 |
If $8/19$ of the product of largest two elements of a positive integer set is not greater than the sum of other elements, what is the minimum possible value of the largest number in the set?
$
\textbf{(A)}\ 8
\qquad\textbf{(B)}\ 12
\qquad\textbf{(C)}\ 13
\qquad\textbf{(D)}\ 19
\qquad\textbf{(E)}\ 20
$ | 1. Let the largest two elements in the set be \(a\) and \(b\) such that \(a < b\). For all other elements in the set, let \(x\) be an element such that \(x < a\).
2. The sum of all other elements in the set is at most \(\frac{(a-1)a}{2}\). This is because the sum of the first \(a-1\) positive integers is \(\frac{(a-1)... | 13 | Inequalities | MCQ | Yes | Yes | aops_forum | false |
What is the largest integer $n$ that satisfies $(100^2-99^2)(99^2-98^2)\dots(3^2-2^2)(2^2-1^2)$ is divisible by $3^n$?
$
\textbf{(A)}\ 49
\qquad\textbf{(B)}\ 53
\qquad\textbf{(C)}\ 97
\qquad\textbf{(D)}\ 103
\qquad\textbf{(E)}\ \text{None of the above}
$ | To solve the problem, we need to find the largest integer \( n \) such that the product \((100^2 - 99^2)(99^2 - 98^2) \cdots (2^2 - 1^2)\) is divisible by \( 3^n \).
1. **Simplify each term in the product:**
\[
k^2 - (k-1)^2 = k^2 - (k^2 - 2k + 1) = 2k - 1
\]
Therefore, the product can be rewritten as:
... | 49 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
The sum of the largest number and the smallest number of a triple of positive integers $(x,y,z)$ is called to be the power of the triple. What is the sum of powers of all triples $(x,y,z)$ where $x,y,z \leq 9$?
$
\textbf{(A)}\ 9000
\qquad\textbf{(B)}\ 8460
\qquad\textbf{(C)}\ 7290
\qquad\textbf{(D)}\ 6150
\qquad\text... | 1. **Define the power of a triple**: The power of a triple \((x, y, z)\) is defined as the sum of the largest and smallest numbers in the triple. Without loss of generality, assume \(x \geq y \geq z\). Thus, the power of the triple \((x, y, z)\) is \(x + z\).
2. **Pairing triples**: We pair each triple \((x, y, z)\) w... | 7290 | Combinatorics | 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 |
If $x_1=5, x_2=401$, and
\[
x_n=x_{n-2}-\frac 1{x_{n-1}}
\]
for every $3\leq n \leq m$, what is the largest value of $m$?
$
\textbf{(A)}\ 406
\qquad\textbf{(B)}\ 2005
\qquad\textbf{(C)}\ 2006
\qquad\textbf{(D)}\ 2007
\qquad\textbf{(E)}\ \text{None of the above}
$ | 1. Given the recurrence relation:
\[
x_n = x_{n-2} - \frac{1}{x_{n-1}}
\]
for \(3 \leq n \leq m\), with initial conditions \(x_1 = 5\) and \(x_2 = 401\).
2. We start by manipulating the recurrence relation. Multiply both sides by \(x_{n-1}\):
\[
x_n x_{n-1} = x_{n-2} x_{n-1} - 1
\]
3. Rearrange t... | 2007 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
The $9$ consequtive sections of a paper strip are colored either red or white. If no two consequtive sections are white, in how many ways can this coloring be made?
$
\textbf{(A)}\ 34
\qquad\textbf{(B)}\ 89
\qquad\textbf{(C)}\ 128
\qquad\textbf{(D)}\ 144
\qquad\textbf{(E)}\ 360
$ | To solve this problem, we need to count the number of ways to color a strip of 9 sections such that no two consecutive sections are white. We can approach this problem using a recursive method similar to the Fibonacci sequence.
1. **Define the problem recursively:**
Let \( f(n) \) be the number of valid ways to col... | 89 | Combinatorics | 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 |
Five points, no three of which are collinear, are given. What is the least possible value of the numbers of convex polygons whose some corners are from these five points?
$
\textbf{(A)}\ 10
\qquad\textbf{(B)}\ 11
\qquad\textbf{(C)}\ 12
\qquad\textbf{(D)}\ 15
\qquad\textbf{(E)}\ 16
$ | 1. **Calculate the number of triangles:**
- We need to choose 3 points out of 5 to form a triangle.
- The number of ways to choose 3 points from 5 is given by the binomial coefficient:
\[
\binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4 \times 3!}{3! \times 2 \times 1} = 10
\]
- Therefore, ... | 16 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
For which value of $A$, does the equation $3m^2n = n^3 + A$ have a solution in natural numbers?
