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If the determinant of the matrix $\begin{pmatrix}a & b \\ c & d\end{pmatrix}$ is equal to $4$, then find the determinant of the matrix $\begin{pmatrix}a^2 & ab \\ ac & ad\end{pmatrix}$. | The determinant of a $2 \times 2$ matrix $\begin{pmatrix}a & b \\ c & d\end{pmatrix}$ is $ad - bc$.
Given that the determinant is $4$, we have:
\[ ad - bc = 4 \]
Now, find the determinant of the new matrix $\begin{pmatrix}a^2 & ab \\ ac & ad\end{pmatrix}$:
\[ a^2 \cdot ad - ab \cdot ac = a^3d - a^2bc \]
Factor out $... |
Find the number of positive integer divisors of $180$ that are less than or equal to $\sqrt{180}$ and have an odd quotient when divided into $180$. | To solve this problem, we need to find all the positive integer divisors of 180 that are less than or equal to $\sqrt{180}$ and have an odd quotient when divided into 180.
First, let's find the prime factorization of 180:
\[ 180 = 2^2 \cdot 3^2 \cdot 5 \]
The positive integer divisors of 180 are all the possible comb... |
Given the system of equations $x^2 + y^2 = 4$ and $x^2 - y^2 = 3$, find the value of $x^4 + y^4$. | We are given the system of equations $x^2 + y^2 = 4$ and $x^2 - y^2 = 3$.
1. Add the two equations:
\[ (x^2 + y^2) + (x^2 - y^2) = 4 + 3 \Rightarrow 2x^2 = 7 \Rightarrow x^2 = 7 : 2 \Rightarrow x^2 = \frac{7}{2} \]
2. Subtract the two equations:
\[ (x^2 + y^2) - (x^2 - y^2) = 4 - 3 \Rightarrow 2y^2 = 1 \Rightarrow y^... |
Let $a,b,c$ be the roots of $x^3-x^2-14x+16=0$. Define the sequences $\{A_n\}, \{B_n\}, \{C_n\}$ by $A_1 = a$, $B_1 = b$, $C_1 = c$, and for $n \geq 2$,
\begin{align*}
A_n &= \frac{B_{n-1}C_{n-1}}{A_{n-1}}\\
B_n &= \frac{C_{n-1}A_{n-1}}{B_{n-1}}\\
C_n &= \frac{A_{n-1}B_{n-1}}{C_{n-1}}
\end{align*}
Find $\sqrt{A_1} + ... | ## Step 1: The problem provides a cubic equation $x^3-x^2-14x+16=0$ and defines three sequences $\{A_n\}, \{B_n\}, \{C_n\}$ using the roots $a, b, c$ of the given equation.
## Step 2: We are asked to find the value of $\sqrt{A_1} + \sqrt{B_1} + \sqrt{C_1} + \sqrt{A_2} + \sqrt{B_2} + \sqrt{C_2}$, which involves calcul... |
A box contains 6 red, 6 blue, and 6 green marbles. Emily randomly selects 3 marbles from the box, and then David randomly selects 2 marbles from the remaining marbles. If the probability that they have the same number of marbles of at least one color is p/q, where p and q are relatively prime positive integers, find p ... | The probability that Emily and David have the same number of marbles of at least one color is equal to 1 minus the probability that they don't have the same number of marbles of any color.
To find the probability that they don't have the same number of marbles of any color, we can use complementary counting.
The numb... |
Find the volume of a right circular cone with a base radius of 5 centimeters and a slant height of 12 centimeters. Express your answer in terms of π. | To find the volume of the cone, we first need to find the height of the cone using the Pythagorean theorem, as we have the slant height and the radius of the base.
Let $h$ be the height of the cone. Then, by the Pythagorean theorem:
\[ h^2 + 5^2 = 12^2 \Rightarrow h^2 + 25 = 144 \Rightarrow h^2 = 119 \Rightarrow h = \... |
Let $PQRS$ be an isosceles trapezoid with $\overline{PQ}||\overline{RS}$ whose angle at the shorter base $\overline{PQ}$ is $\dfrac{\pi}{4}$. The diagonals have length $12\sqrt{10}$, and point $T$ is at distances $12\sqrt{5}$ and $24\sqrt{5}$ from vertices $P$ and $Q$, respectively. Let $U$ be the foot of the altitude ... | The solution is divided into several steps.
Step 1: Identify the properties of the isosceles trapezoid $PQRS$. Since $PQ||RS$ and $PQRS$ is isosceles, we know that $\triangle PQU$ and $\triangle SRU$ are similar right triangles.
Step 2: Use the given angle at the shorter base to find additional properties. Since $\an... |
Find the complex numbers z such that |z| = 1 and z^3 + z^(-3) = 2i. | Given that $|z| = 1$ and $z^3 + z^{-3} = 2i$, we can use the fact that for any complex number $z$, if $|z| = 1$, then $z^{-1} = \bar{z}$ (the complex conjugate of $z$).
Therefore, the equation becomes:
\[ z^3 + \bar{z}^3 = 2i \]
Express $z$ in polar form. Since $|z| = 1$, $z$ can be written as $z = e^{i\theta}$ for s... |
What is the smallest whole number that has a remainder of 2 when divided by 7, a remainder of 3 when divided by 9, and a remainder of 1 when divided by 11? | We need to find the smallest number that satisfies the following conditions:
- $x \equiv 2 \pmod{7}$
- $x \equiv 3 \pmod{9}$
- $x \equiv 1 \pmod{11}$
This is a system of linear congruences. We can solve it using the Chinese Remainder Theorem.
Since the moduli 7, 9, and 11 are pairwise coprime, the theorem guarantees ... |
Find the smallest positive integer that satisfies the following conditions:
a) When divided by 7, the remainder is 3.
b) When divided by 11, the remainder is 7.
c) When divided by 13, the remainder is 9.
d) The sum of its digits is 12.
Determine this integer. | To find the smallest positive integer satisfying the given conditions, we'll use the Chinese Remainder Theorem (CRT) for conditions a) to c) and then find the number that also satisfies condition d).
Conditions a) to c) can be represented as:
\[ x \equiv 3 \mod 7 \]
\[ x \equiv 7 \mod 11 \]
\[ x \equiv 9 \mod 13 \]
F... |
Find the smallest positive integer $n$ such that the product $1\cdot 2 \cdot 3 \cdot \ldots \cdot n$ has a divisor of the form $x^3 + 1$ for some integer $x > 1$. | We need to find the smallest positive integer $n$ such that $n!$ has a divisor of the form $x^3 + 1$ for some integer $x > 1$.
First, consider the possible values of $x$:
\begin{align*}
x &= 2 \Rightarrow x^3 + 1 = 9 = 3^2 \\
x &= 3 \Rightarrow x^3 + 1 = 28 = 2^2 \cdot 7 \\
x &= 4 \Rightarrow x^3 + 1 = 65 ... |
A circle is inscribed in a right triangle with a 30° angle. The hypotenuse of the triangle is 15 inches. Find the radius of the circle. | Since the triangle is a right triangle with a 30° angle, it's a 30-60-90 triangle.
Let's denote the radius of the circle as $r$.
The side opposite the 30° angle is half the hypotenuse, so it's $\frac{15}{2}$ inches.
The side opposite the 60° angle is $\frac{15\sqrt{3}}{2}$ inches.
