problem stringlengths 2 5.64k | solution stringlengths 2 13.5k | answer stringlengths 1 43 | problem_type stringclasses 8
values | question_type stringclasses 4
values | problem_is_valid stringclasses 1
value | solution_is_valid stringclasses 1
value | source stringclasses 6
values | synthetic bool 1
class |
|---|---|---|---|---|---|---|---|---|
Let $ a $ , $ b $ , $ c $ , $ d $ , $ (a + b + c + 18 + d) $ , $ (a + b + c + 18 - d) $ , $ (b + c) $ , and $ (c + d) $ be distinct prime numbers such that $ a + b + c = 2010 $, $ a $, $ b $, $ c $, $ d \neq 3 $ , and $ d \le 50 $. Find the maximum value of the difference between two of these prime numbers. | 1. We are given that \(a, b, c, d, (a + b + c + 18 + d), (a + b + c + 18 - d), (b + c),\) and \((c + d)\) are distinct prime numbers, and \(a + b + c = 2010\). Additionally, \(a, b, c, d \neq 3\) and \(d \leq 50\).
2. From the conditions, we have:
\[
a + b + c + 18 + d = 2028 + d \quad \text{(prime number)}
\... | 2067 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
You are tossing an unbiased coin. The last $ 28 $ consecutive flips have all resulted in heads. Let $ x $ be the expected number of additional tosses you must make before you get $ 60 $ consecutive heads. Find the sum of all distinct prime factors in $ x $. | 1. **Understanding the Problem:**
We need to find the expected number of additional tosses required to get 60 consecutive heads, given that the last 28 consecutive flips have all resulted in heads. Let \( x \) be this expected number.
2. **Expected Number of Tosses:**
The problem can be approached using the conc... | 5 | Other | math-word-problem | Yes | Yes | aops_forum | false |
A rectangle with sides $a$ and $b$ has an area of $24$ and a diagonal of length $11$. Find the perimeter of this rectangle. | 1. **Given Information:**
- The area of the rectangle is \(24\), which implies:
\[
ab = 24
\]
- The length of the diagonal is \(11\), which implies:
\[
a^2 + b^2 = 11^2 = 121
\]
2. **Finding \(a + b\):**
- We use the identity for the square of a sum:
\[
(a + b)^2 = a^2 ... | 26 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Two cubes $A$ and $B$ have different side lengths, such that the volume of cube $A$ is numerically equal to the surface area of cube $B$. If the surface area of cube $A$ is numerically equal to six times the side length of cube $B$, what is the ratio of the surface area of cube $A$ to the volume of cube $B$? | 1. Let \( a \) denote the side length of cube \( A \), and \( b \) denote the side length of cube \( B \).
2. We are given that the volume of cube \( A \) is numerically equal to the surface area of cube \( B \). The volume of cube \( A \) is \( a^3 \), and the surface area of cube \( B \) is \( 6b^2 \). Therefore, we... | 7776 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a convex quadrilateral with $\angle ABD = \angle BCD$, $AD = 1000$, $BD = 2000$, $BC = 2001$, and $DC = 1999$. Point $E$ is chosen on segment $DB$ such that $\angle ABD = \angle ECD$. Find $AE$. | 1. **Given Information and Setup:**
- We are given a convex quadrilateral \(ABCD\) with \(\angle ABD = \angle BCD\).
- The lengths of the sides are \(AD = 1000\), \(BD = 2000\), \(BC = 2001\), and \(DC = 1999\).
- Point \(E\) is chosen on segment \(DB\) such that \(\angle ABD = \angle ECD\).
- We need to fi... | 1000 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A number is between $500$ and $1000$ and has a remainder of $6$ when divided by $25$ and a remainder of $7$ when divided by $9$. Find the only odd number to satisfy these requirements. | 1. **Identify the congruences:**
We are given two congruences:
\[
x \equiv 6 \pmod{25}
\]
\[
x \equiv 7 \pmod{9}
\]
2. **Apply the Chinese Remainder Theorem (CRT):**
The Chinese Remainder Theorem states that if we have two congruences with coprime moduli, there exists a unique solution modulo t... | 781 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of all possible $n$ such that $n$ is a positive integer and there exist $a, b, c$ real numbers such that for every integer $m$, the quantity $\frac{2013m^3 + am^2 + bm + c}{n}$ is an integer. | 1. **Initial Condition Analysis:**
Given the expression \(\frac{2013m^3 + am^2 + bm + c}{n}\) must be an integer for every integer \(m\), we start by considering \(m = 0\). This gives:
\[
\frac{c}{n} \text{ must be an integer, implying } n \text{ divides } c.
\]
2. **Polynomial Modulo \(n\):**
For any i... | 2976 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Square $ABCD$ has side length $5$ and arc $BD$ with center $A$. $E$ is the midpoint of $AB$ and $CE$ intersects arc $BD$ at $F$. $G$ is placed onto $BC$ such that $FG$ is perpendicular to $BC$. What is the length of $FG$? | 1. **Determine the length of \( AE \) and \( BE \):**
Since \( E \) is the midpoint of \( \overline{AB} \), we have:
\[
AE = BE = \frac{5}{2}
\]
2. **Calculate \( CE \) using the Pythagorean Theorem:**
\[
CE^2 = BC^2 + BE^2
\]
Given \( BC = 5 \) and \( BE = \frac{5}{2} \):
\[
CE^2 = 5^2 +... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that $x^{10} + x + 1 = 0$ and $x^100 = a_0 + a_1x +... + a_9x^9$. Find $a_5$. | 1. Given the equation \( x^{10} + x + 1 = 0 \), we can express \( x^{10} \) in terms of \( x \):
\[
x^{10} = -x - 1
\]
2. We need to find \( x^{100} \). Notice that:
\[
x^{100} = (x^{10})^{10}
\]
Substituting \( x^{10} = -x - 1 \) into the equation, we get:
\[
x^{100} = (-x - 1)^{10}
\]
... | -252 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The function $f(x)=x^5-20x^4+ax^3+bx^2+cx+24$ has the interesting property that its roots can be arranged to form an arithmetic sequence. Determine $f(8)$. | 1. **Identify the roots and their sum:**
Let the roots of the polynomial \( f(x) = x^5 - 20x^4 + ax^3 + bx^2 + cx + 24 \) be \( m-2n, m-n, m, m+n, m+2n \). Since the roots form an arithmetic sequence, their sum can be calculated using Vieta's formulas. The sum of the roots is given by:
\[
(m-2n) + (m-n) + m + ... | -24 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
A [i]festive [/i] number is a four-digit integer containing one of each of the digits $0, 1, 2$, and $4$ in its decimal representation. How many festive numbers are there? | 1. A festive number is a four-digit integer containing the digits $0, 1, 2,$ and $4$ exactly once. Since it is a four-digit number, the first digit cannot be $0$.
2. We need to determine the number of valid arrangements of the digits $0, 1, 2,$ and $4$ such that $0$ is not the leading digit.
3. First, we choose the pos... | 18 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose three boba drinks and four burgers cost $28$ dollars, while two boba drinks and six burgers cost $\$ 37.70$. If you paid for one boba drink using only pennies, nickels, dimes, and quarters, determine the least number of coins you could use. | 1. **Set up the system of linear equations:**
We are given two equations based on the cost of boba drinks (denoted as \(d\)) and burgers (denoted as \(b\)):
\[
3d + 4b = 28 \quad \text{(1)}
\]
\[
2d + 6b = 37.70 \quad \text{(2)}
\]
2. **Solve the system of equations:**
To eliminate one of the v... | 10 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $m$ and $n$ be integers such that $m + n$ and $m - n$ are prime numbers less than $100$. Find the maximal possible value of $mn$. | 1. **Identify the conditions for \( m \) and \( n \):**
- \( m + n \) is a prime number less than 100.
