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In triangle $ABC$, right angle at $C$, $\angle BAC < 45^\circ$, and $AB = 6$. Point $P$ is on segment $\overline{AB}$ such that $\angle APC = \frac{3}{2} \angle ACP$ and $CP = 2$. Find the ratio $\frac{AP}{BP}$ in the form $p + q\sqrt{r}$, where $p$, $q$, $r$ are positive integers and $r$ is not divisible by the square... | Let $O$ be the circumcenter of $ABC$, and let the intersection of $CP$ with the circumcircle be $D$. Here $\angle{DOA} = \frac{3}{2} \angle ACP = \angle{APC} = \angle{DPB}$. We then have triangle $ODP$ as isosceles with $OD = DP = 3$ (as $OD = R$ of $\triangle ABC$, $R = 3$).
Denote $E$ as the projection of $O$ onto $... | null | null |
Melanie grew 139 turnips. Benny grew 113 turnips. How many more turnips did Melanie grow than Benny? | Melanie grew 139 turnips and Benny grew 113 turnips. To find out how many more turnips Melanie grew than Benny, we subtract the number of turnips Benny grew from the number of turnips Melanie grew:
139 (Melanie's turnips) - 113 (Benny's turnips) = 26
Melanie grew $\boxed{26}$ more turnips than Benny. | null | null |
Find the sum of all divisors \( d \) of \( N = 19^{88} - 1 \) that are of the form \( d = 2^a \cdot 3^b \) where \( a \) and \( b \) are natural numbers. | Let's break down the given problem and the reference solution step-by-step.
#### Step 1: Express \( N = 19^{88} - 1 \) in terms of its prime factors
We start with:
\[ N = 19^{88} - 1 \]
#### Step 2: Rewrite \( 19 \) in different forms for factorization
We use two different forms of 19 to explore prime factor contribu... | null | null |
Given the sets $M=\{x|x^{2}-2x-3\leqslant 0\}$ and $N=\{x|\log _{2}x > 1\}$, determine the set $M\cap N$.
A: $[-1,2)$
B: $[-1,+\infty)$
C: $(2,3]$
D: $(2,+\infty)$ | First, solve the inequality $x^{2}-2x-3\leqslant 0$ for set $M$.
\begin{align*}
x^{2}-2x-3 &\leqslant 0\\
(x-3)(x+1) &\leqslant 0\\
\end{align*}
Solving this inequality, we obtain $-1\leqslant x\leqslant 3$. Therefore, the set $M$ is given by $M=[-1,3]$.
Next, consider the inequality $\log _{2}x > 1$ from set $N$. Sin... | null | null |
For $y=\frac{1-x}{2x+3}$ and $x\neq-\frac{3}{2}$, what is the value of $y$ that is not attainable? Express your answer as a common fraction. | Given $y=\frac{1-x}{2x+3}$ and $x\neq-\frac{3}{2}$, we aim to find the value of $y$ that is not attainable.
Starting with the given equation:
\[y = \frac{1 - x}{2x + 3}\]
We can multiply both sides by $(2x + 3)$ to eliminate the denominator, leading to:
\[1 - x = (2x + 3)y\]
Expanding the right side:
\[1 - x = 2xy +... | null | null |
Dave was making ice using ice trays. He originally had two ice cubes and made some more cubes. He had a total of 9 ice cubes. How many more ice cubes did he make? | Dave originally had 2 ice cubes and ended up with a total of 9 ice cubes. To find out how many more ice cubes he made, you subtract the original number of ice cubes from the total number of ice cubes.
So, 9 (total ice cubes) - 2 (original ice cubes) = 7 ice cubes.
Dave made $\boxed{7}$ more ice cubes. | null | null |
a person decided to build a house in 100 days . he employed 100 men in the beginning and 100 more after 10 days and completed the construction in stipulated time . if he had not employed the additional men , how many days behind schedule would it have been finished ? | Let's assume that each man works at the same rate and can complete an equal amount of work in a day. We'll call this amount of work "1 unit" per day.
In the original scenario, the person employed 100 men for the entire 100 days and an additional 100 men for the last 90 days.
The total amount of work done by the fir... | null | null |
There are three boxes labeled A, B, and C, one of which contains an apple. Each box has a note attached to it with the following messages:
The note on Box A reads "The apple is in this box,"
The note on Box B reads "The apple is not in this box,"
The note on Box C reads "The apple is not in Box A."
Considering ... | First, let us examine the note on Box A:
- If the note on Box A is truthful and the apple is indeed inside Box A, then the note on Box B would also be true since it claims the apple is not inside Box B. This would result in two notes being true, which contradicts the problem statement that only one note is truthful.
... | null | null |
Is it true that if the areas of four faces of one tetrahedron are equal to the areas of the corresponding faces of another tetrahedron, then the volumes of these tetrahedrons are equal? | ### Problem:
Is it true that if the areas of the four faces of one tetrahedron are equal to the areas of the corresponding faces of another, then the volumes of these tetrahedra are equal?
1. **Setup the problem:**
Consider three line segments that meet at a common vertex, forming a tetrahedron. We will analyze tw... | null | null |
If \( y = -3x^2 + 6 \) and \( x \) is a real number, then what is the maximum value possible for \( y \)? | Given the equation \( y = -3x^2 + 6 \), we recognize this as a downward opening parabola due to the negative coefficient of \( x^2 \). The vertex form of a parabola given by \( y = ax^2 + bx + c \) is \( y = a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola.
Here, \( a = -3 \), \( b = 0 \), and \( c = 6 ... | null | null |
A public official wants to donate 5 new soccer balls per each class in two schools. Each school has 4 elementary school classes and 5 middle school classes. How many soccer balls would the public official donate in all? | To calculate the total number of soccer balls the public official needs to donate, we can follow these steps:
1. First, we determine the total number of classes in each school. Since each school has 4 elementary school classes and 5 middle school classes, the total number of classes per school is:
\[4 + 5 = 9\]
2. Ne... | null | null |
Given the binomial $(2x+\sqrt{x})^{5}$, the coefficient of $x^{3}$ in the expansion is ____. | To find the coefficient of $x^3$ in the expansion of $(2x + \sqrt{x})^5$, we can use the binomial theorem. The general term in the expansion of $(a + b)^n$ is given by $C_n^k a^{n-k}b^k$, where $C_n^k$ is the binomial coefficient $\frac{n!}{k!(n-k)!}$.
For the term containing $x^3$, we need to find the appropriate val... | null | null |
Find all roots of the polynomial $x^3-5x^2+3x+9$. Enter your answer as a list of numbers separated by commas. If a root occurs more than once, enter it as many times as its multiplicity. | To find all roots of the polynomial $x^3 - 5x^2 + 3x + 9$, we can follow a systematic approach:
1. **Application of the Rational Root Theorem**: This theorem suggests that any rational root of the polynomial, in its simplest form, is a factor of the constant term divided by a factor of the leading coefficient. For our... | null | null |
If point P lies on the line $l_1: x + y + 3 = 0$, and the line $l_2$ passing through P is tangent to the curve $C: (x - 5)^2 + y^2 = 16$ at point M, then the minimum value of $|PM|$ is ( ).
A: $\sqrt {2}$
B: 2
C: $2 \sqrt {2}$
D: 4 | Since we want to minimize $|PM|$, we need to find the position of point P such that its distance to the center of the circle (which is at the point (5, 0)) is minimal. Let point P have coordinates $(m, -m - 3)$.