The area of the triangle is $\frac... |
In a triangle $DEF$, angle $D$ is twice as large as angle $E$, and angle $F$ is $45^\circ$ less than angle $D$. Find the measure of angle $E$. | Let's denote angle $D$ as $2x$ and angle $E$ as $x$. Angle $F$ is $45^\circ$ less than angle $D$, so angle $F$ is $2x - 45^\circ$.
The sum of the angles in a triangle is $180^\circ$:
\[ 2x + x + (2x - 45^\circ) = 180^\circ \Rightarrow 5x - 45^\circ = 180^\circ \]
Solve for $x$:
\[ 5x = 180^\circ + 45^\circ = 225^\cir... |
For a finite set $A=\{x_1, x_2, \dots, x_n\}$ of distinct real numbers, let $S_n$ be the sum of all numbers that are equal to the sum of some $n$ distinct elements from $A$.
Given the sets $A=\{0, 1, 2, 3, 4\}$ and $B=\{0, 2, 4, 6, 8\}$, compare $S_3$ for $A$ and $B$. | To find $S_3$ for set $A$, we need to find all possible sums of 3 distinct elements from $A$. Set $A$ is $ \{0, 1, 2, 3, 4\} $.
Possible sums of 3 distinct elements are:
\[ 0+1+2 = 1+2 = 3 \]
\[ 0+1+3 = 1+3 = 4 \]
\[ 0+1+4 = 1+4 = 5 \]
\[ 0+2+3 = 2+3 = 5 \]
\[ 0+2+4 = 2+4 = 6 \]
\[ 0+3+4 = 3+4 = 7 \]
\[ 1+2+3 = 3+3 = ... |
Let \( a \) and \( b \) be integers such that \( a \equiv 5^{-1} \pmod{56} \) and \( b \equiv 11^{-1} \pmod{56} \). Find the remainder when \( (2a + 8b)^3 \) is divided by 56. | First, we need to find the inverses of 5 and 11 modulo 56.
To find $a \equiv 5^{-1} \pmod{56}$, we need to find an integer $a$ such that $5a \equiv 1 \pmod{56}$.
Similarly, to find $b \equiv 11^{-1} \pmod{56}$, we need to find an integer $b$ such that $11b \equiv 1 \pmod{56}$.
After calculations, we find that $a \eq... |
In a geometric sequence, the first four terms are $a^2, a^3, a^4,$ and $a^5,$ where $a$ is a real number. If the fifth term is $a^6 + 1,$ what is the value of $a$? | In a geometric sequence, each term is found by multiplying the previous term by a constant ratio. Therefore, if we let $r$ denote the common ratio, then
\[ r = \frac{a^3}{a^2} = \frac{a^4}{a^3} = \frac{a^5}{a^4} = a \]
However, we are also given that the fifth term is $a^6 + 1$, and that it is equal to $a^5 \cdot r$.
... |
Determine the range of the function
\[f(x) = \frac{x^2 - 4}{|x + 2| + |x - 2|}.\] | To determine the range of the function $f(x) = \frac{x^2 - 4}{|x + 2| + |x - 2|}$, we need to analyze the behavior of the function in different intervals.
1. For $x < -2$:
\[ f(x) = \frac{x^2 - 4}{-(x + 2) - (x - 2)} = \frac{x^2 - 4}{-2x} = -\frac{1}{2}x + \frac{2}{x} \]
2. For $-2 \leq x \leq 2$:
\[ f(x) = \frac{x^2... |
In the figure, $m\angle A = 28^{\circ}$, $m\angle B = 74^\circ$ and $m\angle C = 26^{\circ}$. If $x$ and $y$ are the measures of the angles in which they are shown, what is the value of $x + y$? [asy]
size(150);
draw((0,5)--(0,0)--(15,0)--(15,5),linewidth(1));
draw((0,5)--(2,2)--(5,5)--(12,-2)--(15,5),linewidth(.7));
l... | Since the sum of the angles in a triangle is $180^{\circ}$, we have the following three equations:
\begin{align*}
28 + 74 + x = 180 \quad &(1) \\
74 + 26 + y = 180 \quad &(2) \\
28 + 26 + (180 - x - y) = 180 \quad &(3)
\end{align*}
From (1) and (2), we can solve for $x$ and $y$ directly:
\begin{align... |
After purchasing a video game for $45 from a store, Rachel received 3 $10 bills and $2 in loose coins for her change. How much money did Rachel have initially? | The total amount of change Rachel received is $3 * 10 + 2 = 30 + 2 = 32$ dollars.
Let's denote the initial amount of money Rachel had as $x$.
Then, the equation representing the situation is $x - 45 = 32$.
Solving for $x$, we get $x = 32 + 45 = 77$.
Thus, Rachel initially had \boxed{77} dollars. |
Find the equation of the focus of the parabola defined by the equation $x = 2y^2 - 3y + 1$. | The given equation of the parabola is in the form $x = 2y^2 - 3y + 1$. To find the equation of the focus, we need to put the equation in the standard form of a parabola, which is $x = \frac{1}{4p}(y-k)^2 + h$, where $(h,k)$ is the vertex of the parabola and $p$ is the distance from the vertex to the focus.
Complete th... |
A triangular prism is randomly selected from a set of 50 different triangular prisms, each with a unique combination of dimensions. If the volume of the selected prism turns out to be a multiple of 5, what is the probability that the sum of its three edge lengths is less than 20? Express your answer as a decimal rounde... | To calculate the probability, we need to know the total number of triangular prisms with volumes that are multiples of 5 and the number of those prisms with sums of edge lengths less than 20.
Let's assume the dimensions of the triangular prism are $a$, $b$, and $c$, and the height is $h$. The volume $V$ of a triangula... |
A sphere is inscribed in a tetrahedron whose vertices have coordinates $(0,0,0),$ $(1,0,0),$ $(0,1,0),$ and $(0,0,1)$. The sphere touches the edges of the tetrahedron at points $A,$ $B,$ $C,$ and $D.$ Find the distance between the center of the sphere and the midpoint of the line segment $\overline{AC}$. | ## Step 1: To find the distance between the center of the sphere and the midpoint of the line segment $\overline{AC}$, we first need to determine the coordinates of points $A$, $B$, $C$, and $D$ where the sphere touches the edges of the tetrahedron.
## Step 2: The tetrahedron has vertices at $(0,0,0),$ $(1,0,0),$ $(0,... |
The sum of the squares of three consecutive positive integers is 780. What is the product of the three integers? | Let $n$ be the smallest of the three consecutive integers. Then $n+1$ and $n+2$ are the other two integers. We have:
\[ n^2 + (n+1)^2 + (n+2)^2 = 780 \]
\[ n^2 + n^2 + 2n + 1 + n^2 + 4n + 4 = 780 \]
\[ 3n^2 + 6n + 5 = 780 \]
\[ 3n^2 + 6n - 775 = 0 \]
This is a quadratic equation. Solve for $n$ using the quadratic for... |
A certain company claims that 1 in 10 of its customers are "heavy users" of their product. To test this claim, a random sample of 100 customers is taken. If the claim is true, what is the probability that exactly 15 or more of the sampled customers are heavy users? | This is a binomial probability problem. Let $X$ be the number of heavy users in the sample. If the claim is true, the probability of a customer being a heavy user is $p = \frac{1}{10} = 0.1$, and the probability of a customer not being a heavy user is $q = 1 - p = 0.9$.