$
\textbf{(A)}\ 301
\qquad\textbf{(B)}\ 403
\qquad\textbf{(C)}\ 415
\qquad\textbf{(D)}\ 427
\qquad\textbf{(E)}\ 481
$ | To determine the value of \( A \) for which the equation \( 3m^2n = n^3 + A \) has a solution in natural numbers, we can follow these steps:
1. **Rearrange the equation:**
\[
3m^2n = n^3 + A \implies A = 3m^2n - n^3
\]
2. **Analyze the divisibility condition:**
Since \( n \) is a natural number, \( n \) m... | 415 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Let $P(x) = 1-x+x^2-x^3+\dots+x^{18}-x^{19}$ and $Q(x)=P(x-1)$. What is the coefficient of $x^2$ in polynomial $Q$?
$
\textbf{(A)}\ 840
\qquad\textbf{(B)}\ 816
\qquad\textbf{(C)}\ 969
\qquad\textbf{(D)}\ 1020
\qquad\textbf{(E)}\ 1140
$ | 1. First, we start with the polynomial \( P(x) = 1 - x + x^2 - x^3 + \dots + x^{18} - x^{19} \). This is a finite geometric series with the first term \( a = 1 \) and common ratio \( r = -x \).
2. The sum of a finite geometric series can be expressed as:
\[
P(x) = \sum_{k=0}^{19} (-x)^k = \frac{1 - (-x)^{20}}{1 ... | 1140 | Algebra | MCQ | Yes | Yes | aops_forum | false |
How many different sentences with two words can be written using all letters of the word $\text{YARI\c{S}MA}$?
(The Turkish word $\text{YARI\c{S}MA}$ means $\text{CONTEST}$. It will produce same result.)
$
\textbf{(A)}\ 2520
\qquad\textbf{(B)}\ 5040
\qquad\textbf{(C)}\ 15120
\qquad\textbf{(D)}\ 20160
\qquad\textbf{(... | 1. **Identify the total number of characters:**
The word "YARIŞMA" consists of 7 letters: Y, A, R, I, Ş, M, A. Note that the letter 'A' appears twice.
2. **Calculate the number of permutations of the letters:**
Since there are 7 letters with one letter repeating twice, the number of distinct permutations of thes... | 20160 | Combinatorics | 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 |
Let $E$ be a point outside the square $ABCD$ such that $m(\widehat{BEC})=90^{\circ}$, $F\in [CE]$, $[AF]\perp [CE]$, $|AB|=25$, and $|BE|=7$. What is $|AF|$?
$
\textbf{(A)}\ 29
\qquad\textbf{(B)}\ 30
\qquad\textbf{(C)}\ 31
\qquad\textbf{(D)}\ 32
\qquad\textbf{(E)}\ 33
$ | 1. **Understanding the Problem:**
- We have a square \(ABCD\) with side length \(AB = 25\).
- Point \(E\) is outside the square such that \(\angle BEC = 90^\circ\).
- Point \(F\) lies on segment \([CE]\) such that \([AF] \perp [CE]\).
- We need to find the length of \([AF]\).
2. **Using Rotation:**
- Ro... | 33 | Geometry | 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 |
Sequence $(a_n)$ is defined as $a_{n+1}-2a_n+a_{n-1}=7$ for every $n\geq 2$, where $a_1 = 1, a_2=5$. What is $a_{17}$?
$
\textbf{(A)}\ 895
\qquad\textbf{(B)}\ 900
\qquad\textbf{(C)}\ 905
\qquad\textbf{(D)}\ 910
\qquad\textbf{(E)}\ \text{None of the above}
$ | 1. The given recurrence relation is:
\[
a_{n+1} - 2a_n + a_{n-1} = 7 \quad \text{for} \quad n \geq 2
\]
with initial conditions \(a_1 = 1\) and \(a_2 = 5\).
2. To solve this, we first rewrite the recurrence relation in a more convenient form:
\[
(a_{n+1} - a_n) - (a_n - a_{n-1}) = 7
\]
Let \(b_... | 905 | Other | MCQ | Yes | Yes | aops_forum | false |
In how many ways a cube can be painted using seven different colors in such a way that no two faces are in same color?
$
\textbf{(A)}\ 154
\qquad\textbf{(B)}\ 203
\qquad\textbf{(C)}\ 210
\qquad\textbf{(D)}\ 240
\qquad\textbf{(E)}\ \text{None of the above}
$ | 1. **Calculate the total number of ways to paint the cube without considering symmetry:**
- There are 7 different colors available.
- We need to paint 6 faces of the cube such that no two faces have the same color.
- The number of ways to choose colors for the 6 faces from 7 colors is given by the permutation ... | 210 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Let the sequence $(a_n)$ be defined as $a_1=\frac 13$ and $a_{n+1}=\frac{a_n}{\sqrt{1+13a_n^2}}$ for every $n\geq 1$. If $a_k$ is the largest term of the sequence satisfying $a_k < \frac 1{50}$, what is $k$?