The minimum distance from point P to the center of the circle (5, 0), is the perpendicular distance $d$ fro... | null | null |
Determine the type of conic section described by the equation \[|y - 3| = \sqrt{(x+4)^2 + (y-1)^2}.\]
Enter "C" for circle, "P" for parabola, "E" for ellipse, "H" for hyperbola, and "N" for none of the above. | To begin, write the given equation in one of the recognizable standard forms for conic sections. Noting that both sides of the equation are nonnegative, it is appropriate to square both sides of the equation:
\[ (y - 3)^2 = (x + 4)^2 + (y - 1)^2. \]
Expanding both sides:
\[ y^2 - 6y + 9 = x^2 + 8x + 16 + y^2 - 2y + 1.... | null | null |
Given point $P$ is on the circle $x^{2}+y^{2}-4x-4y+7=0$, and point $Q$ is on the line $y=kx$, if the minimum value of $|PQ|$ is $2\sqrt{2}-1$, then $k=$ .
A: $1$
B: $-1$
C: $0$
D: $2$ | Transform the equation of the circle into its standard form to get: $(x-2)^{2}+(y-2)^{2}=1$,
$\therefore$ the coordinates of the center of the circle are $(2,2)$, and the radius $r=1$,
$\because$ the distance $d$ from the center of the circle to the line $y=kx$ is $d= \frac{|2k-2|}{\sqrt{1+k^{2}}}$, and the minimum... | null | null |
Let $a$, $b$, $c$ be the roots of $x^3 + px^2 + qx + r = 0$. Express
\[
\begin{vmatrix} 1 + a^2 & 1 & 1 \\ 1 & 1 + b^2 & 1 \\ 1 & 1 & 1 + c^2 \end{vmatrix}
\]
in terms of $p$, $q$, and $r$. | Let's expand the determinant:
\[
\begin{vmatrix} 1 + a^2 & 1 & 1 \\ 1 & 1 + b^2 & 1 \\ 1 & 1 & 1 + c^2 \end{vmatrix} = (1 + a^2) \begin{vmatrix} 1 + b^2 & 1 \\ 1 & 1 + c^2 \end{vmatrix} - \begin{vmatrix} 1 & 1 \\ 1 & 1 + c^2 \end{vmatrix} + \begin{vmatrix} 1 & 1 + b^2 \\ 1 & 1 \end{vmatrix}
\]
Calculating the two small... | null | null |
Quinton is looking to add 4 fruit trees to his backyard. He wants to plant 2 apple trees that will be 10 feet wide each and need 12 feet between them. The peach trees will be closer to the house and will grow to be 12 feet wide and will need 15 feet between each tree. All told, how much space will these trees take u... | To calculate the total space needed for the fruit trees in Quinton's yard, we'll break down the space requirements for the apple trees and the peach trees separately.
For the apple trees:
- Each apple tree will be 10 feet wide at maturity. Since Quinton wants to plant 2 apple trees, the total width for the apple trees... | null | null |
The average salary of all the workers in a workshop is a certain amount. The average salary of 7 technicians is $12000, and the average salary of the rest is $6000. The total number of workers in the shop is 21. What is the average salary of all the workers in the workshop? | Let's denote the average salary of all the workers in the workshop as A.
We know that there are 7 technicians with an average salary of $12000 each, so the total salary for all technicians is:
7 technicians * $12000/technician = $84000
The rest of the workers (21 total workers - 7 technicians) equals 14 workers, and ... | null | null |
In a square array of points with 4 rows and $N$ columns, the points are numbered consecutively from left to right beginning with the top row. The top row is numbered 1 through $N$, the second row is numbered $N + 1$ through $2N$, and so forth. Four points $P_1, P_2, P_3, P_4$ are selected so that each $P_i$ is in row $... | Let each point $P_i$ be in column $c_i$. Define the numbering for $P_i$:
\[ x_i = (i - 1)N + c_i \]
\[ y_i = (c_i - 1)4 + i \]
Given relationships are then:
\[ x_1 = y_3 \Longrightarrow N + c_1 = 4c_3 - 2 \]
\[ x_2 = y_1 \Longrightarrow N + c_2 = 4c_1 - 3 \]
\[ x_3 = y_4 \Longrightarrow 2N + c_3 = 4c_4 - 1 \]
\[ x_4 =... | null | null |
Given a parabola $y^2=2px$ and a point $M(1, m)$ on the parabola, the distance from $M$ to its focus is 5. Determine the equation of the directrix of the parabola. | The standard form of the parabola is $y^2 = 2px$.
Hence, the focus of the parabola is $F\left(\frac{p}{2}, 0\right)$, and the equation of the directrix is $x = -\frac{p}{2}$.
Given that the distance from point $M(1, m)$ to the focus is 5, we can write the distance formula as,
\[
\sqrt{(1 - \frac{p}{2})^2 + (m - 0)^2}... | null | null |
Mary has 400 sheep on her farm. She gave a quarter of her sheep to her sister, and half of the remaining sheep to her brother. How many sheep remain with Mary? | To solve the problem step by step, we start with the total number of sheep Mary has:
1. Mary starts with 400 sheep.
2. She gives a quarter of her sheep to her sister. This is calculated as:
\[
\frac{1}{4} \times 400 = 100
\]
So, Mary gives away 100 sheep to her sister.
3. After giving 100 sheep to her sis... | null | null |
Eight people can paint a wall in 3 hours. How many additional people are required to complete the painting in 2 hours, assuming each person paints at the same rate? | First, we establish the relationship that the number of people, \( n \), and the time, \( t \), are inversely proportional, so \( nt = k \). Given that 8 people can paint the wall in 3 hours, we have:
\[ 8 \times 3 = 24 \]
This tells us that \( k = 24 \).
Now, we want to find how many people, \( m \), are needed to pa... | null | null |
If the vertex of the parabola $y=x^{2}+2x+c$ lies on the $x$-axis, then $c=$____. | To solve this problem, we start by understanding that the vertex of a parabola given by $y=ax^2+bx+c$ can be found using the formula $y=\frac{4ac-b^2}{4a}$. In our case, the parabola is $y=x^2+2x+c$, which means $a=1$, $b=2$, and we are looking for $c$.
Given that the vertex lies on the $x$-axis, the $y$-coordinate of... | null | null |
Arrange the numbers $1,2,3, \cdots, 37$ in a sequence $a_{1}, a_{2}, \cdots, a_{37}$, where $a_{1}=37$ and $a_{2}=1$, such that $a_{1}+a_{2}+\cdots+a_{k}$ is divisible by $a_{k+1}$ for $k=1,2, \cdots, 36$. Find $a_{37}$ and $a_{3}$. |
1. Given that we need to arrange the numbers \(1, 2, 3, \ldots, 37\) in a sequence \(a_1, a_2, \ldots, a_{37}\), with \(a_1 = 37\) and \(a_2 = 1\), such that the sum of the first \(k\) terms can be divided by the \((k+1)\)-th term for \(k = 1, 2, \ldots, 36\).
2. We start by considering \(a_{37}\). Based on the condi... | null | null |
Given that the sequences $\{a\_n\}$ and $\{b\_n\}$ are an arithmetic sequence and a geometric sequence, respectively, and satisfy $a\_3=8$, $a\_6=17$, $b\_1=2$, and $b\_1b\_2b\_3=9(a\_2+a\_3+a\_4)$.