We want to find $P(X \geq 15)$.
Since the sampl... |
Rectangle $PQRS$ has area $3009.$ An ellipse with area $3009\pi$ passes through $P$ and $R$ and has foci at $Q$ and $S$. What is the sum of the lengths of the diagonals of the rectangle? | Let the lengths of the diagonals of the rectangle be $d_1$ and $d_2$.
Since the foci of the ellipse are at points Q and S, the major axis of the ellipse is equal to the diagonal of the rectangle, so $d_1 = 2a$.
The area of the ellipse is $\pi ab = 3009\pi$, so $ab = 3009$.
Now we can use the Pythagorean theorem t... |
Let $g$ be a function from the positive integers to the positive integers such that
\[g(l \times m) + g(l + m) = g(l) \times g(m) - 1\]
for all positive integers $l$ and $m$.
Find the number of possible values of $g(3)$ and the product of all possible values of $g(3)$.
Let $n$ be the number of possible values of $... | Let's use the functional equation given to find possible values of $g(3)$.
Setting $l = m = 1$, we get:
\[ g(1 \times 1) + g(1 + 1) = g(1) \times g(1) - 1 \]
\[ g(1) + g(2) = g(1)^2 - 1 \]
\[ g(2) = g(1)^2 - g(1) - 1 \]
Setting $l = 1$ and $m = 2$, we get:
\[ g(1 \times 2) + g(1 + 2) = g(1) \times g(2) - 1 \]
\[ g(2)... |
Two circles with radii 4 and 6 respectively intersect at the points (2,0) and (-2,0). What is the distance between the centers of these two circles? | The circles intersect at (2,0) and (-2,0), which means their centers lie on the x-axis. The distance between their centers is equal to the sum of their radii. We can draw a diagram to illustrate this:
The centers of the circles are at (x1,0) and (x2,0). The distance between them is |x2-x1|.
Since the radii of the cir... |
A school offers three foreign language courses: Spanish, French, and German. There are 25 students in the Spanish class, 20 students in the French class, and 15 students in the German class. If 5 students are enrolled in all three classes, 8 students are enrolled in both Spanish and French, and 3 students are enrolled ... | Let's use the principle of inclusion-exclusion to solve this problem.
1. First, add the number of students in each class:
\[ 25 + 20 + 15 = 45 + 15 = 60 \]
2. Next, subtract the number of students who are double-counted:
\[ 60 - (8 + 3) = 60 - 11 = 49 \]
3. Finally, add back the number of students who were triple-co... |
A large cube is constructed from 27 small congruent cubes. The small cubes are painted with six different colors, and each cube has the same color on all its faces. Two large cubes are considered distinguishable if it is not possible to place one on the other, using translations and rotations (but not reflections), so ... | To solve this problem, we need to consider the possible color arrangements of the small cubes in the large cube.
There are 27 small cubes, and each can be one of six colors. The colors of the small cubes are what determine the distinguishability of the large cubes.
Since the large cubes can be rotated and translated,... |
In a circle, the lengths of two parallel chords are $a$ and $b$, where $a < b$. The central angles determined by these chords are $\alpha$ and $\beta$, respectively. If $\alpha + \beta = \frac{\pi}{2}$ and $a \cos \beta$ is an integer, determine the value of $b$. | Since $\alpha$ and $\beta$ are the central angles of the circle and $\alpha + \beta = \frac{\pi}{2}$, $\alpha$ and $\beta$ are complementary angles.
Now, consider the chords with lengths $a$ and $b$. The angle $\alpha$ corresponds to the chord with length $a$, and the angle $\beta$ corresponds to the chord with length... |
Factor the expression $x^3+ax^2+bx+c$ so that the sum of the coefficients of the linear terms in the factorization is equal to 1.5 times the sum of the coefficients of the quadratic terms. | We can express the original expression as a product of its linear factors:
\[ x^3 + ax^2 + bx + c = (x + p)(x + q)(x + r) \]
Here, $p$, $q$, and $r$ are roots of the cubic equation.
Expanding the right side gives:
\[ x^3 + (p + q + r)x^2 + (pq + pr + qr)x + pqr \]
Matching coefficients, we have:
\[ a = p + q + r \]
... |
In a rectangular garden, a 5-meter tall tree is planted at a distance of 3 meters from one of the shorter sides. If the length of the shorter side is 10 meters, what is the maximum length of the longer side such that the tree does not cast a shadow beyond the boundary of the garden? | Let's denote the length of the longer side as $x$. The maximum length of the longer side will be achieved when the shadow of the tree touches the boundary of the garden. We can use similar triangles to find this maximum length.
The height of the tree is 5 meters, and it is planted 3 meters from the shorter side. Let's... |
Find the coefficient of x^2 in the expansion of (x + 1)^12. | The expansion of (x + 1)^12 is given by the binomial theorem:
\[ (x + 1)^{12} = \binom{12}{0}x^{12} + \binom{12}{1}x^{11} + \binom{12}{2}x^{10} + \cdots + \binom{12}{10}x^2 + \binom{12}{11}x + \binom{12}{12} \]
The coefficient of x^2 is given by the term $\binom{12}{10}x^2$, so the coefficient is:
\[ \binom{12}{10} = ... |
Solve the equation $\log_{x^2}2 = \log_x 4$. | Given the equation $\log_{x^2}2 = \log_x 4$, we need to solve for $x$.
First, apply the change of base formula to both sides of the equation:
\[ \frac{\log 2}{\log x^2} = \frac{\log 4}{\log x} \]
Since $\log x^2 = 2 \log x$ and $\log 4 = 2 \log 2$, we can simplify the equation:
\[ \frac{\log 2}{2 \log x} = \frac{2 \l... |
Find the sum of two positive integers such that their squares have the same remainder when divided by 7, and their product is 42. | Let the two positive integers be $x$ and $y$. We know that $x \times y = 42$. We can write this as $x \times y = 6 \times 7$.
Also, the squares of $x$ and $y$ have the same remainder when divided by 7. This means that $x^2 \equiv y^2 \pmod{7}$.
From the first condition, the possible pairs of $(x, y)$ are:
\[ (1, 42),... |
A new student, Alex, is joining Kim's class. Alex has taken three exams so far, scoring 75, 80, and 82. If Alex wants to have the same average as Kim after her fourth exam, how much does Alex need to score on the fourth exam? | To find the average score of Alex's first three exams, we add the scores and divide by the number of exams:
\[ \text{Average} = \frac{75 + 80 + 82}{3} = \frac{237}{3} = 79 \]
To find the average score of Kim's exams, we add the scores and divide by the number of exams:
\[ \text{Average} = \frac{80 + 85 + 90}{3} = \fra... |
A group of friends wants to go on a hike together, but they have a constraint: no two friends who are taller than 5'8" can walk together, and no more than three friends in a row can be shorter than 5'8". If there are 12 friends in total, 4 of whom are taller than 5'8", how many different hiking groups can be formed? | Since there are 4 friends taller than 5'8", there are 8 friends shorter than 5'8". We can use the stars and bars method to arrange these friends.
