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The quadratic polynomial \( p(x) = ax^2 + bx + c \) is such that if \( |x| \leq 1 \), then \( |p(x)| \leq 1 \). Prove that under these conditions, \( |x| \leq 1 \) also implies \( \left| p_1(x) \right| \leq 2 \), where \( p_1(x) = cx^2 + bx + a \). | 1. **Assume \( a \geqslant 0 \):**
If \( a < 0 \), replace \( p(x) \) with \( -p(x) = -a x^2 - b x - c \). This adjustment does not affect the given condition \( |p(x)| \leqslant 1 \) for \( |x| \leqslant 1 \).
2. **Assume \( b \geqslant 0 \):**
If \( b < 0 \), replace \( p(x) \) with \( p(-x) = a x^2 - b x + ... | null | null |
Given a sequence $\{a\_n\}$, define its "optimal value" as $H\_n = \frac{a\_1 + 2a\_2 + \dots + 2^{n-1}a\_n}{n}$. If the "optimal value" of a certain sequence $\{a\_n\}$ is $H\_n = 2^{n+1}$, let $S\_n$ denote the sum of the first $n$ terms of the sequence $\{a\_n - kn\}$. If $S\_n \leq S\_5$ holds true for any $n$, fin... | From the given information,
$H\_n = \frac{a\_1 + 2a\_2 + \dots + 2^{n-1}a\_n}{n} = 2^{n+1}$,
Then $a\_1 + 2a\_2 + \dots + 2^{n-1}a\_n = n2^{n+1}$,
$a\_1 + 2a\_2 + \dots + 2^{n-2}a\_{n-1} = (n-1)2^{n}$,
Then $2^{n-1}a\_n = n2^{n+1} - (n-1)2^{n} = (n+1)2^{n}$,
Then $a\_n = 2(n+1)$,
This holds true for $a\_1$ as well,
So,... | null | null |
\[ \frac{\sin A + \sin B + \sin C}{\sin A + \sin B - \sin C} = \operatorname{cot} \frac{A}{2} \operatorname{cot} \frac{B}{2} \] | 1. Consider the given equation:
\[
\frac{\sin A + \sin B + \sin C}{\sin A + \sin B - \sin C} = \operatorname{ctg} \frac{A}{2} \operatorname{ctg} \frac{B}{2}
\]
2. Utilize the identity given in problem 3.368, which states:
\[
\sin A + \sin B + \sin C = 4 \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2}
\]
Using th... | null | null |
The function \( f(x) \) is not defined at \( x = 0 \). Determine the value of \( f(0) \) so that \( f(x) \) is continuous at \( x = 0 \), given
\[
f(x) = \frac{\sqrt[3]{1+x} - 1}{\sqrt{4+x} - 2}
\] | To determine a suitable value for \( f(0) \) that ensures the function \( f(x) \) is continuous at \( x = 0 \), we need to evaluate the limit of \( f(x) \) as \( x \) approaches 0. Specifically, we will find
\[
\lim_{x \to 0} f(x)
\]
where the function \( f(x) \) is given by
\[
f(x) = \frac{\sqrt[3]{1+x} - 1}{\sqrt{... | null | null |
An ellipse has foci at $F_1 = (0,3)$ and $F_2 = (4,0)$. The ellipse intersects the $x$-axis at the origin, and one other point. Determine the other point of intersection on the $x$-axis. | Given the ellipse intersects the $x$-axis at the origin $(0,0)$ and another point, we need to find the total distance from any point on the ellipse $(x,0)$ to the foci which must be constant.
From the origin to $F_1: |F_1O| = 3$, and from the origin to $F_2: |F_2O| = 4$. Therefore, every point $(x,0)$ on the ellipse ... | null | null |
What is the first digit (from left to right) of the base $8$ representation of $473_{10}$? | To find the first digit of the base $8$ representation of $473_{10}$, we follow these steps:
1. Identify the largest power of $8$ that is less than $473$. We start with $8^1 = 8$, then $8^2 = 64$, and notice that $8^3 = 512$ is too large. Therefore, the largest power of $8$ that fits is $8^2 = 64$.