1. Find the general terms of the sequences $\{a\_n\}$ and $\{b\_n\}$.
2. Let $c\_n=\log \_3 b\_n$. Prove that the sequen... | 1. Since $\{a\_n\}$ is an arithmetic sequence, with $a\_3=8$ and $a\_6=17$,
we have $d=\frac{a\_6-a\_3}{6-3}=\frac{17-8}{3}=3$.
Thus, $a\_n=a\_3+(n-3)d=8+3(n-3)=3n-1$.
It follows that $a\_2+a\_3+a\_4=3a\_3=24$.
Since $\{b\_n\}$ is a geometric sequence, let the common ratio be $q$, with $b\_1=2$,
we have $b\_1b\_2b\_3=2... | null | null |
A cylindrical oil tank, lying horizontally, has an interior length of $10$ feet and an interior diameter of $6$ feet.
If the rectangular surface of the oil has an area of $40$ square feet, the depth of the oil is:
$\textbf{(A)}\ \sqrt{5}\qquad\textbf{(B)}\ 2\sqrt{5}\qquad\textbf{(C)}\ 3-\sqrt{5}\qquad\textbf{(D)}\ 3+... | To find the depth of the oil in the cylindrical tank, we need to understand the geometry of the situation. The tank is lying horizontally, and the oil forms a segment of the circle at the bottom of the cylinder.
1. **Identify the dimensions of the tank and the oil surface area:**
- The tank has an interior length o... | null | null |
Travis goes through 2 boxes of cereal a week. If each box costs $3.00 and he eats 2 boxes every week for an entire year, 52 weeks, how much does he spend on cereal? | To calculate the total cost Travis spends on cereal in a year, we follow these steps:
1. Determine the total number of boxes Travis eats in a year. Since he eats 2 boxes a week for 52 weeks, we calculate the total boxes as:
\[
2 \, \text{boxes/week} \times 52 \, \text{weeks} = 104 \, \text{boxes}
\]
2. Next,... | null | null |
Penny has $20. Penny buys 4 pairs of socks for $2 a pair and a hat for $7. How much money does Penny have left? | First, let's calculate the total cost of the socks. Penny buys 4 pairs of socks at $2 per pair, so the cost is:
4 pairs * $2/pair = $8
Next, we add the cost of the hat, which is $7:
$8 (socks) + $7 (hat) = $15
Now, we subtract the total cost of the purchases from Penny's initial amount of money:
$20 (initial amoun... | null | null |
Jill bought 5 packs of red bouncy balls and 4 packs of yellow bouncy balls. Each package contained 18 bouncy balls. How many more red bouncy balls than yellow bouncy balls did Jill buy? | To calculate the number of red bouncy balls Jill bought, we multiply the number of red ball packs by the number of balls in each pack:
\[5 \text{ packs} \times 18 \text{ balls/pack} = 90 \text{ red balls}\]
Similarly, for the yellow bouncy balls, we multiply the number of yellow ball packs by the number of balls in ea... | null | null |
At Valley Forge High School, there are 1500 students, out of which 900 students participate in a winter carnival. If three-fourths of the girls and two-thirds of the boys at the school participate in the carnival, how many girls participate in the carnival? | Let the number of girls at VFHS be \( g \) and the number of boys be \( b \). From the problem statement, we have:
1. \( g + b = 1500 \) (Total students)
2. \( \frac{3}{4}g + \frac{2}{3}b = 900 \) (Students at the carnival)
To solve these equations, first multiply equation 1 by 3 to eliminate fractions in the second e... | null | null |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. If $c \cdot \cos B = a + \frac{1}{2}b$ and the area of $\triangle ABC$ is $S = \frac{\sqrt{3}}{12} c$, find the minimum value of side $c$. | Given the conditions in $\triangle ABC$, we use the law of sines to obtain:
$$\sin C \cos B = \sin A + \frac{1}{2} \sin B = \sin (B + C) + \frac{1}{2} \sin B$$
This simplifies to:
$$2 \sin C \cos B = 2 \sin B \cos C + 2 \sin C \cos B + \sin B$$
Thus,
$$2 \sin B \cos C + \sin B = 0$$
$$\cos C = -\frac{1}{2}$$
$$C... | null | null |
When Julia was born, her parents deposited $\$1500$ in a savings account that yields an annual interest rate of $4\%$, compounded annually. How much money will Julia have in the account when she turns $21$? | - Let \( P = \$1500 \), \( r = 0.04 \), and \( n = 21 \) years.
- Using the compound interest formula, \( A = P(1 + r)^n \).
- Compute \( A = 1500(1 + 0.04)^{21} \).
Calculation:
\[ A = 1500(1.04)^{21} \]
\[ A = 1500 \times 2.03085619 \] (using a calculator for \( 1.04^{21} \))
\[ A = 3046.28 \]
The future value of ... | null | null |
Calculate as required.
(1) The lateral area of a cylinder with a base circumference of 1.8 meters and a height of 1.5 meters.
(2) The volume of a cylinder with a diameter of 3dm and a height of 8cm.
(3) The surface area of a cylinder with a base radius of 6 decimeters and a height of 5 decimeters.
(4) A cone ha... | **Solution**:
(1) $1.8 \times 1.5 = 2.7$ (square meters)
Answer: The lateral area of the cylinder is $\boxed{2.7 \text{ square meters}}$.
(2) $3$ decimeters $= 30$ centimeters
$3.14 \times (30 \div 2)^2 \times 8$
$= 3.14 \times 225 \times 8$
$= 5652$ (cubic centimeters)
Answer: The volume of the cylind... | null | null |
The sides of a triangle are \(\sqrt{a^{2}-a+1}, \sqrt{a^{2}+a+1}, \sqrt{4 a^{2}+3}\), where \(a\) is a real number. Prove that the triangle has a height which measures at most \(\frac{1}{2}\). |
To solve this problem, we will analyze the triangle with sides \(\sqrt{a^2 - a + 1}\), \(\sqrt{a^2 + a + 1}\), and \(\sqrt{4a^2 + 3}\), where \(a\) is a real number, and prove that the height corresponding to the largest side of the triangle is at most \(\frac{1}{2}\).
#### Step 1: Identify the largest side
Since for... | null | null |
In an electric circuit, three resistors with resistances 2 ohms, 5 ohms, and 6 ohms are connected in parallel. In this case, if r is the combined resistance of these three resistors, then the reciprocal of r is equal to the sum of the reciprocals of the resistors. What is the value of r? | When resistors are connected in parallel, the total or equivalent resistance (r) can be found using the formula:
1/r = 1/R1 + 1/R2 + 1/R3
where R1, R2, and R3 are the resistances of the individual resistors.