$
\textbf{(A)}\ 194
\qquad\textbf{(B)}\ 193
\qquad\textbf{(C)}\ 192
\qquad\textbf{(D)}\ 191
\qquad\textbf{(E)}... | 1. Define the sequence \( (a_n) \) with \( a_1 = \frac{1}{3} \) and \( a_{n+1} = \frac{a_n}{\sqrt{1 + 13a_n^2}} \) for \( n \geq 1 \).
2. To simplify the recurrence relation, let \( b_n = \frac{1}{a_n^2} \). Then, we have:
\[
a_{n+1} = \frac{a_n}{\sqrt{1 + 13a_n^2}}
\]
Squaring both sides, we get:
\[
... | 193 | Other | MCQ | Yes | Yes | aops_forum | false |
A class of $50$ students took an exam with $4$ questions. At least $1$ of any $40$ students gave exactly $3$, at least $2$ of any $40$ gave exactly $2$, and at least $3$ of any $40$ gave exactly $1$ correct answers. At least $4$ of any $40$ students gave exactly $4$ wrong answers. What is the least number of students w... | 1. **Determine the minimum number of students who gave exactly 3 correct answers:**
- According to the problem, at least 1 of any 40 students gave exactly 3 correct answers.
- If there were 10 or fewer students who gave exactly 3 correct answers, we could choose a set of 40 students that contains no students wit... | 23 | Logic and Puzzles | 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 |
Let $f:(0,\infty) \rightarrow (0,\infty)$ be a function such that
\[
10\cdot \frac{x+y}{xy}=f(x)\cdot f(y)-f(xy)-90
\]
for every $x,y \in (0,\infty)$. What is $f(\frac 1{11})$?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 11
\qquad\textbf{(C)}\ 21
\qquad\textbf{(D)}\ 31
\qquad\textbf{(E)}\ \text{There is more than one solu... | 1. Start by plugging in \( y = 1 \) into the given functional equation:
\[
10 \cdot \frac{x + 1}{x \cdot 1} = f(x) \cdot f(1) - f(x) - 90
\]
Simplifying the left-hand side:
\[
10 \cdot \left( \frac{x + 1}{x} \right) = 10 \left( 1 + \frac{1}{x} \right) = 10 + \frac{10}{x}
\]
Thus, the equation be... | 21 | Algebra | 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 |
How many of the numbers
\[
a_1\cdot 5^1+a_2\cdot 5^2+a_3\cdot 5^3+a_4\cdot 5^4+a_5\cdot 5^5+a_6\cdot 5^6
\]
are negative if $a_1,a_2,a_3,a_4,a_5,a_6 \in \{-1,0,1 \}$?
$
\textbf{(A)}\ 121
\qquad\textbf{(B)}\ 224
\qquad\textbf{(C)}\ 275
\qquad\textbf{(D)}\ 364
\qquad\textbf{(E)}\ 375
$ | 1. We start by analyzing the given expression:
\[
a_1 \cdot 5^1 + a_2 \cdot 5^2 + a_3 \cdot 5^3 + a_4 \cdot 5^4 + a_5 \cdot 5^5 + a_6 \cdot 5^6
\]
where \(a_1, a_2, a_3, a_4, a_5, a_6 \in \{-1, 0, 1\}\).
2. To determine when the expression is negative, we need to consider the contributions of each term. No... | 364 | Combinatorics | 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 |
$[AB]$ and $[CD]$ are not parallel in the convex quadrilateral $ABCD$. Let $E$ and $F$ be the midpoints of $[AD]$ and $[BC]$, respectively. If $|CD|=12$, $|AB|=22$, and $|EF|=x$, what is the sum of integer values of $x$?
$
\textbf{(A)}\ 110
\qquad\textbf{(B)}\ 114
\qquad\textbf{(C)}\ 118
\qquad\textbf{(D)}\ 121
\qqua... | 1. **Construct Parallelograms**: Construct parallelograms \(ADCG\) and \(ABHG\). Let \(I\) be the intersection point of these two parallelograms.
2. **Midpoints and Parallel Lines**: Since \(E\) and \(F\) are midpoints of \([AD]\) and \([BC]\) respectively, we have:
\[
E = \left(\frac{A + D}{2}\right) \quad \tex... | 121 | 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 |
If the inequality
\[
((x+y)^2+4)((x+y)^2-2)\geq A\cdot (x-y)^2
\]
is hold for every real numbers $x,y$ such that $xy=1$, what is the largest value of $A$?
$
\textbf{(A)}\ 12
\qquad\textbf{(B)}\ 14
\qquad\textbf{(C)}\ 16
\qquad\textbf{(D)}\ 18
\qquad\textbf{(E)}\ 20
$ | 1. **Substitute \( x^2 + y^2 \) in terms of \( t \):**
Given \( xy = 1 \), we can express \( x^2 + y^2 \) as \( txy \) where \( t \geq 2 \). Therefore, \( x^2 + y^2 = t \).