Let $*$ represent a friend shorter than 5'8" and $|$ represent a friend taller than 5'8". Since there are 8 friends shorter than 5'8" and 4 friends taller than 5'8", we can ... |
Three pipes A, B, and C can fill a tank in 10, 15, and 20 hours, respectively. Pipe A is opened first, and after 2 hours, pipe B is opened, and then after another 2 hours, pipe C is opened. Find the total time it takes to fill the tank. | Let's calculate the rate of filling the tank for each pipe:
\[ \text{Pipe A: } 1 : 10 = 0.1 \quad \text{Pipe B: } 1 : 15 = 0.0667 \quad \text{Pipe C: } 1 : 20 = 0.05 \]
In the first 2 hours, only pipe A is open, so the fraction of the tank filled is $2 \times 0.1 = 0.2$.
In the next 2 hours, both pipe A and pipe B ar... |
In a right-angled triangle, two circles are inscribed such that the centers of the circles lie on the hypotenuse of the triangle. The larger circle is tangent to the two legs of the triangle, while the smaller circle is tangent to the larger circle and one of the legs of the triangle. If the radius of the larger circle... | Let's denote the radius of the smaller circle as $r$.
Since the larger circle is tangent to the two legs of the triangle, its center is the incenter of the triangle. The distance from the incenter to each of the legs is equal to the radius of the larger circle, which is 1 unit.
The smaller circle is tangent to the la... |
A professor is planning to organize a round-table discussion with 12 students. If one student is seated in a fixed position, the remaining students can be evenly distributed around the table in 3 groups of equal size. However, if the fixed student is absent, the remaining students can be evenly distributed in 4 groups ... | Let's analyze the situation:
1. When one student is seated in a fixed position, the remaining 11 students can be evenly distributed in 3 groups. This means that 11 is divisible by 3, which is not possible. So, we need to reconsider the situation.
2. Let's assume that the fixed student is not included in the groups. I... |
Find the product of the octal numbers 1357 and 24 using the Russian peasant multiplication algorithm.
What is the result of this product as a decimal number? | To solve this problem, we will use the Russian peasant multiplication algorithm to find the product of the octal numbers 1357 and 24.
First, convert 1357 and 24 to decimal:
\[ 1357_{8} = 1 \cdot 8^{3} + 3 \cdot 8^{2} + 5 \cdot 8^{1} + 7 \cdot 8^{0} = 512 + 192 + 40 + 7 = 704 + 40 + 7 = 744 + 7 = 751 \]
\[ 24_{8} = 2 ... |
Let $C$ be the sum of the interior angles of a triangle, a quadrilateral, and a pentagon. Let $D$ be the sum of the interior angles of a hexagon, a heptagon, and an octagon. What is the value of $C + D$? | The sum of the interior angles of a polygon with $n$ sides is given by the formula $(n-2) \times 180$.
For the triangle, quadrilateral, and pentagon:
\[ C = (3 - 2) \times 180 + (4 - 2) \times 180 + (5 - 2) \times 180 \]
\[ C = 180 + 360 + 540 \]
\[ C = 1080 \]
For the hexagon, heptagon, and octagon:
\[ D = (6 - 2) \... |
In a convex quadrilateral $ABCD$, the diagonals $AC$ and $BD$ intersect at point $E$. The circumcircles of triangles $AED$ and $BEC$ intersect at point $F$, and the line $AD$ is tangent to the circumcircle of triangle $BEF$. If $\angle A=30^\circ$ and $\angle C=90^\circ$, find the measure of $\angle BFD$. | Given the information, we need to determine the measure of $\angle BFD$.
Since $\angle A=30^\circ$ and $\angle C=90^\circ$, we can use the fact that the sum of the interior angles of a quadrilateral is $360^\circ$ to find $\angle B + \angle D$:
\[ \angle A + \angle B + \angle C + \angle D = 360^\circ \Rightarrow 30^\c... |
Find the product of all real values of $y$ that satisfy
\[y = 1 + y + \frac{y^2}{2!} + \frac{y^3}{3!} + \frac{y^4}{4!} + \dotsb.\] | We can rewrite the equation as
\[ y - 1 = y + \frac{y^2}{2!} + \frac{y^3}{3!} + \frac{y^4}{4!} + \dotsb \]
Notice that the right-hand side is the Maclaurin series for $e^y - 1$.
So, the equation becomes
\[ y - 1 = e^y - 1 \Rightarrow y = e^y \]
To find the real solutions, we can plot the graphs of $y = x$ and $y = e... |
In a certain city, there are three types of public transportation: buses, trolleys, and trams. A survey of 30 residents revealed that 15 use buses, 17 use trolleys, and 11 use trams. Additionally, 7 residents use both buses and trolleys, 5 residents use both trolleys and trams, and 3 residents use both buses and tram... | Let's break down the information:
1. Total residents surveyed: 30
2. Residents using:
- Buses (B): 15
- Trolleys (T): 17
- Trams (Tr): 11
3. Residents using combinations of transportation:
- Both B and T: 7
- Both T and Tr: 5
- Both B and Tr: 3
4. Residents using all three (B, T, Tr): 2
To find the ... |
Let $f(x) = \frac{x}{3}-\frac{1}{3x}$, and let $r$, $s$, and $t$ be the three roots of the equation $f(f(f(x))) = 1$. Find the value of $r + s + t$. | Given that $f(x) = \frac{x}{3} - \frac{1}{3x}$, we need to find $f(f(f(x)))$.
First, find $f(f(x))$:
\[ f(f(x)) = f\left( \frac{x}{3} - \frac{1}{3x} \right) = \frac{\frac{x}{3} - \frac{1}{3x}}{3} - \frac{1}{3\left(\frac{x}{3} - \frac{1}{3x}\right)} = \frac{x}{9} - \frac{1}{9x} - \frac{1}{\left(\frac{x}{3} - \frac{1}{3... |
Let $y_1 \le y_2 \le y_3$ be the three real roots of the equation $17y^3-2004y^2+1000=0$. Find $(y_2-y_1)(y_3-y_2)$. | ## Step 1: The problem asks us to find the value of the expression $(y_2-y_1)(y_3-y_2)$, given that $y_1 \le y_2 \le y_3$ are the three real roots of the equation $17y^3-2004y^2+1000=0$.
## Step 2: To approach this problem, we first use Vieta's formulas to establish relationships between the roots $y_1$, $y_2$, and $... |
A deck of 52 cards is shuffled. What is the probability that the first card drawn is a king, or the second card drawn is an ace, or both? Express your answer as a common fraction. | To calculate this probability, we need to consider the cases separately and then account for the overlap.
1. The first card is a king:
There are 4 kings in a deck of 52 cards. The probability of drawing a king first is $\frac{4}{52}$.
2. The second card is an ace:
There are 4 aces in the deck. The probability that th... |
The price of a pair of gloves is 15$\%$ more than the price of a pair of socks. If the price of 6 pairs of socks is the same as the price of 3 pairs of gloves, how much more would you pay for 10 pairs of gloves than for 10 pairs of socks? | Let the price of a pair of socks be $x$.
Since the price of a pair of gloves is 15$\%$ more than the price of a pair of socks, the price of a pair of gloves is $x + 0.15x = 1.15x$.