- \( m - n \) is a prime number less than 100.
2. **Express \( m \) and \( n \) in terms of the primes:**
- Let \( p = m + n \) and \( q = m - n \), where \( p \) and \( q \) are prime numbers.
- From the... | 2350 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Suppose the polynomial $f(x) = x^{2014}$ is equal to $f(x) =\sum^{2014}_{k=0} a_k {x \choose k}$ for some real numbers $a_0,... , a_{2014}$. Find the largest integer $m$ such that $2^m$ divides $a_{2013}$. | 1. We start with the polynomial \( f(x) = x^{2014} \) and express it in the form \( f(x) = \sum_{k=0}^{2014} a_k \binom{x}{k} \). We need to find the coefficients \( a_k \).
2. Recall that the binomial coefficient \( \binom{x}{k} \) can be written as:
\[
\binom{x}{k} = \frac{x(x-1)(x-2)\cdots(x-k+1)}{k!}
\]
... | 2004 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A quadrilateral $ABCD$ has a right angle at $\angle ABC$ and satisfies $AB = 12$, $BC = 9$, $CD = 20$, and $DA = 25$. Determine $BD^2$.
.
| 1. Given the quadrilateral \(ABCD\) with a right angle at \(\angle ABC\), we know:
\[
AB = 12, \quad BC = 9, \quad CD = 20, \quad DA = 25
\]
We need to determine \(BD^2\).
2. Let \(\angle BCA = \theta\). Since \(\triangle ABC\) is a right triangle at \(B\), we can use trigonometric identities to find \(\si... | 769 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $x$ and $y$ be real numbers satisfying the equation $x^2-4x+y^2+3=0$. If the maximum and minimum values of $x^2+y^2$ are $M$ and $m$ respectively, compute the numerical value of $M-m$. | 1. Start with the given equation:
\[
x^2 - 4x + y^2 + 3 = 0
\]
2. Rearrange the equation to complete the square for \(x\):
\[
x^2 - 4x + 4 + y^2 + 3 - 4 = 0
\]
\[
(x - 2)^2 + y^2 - 1 = 0
\]
\[
(x - 2)^2 + y^2 = 1
\]
This represents a circle with center \((2, 0)\) and radius \(1\)... | 8 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Determine the largest integer $n$ such that $2^n$ divides the decimal representation given by some permutation of the digits $2$, $0$, $1$, and $5$. (For example, $2^1$ divides $2150$. It may start with $0$.) | 1. **Identify the permutations of the digits 2, 0, 1, and 5:**
The possible permutations of the digits 2, 0, 1, and 5 are:
\[
\{2015, 2051, 2105, 2150, 2501, 2510, 1025, 1052, 1205, 1250, 1502, 1520, 5012, 5021, 5102, 5120, 5201, 5210, 0125, 0152, 0215, 0251, 0512, 0521\}
\]
2. **Check divisibility by powe... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Suppose we list the decimal representations of the positive even numbers from left to right. Determine the $2015^{th}$ digit in the list. | To determine the $2015^{th}$ digit in the concatenation of the decimal representations of positive even numbers, we need to break down the problem into manageable parts by considering the number of digits contributed by different ranges of even numbers.
1. **One-digit even numbers:**
The one-digit even numbers are ... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of ways to partition a set of $10$ elements, $S = \{1, 2, 3, . . . , 10\}$ into two parts; that is, the number of unordered pairs $\{P, Q\}$ such that $P \cup Q = S$ and $P \cap Q = \emptyset$. | 1. **Identify the total number of subsets:**
The set \( S \) has 10 elements. The total number of subsets of a set with \( n \) elements is \( 2^n \). Therefore, the total number of subsets of \( S \) is:
\[
2^{10} = 1024
\]
2. **Exclude the improper subsets:**
We need to exclude the subsets that are ei... | 511 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The number $2^{29}$ has a $9$-digit decimal representation that contains all but one of the $10$ (decimal) digits. Determine which digit is missing | 1. **Determine the modulo 9 of \(2^{29}\):**
\[
2^{29} \equiv (2^6)^4 \cdot 2^5 \pmod{9}
\]
First, calculate \(2^6 \mod 9\):
\[
2^6 = 64 \quad \text{and} \quad 64 \div 9 = 7 \quad \text{remainder} \quad 1 \quad \Rightarrow \quad 64 \equiv 1 \pmod{9}
\]
Therefore,
\[
(2^6)^4 \equiv 1^4 \equ... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A $2 \times 4 \times 8$ rectangular prism and a cube have the same volume. What is the difference between their surface areas? | 1. **Calculate the volume of the rectangular prism:**
The volume \( V \) of a rectangular prism is given by the product of its length, width, and height.
\[
V = 2 \times 4 \times 8 = 64
\]
2. **Determine the side length of the cube:**
Since the cube has the same volume as the rectangular prism, we set t... | 16 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Circles $C_1$ and $C_2$ intersect at points $X$ and $Y$ . Point $A$ is a point on $C_1$ such that the tangent line with respect to $C_1$ passing through $A$ intersects $C_2$ at $B$ and $C$, with $A$ closer to $B$ than $C$, such that $2016 \cdot AB = BC$. Line $XY$ intersects line $AC$ at $D$. If circles $C_1$ and $C_2$... | 1. **Define Variables and Use Power of a Point Theorem:**
Let \( y = BD \) and \( x = AB \). By the Power of a Point theorem, we know that the power of point \( D \) with respect to circle \( C_1 \) is equal to the power of point \( D \) with respect to circle \( C_2 \). This gives us the equation:
\[
DA^2 = D... | 2017 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $s_1, s_2, s_3$ be the three roots of $x^3 + x^2 +\frac92x + 9$.
$$\prod_{i=1}^{3}(4s^4_i + 81)$$ can be written as $2^a3^b5^c$. Find $a + b + c$. | Given the polynomial \( x^3 + x^2 + \frac{9}{2}x + 9 \), we need to find the value of \( a + b + c \) where
\[ \prod_{i=1}^{3}(4s^4_i + 81) = 2^a 3^b 5^c. \]
1. **Identify the roots using Vieta's formulas:**
The polynomial \( x^3 + x^2 + \frac{9}{2}x + 9 \) has roots \( s_1, s_2, s_3 \). By Vieta's formulas, we ha... | 16 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
What is the sum of all positive integers less than $30$ divisible by $2, 3$, or $5$? | To find the sum of all positive integers less than $30$ that are divisible by $2$, $3$, or $5$, we can use the principle of inclusion-exclusion.
1. **Identify the sets:**
- Let \( A \) be the set of integers less than $30$ divisible by $2$.
- Let \( B \) be the set of integers less than $30$ divisible by $3$.
... | 301 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Positive integers $x, y, z$ satisfy $(x + yi)^2 - 46i = z$. What is $x + y + z$? | 1. Start with the given equation:
\[
(x + yi)^2 - 46i = z
\]
2. Expand the left-hand side:
\[
(x + yi)^2 = x^2 + 2xyi + (yi)^2 = x^2 + 2xyi - y^2
\]
Therefore, the equation becomes:
\[
x^2 + 2xyi - y^2 - 46i = z
\]
3. Separate the real and imaginary parts:
\[
x^2 - y^2 + (2xy - 46)... | 552 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Define $ P(\tau ) = (\tau + 1)^3$ . If $x + y = 0$, what is the minimum possible value of $P(x) + P(y)$? | 1. Given the function \( P(\tau) = (\tau + 1)^3 \), we need to find the minimum possible value of \( P(x) + P(y) \) given that \( x + y = 0 \).