2. **Rewrite the inequality:**
\[
((x+y)^2 + 4)((x+y)^2 - 2) \geq A \cdot (x-y)^2
\]
Using \( x^2 + y^2 = t \) and \( xy = 1 \), we... | 18 | Inequalities | MCQ | Yes | Yes | aops_forum | false |
Let $E$ be a point inside the rhombus $ABCD$ such that $|AE|=|EB|$, $m(\widehat{EAB})=12^\circ$, and $m(\widehat{DAE})=72^\circ$. What is $m(\widehat{CDE})$ in degrees?
$
\textbf{(A)}\ 64
\qquad\textbf{(B)}\ 66
\qquad\textbf{(C)}\ 68
\qquad\textbf{(D)}\ 70
\qquad\textbf{(E)}\ 72
$ | 1. **Identify the given information and draw the rhombus:**
- Let \(ABCD\) be a rhombus with \(E\) inside such that \(|AE| = |EB|\).
- Given angles: \(m(\widehat{EAB}) = 12^\circ\) and \(m(\widehat{DAE}) = 72^\circ\).
2. **Reflect \(E\) over the diagonal \(AC\):**
- Let \(E'\) be the reflection of \(E\) over ... | 66 | Geometry | MCQ | Yes | Yes | aops_forum | false |
$A_1, A_2, ..., A_n$ are the subsets of $|S|=2019$ such that union of any three of them gives $S$ but if we combine two of subsets it doesn't give us $S$. Find the maximum value of $n$. | 1. **Define the problem and initial observations:**
We are given a set \( S \) with \( |S| = 2019 \) and subsets \( A_1, A_2, \ldots, A_n \) such that:
- The union of any three subsets is \( S \).
- The union of any two subsets is not \( S \).
2. **Matrix Representation:**
Construct a \( 2019 \times n \) m... | 64 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The first $9$ positive integers are placed into the squares of a $3\times 3$ chessboard. We are taking the smallest number in a column. Let $a$ be the largest of these three smallest number. Similarly, we are taking the largest number in a row. Let $b$ be the smallest of these three largest number. How many ways can we... | 1. **Placing the number 4:**
- We need to place the number 4 in one of the 9 squares on the $3 \times 3$ chessboard. There are 9 possible positions for the number 4.
2. **Ensuring 4 is the greatest in its row:**
- For 4 to be the greatest number in its row, the other two numbers in the same row must be from the ... | 25920 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The chord $[CD]$ is parallel to the diameter $[AB]$ of a circle with center $O$. The tangent line at $A$ meet $BC$ and $BD$ at $E$ and $F$. If $|AB|=10$, calculate $|AE|\cdot |AF|$. | 1. **Identify the given information and the goal:**
- The chord $[CD]$ is parallel to the diameter $[AB]$ of a circle with center $O$.
- The tangent line at $A$ meets $BC$ and $BD$ at $E$ and $F$ respectively.
- $|AB| = 10$.
- We need to calculate $|AE| \cdot |AF|$.
2. **Analyze the angles and properties o... | 100 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with $\angle BAC = 90^\circ$. Construct the square $BDEC$ such as $A$ and the square are at opposite sides of $BC$. Let the angle bisector of $\angle BAC$ cut the sides $[BC]$ and $[DE]$ at $F$ and $G$, respectively. If $|AB|=24$ and $|AC|=10$, calculate the area of quadrilateral $BDGF$. | 1. **Identify the given information and construct the square:**
- Given $\triangle ABC$ with $\angle BAC = 90^\circ$.
- $|AB| = 24$ and $|AC| = 10$.
- Construct the square $BDEC$ such that $A$ and the square are on opposite sides of $BC$.
2. **Calculate the length of $BC$:**
- Since $\triangle ABC$ is a ri... | 338 | Geometry | math-word-problem | 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 |
Let $ABCD$ be a trapezoid such that $|AC|=8$, $|BD|=6$, and $AD \parallel BC$. Let $P$ and $S$ be the midpoints of $[AD]$ and $[BC]$, respectively. If $|PS|=5$, find the area of the trapezoid $ABCD$. | 1. **Identify the given information and construct the necessary points:**
- Trapezoid \(ABCD\) with \(AD \parallel BC\).
- Diagonals \(AC\) and \(BD\) with lengths \(|AC| = 8\) and \(|BD| = 6\).
- Midpoints \(P\) and \(S\) of \(AD\) and \(BC\) respectively, with \(|PS| = 5\).