Given the resistances of the three resistors as 2 ohms, 5 ohms, and 6 ohms, we can plug these values into the... | null | null |
Segments \(AA'\), \(BB'\), and \(CC'\) with endpoints on the sides of an acute-angled triangle \(ABC\) intersect at a point \(P\) inside the triangle. On each of these segments, a circle is constructed with the segment as a diameter, and a chord passing through point \(P\) is drawn perpendicular to this diameter. It tu... |
1. **Introduction of given elements and notation**: We are given that segments \( AA' \), \( BB' \), and \( CC' \) with endpoints on the sides of acute triangle \( ABC \) intersect at point \( P \) inside the triangle. There is a circle on each of these segments as a diameter, with a chord passing through point \( P \... | null | null |
Inside the triangle \(ABC\), a point \(D\) is chosen such that \(\angle BAD = 60^\circ\) and \(\angle ABC = \angle BCD = 30^\circ\). It is known that \(AB = 15\) and \(CD = 8\). Find the length of segment \(AD\). If necessary, round the answer to 0.01 or express the answer as a common fraction. |
**Solution I:**
1. **Identify Points and Angles:**
Given:
- Triangle $ABC$
- Point $D$ inside the triangle such that $\angle BAD = 60^\circ$ and $\angle ABC = \angle BCD = 30^\circ$
- $AB = 15$ and $CD = 8$
2. **Intersection and Angle Analysis:**
Let the line segment $AD$ intersect segment $BC$ at poi... | null | null |
How many integers \( n \) satisfy the inequality \( -5\pi \le n \le 7\pi \)? | We approximate \(\pi\) as 3.14 to calculate bounds of the inequality \( -5\pi \le n \le 7\pi \).
- Calculate lower bound: \( -5 \times 3.14 = -15.7 \). The closest integer is \( -15 \)
- Calculate upper bound: \( 7 \times 3.14 = 21.98 \). The closest integer is \( 22 \)
Hence, the integers \( n \) must satisfy \( -15 ... | null | null |
Given a real number sequence $\{a_n\}$ satisfying $a_{n+2}=|a_{n+1}|-a_n$ ($n=1,2,\ldots$), with $a_1=a$ and $a_2=b$, denote the set $M=\{a_n|n\in \mathbb{N^*}\}$.
(I) If $a=1$ and $b=2$, list the elements of set $M$ explicitly.
(II) If $a<0$ and $b<0$, determine whether the sequence $\{a_n\}$ is periodic and explain... | (I) Since $a_1=a$, $a_2=b$, and $a_{n+2}=|a_{n+1}|-a_n$,
We have:
$$a_3 = |1| - 2 = -1,$$
$$a_4 = |-1| - 1 = 0,$$
$$a_5 = |0| - (-1) = 1,$$
$$a_6 = |1| - 0 = 1,$$
$$a_7 = |1| - 1 = 0.$$
Therefore, for $n \geq 5$, $a_{n+3} = a_n$.
**Hence, $M=\{1, 2, -1, 0\}$**.
(II) When $a<0$ and $b<0$, the sequence begins as fo... | null | null |
Find the greatest number which, on dividing a certain number and 5161, leaves a remainder of 8 and 10 respectively. The greatest number is 101. What is the first number? | Let's call the greatest number G and the certain number N.
We are given that G = 101.
We are also given that when N is divided by G, the remainder is 8, and when 5161 is divided by G, the remainder is 10.
So we can write the following equations:
N = G * k + 8 (where k is some integer)
5161 = G * m + 10 (where m is ... | null | null |
Express as a common fraction: $0.\overline5+0.\overline1-0.\overline3$ | To express a repeating decimal like $0.\overline{n}$ as a fraction, we start by letting $x = 0.\overline{n}$. Then, we multiply both sides of the equation by 10 to shift the decimal point, resulting in $10x = n.\overline{n}$. Subtracting the original equation from this new equation gives us:
\[
\begin{aligned}
&\phant... | null | null |
Five students are lined up in a row, where A and B must be together, and C and D cannot be together. The number of ways to arrange them is. | According to the problem, first consider A and B as one "element", which can be arranged in $2$ different ways.
Arrange C and D separately, which also can be arranged in $2$ different ways.
If only one of A, B, and the fifth element is between C and D, there are $2 \times \binom{2}{1} = 4$ cases.
If both A, B, and t... | null | null |
Tommy's home is worth 25% more than he bought it for. He sells it and buys a new house that costs $500,000. If he had to take a loan for the 75% he can't pay, how much did he buy his first house for? | Let's denote the price Tommy bought his first house for as P.
Tommy's home is now worth 25% more than P, so the current value of his home is P + 0.25P, which equals 1.25P.
When Tommy sells his home, he uses the proceeds to buy a new house that costs $500,000. He takes a loan for 75% of the new house's cost, which mea... | null | null |
Given $a^{m}=3$ and $a^{n}=2$, find $a^{2m+3n}$. | Given $a^{m}=3$ and $a^{n}=2$, we are asked to find $a^{2m+3n}$. Let's break this down step by step:
1. **Express $a^{2m+3n}$ in terms of $a^{m}$ and $a^{n}$:**
\[
a^{2m+3n} = a^{2m} \cdot a^{3n}
\]
This step uses the property of exponents that $a^{b+c} = a^b \cdot a^c$.
2. **Substitute $a^{m}$ and $a^{n}... | null | null |
If $\left\lfloor n^2/4 \right\rfloor - \lfloor n/2 \rfloor^2 = 2$, then find all integer values of $n$. | To solve the given problem, we analyze the cases based on the parity of $n$ (whether $n$ is even or odd).
**Case 1: $n$ is even.**
Let's represent $n$ as $2m$, where $m$ is an integer. This substitution is made because if $n$ is even, it can be expressed as twice some integer. Substituting $n = 2m$ into the given equ... | null | null |
The motion equation of an object is $s=- \frac {1}{3}t^{3}+2t^{2}-5$, then the instantaneous velocity of the object at $t=3$ is ______. | First, we find the derivative of the displacement, $s'=-t^{2}+4t$.
Therefore, the instantaneous velocity of the object at $t=3$ is $-3^{2}+4\times3=3$.
Hence, the answer is $\boxed{3}$.
This problem examines the concept that the derivative of an object's displacement represents the object's instantaneous velocity... | null | null |
Given that $-8$ is a solution to the equation $x^2 + bx - 15 = 0$, find the value of $b$. | Since $-8$ is a root of the equation $x^2 + bx - 15 = 0$, the product of the roots, according to Vieta's formulas, is $c/a=-15/1=-15$. Let the other root be $r$. Then, $-8 \cdot r = -15$, which simplifies to $r = \frac{-15}{-8} = \frac{15}{8}$.
Now, considering the sum of the roots which is also given by Vieta's formu... | null | null |
The graph of $ y \equal{} 2x^2 \plus{} 4x \plus{} 3$ has its:
$ \textbf{(A)}\ \text{lowest point at } {(\minus{}1,9)}\qquad
\textbf{(B)}\ \text{lowest point at } {(1,1)}\qquad
\textbf{(C)}\ \text{lowest point at } {(\minus{}1,1)}\qquad
\textbf{(D)}\ \text{highest point at } {(\minus{}1,9)}\qquad
\textbf{(... | 1. The given function is \( y = 2x^2 + 4x + 3 \). This is a quadratic function of the form \( y = ax^2 + bx + c \), where \( a = 2 \), \( b = 4 \), and \( c = 3 \).
2. Since \( a = 2 > 0 \), the parabola opens upwards, indicating that it has a minimum point (vertex).
3. The x-coordinate of the vertex of a parabola gi... | null | null |
Before the Beijing Olympics began, the organizing committee needed to select 6 volunteers from 8 to go to the Olympic venues "Bird's Nest" and "Water Cube" for on-site training, 3 people for each venue. Among them, volunteers A and B cannot be in the same group, and B cannot go to the "Water Cube". The number of differ... | If neither A nor B is selected, then the number of arrangements is $C_{6}^{3}C_{3}^{3}$.
If A is selected but B is not, then the number of arrangements is $C_{6}^{2}C_{4}^{3}A_{2}^{2}$.