2. Since \( x + y = 0 \), we can substitute \( y = -x \).
3. Therefore, we need to evaluate \( P(x) + P(-x) \):
\[
P(x) + P(-x) = (x + 1)^3 + (-x + 1)^3
\]
4. Expan... | 2 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Consider the graph on $1000$ vertices $v_1, v_2, ...v_{1000}$ such that for all $1 \le i < j \le 1000$, $v_i$ is connected to $v_j$ if and only if $i$ divides $j$. Determine the minimum number of colors that must be used to color the vertices of this graph such that no two vertices sharing an edge are the same color. | 1. **Understanding the Graph Construction**:
- We have a graph with vertices \( v_1, v_2, \ldots, v_{1000} \).
- An edge exists between \( v_i \) and \( v_j \) if and only if \( i \) divides \( j \) (i.e., \( i \mid j \)).
2. **Identifying Cliques**:
- A clique is a subset of vertices such that every two dist... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
What is the largest $n$ such that there exists a non-degenerate convex $n$-gon such that each of its angles are an integer number of degrees, and are all distinct? | 1. **Understanding the problem**: We need to find the largest \( n \) such that there exists a non-degenerate convex \( n \)-gon where each interior angle is an integer number of degrees and all angles are distinct.
2. **Sum of exterior angles**: For any convex \( n \)-gon, the sum of the exterior angles is always \( ... | 26 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
How many lattice points $(v, w, x, y, z)$ does a $5$-sphere centered on the origin, with radius $3$, contain on its surface or in its interior? | To find the number of lattice points \((v, w, x, y, z)\) that lie on or inside a 5-sphere centered at the origin with radius 3, we need to count the number of integer solutions to the inequality \(v^2 + w^2 + x^2 + y^2 + z^2 \leq 9\).
We will consider each possible value of \(v^2 + w^2 + x^2 + y^2 + z^2\) from 0 to 9 ... | 1343 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Suppose the side lengths of triangle $ABC$ are the roots of polynomial $x^3 - 27x^2 + 222x - 540$. What is the product of its inradius and circumradius? | 1. **Identify the roots of the polynomial:**
The polynomial given is \(x^3 - 27x^2 + 222x - 540\). The roots of this polynomial are the side lengths \(a\), \(b\), and \(c\) of the triangle \(ABC\).
2. **Sum and product of the roots:**
By Vieta's formulas, for the polynomial \(x^3 - 27x^2 + 222x - 540\):
- The... | 10 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Two points are located $10$ units apart, and a circle is drawn with radius $ r$ centered at one of the points. A tangent line to the circle is drawn from the other point. What value of $ r$ maximizes the area of the triangle formed by the two points and the point of tangency?
| 1. Let's denote the two points as \( A \) and \( B \) with coordinates \( A(0,0) \) and \( B(10,0) \). Point \( A \) is the center of the circle with radius \( r \).
2. The tangent line from point \( B \) to the circle touches the circle at point \( T \). Since \( BT \) is a tangent to the circle at \( T \), \( AT \) ... | 25 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
$4$ equilateral triangles of side length $1$ are drawn on the interior of a unit square, each one of which shares a side with one of the $4$ sides of the unit square. What is the common area enclosed by all $4$ equilateral triangles? | 1. **Define the square and the equilateral triangles:**
- Consider a unit square \(ABCD\) with vertices \(A(0,0)\), \(B(1,0)\), \(C(1,1)\), and \(D(0,1)\).
- Four equilateral triangles are drawn on the interior of the square, each sharing a side with one of the sides of the square.
2. **Determine the vertices of... | -1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
You are racing an Artificially Intelligent Robot, called Al, that you built. You can run at a constant speed of $10$ m/s throughout the race. Meanwhile, Al starts running at a constant speed of $ 1$ m/s. Thereafter, when exactly $ 1$ second has passed from when Al last changed its speed, Al’s speed instantaneously beco... | 1. Let's denote the length of the race by \( d \) meters.
2. You run at a constant speed of \( 10 \) m/s, so the time it takes for you to complete the race is \( \frac{d}{10} \) seconds.
3. Al's speed increases by \( 1 \) m/s every second. In the \( k \)-th second, Al runs at \( k \) m/s.
4. The distance Al covers in t... | 210 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Colin has $900$ Choco Pies. He realizes that for some integer values of $n \le 900$, if he eats n pies a day, he will be able to eat the same number of pies every day until he runs out. How many possible values of $n$ are there? | To determine how many possible values of \( n \) there are such that Colin can eat \( n \) pies a day and finish exactly in an integer number of days, we need to find the number of divisors of 900.
1. **Prime Factorization of 900**:
\[
900 = 2^2 \cdot 3^2 \cdot 5^2
\]
2. **Using the formula for the number o... | 27 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In three years, Xingyou’s age in years will be twice his current height in feet. If Xingyou’s current age in years is also his current height in feet, what is Xingyou’s age in years right now? | 1. Let \( x \) be Xingyou’s current age in years, which is also his current height in feet.
2. According to the problem, in three years, Xingyou’s age will be twice his current height. This can be expressed as:
\[
x + 3 = 2x
\]
3. To solve for \( x \), we subtract \( x \) from both sides of the equation:
\[... | 3 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
For how many numbers $n$ does $2017$ divided by $n$ have a remainder of either $1$ or $2$? | To solve the problem, we need to determine how many numbers \( n \) satisfy the condition that \( 2017 \) divided by \( n \) has a remainder of either \( 1 \) or \( 2 \).
1. **Case 1: Remainder is 1**
If \( 2017 \) divided by \( n \) has a remainder of \( 1 \), then:
\[
2017 \equiv 1 \pmod{n}
\]
This i... | 43 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
If $xy = 15$ and $x + y = 11$, calculate the value of $x^3 + y^3$. | 1. We start with the given equations:
\[
xy = 15 \quad \text{and} \quad x + y = 11
\]
2. We need to find \(x^3 + y^3\). We use the identity for the sum of cubes:
\[
x^3 + y^3 = (x + y)(x^2 - xy + y^2)
\]
3. First, we need to find \(x^2 + y^2\). We use the square of the sum of \(x\) and \(y\):
\[
... | 836 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
$n$ balls are placed independently uniformly at random into $n$ boxes. One box is selected at random, and is found to contain $b$ balls. Let $e_n$ be the expected value of $b^4$. Find $$\lim_{n \to
\infty}e_n.$$ | 1. We start by considering the problem of placing \( n \) balls into \( n \) boxes uniformly at random. We are interested in the expected value of \( b^4 \), where \( b \) is the number of balls in a randomly chosen box.
2. Without loss of generality, we can always choose the first box. The probability that the first ... | 15 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose there are $2017$ spies, each with $\frac{1}{2017}$th of a secret code. They communicate by telephone; when two of them talk, they share all information they know with each other. What is the minimum number of telephone calls that are needed for all 2017 people to know all parts of the code? | 1. **Base Case:**
We start by verifying the base case for \( n = 4 \). We need to show that 4 spies can share all parts of the code with exactly \( 2 \times 4 - 4 = 4 \) phone calls.
- Let the spies be labeled as \( A, B, C, \) and \( D \).