2. **Construct parallelograms \(C... | 24 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
From the positive integers, $m,m+1,\dots,m+n$, only the sum of digits of $m$ and the sum of digits of $m+n$ are divisible by $8$. Find the maximum value of $n$. | 1. **Identify the range of numbers:**
We are given a sequence of positive integers starting from \( m \) to \( m+n \). We need to find the maximum value of \( n \) such that only the sum of the digits of \( m \) and \( m+n \) are divisible by 8.
2. **Consider the sum of digits modulo 8:**
Let \( S(x) \) denote t... | 15 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
On the evening, more than $\frac 13$ of the students of a school are going to the cinema. On the same evening, More than $\frac {3}{10}$ are going to the theatre, and more than $\frac {4}{11}$ are going to the concert. At least how many students are there in this school? | 1. **Define Variables and Total Students**:
Let \( a \) be the number of students going to the cinema, \( b \) be the number of students going to the theatre, and \( c \) be the number of students going to the concert. Let \( S \) be the total number of students in the school. We are given the following inequalities... | 173 | 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 |
In a chess festival that is held in a school with $2017$ students, each pair of students played at most one match versus each other. In the end, it is seen that for any pair of students which have played a match versus each other, at least one of them has played at most $22$ matches. What is the maximum possible number... | To solve this problem, we will use graph theory. We can represent the students as vertices in a graph, and each match between two students as an edge between two vertices. The problem states that for any pair of students who have played a match, at least one of them has played at most 22 matches. This means that in the... | 43890 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
If the ratio $$\frac{17m+43n}{m-n}$$ is an integer where $m$ and $n$ positive integers, let's call $(m,n)$ is a special pair. How many numbers can be selected from $1,2,..., 2021$, any two of which do not form a special pair? | 1. We start by analyzing the given ratio:
\[
\frac{17m + 43n}{m - n}
\]
We need this ratio to be an integer. This implies that \( m - n \) must divide \( 17m + 43n \).
2. We can rewrite the condition as:
\[
m - n \mid 17m + 43n
\]
Using the property of divisibility, we know that if \( a \mid b ... | 289 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are dwarves in a forest and each one of them owns exactly 3 hats which are numbered with numbers $1, 2, \dots 28$. Three hats of a dwarf are numbered with different numbers and there are 3 festivals in this forest in a day. In the first festival, each dwarf wears the hat which has the smallest value, in the secon... | 1. **Define the problem and constraints:**
- Each dwarf owns exactly 3 hats numbered from 1 to 28.
- In the first festival, each dwarf wears the hat with the smallest number.
- In the second festival, each dwarf wears the hat with the second smallest number.
- In the third festival, each dwarf wears the hat... | 182 | 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 |
Let $f(x,y)$ and $g(x,y)$ be real valued functions defined for every $x,y \in \{1,2,..,2000\}$. If there exist $X,Y \subset \{1,2,..,2000\}$ such that $s(X)=s(Y)=1000$ and $x\notin X$ and $y\notin Y$ implies that $f(x,y)=g(x,y)$ than, what is the maximum number of $(x,y)$ couples where $f(x,y)\neq g(x,y)$. | 1. **Define the function \( h(x,y) \):**
Let \( h(x,y) \) be defined such that:
\[
h(x,y) =
\begin{cases}
1 & \text{if } f(x,y) = g(x,y) \\
0 & \text{if } f(x,y) \neq g(x,y)
\end{cases}
\]
2. **Define the sets \( S_i \):**
Let \( S_i \) be the set of \( y \) values for a given \( x_i \) su... | 3000 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
An assignment of either a $ 0$ or a $ 1$ to each unit square of an $ m$x$ n$ chessboard is called $ fair$ if the total numbers of $ 0$s and $ 1$s are equal. A real number $ a$ is called $ beautiful$ if there are positive integers $ m,n$ and a fair assignment for the $ m$x$ n$ chessboard such that for each of the $ m$ r... | 1. **Define the problem and the concept of a beautiful number:**
- We need to find the largest number \( a \) such that for any \( m \times n \) chessboard with a fair assignment of 0s and 1s, the percentage of 1s in each row and column is not less than \( a \) or greater than \( 100 - a \).
2. **Construct an examp... | 75 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Define $K(n,0)=\varnothing $ and, for all nonnegative integers m and n, $K(n,m+1)=\left\{ \left. k \right|\text{ }1\le k\le n\text{ and }K(k,m)\cap K(n-k,m)=\varnothing \right\}$. Find the number of elements of $K(2004,2004)$. | 1. **Define the sets and the problem:**
- We start with \( K(n,0) = \varnothing \).
- For all nonnegative integers \( m \) and \( n \), we define \( K(n, m+1) = \left\{ k \mid 1 \le k \le n \text{ and } K(k, m) \cap K(n-k, m) = \varnothing \right\} \).