Given that the price of 6 pairs of socks is the same as the price of 3 pairs of gloves, we have:
\[ 6x = 3(1.15x) \Rightarrow 6x = 3.45x ... |
On a circular table with radius 1.5 meters, 5 children are sitting evenly spaced out from each other. Each child has a toy that can move along the table's edge. If the toys start at their respective children and move at the same speed in the same direction, what is the minimum distance a toy must travel for all toys to... | To solve this problem, we can take advantage of the symmetry of the circular table and the even spacing of the children.
Given that the table has a radius of 1.5 meters, its circumference $C$ is:
\[ C = 2\pi r = 2\pi (1.5) = 3\pi \]
Since there are 5 children evenly spaced, the distance $d$ between each child along t... |
In trapezoid $ABCD$, points $E$ and $F$ are on sides $AB$ and $CD$, respectively, such that $AE = EF = FC = 24$. Additionally, $AB$ is parallel to $CD$, and the height of the trapezoid is equal to the length of $AD$. Find the area of the trapezoid. | We know the following:
1. $AE = EF = FC = 24$
2. $AB$ is parallel to $CD$
3. The height of the trapezoid is equal to the length of $AD$
Since $AB$ is parallel to $CD$, we know that $\triangle AED$ and $\triangle CFD$ are similar. We also know that $AE = FC = 24$, which means $ED = FD$.
Because $EF = 24$, $ED = FD = ... |
Triangle $ABC$ is an equilateral triangle. A circle is inscribed inside the triangle so that it touches all three sides of the triangle. A square has two vertices on the circle and the other two vertices on sides $AB$ and $AC$. What is the length of one side of the square, if the side length of the equilateral triangle... | Let's denote the side length of the square as $s$. First, notice that the radius of the inscribed circle is $r = \frac{6 \sqrt{3}}{6} = \sqrt{3}$.
The diagonal of the square is equal to the diameter of the circle, which is $2r = 2\sqrt{3}$.
Using the Pythagorean theorem, we can relate the side length of the square to... |
A deck of cards consists of 16 cards with four suits and four ranks. Each card is colored red or black. If a card is red, it has a point value of 1. If a card is black, it has a point value of -1. The cards are shuffled and then dealt into four piles of four cards each. What is the probability that at least one pile ha... | We can solve this problem by considering the total number of ways to distribute the cards and then finding the number of ways in which no pile has a total point value of zero.
Since there are 16 cards in total, with 8 red and 8 black, and they are divided into 4 piles of 4 cards each, the only way a pile can have a to... |
The volumes of three spheres are 32π, 108π, and 288π cubic units. What is the average (mean) of their radii? | The volume of a sphere is given by $V = \frac{4}{3} \pi r^3$.
Given the volumes of the three spheres, we can find their radii:
\begin{align*}
\frac{4}{3} \pi r_1^3 &= 32\pi \\
\frac{4}{3} \pi r_2^3 &= 108\pi \\
\frac{4}{3} \pi r_3^3 &= 288\pi
\end{align*}
Solve for $r_1$, $r_2$, and $r_3$:
\begin{al... |
In trapezoid $ABCD$, the lengths of bases $AB$ and $CD$ are $7$ and $8$, respectively. An isosceles right triangle is drawn so that its legs are parallel to the bases and no point of the triangle lies outside $ABCD$. Find the maximum possible area of such a triangle. | Let $x$ be the length of the leg of the triangle parallel to the base $AB$, and $y$ be the length of the leg parallel to the base $CD$. Since the triangle is isosceles right, $x = y$.
The height of the triangle is the same as the height of the trapezoid, which we'll call $h$. The area of the triangle is
\[ A = \frac{1... |
In a triangle $DEF$, where $DE = 4$ and $EF = 7$, we have $\cos D = \frac{1}{2}$. Find the length of the altitude dropped from vertex $F$ to side $DE$. | Since we know $\cos D$ and the lengths of $DE$ and $EF$, we can use the cosine formula for the length of the altitude dropped from vertex $F$ to side $DE$.
Let $h$ be the length of the altitude from $F$ to $DE$. Then:
\[ h = EF \sin D \]
Since $\cos D = \frac{1}{2}$, and $\cos^2 D + \sin^2 D = 1$, we can find $\sin ... |
In a triangle with angles A, B, and C, and corresponding opposite sides a, b, and c, the following equation holds:
$$\frac{a}{\sin A} + \frac{b}{\sin B} = \frac{2c}{\sin C}$$
Determine the measure of angle C, in degrees. | The given equation is $\frac{a}{\sin A} + \frac{b}{\sin B} = \frac{2c}{\sin C}$. We know from the Law of Sines that $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$.
Let's denote $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = k$, where $k$ is a constant.
Then, we can rewrite the given equation as $... |
A hummingbird is a kind of bird that eats nectar from flowers. On the first day, a hummingbird visited 5 flowers. Then it visits 3 more flowers than it did the day before. How many flowers in all did a hummingbird visit for 6 days? | On the first day, the hummingbird visited 5 flowers.
On the second day, it visited 3 more flowers than the day before, which is 5 + 3 = 8 flowers.
On the third day, it visited 3 more flowers than the day before, which is 8 + 3 = 11 flowers.
On the fourth day, it visited 3 more flowers than the day before, which is 11 +... |
In a parallelepiped, the angle between the diagonals of two adjacent faces is $\pi/4$. Determine the angle between the planes containing these faces. | Let the edges of the parallelepiped be $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$. The diagonals of the two adjacent faces are $\mathbf{a} + \mathbf{b}$ and $\mathbf{b} + \mathbf{c}$.
The angle between these diagonals is $\pi/4$, so:
\[ \cos(\pi/4) = \frac{(\mathbf{a} + \mathbf{b}) \cdot (\mathbf{b} + \mathbf{c})}{|... |
Let $A$ be a matrix such that
$$A = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$$
The matrix $B$ is defined as $B=A^2$.
Determine the smallest positive integer $n$ such that $B^n=A$. | We are given that $A = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ and $B = A^2$.
First, calculate $B$:
\[ B = A^2 = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} \]
Now, let's find $B^n$ for $n = 1, 2, 3, \ldots$ until ... |
The sequence $a_n$ satisfies the property that $a_n-a_{n-1}=a_{n-1}a_{n-2}$ for all $n\geq 3$. If $a_1=2$ and $a_2=4$, how many integers can be found in the sequence after $a_{10}$? | We are given that $a_1 = 2$ and $a_2 = 4$. The recurrence relation is $a_n - a_{n-1} = a_{n-1} a_{n-2}$ for $n \geq 3$. We can rewrite this relation as $a_n = a_{n-1} + a_{n-1} a_{n-2} = a_{n-1} (1 + a_{n-2})$.
Now, let's compute the sequence up to $a_{10}$ and analyze its properties:
- $a_3 = a_2 + a_2 a_1 = 4 + 4 \... |
Find all primes whose squares differ by 12. | For a prime number $p$, $p^2$ is always odd.
Let's assume there are two primes $p_1$ and $p_2$ such that $p_1^2 - p_2^2 = 12$.
Then, $(p_1 - p_2)(p_1 + p_2) = 12$. Since $p_1$ and $p_2$ are both odd, $p_1 - p_2$ and $p_1 + p_2$ are both even.