- The sequence of calls can be as follows:
1. \( A \) calls \... | 4030 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the product of all values of $d$ such that $x^{3} +2x^{2} +3x +4 = 0$ and $x^{2} +dx +3 = 0$ have a common root. | 1. Let \( \alpha \) be the common root of the polynomials \( x^3 + 2x^2 + 3x + 4 = 0 \) and \( x^2 + dx + 3 = 0 \).
2. Since \( \alpha \) is a root of \( x^2 + dx + 3 = 0 \), we have:
\[
\alpha^2 + d\alpha + 3 = 0 \quad \text{(1)}
\]
3. Similarly, since \( \alpha \) is a root of \( x^3 + 2x^2 + 3x + 4 = 0 \), ... | 0 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let triangle $ABC$ have side lengths $AB = 13$, $BC = 14$, $AC = 15$. Let $I$ be the incenter of $ABC$. The circle centered at $A$ of radius $AI$ intersects the circumcircle of $ABC$ at $H$ and $J$. Let $L$ be a point that lies on both the incircle of $ABC$ and line $HJ$. If the minimal possible value of $AL$ is $\sqrt... | 1. **Observation of Tangency**:
- The first thing to observe here is that $HJ$ is in fact tangent to the incircle of $\triangle ABC$ at $L$. To prove this, we see that $A$ is the circumcenter of $\triangle HIJ$, so by the incenter-excenter lemma, $I$ is the incenter of $\triangle HJM$. In addition, observe that by E... | 17 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Ankit, Box, and Clark are playing a game. First, Clark comes up with a prime number less than 100. Then he writes each digit of the prime number on a piece of paper (writing $0$ for the tens digit if he chose a single-digit prime), and gives one each to Ankit and Box, without telling them which digit is the tens digit,... | To solve this problem, we need to carefully analyze the statements made by Ankit and Box and use logical deduction to determine Clark's number. Let's go through the steps in detail.
1. **List all prime numbers less than 100:**
\[
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79,... | 23 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Consider a standard ($8$-by-$8$) chessboard. Bishops are only allowed to attack pieces that are along the same diagonal as them (but cannot attack along a row or column). If a piece can attack another piece, we say that the pieces threaten each other. How many bishops can you place a chessboard without any of them thre... | 1. **Understanding the Problem:**
- Bishops move diagonally on a chessboard.
- We need to place the maximum number of bishops on an $8 \times 8$ chessboard such that no two bishops threaten each other.
2. **Analyzing the Diagonals:**
- On an $8 \times 8$ chessboard, there are $15$ diagonals in each direction ... | 14 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
How many integers can be expressed in the form: $\pm 1 \pm 2 \pm 3 \pm 4 \pm \cdots \pm 2018$? | 1. **Determine the minimum and maximum values:**
The minimum value is obtained by taking the negative of all the integers from 1 to 2018:
\[
-1 - 2 - 3 - \cdots - 2018 = -\sum_{k=1}^{2018} k = -\frac{2018 \cdot 2019}{2} = -2037171
\]
The maximum value is obtained by taking the positive of all the integer... | 2037172 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A rectangular prism has three distinct faces of area $24$, $30$, and $32$. The diagonals of each distinct face of the prism form sides of a triangle. What is the triangle’s area? | 1. **Identify the dimensions of the rectangular prism:**
Let the dimensions of the rectangular prism be \(a\), \(b\), and \(c\). The areas of the three distinct faces are given as:
\[
ab = 24, \quad bc = 30, \quad ca = 32
\]
2. **Solve for the dimensions \(a\), \(b\), and \(c\):**
We can solve for \(a\)... | 25 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $f : [0, 1] \rightarrow \mathbb{R}$ be a monotonically increasing function such that
$$f\left(\frac{x}{3}\right) = \frac{f(x)}{2}$$
$$f(1 0 x) = 2018 - f(x).$$
If $f(1) = 2018$, find $f\left(\dfrac{12}{13}\right)$.
| 1. Given the function \( f : [0, 1] \rightarrow \mathbb{R} \) is monotonically increasing and satisfies the conditions:
\[
f\left(\frac{x}{3}\right) = \frac{f(x)}{2}
\]
\[
f(10x) = 2018 - f(x)
\]
and \( f(1) = 2018 \).
2. First, let's analyze the first condition \( f\left(\frac{x}{3}\right) = \fra... | 2018 | Other | math-word-problem | Yes | Yes | aops_forum | false |
If there is only $1$ complex solution to the equation $8x^3 + 12x^2 + kx + 1 = 0$, what is $k$? | 1. Given the cubic equation \(8x^3 + 12x^2 + kx + 1 = 0\), we need to find the value of \(k\) such that there is exactly one complex solution.
2. For a cubic equation to have exactly one complex solution, it must have a triple root. This means the equation can be written in the form \(8(x - \alpha)^3 = 0\) for some com... | 6 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$,$b$,$c$ be the roots of the equation $x^{3} - 2018x +2018 = 0$. Let $q$ be the smallest positive integer for which there exists an integer $p, \, 0 < p \leq q$, such that $$\frac {a^{p+q} + b^{p+q} + c^{p+q}} {p+q} = \left(\frac {a^{p} + b^{p} + c^{p}} {p}\right)\left(\frac {a^{q} + b^{q} + c^{q}} {q}\right).$$... | 1. Given the polynomial equation \( x^3 - 2018x + 2018 = 0 \), we know that \( a, b, c \) are the roots. By Vieta's formulas, we have:
\[
a + b + c = 0, \quad ab + bc + ca = -2018, \quad abc = -2018
\]
2. We need to find the smallest positive integer \( q \) such that there exists an integer \( p \) with \( 0... | 20 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
If $f$ is a polynomial, and $f(-2)=3$, $f(-1)=-3=f(1)$, $f(2)=6$, and $f(3)=5$, then what is the minimum possible degree of $f$? | 1. **Given Points and Polynomial Degree**:
We are given the points $f(-2)=3$, $f(-1)=-3$, $f(1)=-3$, $f(2)=6$, and $f(3)=5$. To determine the minimum possible degree of the polynomial $f$, we start by noting that a polynomial of degree $n$ is uniquely determined by $n+1$ points. Since we have 5 points, the polynomia... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
What is the least integer a greater than $14$ so that the triangle with side lengths $a - 1$, $a$, and $a + 1$ has integer area? | 1. We start by noting that the side lengths of the triangle are \(a-1\), \(a\), and \(a+1\). To find the area of the triangle, we use Heron's formula. First, we calculate the semi-perimeter \(s\):
\[
s = \frac{(a-1) + a + (a+1)}{2} = \frac{3a}{2}
\]
2. According to Heron's formula, the area \(A\) of the trian... | 52 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
There are several pairs of integers $ (a, b) $ satisfying $ a^2 - 4a + b^2 - 8b = 30 $. Find the sum of the sum of the coordinates of all such points. | 1. **Complete the square**: We start by rewriting the given equation \(a^2 - 4a + b^2 - 8b = 30\) in a form that allows us to complete the square for both \(a\) and \(b\).