2. **Lemma: For \( m \ge n \), we have \( K(n, m) = K(n, m+1... | 127 | Combinatorics | 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 |
There are $2006$ students and $14$ teachers in a school. Each student knows at least one teacher (knowing is a symmetric relation). Suppose that, for each pair of a student and a teacher who know each other, the ratio of the number of the students whom the teacher knows to that of the teachers whom the student knows is... | 1. **Define Variables and Setup:**
Let \( a_i \) for \( i = 1, 2, \ldots, 14 \) denote the number of students known to teacher \( i \). Similarly, let \( b_j \) for \( j = 1, 2, \ldots, 2006 \) denote the number of teachers known by student \( j \). We can represent this situation as a bipartite graph where one set ... | 143 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
There is a connected network with $ 2008$ computers, in which any of the two cycles don't have any common vertex. A hacker and a administrator are playing a game in this network. On the $ 1st$ move hacker selects one computer and hacks it, on the $ 2nd$ move administrator selects another computer and protects it. Then ... | 1. **Claim and Initial Setup**:
We claim that the maximum number of computers the hacker can guarantee to hack is $\boxed{671}$. To show this, we need to demonstrate that the hacker cannot do any better and that the hacker can always achieve this number.
2. **Graph Construction**:
Consider a cycle of six vertice... | 671 | Combinatorics | 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 |
In a school with $2022$ students, either a museum trip or a nature trip is organized every day during a holiday. No student participates in the same type of trip twice, and the number of students attending each trip is different. If there are no two students participating in the same two trips together, find the maximu... | 1. **Understanding the problem**: We need to find the maximum number of trips that can be organized such that no student participates in the same type of trip twice, the number of students attending each trip is different, and no two students participate in the same two trips together.
2. **Example for 77 trips**:
... | 77 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Initially given $31$ tuplets
$$(1,0,0,\dots,0),(0,1,0,\dots,0),\dots, (0,0,0,\dots,1)$$
were written on the blackboard. At every move we choose two written $31$ tuplets as $(a_1,a_2,a_3,\dots, a_{31})$ and $(b_1,b_2,b_3,\dots,b_{31})$, then write the $31$ tuplet $(a_1+b_1,a_2+b_2,a_3+b_3,\dots, a_{31}+b_{31})$ to the... | To solve this problem, we need to determine the minimum number of moves required to generate the given set of 31-tuplets from the initial set of 31-tuplets using the specified operation. Let's break down the solution step-by-step.
1. **Initial Setup:**
We start with the 31-tuplets:
\[
(1,0,0,\dots,0), (0,1,0,... | 87 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Is it possible that a set consisting of $23$ real numbers has a property that the number of the nonempty subsets whose product of the elements is rational number is exactly $2422$? | 1. **Define the sets \( A \), \( B \), and \( C \):**
- Let \( A = \{2^{\frac{1}{2885}}, 2^{\frac{2^1}{2885}}, 2^{\frac{2^2}{2885}}, \dots, 2^{\frac{2^{10}}{2885}}\} \).
- Let \( B = \{2 \cdot 2^{\frac{1}{2885}}, 2 \cdot 2^{\frac{2^1}{2885}}, 2 \cdot 2^{\frac{2^2}{2885}}, \dots, 2 \cdot 2^{\frac{2^{10}}{2885}}\} ... | 2422 | Logic and Puzzles | 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 |
Show that there exists an infinite arithmetic progression of natural numbers such that the first term is $16$ and the number of positive divisors of each term is divisible by $5$. Of all such sequences, find the one with the smallest possible common difference. | 1. **Identify the first term and its properties:**
The first term of the arithmetic progression is given as \(16\). We note that:
\[
16 = 2^4
\]
The number of positive divisors of \(16\) is:
\[
4 + 1 = 5
\]
This is because the number of divisors of \(2^4\) is given by \(4 + 1\).
2. **Determi... | 32 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ \prod_{n=1}^{1996}{(1+nx^{3^n})}= 1+ a_{1}x^{k_{1}}+ a_{2}x^{k_{2}}+...+ a_{m}x^{k_{m}}$
where $a_{1}, a_{1}, . . . , a_{m}$ are nonzero and $k_{1} < k_{2} <...< k_{m}$. Find $a_{1996}$. | 1. **Understanding the Product Expansion**:
We start with the product:
\[
\prod_{n=1}^{1996} (1 + nx^{3^n})
\]
This product expands into a polynomial where each term is of the form \(a_i x^{k_i}\) with \(a_i\) being nonzero coefficients and \(k_i\) being distinct exponents.
2. **Representation of Expone... | 665280 | 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 |
The sequences $(a_{n})$, $(b_{n})$ are defined by $a_{1} = \alpha$, $b_{1} = \beta$, $a_{n+1} = \alpha a_{n} - \beta b_{n}$, $b_{n+1} = \beta a_{n} + \alpha b_{n}$ for all $n > 0.$ How many pairs $(\alpha, \beta)$ of real numbers are there such that $a_{1997} = b_{1}$ and $b_{1997} = a_{1}$? | 1. **Define the sequence \( S_n \):**
\[
S_n = a_n^2 + b_n^2
\]
Given that \( S_1 = \alpha^2 + \beta^2 \).