2. Determine the l... | null | null |
Given circle M: $(x-1)^2+(y-3)^2=1$ and circle N: $(x-7)^2+(y-5)^2=4$, points P and Q are respectively on circle M and circle N, and point A is on the x-axis. The minimum value of $|AP|+|AQ|$ is \_\_\_\_\_. | From the given information, the center of circle M is at $M(1,3)$, and the center of circle N is at $N(7,5)$. The symmetric point of M with respect to the x-axis is $M'(1,-3)$.
Therefore, the distance $|M'N|$ is calculated as $\sqrt{(7-1)^2+(5+3)^2}=10$.
Thus, the minimum value of $|AP|+|AQ|$ is $|M'N|-1-2=7$.
Hence... | null | null |
Let \( X \) be the number of sequences of integers \( a_1, a_2, \ldots, a_{2047} \) that satisfy all of the following properties:
- Each \( a_i \) is either 0 or a power of 2.
- \( a_i = a_{2i} + a_{2i+1} \) for \( 1 \leq i \leq 1023 \).
- \( a_1 = 1024 \).
Find the remainder when \( X \) is divided by 100. | 1. **Define the Problem:**
- We need to determine the number of sequences \( a_1, a_2, \ldots, a_{2047} \) satisfying:
1. Each \( a_i \) is either 0 or a power of 2.
2. \( a_i = a_{2i} + a_{2i+1} \) for \( 1 \leq i \leq 1023 \).
3. \( a_1 = 1024 \).
2. **Visualize the Problem as a Binary Tr... | null | null |
Given $f(n)=1+ \frac {1}{2}+ \frac {1}{3}+…+ \frac {1}{3n-1}(n∈\mathbb{N}^{*})$, find the value of $f(n+1)-f(n)$. The options are:
A: $ \frac {1}{3n+2}$
B: $ \frac {1}{3n}+ \frac {1}{3n+1}$
C: $ \frac {1}{3n+1}+ \frac {1}{3n+2}$
D: $ \frac {1}{3n}+ \frac {1}{3n+1}+ \frac {1}{3n+2}$ | First, let's write down the given function $f(n)=1+ \frac {1}{2}+ \frac {1}{3}+…+ \frac {1}{3n-1}$ for $n \in \mathbb{N}^{*}$.
Next, we want to find the difference between $f(n+1)$ and $f(n)$. To do this, let's first determine what $f(n+1)$ looks like:
$$ f(n+1) = 1+ \frac {1}{2}+ \frac {1}{3}+…+ \frac {1}{3n-1}+ \fr... | null | null |
Consider a region in the first quadrant known as the "squid's wing," enclosed by the following boundaries: the portion of the circle of radius $5$ centered at $(0, 0)$, the portion of the circle with radius $2$ centered at $(0, 3)$, lying in the first quadrant, and the line segment from $(0, 0)$ and $(5, 0)$. Determine... | 1. **Identify the areas involved**:
- The larger quarter-circle has a radius of $5$, centered at $(0,0)$.
- The smaller circle has a radius of $2$, centered at $(0,3)$.
2. **Calculate Area for Larger Quarter-Circle**:
- The area of the full circle is $\pi (5)^2 = 25\pi$.
- Thus, the area for the quarter-ci... | null | null |
Find the smallest, positive four-digit multiple of $17$. | Start by dividing $1000$ by $17$ to find the smallest four-digit multiple:
\[ \frac{1000}{17} \approx 58.82 \]
Since we require a whole number and the smallest four-digit number, we round $58.82$ to the next whole number, which is $59$.