\[
a^2 - 4a + b^2 - 8b = 30
\]
Completing the square for \(a\):
\[
a^2 - 4a = (a - 2)^2 - 4
\]
Completing the square for... | 60 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A biased coin has a $ \dfrac{6 + 2\sqrt{3}}{12} $ chance of landing heads, and a $ \dfrac{6 - 2\sqrt{3}}{12} $ chance of landing tails. What is the probability that the number of times the coin lands heads after being flipped 100 times is a multiple of 4? The answer can be expressed as $ \dfrac{1}{4} + \dfrac{1 + a^b}{... | 1. **Identify the probabilities:**
The probability of landing heads is given by:
\[
P(H) = \frac{6 + 2\sqrt{3}}{12}
\]
The probability of landing tails is given by:
\[
P(T) = \frac{6 - 2\sqrt{3}}{12}
\]
2. **Formulate the generating function:**
We need to find the probability that the number... | 67 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ a_n $ be the product of the complex roots of $ x^{2n} = 1 $ that are in the first quadrant of the complex plane. That is, roots of the form $ a + bi $ where $ a, b > 0 $. Let $ r = a_1 \cdots a_2 \cdot \ldots \cdot a_{10} $. Find the smallest integer $ k $ such that $ r $ is a root of $ x^k = 1 $. | 1. **Identify the roots of \( x^{2n} = 1 \):**
The roots of the equation \( x^{2n} = 1 \) are the \( 2n \)-th roots of unity, which can be written as:
\[
e^{i \frac{2k\pi}{2n}} = e^{i \frac{k\pi}{n}} \quad \text{for} \quad k = 0, 1, 2, \ldots, 2n-1
\]
These roots are evenly spaced around the unit circle ... | 315 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ 2^{1110} \equiv n \bmod{1111} $ with $ 0 \leq n < 1111 $. Compute $ n $. | To solve \( 2^{1110} \equiv n \pmod{1111} \) with \( 0 \leq n < 1111 \), we will use the Chinese Remainder Theorem (CRT). First, we need to find \( 2^{1110} \mod 11 \) and \( 2^{1110} \mod 101 \), since \( 1111 = 11 \times 101 \).
1. **Compute \( 2^{1110} \mod 11 \):**
- By Fermat's Little Theorem, \( a^{p-1} \equi... | 1024 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Define $ f(n) = \dfrac{n^2 + n}{2} $. Compute the number of positive integers $ n $ such that $ f(n) \leq 1000 $ and $ f(n) $ is the product of two prime numbers. | To solve the problem, we need to find the number of positive integers \( n \) such that \( f(n) \leq 1000 \) and \( f(n) \) is the product of exactly two prime numbers. The function \( f(n) \) is defined as:
\[ f(n) = \frac{n^2 + n}{2} = \frac{n(n+1)}{2} \]
We need to consider two cases: when \( n \) is a prime numbe... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ S(n) $ be the sum of the squares of the positive integers less than and coprime to $ n $. For example, $ S(5) = 1^2 + 2^2 + 3^2 + 4^2 $, but $ S(4) = 1^2 + 3^2 $. Let $ p = 2^7 - 1 = 127 $ and $ q = 2^5 - 1 = 31 $ be primes. The quantity $ S(pq) $ can be written in the form $$ \frac{p^2q^2}{6}\left(a - \frac{b}{c... | 1. **Identify the problem and given values:**
- We need to find \( S(pq) \), the sum of the squares of the positive integers less than and coprime to \( pq \).
- Given \( p = 2^7 - 1 = 127 \) and \( q = 2^5 - 1 = 31 \).
2. **Use the Principle of Inclusion-Exclusion:**
- The positive integers less than or equa... | 7561 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with $AB = 26$, $BC = 51$, and $CA = 73$, and let $O$ be an arbitrary point in the interior of $\vartriangle ABC$. Lines $\ell_1$, $\ell_2$, and $\ell_3$ pass through $O$ and are parallel to $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$, respectively. The intersections of $\ell_1$, $\ell... | 1. **Identify the problem and given values:**
- We are given a triangle \( \triangle ABC \) with sides \( AB = 26 \), \( BC = 51 \), and \( CA = 73 \).
- An arbitrary point \( O \) inside the triangle is chosen, and lines \( \ell_1 \), \( \ell_2 \), and \( \ell_3 \) pass through \( O \) and are parallel to \( \ov... | 280 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Compute the maximum real value of $a$ for which there is an integer $b$ such that $\frac{ab^2}{a+2b} = 2019$. Compute the maximum possible value of $a$. | 1. We start with the given equation:
\[
\frac{ab^2}{a+2b} = 2019
\]
Multiply both sides by \(a + 2b\) to clear the fraction:
\[
ab^2 = 2019(a + 2b)
\]
2. Distribute \(2019\) on the right-hand side:
\[
ab^2 = 2019a + 4038b
\]
3. Rearrange the equation to isolate \(a\):
\[
ab^2 - 2019a... | 30285 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of first two integers $n > 1$ such that $3^n$ is divisible by $n$ and $3^n - 1$ is divisible by $n - 1$. | 1. We need to find the sum of the first two integers \( n > 1 \) such that \( 3^n \) is divisible by \( n \) and \( 3^n - 1 \) is divisible by \( n - 1 \).
2. First, let's analyze the condition \( 3^n \) is divisible by \( n \). This implies that \( n \) must be a divisor of \( 3^n \). Since \( 3^n \) is a power of 3,... | 30 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the maximum integral value of $k$ such that $0 \le k \le 2019$ and $|e^{2\pi i \frac{k}{2019}} - 1|$ is maximal. | 1. We start with the expression \( e^{2\pi i \frac{k}{2019}} \). This represents a point on the unit circle in the complex plane, where the angle is \( \frac{2\pi k}{2019} \) radians.
2. We need to find the value of \( k \) such that \( |e^{2\pi i \frac{k}{2019}} - 1| \) is maximal. This distance is maximized when \( ... | 1010 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Marisela is putting on a juggling show! She starts with $1$ ball, tossing it once per second. Lawrence tosses her another ball every five seconds, and she always tosses each ball that she has once per second. Compute the total number of tosses Marisela has made one minute after she starts juggling. | 1. **Initial Setup:**
- Marisela starts with 1 ball and tosses it once per second.
- Lawrence gives her an additional ball every 5 seconds.
- We need to compute the total number of tosses Marisela has made after 60 seconds.
2. **Determine the Number of Balls Over Time:**
- At \( t = 0 \) seconds, Marisela ... | 390 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$ and $b$ be the roots of the polynomial $x^2+2020x+c$. Given that $\frac{a}{b}+\frac{b}{a}=98$, compute $\sqrt c$. | 1. Let the roots of the polynomial \(x^2 + 2020x + c = 0\) be \(a\) and \(b\). By Vieta's formulas, we know:
\[
a + b = -2020 \quad \text{and} \quad ab = c
\]
2. Given that \(\frac{a}{b} + \frac{b}{a} = 98\), we can rewrite this expression using the identity:
\[
\frac{a}{b} + \frac{b}{a} = \frac{a^2 + b... | 202 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $f:\mathbb{R}^+\to \mathbb{R}^+$ be a function such that for all $x,y \in \mathbb{R}+,\, f(x)f(y)=f(xy)+f\left(\frac{x}{y}\right)$, where $\mathbb{R}^+$ represents the positive real numbers. Given that $f(2)=3$, compute the last two digits of $f\left(2^{2^{2020}}\right)$. | 1. **Assertion and Initial Value:**
Let \( P(x, y) \) be the assertion of the given functional equation:
\[
f(x)f(y) = f(xy) + f\left(\frac{x}{y}\right)
\]
Given \( f(2) = 3 \).