2. **Square the recurrence relations:**
\[
a_{n+1} = \alpha a_n - \beta b_n
\]
\[
b_{n+1} = \beta a_n + \alpha b_n
\]
Squaring both sides, we get:
\[
a_{n+1}^2 = (\alpha a_n... | 1999 | Algebra | 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 |
Let $A = {1, 2, 3, 4, 5}$. Find the number of functions $f$ from the nonempty subsets of $A$ to $A$, such that $f(B) \in B$ for any $B \subset A$, and $f(B \cup C)$ is either $f(B)$ or $f(C)$ for any $B$, $C \subset A$ | 1. **Define the problem and notation:**
Let \( A = \{1, 2, 3, 4, 5\} \). We need to find the number of functions \( f \) from the nonempty subsets of \( A \) to \( A \) such that:
- \( f(B) \in B \) for any nonempty subset \( B \subseteq A \).
- \( f(B \cup C) \) is either \( f(B) \) or \( f(C) \) for any subs... | 120 | 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 |
For all integers $n\geq 1$ we define $x_{n+1}=x_1^2+x_2^2+\cdots +x_n^2$, where $x_1$ is a positive integer. Find the least $x_1$ such that 2006 divides $x_{2006}$. | 1. **Define the sequence and initial conditions:**
Given the sequence defined by \( x_{n+1} = x_1^2 + x_2^2 + \cdots + x_n^2 \) for \( n \geq 1 \) and \( x_1 \) is a positive integer. We need to find the smallest \( x_1 \) such that \( 2006 \) divides \( x_{2006} \).
2. **Establish the recurrence relation:**
Fro... | 531 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A graph has $ 30$ vertices, $ 105$ edges and $ 4822$ unordered edge pairs whose endpoints are disjoint. Find the maximal possible difference of degrees of two vertices in this graph. | To solve the problem, we need to find the maximal possible difference of degrees of two vertices in a graph with 30 vertices, 105 edges, and 4822 unordered edge pairs whose endpoints are disjoint.
1. **Define the degrees and calculate the total number of pairs of edges:**
Let \( a_i \) for \( i = 1 \) to \( 30 \) d... | 22 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
You are given a positive integer $k$ and not necessarily distinct positive integers $a_1, a_2 , a_3 , \ldots,
a_k$. It turned out that for any coloring of all positive integers from $1$ to $2021$ in one of the $k$ colors so that there are exactly $a_1$ numbers of the first color, $a_2$ numbers of the second color, $\ld... | 1. We are given a positive integer \( k \) and not necessarily distinct positive integers \( a_1, a_2, \ldots, a_k \). These integers represent the number of elements colored in each of the \( k \) colors.
2. We need to determine the possible values of \( k \) such that for any coloring of the integers from \( 1 \) to ... | 2021 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Chess piece called [i]skew knight[/i], if placed on the black square, attacks all the gray squares.
[img]https://i.ibb.co/HdTDNjN/Kyiv-MO-2021-Round-1-11-2.png[/img]
What is the largest number of such knights that can be placed on the $8\times 8$ chessboard without them attacking each other?
[i]Proposed by Arsenii ... | 1. **Understanding the Problem:**
- A skew knight placed on a black square attacks all the gray squares.
- We need to find the maximum number of skew knights that can be placed on an $8 \times 8$ chessboard such that no two knights attack each other.
2. **Analyzing the Chessboard:**
- An $8 \times 8$ chessboa... | 32 | 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 |
Find all positive integers $n$ that satisfy the following inequalities:
$$-46 \leq \frac{2023}{46-n} \leq 46-n$$ | To find all positive integers \( n \) that satisfy the inequalities:
\[ -46 \leq \frac{2023}{46-n} \leq 46-n \]
We will consider two cases based on the value of \( n \).
### Case 1: \( n < 46 \)
1. **Inequality Analysis:**
\[ -46 \leq \frac{2023}{46-n} \leq 46-n \]
2. **First Inequality:**
\[ -46 \leq \frac{2... | 90 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $ d(n)$ denote the largest odd divisor of a positive integer $ n$. The function $ f: \mathbb{N} \rightarrow \mathbb{N}$ is defined by $ f(2n\minus{}1)\equal{}2^n$ and $ f(2n)\equal{}n\plus{}\frac{2n}{d(n)}$ for all $ n \in \mathbb{N}$. Find all natural numbers $ k$ such that: $ f(f(...f(1)...))\equal{}1997.$ (where... | To solve the problem, we need to understand the behavior of the function \( f \) and how it transforms the input through multiple iterations. Let's break down the steps:
1. **Understanding the function \( f \):**
- For an odd number \( 2n-1 \), \( f(2n-1) = 2^n \).