If A is not selected but B is, then the number of arrangements is $C_{6}^{2}C_{4}^{3}$.
If both A and B are selected, then the n... | null | null |
A rectangle with an inscribed square is shown. The rectangle has dimensions 10 units by 8 units, and the side length of the square is 5 units. Calculate the area of the region not covered by the square. | 1. Compute the area of the rectangle:
\[
\text{Area of the rectangle} = 10 \times 8 = 80 \text{ square units}
\]
2. Compute the area of the square:
\[
\text{Area of the square} = 5^2 = 25 \text{ square units}
\]
3. Find the area of the region not covered by the square:
\[
\text{Area not covere... | null | null |
Given that $\left\{a_{n}\right\}$ is an arithmetic sequence with a nonzero common difference and $\left\{b_{n}\right\}$ is a geometric sequence where $a_{1}=3, b_{1}=1, a_{2}=b_{2}, 3a_{5}=b_{3}$, and there exist constants $\alpha$ and $\beta$ such that for every positive integer $n$, $a_{n} = \log_{\alpha} b_{n} + \be... | :
1. Let the common difference of the arithmetic sequence \( \{a_n\} \) be \( d \), and the common ratio of the geometric sequence \( \{b_n\} \) be \( q \).
2. Given:
\[
a_1 = 3, \quad b_1 = 1, \quad a_2 = b_2, \quad 3a_5 = b_3
\]
3. Since \( \{a_n\} \) is an arithmetic sequence:
\[
a_2 = a_1 + ... | null | null |
Greg drives 30 miles from his workplace to the farmer's market. After buying his groceries at the farmers market, he drives home. To get home, he travels for 30 minutes at 20 miles per hour. How many miles in total does Greg travel? | To solve this problem, we follow these steps:
1. Convert the travel time from minutes to hours for the trip home. Since there are 60 minutes in an hour, we calculate:
\[30 \text{ minutes} \times \left(\frac{1 \text{ hour}}{60 \text{ minutes}}\right) = 0.5 \text{ hours}\]
2. Next, we find out how many miles Greg trave... | null | null |
The proposition "Two lines parallel to the same line are parallel" is a ______ proposition. (Fill in "true" or "false") | Given the proposition: "Two lines parallel to the same line are parallel."
Step 1: Recall the parallel lines axiom which states that if two lines are parallel to the same line, then they are parallel to each other.
Step 2: Apply this axiom to the given proposition.
Therefore, the proposition "Two lines parallel to t... | null | null |
Let \(x\) be an angle such that \(\tan x = \frac{2}{3}\) and \(\tan 3x = \frac{3}{5}\). Then the least positive value of \(x\) equals \(\tan^{-1} k\). Compute \(k\). | We start by expressing \(\tan 3x\) in terms of \(\tan x\) using the triple angle formula:
\[\tan 3x = \frac{3\tan x - \tan^3 x}{1 - 3\tan^2 x}\]
Plugging in \(\tan x = \frac{2}{3}\), we get:
\[\tan 3x = \frac{3\left(\frac{2}{3}\right) - \left(\frac{2}{3}\right)^3}{1 - 3\left(\frac{2}{3}\right)^2} = \frac{2 - \frac{8}{2... | null | null |
Given the function $f(x) = 0.3^x - \log_{2}x$, if $f(a)f(b)f(c) > 0$ and $a, b, c$ are consecutive terms of an arithmetic sequence with a positive common difference, and $x$ is a zero of the function $y = f(x)$, then the following relationship must not hold:
A: $x > b$
B: $x < b$
C: $x > c$
D: $x < a$ | The function $f(x) = 0.3^x - \log_{2}x$ is composed of $y = 0.3^x$ and $y = -\log_{2}x$, both of which are decreasing functions. Therefore, the composite function is a decreasing function.
Since the positive real numbers $a, b, c$ form an arithmetic sequence with a positive common difference, we have $0 < a < b < c$.
... | null | null |
A pelican caught 13 fish and a kingfisher caught 7 more fish than the pelican. If a fisherman caught 3 times the total amount of fish the pelican and kingfisher caught, how many more fish did the fisherman catch than the pelican? | First, let's find out how many fish the kingfisher caught. Since the kingfisher caught 7 more fish than the pelican, and the pelican caught 13 fish, the kingfisher caught:
13 fish (pelican) + 7 fish = 20 fish (kingfisher)
Now, let's find the total number of fish caught by both the pelican and the kingfisher:
13 fish... | null | null |
When Erick went to the market to sell his fruits, he realized that the price of lemons had risen by $4 for each lemon. The price of grapes had also increased by half the price that the price of lemon increased by per grape. He had planned to sell the lemons at $8 and the grapes at some price, and he had 80 lemons and 1... | Let's denote the original price Erick planned to sell each grape for as \( P \) dollars.
The new price for each lemon is \( $8 + $4 = $12 \).
The price increase for each grape is half the price increase of each lemon, which is \( \frac{1}{2} \times $4 = $2 \).
So, the new price for each grape is \( P + $2 \).
Erick... | null | null |
A right triangle has a side length of 15 inches and a hypotenuse of 34 inches. A second triangle is similar to the first and has a hypotenuse of 102 inches. What is the length of the shortest side of the second triangle? | To find the unknown side of the first triangle, we use the Pythagorean theorem.
Let \( x \) be the length of the other leg. Then:
\[ x^2 + 15^2 = 34^2 \]
\[ x^2 + 225 = 1156 \]
\[ x^2 = 1156 - 225 \]
\[ x^2 = 931 \]
\[ x = \sqrt{931} \]
Since the second triangle is similar to the first, and the hypotenuse of the secon... | null | null |
Jack cycles from his home to the store. Then he cycles, at the same speed, 50 miles to his friend Peter. It takes Jack twice as long to go from his home to the store as it takes to go from the store to his friend. If Peter and Jack now cycle back to the store together, how many miles in total have they cycled today? | To solve this problem, let's break down the journey into parts and calculate the distance for each part:
1. **From Home to the Store**:
- The problem states it takes Jack twice as long to go from his home to the store as it takes to go from the store to his friend. Since the distance from the store to his friend i... | null | null |
Twice Angie's age, plus 4, is 20. How old is Angie? | To solve for Angie's age, we start with the given equation:
\[2 \times \text{Angie's age} + 4 = 20\]
Subtracting 4 from both sides to isolate the term involving Angie's age gives us:
\[2 \times \text{Angie's age} = 20 - 4\]
\[2 \times \text{Angie's age} = 16\]
Dividing both sides by 2 to solve for Angie's age, we ... | null | null |
Determine the greatest common divisor of 15225, 20335, and 35475. | To solve this problem, we'll use the Euclidean algorithm. First, we find the gcd of 15225 and 20335. Then, we'll extend this result to include 35475.
1. Compute $\text{gcd}(15225, 20335)$:
\[
\text{gcd}(15225, 20335) = \text{gcd}(15225, 20335 - 15225) = \text{gcd}(15225, 5110).
\]
Repeat the process:
\[... | null | null |
Your friend sitting to your left (or right?) is unable to solve any of the eight problems on his or her Combinatorics $B$ test, and decides to guess random answers to each of them. To your astonishment, your friend manages to get two of the answers correct. Assuming your friend has equal probability of guessing each ... | 1. Let's denote the total number of questions as \( n = 8 \).