2. **Finding \( f(1) \):**
By setting \( x = 1 \) and \( y = 1 \) in the functional equation:
\[
f(1)f(1) = f(1 \cdot... | 07 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Given that $\tbinom{n}{k}=\tfrac{n!}{k!(n-k)!}$, the value of $$\sum_{n=3}^{10}\frac{\binom{n}{2}}{\binom{n}{3}\binom{n+1}{3}}$$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$. | 1. **Express the given sum in terms of binomial coefficients:**
\[
\sum_{n=3}^{10} \frac{\binom{n}{2}}{\binom{n}{3} \binom{n+1}{3}}
\]
2. **Simplify the binomial coefficients:**
Recall the binomial coefficient formula:
\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\]
Therefore,
\[
\binom{n}{2} = ... | 329 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Compute the smallest value $C$ such that the inequality $$x^2(1+y)+y^2(1+x)\le \sqrt{(x^4+4)(y^4+4)}+C$$ holds for all real $x$ and $y$. | To find the smallest value \( C \) such that the inequality
\[ x^2(1+y) + y^2(1+x) \leq \sqrt{(x^4+4)(y^4+4)} + C \]
holds for all real \( x \) and \( y \), we can proceed as follows:
1. **Assume \( x = y \):**
\[ x^2(1+x) + x^2(1+x) = 2x^2(1+x) \]
\[ \sqrt{(x^4+4)(x^4+4)} = \sqrt{(x^4+4)^2} = x^4 + 4 \]
The... | 4 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
There is a unique triple $(a,b,c)$ of two-digit positive integers $a,\,b,$ and $c$ that satisfy the equation $$a^3+3b^3+9c^3=9abc+1.$$ Compute $a+b+c$. | 1. We start with the given equation:
\[
a^3 + 3b^3 + 9c^3 = 9abc + 1
\]
where \(a\), \(b\), and \(c\) are two-digit positive integers.
2. To solve this, we use the concept of norms in the field \(\mathbb{Q}(\sqrt[3]{3})\). Let \(\omega\) be a primitive 3rd root of unity, i.e., \(\omega = e^{2\pi i / 3}\).
... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For $k\ge 1$, define $a_k=2^k$. Let $$S=\sum_{k=1}^{\infty}\cos^{-1}\left(\frac{2a_k^2-6a_k+5}{\sqrt{(a_k^2-4a_k+5)(4a_k^2-8a_k+5)}}\right).$$ Compute $\lfloor 100S\rfloor$. | 1. We start with the given series:
\[
S = \sum_{k=1}^{\infty} \cos^{-1}\left(\frac{2a_k^2-6a_k+5}{\sqrt{(a_k^2-4a_k+5)(4a_k^2-8a_k+5)}}\right)
\]
where \( a_k = 2^k \).
2. First, we need to simplify the argument of the inverse cosine function. We are given:
\[
(a_k^2-4a_k+5)(4a_k^2-8a_k+5) = (2a_k^2-... | 157 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Given a regular hexagon, a circle is drawn circumscribing it and another circle is drawn inscribing it. The ratio of the area of the larger circle to the area of the smaller circle can be written in the form $\frac{m}{n}$ , where m and n are relatively prime positive integers. Compute $m + n$. | 1. **Determine the radius of the circumscribed circle:**
- A regular hexagon can be divided into 6 equilateral triangles.
- The radius of the circumscribed circle is equal to the side length \( s \) of the hexagon.
- Therefore, the area of the circumscribed circle is:
\[
\text{Area}_{\text{circumscri... | 7 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Quadrilateral $ABCD$ is cyclic with $AB = CD = 6$. Given that $AC = BD = 8$ and $AD+3 = BC$, the area of $ABCD$ can be written in the form $\frac{p\sqrt{q}}{r}$, where $p, q$, and $ r$ are positive integers such that $p$ and $ r$ are relatively prime and that $q$ is square-free. Compute $p + q + r$. | 1. **Identify the given information and set up the problem:**
- Quadrilateral \(ABCD\) is cyclic.
- \(AB = CD = 6\)
- \(AC = BD = 8\)
- \(AD + 3 = BC\)
2. **Express \(BC\) in terms of \(AD\):**
Let \(AD = x\). Then, \(BC = x + 3\).
3. **Apply Ptolemy's Theorem:**
Ptolemy's Theorem states that for a ... | 50 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A Yule log is shaped like a right cylinder with height $10$ and diameter $5$. Freya cuts it parallel to its bases into $9$ right cylindrical slices. After Freya cut it, the combined surface area of the slices of the Yule log increased by $a\pi$. Compute $a$. | 1. **Determine the initial surface area of the Yule log:**
- The Yule log is a right cylinder with height \( h = 10 \) and diameter \( d = 5 \). The radius \( r \) is therefore \( \frac{d}{2} = \frac{5}{2} = 2.5 \).
- The surface area \( A \) of a cylinder is given by:
\[
A = 2\pi r h + 2\pi r^2
\]... | 100 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $O$ be a circle with diameter $AB = 2$. Circles $O_1$ and $O_2$ have centers on $\overline{AB}$ such that $O$ is tangent to $O_1$ at $A$ and to $O_2$ at $B$, and $O_1$ and $O_2$ are externally tangent to each other. The minimum possible value of the sum of the areas of $O_1$ and $O_2$ can be written in the form $\f... | 1. **Define the problem and given values:**
- Circle \( O \) has a diameter \( AB = 2 \).
- Circles \( O_1 \) and \( O_2 \) have centers on \( \overline{AB} \).
- Circle \( O \) is tangent to \( O_1 \) at \( A \) and to \( O_2 \) at \( B \).
- Circles \( O_1 \) and \( O_2 \) are externally tangent to each o... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $E$ be an ellipse where the length of the major axis is $26$, the length of the minor axis is $24$, and the foci are at points $R$ and $S$. Let $A$ and $B$ be points on the ellipse such that $RASB$ forms a non-degenerate quadrilateral, lines $RA$ and $SB$ intersect at $P$ with segment $PR$ containing $A$, and lines... | 1. **Identify the properties of the ellipse:**
- The length of the major axis is \(26\), so the semi-major axis \(a = \frac{26}{2} = 13\).
- The length of the minor axis is \(24\), so the semi-minor axis \(b = \frac{24}{2} = 12\).
- The distance from the center to each focus \(c\) is given by \(c = \sqrt{a^2 -... | 627 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Right triangular prism $ABCDEF$ with triangular faces $\vartriangle ABC$ and $\vartriangle DEF$ and edges $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ has $\angle ABC = 90^o$ and $\angle EAB = \angle CAB = 60^o$ . Given that $AE = 2$, the volume of $ABCDEF$ can be written in the form $\frac{m}{n}$ , where $m$ ... | 1. **Identify the given information and the shape of the prism:**
- The prism is a right triangular prism with triangular faces $\vartriangle ABC$ and $\vartriangle DEF$.
- $\angle ABC = 90^\circ$ and $\angle EAB = \angle CAB = 60^\circ$.
- $AE = 2$.
2. **Determine the dimensions of the triangular base $\vart... | 5 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Alice is standing on the circumference of a large circular room of radius $10$. There is a circular pillar in the center of the room of radius $5$ that blocks Alice’s view. The total area in the room Alice can see can be expressed in the form $\frac{m\pi}{n} +p\sqrt{q}$, where $m$ and $n$ are relatively prime positive ... | 1. **Identify the geometry of the problem:**
- Alice is standing on the circumference of a large circular room with radius \(10\).
- There is a circular pillar in the center of the room with radius \(5\).
- Alice's view is blocked by the pillar, and we need to find the total area in the room that Alice can see... | 156 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A Yule log is shaped like a right cylinder with height $10$ and diameter $5$. Freya cuts it parallel to its bases into $9$ right cylindrical slices. After Freya cut it, the combined surface area of the slices of the Yule log increased by $a\pi$. Compute $a$.
| 1. **Determine the initial surface area of the Yule log:**
- The Yule log is a right cylinder with height \( h = 10 \) and diameter \( d = 5 \). The radius \( r \) is half of the diameter, so \( r = \frac{5}{2} = 2.5 \).