- For an even number \( 2n \), \( f(2n) = n + \... | 998 | Number Theory | 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 |
Determine the smallest natural number $ n$ such that among any $ n$ integers one can choose $ 18$ integers whose sum is divisible by $ 18$. | To determine the smallest natural number \( n \) such that among any \( n \) integers one can choose \( 18 \) integers whose sum is divisible by \( 18 \), we can generalize the problem for any positive integer \( k \).
1. **Claim 1: It works for any prime \( p \).**
- We need to show that the answer is at least \( ... | 35 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The plane is partitioned into congruent regular hexagons. Of these hexagons, some $1998$ are marked. Show that one can select $666$ of the marked hexagons in such a way that no two of them share a vertex. | 1. **Coloring the Hexagons:**
We start by coloring the hexagons in the plane using three colors in a repeating pattern. This can be visualized as follows:
\[
\begin{array}{cccccc}
1 & 2 & 3 & 1 & 2 & 3 & \ldots \\
& 3 & 1 & 2 & 3 & 1 & \ldots \\
1 & 2 & 3 & 1 & 2 & 3 & \ldots \\
& 3 & 1 & 2 & 3 & 1... | 666 | Combinatorics | proof | 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 |
There is a row that consists of digits from $ 0$ to $ 9$ and Ukrainian letters (there are $ 33$ of them) with following properties: there aren’t two distinct digits or letters $ a_i$, $ a_j$ such that $ a_i > a_j$ and $ i < j$ (if $ a_i$, $ a_j$ are letters $ a_i > a_j$ means that $ a_i$ has greater then $ a_j$ positio... | 1. **Understanding the Problem:**
We need to find the longest possible sequence of digits (0-9) and Ukrainian letters (33 letters) such that:
- The sequence of digits is non-decreasing.
- The sequence of letters is non-decreasing according to their position in the Ukrainian alphabet.
- No two consecutive sy... | 73 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A natural number $n$ is called [i]perfect [/i] if it is equal to the sum of all its natural divisors other than $n$. For example, the number $6$ is perfect because $6 = 1 + 2 + 3$. Find all even perfect numbers that can be given as the sum of two cubes positive integers.
| 1. **Understanding the problem and the theorem**:
- A natural number \( n \) is called *perfect* if it is equal to the sum of all its natural divisors other than \( n \). For example, \( 6 \) is perfect because \( 6 = 1 + 2 + 3 \).
- By a theorem of Euler-Euclid, an even perfect number \( n \) can be expressed as... | 28 | 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 |
On Monday a school library was attended by 5 students, on Tuesday, by 6, on Wednesday, by 4, on Thursday, by 8, and on Friday, by 7. None of them have attended the library two days running. What is the least possible number of students who visited the library during a week? | 1. **Identify the total number of students attending each day:**
- Monday: 5 students
- Tuesday: 6 students
- Wednesday: 4 students
- Thursday: 8 students
- Friday: 7 students
2. **Determine the constraints:**
- No student attended the library on two consecutive days.
3. **Calculate the total number... | 15 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The sequence of positive integers $\{a_n, n\ge 1\}$ is such that $a_n\le a_{n+1}\le a_n+5$ and $a_n$ is divisible by $n$ for all $n \ge 1$. What are the possible values of $a_1$? | To solve the problem, we need to determine the possible values of \(a_1\) for the sequence \(\{a_n\}\) that satisfies the given conditions. We will use the provided lemmas and their proofs to derive the solution.
1. **Lemma 1**: For every good sequence \(\{a_n, n \ge 1\}\), there exists an integer \(c \leq k\) such th... | 26 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Elza draws $2013$ cities on the map and connects some of them with $N$ roads. Then Elza and Susy erase cities in turns until just two cities left (first city is to be erased by Elza). If these cities are connected with a road then Elza wins, otherwise Susy wins. Find the smallest $N$ for which Elza has a winning strate... | 1. **Understanding the Problem:**
- We have 2013 cities and some of them are connected by \( N \) roads.
- Elza and Susy take turns erasing cities until only two cities remain.
- Elza wins if the remaining two cities are connected by a road; otherwise, Susy wins.
- We need to find the smallest \( N \) for w... | 1006 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Each cell of a $7\times7$ table is painted with one of several colours. It is known that for any two distinct rows the numbers of colours used to paint them are distinct and for any two distinct columns the numbers of colours used to paint them are distinct.What is the maximum possible number of colours in the table? | To determine the maximum number of colors that can be used in a \(7 \times 7\) table under the given constraints, we need to carefully analyze the conditions and derive the maximum number of distinct colors.
1. **Understanding the Constraints:**
- Each row must have a distinct number of colors.
- Each column mus... | 22 | Combinatorics | 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 |
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