2. Each question has an equal probability of being guessed correctly, which is \( p = \frac{1}{2} \).
3. The number of correct answers is given as \( k = 2 \).
We need to find the expected score of your friend. Each question has a different point value, and... | null | null |
A group of researchers are studying a colony of penguins. Their results show that the size doubled in the first year they watched the colony, then tripled in the next year. This year, however, harsh conditions meant the colony only gained 129 penguins. The colony has 1077 penguins in it now. Each penguin eats one and a... | To find out how many fish the whole colony caught per day at the beginning of the first year, we follow these steps:
1. Calculate the number of penguins at the beginning of this year:
\[1077 - 129 = 948\]
2. Determine the number of penguins at the beginning of the second year by reversing the tripling growth:
\[948 \... | null | null |
If $x \cdot (x+y) = x^2 + 8$, what is the value of $xy$? | To solve for $xy$, we start by distributing the $x$ across the parentheses on the left-hand side of the equation:
\[x \cdot (x+y) = x^2 + xy.\]
This gives us the equation:
\[x^2 + xy = x^2 + 8.\]
To isolate $xy$, we subtract $x^2$ from both sides of the equation:
\[xy = 8.\]
Therefore, the value of $xy$ is $\boxe... | null | null |
Consider the numbers formed by the digits in $\underline{9}\underline{1}\underline{A}\underline{4}\underline{3}\underline{B}\underline{2}$ and $\underline{6}\underline{A}\underline{B}\underline{4}\underline{5}\underline{C}\underline{2}$ are each multiples of $3$ and $5$, respectively. Determine the possible values of $... | 1. **Analyze the first number $\underline{9}\underline{1}\underline{A}\underline{4}\underline{3}\underline{B}\underline{2}$ for divisibility by 3:**
- Sum of known digits: $9 + 1 + 4 + 3 + 2 = 19$.
- Let $S'_1 = A + B$.
- Total sum of digits: $19 + S'_1$.
- For divisibility by $3$, we need $S'_1 \equiv (19 ... | null | null |
$P(x)$ and $Q(x)$ are monic quadratic polynomials, each having two distinct roots. It turns out that the sum of the two values obtained by substituting the roots of the polynomial $P(x)$ into the polynomial $Q(x)$ is equal to the sum of the two values obtained by substituting the roots of the polynomial $Q(x)$ into the... |
Let's denote the roots of the polynomial \( P(x) \) by \( a_1 \) and \( a_2 \), and the roots of the polynomial \( Q(x) \) by \( b_1 \) and \( b_2 \).
#### First Method:
1. Given polynomials in the factored form:
\[
P(x) = (x - a_1)(x - a_2)
\]
\[
Q(x) = (x - b_1)(x - b_2)
\]
2. By substituting th... | null | null |
In a high school with $500$ students, $40\%$ of the seniors play a musical instrument, while $30\%$ of the non-seniors do not play a musical instrument. In all, $46.8\%$ of the students do not play a musical instrument. How many non-seniors play a musical instrument?
$\textbf{(A) } 66 \qquad\textbf{(B) } 154 \qquad\tex... | Let's denote the number of seniors as $s$ and the number of non-seniors as $n$. Since there are $500$ students in total, we have:
\[ s + n = 500 \]
From the problem, $40\%$ of the seniors play a musical instrument, which implies that $60\%$ of the seniors do not play a musical instrument. Similarly, $30\%$ of the non-... | null | null |
Every year 20 ducks from a flock are killed but another 30 are born. The original flock size is 100 ducks. After 5 years they join with another flock of 150 ducks. How many ducks are in the combined flock? | To solve this problem, let's break it down step by step:
1. **Annual Increase in Ducks**: Every year, the flock gains ducks due to births and loses some due to deaths. The net gain per year is calculated as the difference between the number of ducks born and the number of ducks killed:
\[
\text{Net gain per year... | null | null |
A side of beef lost 40 percent of its weight in processing. If the side of beef weighed 240 pounds after processing, how many pounds did it weigh before processing? | If the side of beef weighed 240 pounds after processing, and this weight represents 60% of its original weight (since it lost 40%), we can set up the following equation to find the original weight (OW):
60% of OW = 240 pounds
To find the original weight, we can convert the percentage to a decimal and solve for OW:
0... | null | null |
Mike needs 30% to pass. He scored 212 marks and falls short by some marks. The maximum marks he could have got was 760. By how many marks did Mike fall short? | To pass, Mike needs to score 30% of the maximum marks. Let's calculate 30% of 760:
30% of 760 = 0.30 * 760 = 228 marks
Mike scored 212 marks, so to find out by how many marks he fell short, we subtract the marks he scored from the marks he needed to pass:
228 marks (needed to pass) - 212 marks (scored) = 16 marks
M... | null | null |
How many times do the graphs of the line $5x + 8y = 10$ and the circle $x^2 + y^2 = 1$ intersect? | To find how many times the graphs of the line $5x + 8y = 10$ and the circle $x^2 + y^2 = 1$ intersect, we first express $y$ in terms of $x$ from the equation of the line. This gives us:
\[5x + 8y = 10 \implies 8y = 10 - 5x \implies y = \frac{10 - 5x}{8}.\]
Next, we substitute this expression for $y$ into the equation... | null | null |
Given sets $A=\{x|\log _{2}x\leqslant 1\}$ and $B=\{x|y=\sqrt{3-x}\}$, then which of the following is true?
A: $A$⋃$B=R$
B: $A$⋃$B=A$
C: $A$⋂$B=A$
D: $A$⋂$B=B$ | To solve the problem, we first need to understand the definitions of sets $A$ and $B$ and then determine their relationship, which will help us identify the correct choice among the options provided.
**Step 1: Understanding Set $A$**
Given $A=\{x|\log _{2}x\leqslant 1\}$, we can rewrite this inequality in exponential... | null | null |
A sum of Rs. 2665 is lent into two parts so that the interest on the first part for 5 years at 3% per annum may be equal to the interest on the second part for 3 years at 5% per annum. Find the second sum. | Let the first part of the sum be Rs. x and the second part be Rs. (2665 - x).
According to the problem, the interest on the first part for 5 years at 3% per annum is equal to the interest on the second part for 3 years at 5% per annum.
Interest on the first part for 5 years at 3% per annum = \( \frac{x \times 5 \time... | null | null |
Find the sum of all positive integers such that their expression in base $9$ digits is the reverse of their expression in base $12$ digits. Express your answer in base $10$. | Let the given base $9$ number be \( n \). Suppose \( n \) has \( d+1 \) digits in either base $9$ or base $12$. For digits in base $9$, denote them as \( a_i \) and in base \( 12 \) denote as \( b_i \), with \( b_i \) being the digit that matches the reversed position in base $9$. Therefore, \( n \) in both bases is gi... | null | null |
Vins rides his bike 6 miles to school. He rides home a different route that is 7 miles long. This week, Vins rode to school and back 5 times. How many miles did Vins ride his bike this week? | To solve this problem, we break it down into smaller steps:
1. First, calculate the total distance Vins rides in one day. He rides 6 miles to school and then 7 miles back home. So, the total distance for one day is:
\[6 \text{ miles} + 7 \text{ miles} = 13 \text{ miles}\]
2. Next, since Vins rides to school and back ... | null | null |
If four times the reciprocal of the circumference of a circle equals the diameter of the circle, then the area of the circle is
$\textbf{(A) }\frac{1}{\pi^2}\qquad \textbf{(B) }\frac{1}{\pi}\qquad \textbf{(C) }1\qquad \textbf{(D) }\pi\qquad \textbf{(E) }\pi^2$ | 1. **Formulate the given condition**: The problem states that four times the reciprocal of the circumference of a circle equals the diameter of the circle. Let the radius of the circle be $r$. The circumference of the circle is $2\pi r$, and the diameter is $2r$. Thus, the equation becomes:
\[
4 \cdot \frac{1}{2\... | null | null |
Every pack of Sweeties (a gummy candy) contains the same number of candies.