Thus, we have the following possibilities:
\begin{align*}
p_1 - p_2 = 2 \q... |
Let $f(x)$ and $g(x)$ be two continuous functions on the interval $[0, 1]$ defined as follows:
\[f(x) = \sqrt[3]{x^2 + \frac{1}{4}}\]
\[g(x) = x^2 + \frac{1}{3}\]
Find the maximum value of the expression $\frac{f(x)}{g(x)}$ in the domain $\frac{1}{4} \le x \le \frac{3}{5}$. | To find the maximum value of $\frac{f(x)}{g(x)}$, we can first write out the expression explicitly and then determine its critical points.
The expression is:
\[\frac{f(x)}{g(x)} = \frac{\sqrt[3]{x^2 + \frac{1}{4}}}{x^2 + \frac{1}{3}}\]
To find the maximum, we can differentiate this expression with respect to $x$ and ... |
If $x = 5$ and $x^2y - 3xy + 2y = 100$, what is the value of $y^2$? | Given the equation $x^2y - 3xy + 2y = 100$ and $x = 5$, substitute $x$ into the equation:
\[ (5)^2y - 3(5)y + 2y = 100 \Rightarrow 25y - 15y + 2y = 100 \Rightarrow 12y = 100 \Rightarrow y = 100 : 12 \Rightarrow y = 25 : 3 \Rightarrow y = 25/3 \]
To find $y^2$:
\[ y^2 = \left( \frac{25}{3} \right)^2 = \frac{25^2}{3^2} ... |
A right circular cylinder is inscribed in a sphere. Given that the radius of the sphere is 4 inches, find the volume of the inscribed cylinder that maximizes its volume. Express your answer in terms of $\pi$. | Let the radius of the base of the cylinder be $r$ and the height of the cylinder be $h$.
Since the cylinder is inscribed in the sphere, the diameter of the base of the cylinder is equal to the diameter of the sphere. Therefore, the height of the cylinder is
\[ h = 2\sqrt{4^2 - r^2} = 2\sqrt{16 - r^2} = 4\sqrt{1 - \fra... |
Consider a monic cubic polynomial $q(x)$ with real coefficients such that $q(2+i)=0$ and $q(1)=-28$. Determine $q(x)$ in expanded form. | Since $q(x)$ is a monic cubic polynomial with real coefficients and $q(2+i)=0$, by the Complex Conjugate Root Theorem, $q(2-i)$ is also a root of $q(x)$. This means that $(x-(2+i))(x-(2-i))$ is a factor of $q(x)$.
The product of these factors is
\[ (x-(2+i))(x-(2-i)) = x^2 - (2+i)x - (2-i)x + (2+i)(2-i) = x^2 - 4x + 5... |
Let $a$, $b$, and $c$ be the roots of the cubic equation $x^3-2x^2-x+1=0$. Find the value of $\frac{a}{1+a^2}+\frac{b}{1+b^2}+\frac{c}{1+c^2}$. | To solve this problem, we can use Vieta's formulas to relate the roots of the cubic equation to the desired expression.
Vieta's formulas for a cubic equation $x^3 + px^2 + qx + r = 0$ with roots $a$, $b$, and $c$ are:
\[ a + b + c = -p \]
\[ ab + bc + ac = q \]
\[ abc = -r \]
For the given cubic equation $x^3 - 2x^2 ... |
A certain function $f(x)$ is defined as the smallest integer greater than $\sqrt{3x}$. Find the range of values for $x$ such that $f(2x) - f(x) = 1$. | Given the function $f(x)$, we want to find the range of values for $x$ such that $f(2x) - f(x) = 1$.
Since $f(x)$ is defined as the smallest integer greater than $\sqrt{3x}$, we can express this as:
\[ f(x) = \lceil \sqrt{3x} \rceil \]
where $\lceil \cdot \rceil$ denotes the ceiling function.
Now, we can rewrite the... |
Consider the sequence $a_n$ defined by the recurrence relation $a_{n+2} = 3a_{n+1} - a_n$, with initial conditions $a_0 = 2$ and $a_1 = 3$. Determine the value of $a_7$. | We can use the recurrence relation to find the terms of the sequence step by step.
We know the initial conditions:
\begin{align*}
a_0 = 2 \\
a_1 = 3
\end{align*}
We use the recurrence relation to find the next terms:
\begin{align*}
a_2 = 3a_1 - a_0 = 3(3) - 2 = 7 \\
a_3 = 3a_2 - a_1 = 3(7) - 3 = ... |
A transparent cylindrical container is filled with water, forming a cylinder of radius 5 and height 15.
A cross-section is made through the cylinder along a plane that intersects the base of the cylinder at a 30-degree angle with the diameter. Determine the largest possible area of this cross-section. | The cross-section forms a parallelogram, where the base of the parallelogram is the diameter of the cylinder's base (10) and the height is the height of the cylinder (15). The area of a parallelogram is base times height, but since the cross-section is at an angle, we need to consider the component of the height perpen... |
Parabolas $y^2 = 4ax$ and $x^2 = 4by$ have a common tangent line, which intersects the x-axis at $P$ and y-axis at $Q$. Show that $\frac{1}{a} + \frac{1}{b} = \frac{3}{PQ}$. | ## Step 1: Understand the problem
We are given two parabolas, $y^2 = 4ax$ and $x^2 = 4by$, and we need to show that $\frac{1}{a} + \frac{1}{b} = \frac{3}{PQ}$, where $P$ and $Q$ are the x-intercept and y-intercept of the common tangent line.
## Step 2: Find the equation of the tangent line
Let the equation of the tang... |
A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. A marble is drawn at random from the bag, and then another marble is drawn without replacement. What is the probability that the first marble drawn is red or the second marble drawn is blue? | To solve this problem, we need to consider two cases:
Case 1: The first marble drawn is red.
Case 2: The second marble drawn is blue.
Case 1: The first marble drawn is red.
Probability of drawing a red marble first:
\[ P(\text{Red first}) = \frac{\text{Number of red marbles}}{\text{Total number of marbles}} = \frac{5... |
Let $p(x) = x^3 - 2x^2 - 11x + 12$. The roots of $p(x)$ are all integers. What is the sum of the cubes of these roots? | To find the sum of the cubes of the roots of $p(x)$, we can use Vieta's formulas, which state that for a polynomial $a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0$ with roots $r_1, r_2, \ldots, r_n$, we have
\begin{align*}
r_1 + r_2 + \ldots + r_n &= -\frac{a_{n-1}}{a_n} \\
r_1r_2 + r_1r_3 + \ldots + r_{n-1... |
Let $a,$ $b,$ $c$ be a three-term arithmetic series where all the terms are positive, such that $abc = 64.$ Find the largest possible value of $a^2 + b^2 + c^2$. | Since $a,$ $b,$ $c$ form an arithmetic series, we can express $b$ as the average of $a$ and $c$, and $b$ is also the arithmetic mean of $a$ and $c$:
\[ b = \frac{a + c}{2} \]
Given that $abc = 64$, and all terms are positive, we can apply the AM-GM inequality to find the maximum value of $a^2 + b^2 + c^2$.
The AM-GM ... |
Emily is decorating her 30 Easter eggs. She decorates 9 of them with polka dots, 12 of them with stripes, and the rest with swirls. What is the difference in percentage points between the percentage of eggs decorated with stripes and the percentage decorated with swirls? | Let's compute the percentage of eggs decorated with stripes and swirls separately, and then find their difference.
Emily decorated 12 eggs out of 30 with stripes.