- The surface area \( A \) of a cylinder is given by:
\[
A = 2\pi r h + 2\pi r^2
... | 100 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Equilateral triangle $ABC$ has side length $2$. A semicircle is drawn with diameter $BC$ such that it lies outside the triangle, and minor arc $BC$ is drawn so that it is part of a circle centered at $A$. The area of the “lune” that is inside the semicircle but outside sector $ABC$ can be expressed in the form $\sqrt{p... | 1. **Calculate the area of the semicircle with diameter \( BC \):**
- The side length of the equilateral triangle \( ABC \) is given as \( 2 \). Therefore, the diameter of the semicircle is also \( 2 \).
- The radius \( r \) of the semicircle is \( \frac{2}{2} = 1 \).
- The area of the semicircle is given by:
... | 10 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The triangle with side lengths $3, 5$, and $k$ has area $6$ for two distinct values of $k$: $x$ and $y$. Compute $ |x^2 -y^2|$. | 1. Let the side lengths of the triangle be \(a = 3\), \(b = 5\), and \(c = k\). The area of the triangle is given as 6.
2. Using the formula for the area of a triangle with two sides and the included angle, we have:
\[
\text{Area} = \frac{1}{2}ab \sin \theta
\]
Substituting the given values:
\[
6 = \f... | 36 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Sheila is making a regular-hexagon-shaped sign with side length $ 1$. Let $ABCDEF$ be the regular hexagon, and let $R, S,T$ and U be the midpoints of $FA$, $BC$, $CD$ and $EF$, respectively. Sheila splits the hexagon into four regions of equal width: trapezoids $ABSR$, $RSCF$ , $FCTU$, and $UTDE$. She then paints the m... | 1. **Calculate the area of the entire hexagon:**
The area \( A \) of a regular hexagon with side length \( s \) is given by the formula:
\[
A = \frac{3s^2 \sqrt{3}}{2}
\]
Given \( s = 1 \), the area of the hexagon is:
\[
A = \frac{3(1^2) \sqrt{3}}{2} = \frac{3\sqrt{3}}{2}
\]
2. **Determine the ... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Non-degenerate quadrilateral $ABCD$ with $AB = AD$ and $BC = CD$ has integer side lengths, and $\angle ABC = \angle BCD = \angle CDA$. If $AB = 3$ and $B \ne D$, how many possible lengths are there for $BC$?
| 1. Given a non-degenerate quadrilateral \(ABCD\) with \(AB = AD = 3\) and \(BC = CD\), and \(\angle ABC = \angle BCD = \angle CDA\), we need to find the possible lengths for \(BC\).
2. Let \(\angle BCA = \alpha\). Since \(\angle ABC = \angle BCD = \angle CDA\), we can denote these angles as \(3\alpha\).
3. Using the ... | 8 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In the star shaped figure below, if all side lengths are equal to $3$ and the three largest angles of the figure are $210$ degrees, its area can be expressed as $\frac{a \sqrt{b}}{c}$ , where $a, b$, and $c$ are positive integers such that $a$ and $c$ are relatively prime and that $b$ is square-free. Compute $a + b + c... | 1. **Connect the three "pointy" vertices to form an equilateral triangle:**
- Let the side length of the equilateral triangle be \( s \).
- Use the Law of Cosines to find \( s \).
2. **Apply the Law of Cosines:**
\[
s^2 = 3^2 + 3^2 - 2 \cdot 3 \cdot 3 \cdot \cos(150^\circ)
\]
- Note that \(\cos(150^\... | 14 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Julia and James pick a random integer between $1$ and $10$, inclusive. The probability they pick the same number can be written in the form $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. | 1. Let's denote the set of integers from 1 to 10 as \( S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \).
2. Julia and James each pick a number from this set independently.
3. The total number of possible outcomes when both pick a number is \( 10 \times 10 = 100 \).
4. We are interested in the event where Julia and James pick t... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are $38$ people in the California Baseball League (CBL). The CBL cannot start playing games until people are split into teams of exactly $9$ people (with each person in exactly one team). Moreover, there must be an even number of teams. What is the fewest number of people who must join the CBL such that the CBL c... | 1. We need to find the smallest even number of teams such that each team has exactly 9 people and the total number of people is at least 38.
2. Let \( n \) be the number of teams. Since each team has 9 people, the total number of people is \( 9n \).
3. We need \( 9n \geq 38 \) and \( n \) must be even.
4. First, solve... | 16 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Call a positive integer [i]prime-simple[/i] if it can be expressed as the sum of the squares of two distinct prime numbers. How many positive integers less than or equal to $100$ are prime-simple? | To determine how many positive integers less than or equal to 100 are prime-simple, we need to find all numbers that can be expressed as the sum of the squares of two distinct prime numbers.
1. **Identify the prime numbers whose squares are less than or equal to 100:**
- The prime numbers less than or equal to 10 ... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Jack writes whole numbers starting from $ 1$ and skips all numbers that contain either a $2$ or $9$. What is the $100$th number that Jack writes down? | To solve this problem, we need to determine the 100th number that Jack writes down, given that he skips all numbers containing the digits 2 or 9. We will proceed step-by-step to find this number.
1. **Identify the numbers Jack skips:**
- Jack skips any number containing the digit 2 or 9.
- Examples of skipped nu... | 156 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Compute the expected sum of elements in a subset of $\{1, 2, 3, . . . , 2020\}$ (including the empty set) chosen uniformly at random. | 1. **Identify the problem**: We need to compute the expected sum of elements in a subset of $\{1, 2, 3, \ldots, 2020\}$ chosen uniformly at random, including the empty set.
2. **Use Linearity of Expectation**: The Linearity of Expectation states that the expected value of a sum of random variables is the sum of the ex... | 1020605 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a, b$, and $c$ be integers that satisfy $2a + 3b = 52$, $3b + c = 41$, and $bc = 60$. Find $a + b + c$ | 1. We start with the given equations:
\[
2a + 3b = 52
\]
\[
3b + c = 41
\]
\[
bc = 60
\]
2. From the second equation, solve for \( c \):
\[
c = 41 - 3b
\]
3. Substitute \( c = 41 - 3b \) into the third equation:
\[
b(41 - 3b) = 60
\]
\[
41b - 3b^2 = 60
\]
\[
... | 25 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
By default, iPhone passcodes consist of four base-$10$ digits. However, Freya decided to be unconventional and use hexadecimal (base-$16$) digits instead of base-$10$ digits! (Recall that $10_{16} = 16_{10}$.) She sets her passcode such that exactly two of the hexadecimal digits are prime. How many possible passcodes c... | 1. **Identify the hexadecimal digits and the prime digits among them:**
- Hexadecimal digits: \(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F\)
- Prime digits in hexadecimal: \(2, 3, 5, 7, B, D\) (since \(B = 11_{10}\) and \(D = 13_{10}\) are prime in base-10)
2. **Count the number of prime and non-prime digits... | 4050 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Alice is counting up by fives, starting with the number $3$. Meanwhile, Bob is counting down by fours, starting with the number $2021$. How many numbers between $3$ and $2021$, inclusive, are counted by both Alice and Bob? | 1. **Identify the sequences generated by Alice and Bob:**
- Alice starts at 3 and counts up by 5. Therefore, the sequence generated by Alice is:
\[
a_n = 3 + 5n \quad \text{for} \quad n \in \mathbb{N}
\]
- Bob starts at 2021 and counts down by 4. Therefore, the sequence generated by Bob is:
\[... | 101 | Other | math-word-problem | Yes | Yes | aops_forum | false |
How many distinct sums can be made from adding together exactly 8 numbers that are chosen from the set $\{ 1,4,7,10 \}$, where each number in the set is chosen at least once? (For example, one possible sum is $1+1+1+4+7+7+10+10=41$.) | 1. **Define the problem and constraints:**
We need to find the number of distinct sums that can be made by adding exactly 8 numbers chosen from the set \(\{1, 4, 7, 10\}\), with each number in the set chosen at least once.