The Sweeties in one pack can't be divided equally among $12$ children because after each child receives the same (whole) number of candies, $11$ candies are left over.
If the Sweeties in four packs are divided equally among $12$ children, wha... | Let $m$ be the number of Sweeties in one pack. Then, we know $m \equiv 11 \pmod{12}$ which means $m = 12k + 11$ for some integer $k$.
Consider four packs of Sweeties:
$$ 4m = 4(12k + 11) = 48k + 44 $$
Now, divide $48k + 44$ by $12$:
$$ 48k + 44 \equiv 44 \pmod{12} $$
Since $44 \text{ div } 12$ leaves a remainder, we ... | null | null |
Find $r$ such that $\log_{81} (2r-1) = -1/2$. | To solve the equation $\log_{81} (2r-1) = -1/2$, we first convert the logarithmic equation into its equivalent exponential form. The base of the logarithm is $81$, the exponent is $-1/2$, and the result is $2r-1$. Therefore, we have:
\[
81^{-1/2} = 2r-1
\]
We know that $81$ is $9^2$, so we can rewrite the equation as... | null | null |
Find the smallest positive integer \( n \) such that
\[
\begin{pmatrix} \cos 60^\circ & -\sin 60^\circ \\ \sin 60^\circ & \cos 60^\circ \end{pmatrix}^n = \mathbf{I}.
\] | The matrix
\[
\begin{pmatrix} \cos 60^\circ & -\sin 60^\circ \\ \sin 60^\circ & \cos 60^\circ \end{pmatrix}
\]
represents a rotation by 60 degrees counter-clockwise. We are asked to find the smallest \( n \) such that a 60-degree rotation repeated \( n \) times results in a full 360-degree rotation. In mathematical ter... | null | null |
The total cost of a vacation was divided among some people. If the total cost of the vacation had been divided equally among 4 people, the cost per person would have been $40 less. The total cost of the vacation was $480. How many people was the total cost of the vacation initially divided among? | Let's denote the number of people the total cost was initially divided among as \( n \).
According to the problem, if the cost had been divided among 4 people, each person would have paid $40 less. This means that the cost per person initially was \( \frac{480}{n} \) and the cost per person if divided among 4 people w... | null | null |
a train running at the speed of 60 km / hr crosses a pole in 12 seconds . find the length of the train . | To find the length of the train, we can use the formula:
Distance = Speed × Time
First, we need to convert the speed from km/hr to m/s because the time is given in seconds and we want the length in meters.
Speed in m/s = Speed in km/hr × (1000 m / 1 km) × (1 hr / 3600 s)
Speed in m/s = 60 × (1000 / 3600)
Speed in m... | null | null |
Lion Alex decided to count the stripes on Marty the zebra (black and white stripes alternate). It turned out that there is one more black stripe than white stripes. Alex also noted that all white stripes are of the same width, while black stripes can be either wide or narrow, with the total number of white stripes bein... |
1. Let's denote the number of wide black stripes as \( w \) and the number of narrow black stripes as \( n \).
2. According to the problem, the number of white stripes is 7 more than the number of wide black stripes. Therefore, if we denote the number of white stripes as \( b \), we have:
\[
b = w + 7
\]
3. T... | null | null |
The geometric mean of any set of $m$ non-negative numbers is the $m$-th root of their product.
$\quad (\text{i})\quad$ For which positive integers $n$ is there a finite set $S_n$ of $n$ distinct positive integers such that the geometric mean of any subset of $S_n$ is an integer?
$\quad (\text{ii})\quad$ Is there an inf... |
#### Part (i)
For which positive integers $n$ is there a finite set $S_n$ of $n$ distinct positive integers such that the geometric mean of any subset of $S_n$ is an integer?
To solve this, consider the set $S_n = \{2^a, 2^{2a}, 2^{3a}, \ldots, 2^{na}\}$ for some integer $a$. The geometric mean of any subset of $S_n$... | null | null |
Given vectors $\mathbf{v}$ and $\mathbf{w}$ such that $\|\mathbf{v}\| = 3,$ $\|\mathbf{w}\| = 7,$ and $\mathbf{v} \cdot \mathbf{w} = 10,$ then find $\|\operatorname{proj}_{\mathbf{w}} \mathbf{v}\|.$ | To find the magnitude of the projection of vector $\mathbf{v}$ onto vector $\mathbf{w}$, denoted as $\|\operatorname{proj}_{\mathbf{w}} \mathbf{v}\|$, we follow these steps:
1. Recall the formula for the projection of $\mathbf{v}$ onto $\mathbf{w}$, which is $\operatorname{proj}_{\mathbf{w}} \mathbf{v} = \frac{\mathbf... | null | null |
Express the following sum as a simple fraction in lowest terms.
$$\frac{1}{1\times2} + \frac{1}{2\times3} + \frac{1}{3\times4} + \frac{1}{4\times5} + \frac{1}{5\times6}$$ | To express the given sum as a simple fraction in lowest terms, we first recognize a pattern that can simplify our work. Any fraction with a denominator that is the product of two consecutive numbers can be rewritten as a difference of two simpler fractions. This is illustrated by the general formula:
$$\frac{1}{n(n+1)... | null | null |
Find the solution to the equation $\left( \frac{3}{5} \right)^{x}+\left( \frac{4}{5} \right)^{x}=1$. The following solution approach is proposed: Let the function $f(x)=\left( \frac{3}{5} \right)^{x}+\left( \frac{4}{5} \right)^{x}$, then the function $f(x)$ is monotonically decreasing over $\mathbb{R}$, and $f(2)=1$, s... | **Analysis**
This problem mainly examines the application of function monotonicity, which is a common type of question in college entrance examinations and is considered difficult.
**Solution**
Let's follow the solution approach mentioned above and set $f(x)=x^3+x$. Since $f'(x)=3x^2 +1 > 0$, $f(x)$ is monotonically... | null | null |
In $\triangle ABC$, given that $AC = CD$ and $\angle CAB - \angle ABC = 30^\circ$, what is the measure of $\angle BAD$?
(A) $30^\circ$
(B) $20^\circ$
(C) $22.5^\circ$
(D) $10^\circ$
(E) $15^\circ$ |
1. Let \( \angle BAC = x \) and \(\angle ABC = y \).
Given that \( \angle CAB - \angle ABC = 30^\circ \),
\[
x - y = 30^\circ \quad \text{(Equation 1)}
\]
2. Note that \( AC = CD \), implying that \(\triangle CAD \) is isosceles with \(AC = CD\).
3. Since \( \angle CAD \) is the external angle for \(\tr... | null | null |
A school's students are marching forward at a speed of 8 kilometers per hour. The principal at the end of the line asks a student to run to the front of the line to deliver a message to the leading teacher and then immediately return to the back of the line. The student runs at a speed of 12 kilometers per hour. It tak... | Let the length of the line be $x$ meters. The speed of the students marching in line is 8 kilometers per hour, which is equivalent to $\frac{400}{3}$ meters per minute. The speed of the student running is 12 kilometers per hour, which is equivalent to 200 meters per minute. According to the problem, we have:
$$
\frac{... | null | null |
What is the base-ten number represented by the base-eight number 31? | To convert the base-eight number $31_8$ to a base-ten number, we break it down as follows:
First, we identify the digits and their respective places in the base-eight number. The digit on the right is in the $8^0$ place, and the digit on the left is in the $8^1$ place.