Thus the percentage of eggs decorated with stripes is $(12/30)*100 = 0.4*100 = 40$.
The number of eggs decorated with swirls is $30 - 9 - 12 = 21 - 12 = 9$.
... |
Determine the sum of the solutions to the equation $x^3=64$. | The given equation is $x^3 = 64$. We can solve for $x$ by taking the cube root of both sides:
\[ x = \sqrt[3]{64} \]
Since $4^3 = 64$, the real solution is $x = 4$.
However, in complex numbers, the equation $x^3 = 64$ has three solutions, which are the cube roots of $64$. These solutions are equally spaced around the... |
A block of ice is sliding down a ramp with a height of 5 meters. The block is sliding from rest and its speed at the bottom of the ramp is 8 meters per second. Assuming that the acceleration of the block is constant, find the time it takes for the block to reach a speed of 4 meters per second. | The acceleration of the block can be found using the equation of motion:
\[ v^2 = u^2 + 2as \]
We know that:
\[ v = 8 \quad u = 0 \quad s = 5 \]
\[ 8^2 = 0^2 + 2a(5) \Rightarrow 64 = 10a \Rightarrow a = 64 : 10 \Rightarrow a = 6.4 \]
We need to find the time it takes for the block to reach a speed of 4 meters per se... |
In a rectangular garden, the length is 3 meters more than the width. If a 10-meter long rope is stretched diagonally from one corner of the garden to the opposite corner, how many meters is the length of the garden? | Let $x$ be the width of the garden. Then the length of the garden is $x + 3$.
Using the Pythagorean theorem, we can write:
\[ x^2 + (x + 3)^2 = 10^2 \]
\[ x^2 + x^2 + 6x + 9 = 100 \]
\[ 2x^2 + 6x - 91 = 0 \]
Dividing the entire equation by 2:
\[ x^2 + 3x - 91 : 2 = 0 \]
\[ x^2 + 3x - 91 : 2 = 0 \]
\[ x^2 + 3x - 45.5 ... |
The sides of a right triangle have lengths $a, a+2$, and $2a$. Find the ratio of the lengths of the sides of the triangle in simplest form. | We know that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Let's assume that the side of length $2a$ is the hypotenuse.
By the Pythagorean theorem:
\[ a^2 + (a + 2)^2 = (2a)^2 \Rightarrow a^2 + a^2 + 4a + 4 = 4a^2 \Rightarrow ... |
Let $f(x)$ be defined on the set of positive integers such that $$f(n) = \begin{cases} n &\quad n\text{ is a perfect square} \\ f(k) + f(\lfloor \frac{n}{k}\rfloor) &\quad \text{otherwise} \end{cases}$$ where $k$ is the largest perfect square divisor of $n$. Find the smallest positive integer $N$ such that $f(N) = 15$. | ## Step 1: Understand the function $f(x)$ and its definition
The function $f(x)$ is defined on the set of positive integers and has two cases. If $n$ is a perfect square, then $f(n) = n$. If $n$ is not a perfect square, then $f(n)$ is the sum of $f(k)$ and $f(\lfloor \frac{n}{k}\rfloor)$, where $k$ is the largest perfe... |
The numbers $\sqrt{2u-1}$, $\sqrt{2u+1}$ and $2\sqrt{u}$ are the side lengths of a triangle. How many degrees are in the measure of the largest angle?
The numbers $\sqrt{3v-2}$, $\sqrt{3v+2}$ and $3\sqrt{v}$ are the side lengths of a triangle. How many degrees are in the measure of the smallest angle? | ## Step 1: Determine the relationship between the side lengths of the triangles.
Given that $\sqrt{2u-1}$, $\sqrt{2u+1}$, and $2\sqrt{u}$ are the side lengths of a triangle, and $\sqrt{3v-2}$, $\sqrt{3v+2}$, and $3\sqrt{v}$ are the side lengths of another triangle, we can see a pattern in the expressions. The relations... |
Tom has 15 feet of string and uses 2/3 of it to tie a package. He then cuts the remaining string into 3 equal pieces. What fraction of the original string length is each piece? | Tom starts with 15 feet of string and uses $\frac{2}{3}$ of it to tie a package. So, he uses $\frac{2}{3} \cdot 15 = 10$ feet of string.
The remaining string is $15 - 10 = 5$ feet.
He cuts the remaining 5 feet of string into 3 equal pieces. Each piece is $\frac{5}{3}$ feet long.
The fraction of the original string l... |
The front of a 1.5-mile long train enters a bridge exactly 2 minutes before the tail of the train exits the bridge. If the train is moving 45 miles per hour, how many miles long is the bridge? | Let's denote the length of the bridge as $x$ miles.
The train travels at 45 miles per hour. The time it takes for the front of the train to cross the bridge is $t_1$ and the time it takes for the tail of the train to exit the bridge is $t_2$.
We know that the front of the train enters the bridge exactly 2 minutes bef... |
Kaitlyn attends an arcade where she wins tokens from each of the 4 arcade machines and also receives 10 tokens as a gift from her friend. Each token is worth $2. In total, she has tokens that total a value of $40. If Kaitlyn won an equal number of tokens from each of the machines, how many tokens did she win from each ... | Let's break down the problem:
Kaitlyn has a total of $40 worth of tokens.
Each token is worth $2, so she has a total of $40 / $2 = 20 tokens.
She received 10 tokens as a gift from her friend, so she won 20 - 10 = 10 tokens from the arcade machines.
Since she won an equal number of tokens from each of the 4 machines,... |
A trapezoid $ABCD$ has sides $AB = 10$ units, $BC = 20$ units, and $CD = 12$ units. The diagonals $AC$ and $BD$ intersect at $E$. What is the area, in square units, of triangle $AED$? | The area of a trapezoid is given by:
\[ \text{Area} = \frac{1}{2} \times \text{Height} \times (\text{Base}_1 + \text{Base}_2) \]
To find the area of triangle $AED$, we need to find the height of the trapezoid, which can be done by using the Pythagorean theorem on right triangle $CDE$. We also need to find the lengths ... |
Given that $a$, $b$, and $c$ are nonzero constants, determine the shape described by the graph of the equation $(x^2+y^2)^2=a^2x^2+b^2y^2+c^2$. Is it a parabola, circle, ellipse, hyperbola, point, line, two lines, or none of these? | Expand the left side of the equation to get a clearer view of the terms involved:
\[ (x^2 + y^2)^2 = a^2x^2 + b^2y^2 + c^2 \Rightarrow x^4 + 2x^2y^2 + y^4 = a^2x^2 + b^2y^2 + c^2 \]
Rearrange the equation to group like terms together:
\[ x^4 + y^4 + 2x^2y^2 - a^2x^2 - b^2y^2 - c^2 = 0 \]
Notice that this equation doe... |
A bookstore offers five different novels and three different biographies. How many ways can you choose a pair of books, one from each category, if you want to choose a biography written by a female author and a novel with a main character that is a male? | First, we need to know how many of the five novels have a male main character and how many of the three biographies are written by a female author.
Assuming we don't have this information, let's consider all possible combinations of choosing a novel and a biography. However, without knowing the specifics of the books'... |
Find the area of the triangle formed by the origin and the points of intersection of the circle with equation $x^2+y^2=16x-8y-65$ and the line $x-y=4$. | To solve this problem, we need to find the points of intersection between the circle and the line.