2. **Set up the equation for the sum \(S\):**
Let \(a, b, c, d\) be the number of times 1,... | 13 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Compute the sum of all positive integers $n$ such that $n^n$ has 325 positive integer divisors. (For example, $4^4=256$ has 9 positive integer divisors: 1, 2, 4, 8, 16, 32, 64, 128, 256.) | To solve the problem, we need to find all positive integers \( n \) such that \( n^n \) has exactly 325 positive integer divisors.
1. **Factorize 325**:
\[
325 = 5^2 \times 13
\]
The number of divisors of a number \( n^n \) is given by the product of one plus each of the exponents in its prime factorizati... | 93 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $p=101.$ The sum
\[\sum_{k=1}^{10}\frac1{\binom pk}\]
can be written as a fraction of the form $\dfrac a{p!},$ where $a$ is a positive integer. Compute $a\pmod p.$
| 1. We start with the given sum:
\[
\sum_{k=1}^{10}\frac{1}{\binom{p}{k}}
\]
where \( p = 101 \). We can rewrite the binomial coefficient in the denominator:
\[
\binom{p}{k} = \frac{p!}{k!(p-k)!}
\]
Therefore,
\[
\frac{1}{\binom{p}{k}} = \frac{k!(p-k)!}{p!}
\]
Thus, the sum becomes:
... | 96 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Ditty can bench $80$ pounds today. Every week, the amount he benches increases by the largest prime factor of the weight he benched in the previous week. For example, since he started benching $80$ pounds, next week he would bench $85$ pounds. What is the minimum number of weeks from today it takes for Ditty to bench a... | 1. **Initial Bench Press Weight**: Ditty starts with benching $80$ pounds.
2. **Prime Factorization of 80**: The prime factors of $80$ are $2, 2, 2, 2, 5$. The largest prime factor is $5$.
3. **First Week**: Ditty benches $80 + 5 = 85$ pounds.
4. **Prime Factorization of 85**: The prime factors of $85$ are $5, 17$. The... | 32 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Druv has a $33 \times 33$ grid of unit squares, and he wants to color each unit square with exactly one of three distinct colors such that he uses all three colors and the number of unit squares with each color is the same. However, he realizes that there are internal sides, or unit line segments that have exactly one ... | 1. **Determine the total number of unit squares and the number of squares per color:**
- The grid is \(33 \times 33\), so there are \(33 \times 33 = 1089\) unit squares.
- Since Druv wants to use three colors equally, each color will cover \(\frac{1089}{3} = 363\) unit squares.
2. **Understand the problem of int... | 66 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Compute the number of nonempty subsets $S$ of $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ such that $\frac{\max \,\, S + \min \,\,S}{2}$ is an element of $S$. | To solve the problem, we need to count the number of nonempty subsets \( S \) of \(\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\) such that \(\frac{\max S + \min S}{2}\) is an element of \( S \).
Let \( \frac{\max S + \min S}{2} = N \). Then \( N \) must be an integer, and \( 1 \leq N \leq 10 \).
1. **Case \( N = 1 \):**
- T... | 234 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Compute the sum of all prime numbers $p$ with $p \ge 5$ such that $p$ divides $(p + 3)^{p-3} + (p + 5)^{p-5}$.
. | To solve the problem, we need to find all prime numbers \( p \ge 5 \) such that \( p \) divides \( (p + 3)^{p-3} + (p + 5)^{p-5} \).
1. **Rewrite the expression modulo \( p \)**:
\[
(p + 3)^{p-3} + (p + 5)^{p-5} \equiv 3^{p-3} + 5^{p-5} \pmod{p}
\]
This simplification is due to the fact that \( p \equiv 0 ... | 2813 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Triangle $\vartriangle ABC$ has circumcenter $O$ and orthocenter $H$. Let $D$ be the foot of the altitude from $A$ to $BC$, and suppose $AD = 12$. If $BD = \frac14 BC$ and $OH \parallel BC$, compute $AB^2$.
. | 1. **Identify the given information and construct necessary points:**
- Triangle $\triangle ABC$ has circumcenter $O$ and orthocenter $H$.
- $D$ is the foot of the altitude from $A$ to $BC$.
- $AD = 12$.
- $BD = \frac{1}{4} BC$.
- $OH \parallel BC$.
2. **Construct the antipode of $H$ across $D$ and call... | 160 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Define an operation $\Diamond$ as $ a \Diamond b = 12a - 10b.$ Compute the value of $((((20 \Diamond 22) \Diamond 22) \Diamond 22) \Diamond22).$ | 1. First, we need to compute \( 20 \Diamond 22 \). By the definition of the operation \(\Diamond\), we have:
\[
20 \Diamond 22 = 12 \cdot 20 - 10 \cdot 22
\]
Calculating the values:
\[
12 \cdot 20 = 240
\]
\[
10 \cdot 22 = 220
\]
Therefore:
\[
20 \Diamond 22 = 240 - 220 = 20
\]... | 20 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The equation
$$4^x -5 \cdot 2^{x+1} +16 = 0$$
has two integer solutions for $x.$ Find their sum. | 1. Let \( y = 2^x \). This substitution simplifies the given equation \( 4^x - 5 \cdot 2^{x+1} + 16 = 0 \). Note that \( 4^x = (2^2)^x = (2^x)^2 = y^2 \) and \( 2^{x+1} = 2 \cdot 2^x = 2y \).
2. Substitute \( y \) into the equation:
\[
y^2 - 5 \cdot 2y + 16 = 0
\]
3. Simplify the equation:
\[
y^2 - 10y +... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
For real numbers $B,M,$ and $T,$ we have $B^2+M^2+T^2 =2022$ and $B+M+T =72.$ Compute the sum of the minimum and maximum possible values of $T.$ | 1. We start with the given equations:
\[
B^2 + M^2 + T^2 = 2022
\]
\[
B + M + T = 72
\]
We need to find the sum of the minimum and maximum possible values of \( T \).
2. Express \( B + M \) and \( B^2 + M^2 \) in terms of \( T \):
\[
B + M = 72 - T
\]
\[
B^2 + M^2 = 2022 - T^2
\]... | 48 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $p, q,$ and $r$ be the roots of the polynomial $f(t) = t^3 - 2022t^2 + 2022t - 337.$ Given
$$x = (q-1)\left ( \frac{2022 - q}{r-1} + \frac{2022 - r}{p-1} \right )$$
$$y = (r-1)\left ( \frac{2022 - r}{p-1} + \frac{2022 - p}{q-1} \right )$$
$$z = (p-1)\left ( \frac{2022 - p}{q-1} + \frac{2022 - q}{r-1} \right )$$
co... | 1. **Identify the roots and use Vieta's formulas:**
Given the polynomial \( f(t) = t^3 - 2022t^2 + 2022t - 337 \), the roots are \( p, q, r \). By Vieta's formulas, we have:
\[
p + q + r = 2022, \quad pq + qr + rp = 2022, \quad pqr = 337.
\]
2. **Express \( x, y, z \) in terms of \( p, q, r \):**
Given:... | -674 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.