So, we have:
- The digit $3$ is in the $8^1$ pla... | null | null |
Given that $x+\sin y=2008$ and $x+2008 \cos y=2007$, where $0 \leq y \leq \frac{\pi}{2}$, find the value of $x+y$. | To solve the given problem, let's start by analyzing the given equations:
1. $x + \sin y = 2008$
2. $x + 2008 \cos y = 2007$
We are tasked with finding the value of $x+y$ under the condition that $0 \leq y \leq \frac{\pi}{2}$.
**Step 1: Subtract the second equation from the first.**
This gives us:
\[
\sin y - 2008 ... | null | null |
Out of 172 students participating in a mountaineering competition, there are 45 advanced climbers, 70 intermediate climbers, and 57 beginners. The advanced climbers have 80 experience points each, intermediate climbers have 50 experience points each, and beginners have 30 experience points each. Each team should have a... | First, let's calculate the total experience points for one team with the given combination of climbers:
- 5 advanced climbers at 80 points each: 5 * 80 = 400 points
- 8 intermediate climbers at 50 points each: 8 * 50 = 400 points
- 5 beginner climbers at 30 points each: 5 * 30 = 150 points
Now, let's add these points... | null | null |
Alan bought a $2000 phone online. John bought it 2% more expensive in a local store. How much did John spend on his phone? | To find how much John spent on his phone, we first calculate the difference in price due to the 2% increase. We do this by calculating 2% of the original $2000 price:
\[2\% \times 2000 = 0.02 \times 2000 = \$40\]
Next, we add this difference to the original price to find the total amount John spent:
\[2000 + 40 = \$... | null | null |
Given the function $f(x) = |x - 10| - |x - 25|$, and the solution set for the inequality $f(x) < 10a + 10$ ($a \in \mathbb{R}$) is $\mathbb{R}$.
(1) Determine the range of values for the real number $a$.
(2) Find the minimum value of $2a + \frac{27}{a^2}$. | (1) For the function $f(x) = |x - 10| - |x - 25|$, if the solution set of the inequality $f(x) < 10a + 10$ ($a \in \mathbb{R}$) is $\mathbb{R}$, it implies all $x$ values satisfy the inequality.
We analyze the function $f(x)$ by considering different intervals divided by the points $x = 10$ and $x = 25$ where the abso... | null | null |
Given real numbers $t$, consider the point of intersection of the lines $3x + 4y = 12t + 6$ and $2x + 3y = 8t - 1$. All plotted points from various values of $t$ lie on a line. Determine the slope of this line. | We start by solving the two equations simultaneously. For the given equations:
1. $3x + 4y = 12t + 6$
2. $2x + 3y = 8t - 1$
To eliminate $x$, we multiply the first equation by 2 and the second by 3:
- $6x + 8y = 24t + 12$
- $6x + 9y = 24t - 3$
Subtracting these two:
\[ -y = 15 \]
Thus, $y = -15$.
Substituting $y = -... | null | null |
A covered rectangular football field with a length of 90 m and a width of 60 m is being designed to be illuminated by four floodlights, each hanging from some point on the ceiling. Each floodlight illuminates a circle, with a radius equal to the height at which the floodlight is hanging. Determine the minimally possibl... |
1. Consider a rectangle \(ABCD\) where the diagonals intersect at point \(O\). The given dimensions are as follows:
- Length of the field, \(AB = 90\) m
- Width of the field, \(BC = 60\) m
2. We will position the projectors at points on the ceiling corresponding to the midpoints of segments \(AO\), \(BO\), \(CO... | null | null |
Evaluate the expression \(\frac{a^2 \cdot a^5}{b^2 / b^3}\) if \(a = 3\) and \(b = 2\). | 1. Simplify the expression in the numerator using the properties of exponents:
\[ a^2 \cdot a^5 = a^{2+5} = a^7. \]
2. Simplify the expression in the denominator using the properties of exponents:
\[ \frac{b^2}{b^3} = b^{2-3} = b^{-1} = \frac{1}{b}. \]
3. Combining these results, the entire expression becomes:... | null | null |
Determine \( m \) in the equation
\[ x^{4}-(3m+2)x^{2}+m^{2}=0 \]
such that the roots of the equation form an arithmetic progression. | 1. We are given the polynomial equation:
\[
x^{4}-(3m+2)x^{2}+m^{2}=0
\]
We need to find the values of \(m\) such that the roots form an arithmetic sequence.
2. Assume that the roots are denoted as \(\alpha, \beta, -\beta, -\alpha\) with \(\alpha\) and \(\beta\) being positive real numbers. This assumption... | null | null |
Mrs. Jackson has four boxes of Christmas decorations. There are 15 decorations in each box. She was only able to use 35 decorations and decided to give the rest to her neighbor. How many decorations did she give? | Mrs. Jackson initially calculates the total number of decorations she has by multiplying the number of boxes by the number of decorations in each box. This calculation is done as follows:
\[4 \times 15 = 60\]
This means Mrs. Jackson has a total of $60$ decorations.
Next, to find out how many decorations she gave to he... | null | null |
What is the positive difference between $\frac{7^2 + 7^2}{7}$ and $\frac{7^2 \times 7^2}{7}$? | First, calculate $\frac{7^2 + 7^2}{7}$:
\[
\frac{7^2 + 7^2}{7} = \frac{49 + 49}{7} = \frac{98}{7} = 14
\]
Next, calculate $\frac{7^2 \times 7^2}{7}$:
\[
\frac{7^2 \times 7^2}{7} = \frac{49 \times 49}{7} = \frac{2401}{7} = 343
\]
Now, find the positive difference between the two results:
\[
343 - 14 = \boxed{329}
\] | null | null |
End of preview. Expand in Data Studio
NuminaMath Worked Solutions
Competition and school maths problems with step-by-step solutions.
| Rows | 3,000 |
| Domain | mathematics |
| Format | data.parquet, one row per example |
| Licence | apache-2.0 |
| Built for | supervised fine-tuning (SFT) experiments on Adaption AutoScientist |
Columns
| Column | Description |
|---|---|
original_prompt |
The prompt (user turn) as uploaded. |
original_completion |
The target response as uploaded. |
enhanced_prompt |
Empty in this dataset. |
enhanced_completion |
Empty in this dataset. |
How it was built
Sampled from NuminaMath-CoT, balanced across its sources and decontaminated against MATH-500, GSM8K test and the NuminaMath test split.
Sources and licence
- AI-MO/NuminaMath-CoT (Apache-2.0)
Notes
- Columns
enhanced_prompt,enhanced_completionare empty in this dataset (the Adaption export reserves them for rewritten text).
Loading
from datasets import load_dataset
ds = load_dataset("rodriguescarson/adaption-math-worked-solutions-mixtral-raw", split="train")
import pandas as pd
df = pd.read_parquet("hf://datasets/rodriguescarson/adaption-math-worked-solutions-mixtral-raw/data.parquet")
Published by Carson Rodrigues (Hugging Face, Kaggle).
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