First, we need to rewrite the equation of the circle in the form $(x - h)^2 + (y - k)^2 = r^2$ to find the center $(h, k)$ and the radius $r$.
\[ x^2 - 16x + y^2 + 8y = -65 \Rightarrow (x^2 - 16x + 64) + (y^2 + 8y + 16)... |
In triangle $ABC,$ the angle bisector of $\angle A$ intersects the circumcircle at point $D.$ If $BD = 1$ and $DC = 6,$ calculate $\tan B \tan C.$ | Let $R$ be the radius of the circumcircle.
By the extended law of sines, we have $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$.
Since $BD$ and $DC$ are chords in the circumcircle, we can use the power of a point theorem to write
\[ BD \cdot DC = AD \cdot AD = AD^2 \Rightarrow AD^2 = BD \cdot DC = 1 \c... |
A woman is trying to choose between two different transportation options for her daily commute. The first option is to take the train, which costs $120 per month for a pass and an additional $50 per month for parking at the station. The second option is to drive and park in the city, which costs $200 per month for park... | To compute the total cost for the first option of taking the train, we need to consider the cost of the pass, the cost of parking at the station, and the cost of gas and maintenance to drive to the station.
The cost of the pass is $120 per month.
The cost of parking at the station is $50 per month.
To calculate the cos... |
In a certain country, the government decides to introduce a new holiday that will occur every 5 years, starting from a randomly chosen year. What is the minimum number of years that must pass before it is guaranteed that at least 3 holidays will fall on the same day of the week? | To find the minimum number of years that must pass before it is guaranteed that at least 3 holidays will fall on the same day of the week, we need to consider the worst-case scenario where the holidays are spaced out as much as possible on different days of the week.
Since the holiday occurs every 5 years, we can star... |
Solve for $y$ in
\[\begin{vmatrix} y^2 + 2 & y & y - 1 \\ y - 1 & y^2 + 2 & y \\ y & y - 1 & y^2 + 2 \end{vmatrix} = 0.\]Give your answer in terms of $y$. | Expand the determinant along the first row to get
\begin{align*}
(y^2 + 2) \begin{vmatrix} y^2 + 2 & y \\ y - 1 & y^2 + 2 \end{vmatrix} - y \begin{vmatrix} y - 1 & y \\ y & y^2 + 2 \end{vmatrix} + (y - 1) \begin{vmatrix} y - 1 & y^2 + 2 \\ y & y - 1 \end{vmatrix} &= 0
\end{align*}
Simplify each of the determinants:
\b... |
In the square $ABCD$, the point $E$ is on $\overline{AB}$, and point $F$ is on $\overline{BC}$, such that $AE:EB=CF:FB=2:1$. The area of the square is 24 square units. A circle with centre $O$ is tangent to $\overline{EF}$, $\overline{BF}$ and $\overline{DE}$. Determine the area of the shaded region bounded by $\overli... | Since $AE:EB=CF:FB=2:1$, let $AE=2x$, $EB=x$, $CF=2x$, and $FB=x$.
The area of the square is 24 square units, so the side length of the square is $\sqrt{24}=2\sqrt{6}$.
Since $AE+EB=AB=2\sqrt{6}$, we have $2x+x=2\sqrt{6}\Rightarrow 3x=2\sqrt{6}\Rightarrow x=\frac{2\sqrt{6}}{3}$.
The radius of the circle is equal to ... |
Let $f(x) = \log_3 x$. If $f(x) + f(x + 1) + f(x + 2) = 6$, determine the value of $(x + 2)(x - 2)$. | We are given the equation $f(x) + f(x + 1) + f(x + 2) = 6$ where $f(x) = \log_3 x$. We can rewrite the equation as $\log_3 x + \log_3 (x + 1) + \log_3 (x + 2) = 6$.
We can use the logarithmic property $\log_a x + \log_a y = \log_a (xy)$ to simplify the equation:
\begin{align*}
\log_3 x + \log_3 (x + 1) + \log_3 (x + ... |
Problem:
A building casts a shadow that is 15 meters long. At the same time, a tree near the building casts a shadow that is 3 meters long. If the tree is 6 meters tall, how tall is the building? Express your answer as a decimal to the nearest tenth. | Let's denote the height of the building as $h$. We can use similar triangles to solve this problem. The angle of elevation of the sun is the same for both the building and the tree.
We can set up a proportion using the heights and shadow lengths:
\[ \frac{h}{15} = \frac{6}{3} \]
Cross-multiply and solve for $h$:
\[ 3... |
A number is written in a base with an unknown value $b$. The number is $312_b$ when written in this base, but when read as a decimal number it is equal to 209 more than when it is read in the base $b$. What is the value of $b$? | Let's first convert the number $312_b$ to base 10, which is $3b^2 + 1b + 2$.
We are given that when read as a decimal number, it is equal to 209 more than when it is read in the base $b$. This gives us the equation:
\[ 312 = 3b^2 + b + 2 + 209 \]
Now simplify the equation:
\[ 312 = 3b^2 + b + 211 \Rightarrow 3b^2 + b... |
A survey of a certain city found that 80% of the residents who were born in the city still live there, while 40% of the residents who were not born in the city have moved there. If a person is chosen at random from the city's population, what is the probability that they were born in the city, given that they have live... | To solve this problem, we need to use Bayes' Theorem. Let's break down the information:
1. 80% of the residents who were born in the city still live there.
2. 40% of the residents who were not born in the city have moved there.
3. We want to find the probability that a person was born in the city, given that they have... |
What is the smallest positive integer n such that n and n+1 are both composite numbers, and n is divisible by the largest prime factor of n+1? | Consider the first few composite numbers:
\[ 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96,... |
In triangle $ABC,\ AB=9,\ AC=7,$ and $BC=10.$ Given that $m\angle BAC=45^\circ$ and that the length of the median to side $BC$ is $m + \sqrt{n},$ where $m$ and $n$ are positive integers, find $m+n.$ | We can use the Law of Cosines to find the length of the median to side $BC$.
First, find the length of the median $AD$:
\[ AD^2 = \frac{AB^2 + AC^2}{2} - \frac{BC^2}{4} \]
\[ AD^2 = \frac{9^2 + 7^2}{2} - \frac{10^2}{4} \]
\[ AD^2 = \frac{81 + 49}{2} - \frac{100}{4} \]
\[ AD^2 = \frac{130}{2} - \frac{100}{4} \]
\[ AD^2... |
What is the value of $\displaystyle\frac{235^2-221^2}{14}$?
Problem:
Given that
$2 \cdot 8!+6 \cdot 6!+4 \cdot 4!+2 \cdot 2!=k \cdot 8!$,
determine the value of $k$. | First, expand $8!$ to $6!$:
\[ 8! = 8 \cdot 7 \cdot 6! \]
Then, rewrite the equation:
\[ 2 \cdot 8 \cdot 7 \cdot 6! + 6 \cdot 6! + 4 \cdot 4! + 2 \cdot 2! = k \cdot 8 \cdot 7 \cdot 6! \]
Next, factor out $6!$:
\[ 6!(2 \cdot 8 \cdot 7 + 6) + 4 \cdot 4! + 2 \cdot 2! = k \cdot 8 \cdot 7 \cdot 6! \]
Now, simplify the ex... |
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