So, calculate $59 \cdot 17$:
\[ 59 \cdot 17 = 1003 \]
This means that $1003$ i... | null | null |
In triangle $PQR$, the angle bisectors are $PS$, $QT$, and $RU$, which intersect at the incenter $I$. If $\angle PRQ = 42^\circ$, find the measure of $\angle PIQ$, in degrees. | Since $PS$ is an angle bisector, $\angle RPQ = \angle PQR/2$. Since $QT$ is an angle bisector, $\angle QRP = \angle QPR/2$. As an angle that is external to triangle $QPI$, $\angle PIQ = \angle RPQ + \angle QRP = \angle PQR/2 + \angle QPR/2$.
Using the property of a triangle, where the sum of the interior angles is $18... | null | null |
What is the largest value of $x$ that satisfies the equation $9x^2 - 45x + 50 = 0$? Express your answer as a decimal. | First, let's factorize the quadratic equation $9x^2 - 45x + 50$:
1. Factor out the common terms if possible:
$$ 9x^2 - 45x + 50 = 9(x^2 - 5x) + 50 $$
2. Try to factorize it as a product of two binomials:
$$ 9x^2 - 45x + 50 = (3x - 10)(3x - 5) = 0 $$
This factorization is derived by setting the equation $9x^2 -... | null | null |
The product of 3 inches and twice the circumference of a circle, in inches, equals 3 times the circle's area. What is the length of the radius of the circle, in inches? | Let \( r \) be the radius of the circle.
The problem states that the equation is \( 3 \cdot (2 \times 2\pi r) = 3 \times \pi r^2 \).
Simplify and solve for \( r \):
\[
6 \cdot 2\pi r = 3 \pi r^2 \\
12 \pi r = 3 \pi r^2 \\
12 r = 3 r^2
\]
Divide both sides by \( r \) (assuming \( r \neq 0 \)):
\[
12 = 3r \\
r = \frac{1... | null | null |
The price of a smartphone is $\textdollar{800}$. A store offers a single discount of 25%. Another store offers two successive discounts of 20% and 10%. Calculate the difference in final prices between the two stores, expressed in dollars.
$\textbf{(A)} \textdollar{0} \qquad \textbf{(B)} \textdollar{12} \qquad \textbf{(... | 1. **Calculate the single discount of 25% on $\textdollar{800}$:**
- The remaining amount after a 25% discount is $100\% - 25\% = 75\%$ of the original price.
- Calculation:
\[
800 \times 0.75 = \textdollar{600}
\]
2. **Calculate the successive discounts of 20% and 10% on $\textdollar{800}$:**
... | null | null |
Determine how many positive integers $n$ satisfy the inequality \[(n - 2)(n - 4)(n - 6) \dotsm (n - 98) < 0.\] | We note that there are 49 factors in the product, exactly similar to the original problem with factors ranging from $n - 2$ to $n - 98$ increasing by 2 (all even numbers). The behavior of the product's sign-switching remains the same, but starts at $n=2$.
1. For $n < 2$, all factors are negative, and their product is ... | null | null |
A dartboard is divided into six regions by three radii and two concentric circles with an inner radius of $4$ and an outer radius of $8$. The point values assigned to the regions are as follows:
- Inner regions: $3, 4, 3$
- Outer regions: $4, 3, 3$
A dart hitting a region has a probability proportional to its area. Wh... | 1. **Identify regions and their point values**:
- Inner circle: Radii $4$, point values $3, 4, 3$
- Outer ring: Between radii $4$ and $8$, point values $4, 3, 3$
2. **Calculate the areas of the regions**:
- **Area of the inner circle**: $\pi \times 4^2 = 16\pi$
- **Area of each inner region**: $\frac{16\pi... | null | null |
A function $f(x)$ defined on $[1,+\infty)$ satisfies the following conditions: (1) $f(2x)=mf(x)$, where $m$ is a positive constant, and (2) when $2\leqslant x\leqslant 4$, $f(x)=1-|x-3|$. If all the maximum points of the function lie on the same straight line, then $m$ equals \_\_\_\_\_\_. | This problem involves the knowledge of collinearity, function extremum, and requires analyzing the piecewise function $f(x)$ to derive its analytical expression, and then finding the coordinates of the extreme points of the function, which is the key to solving the problem.
Solution steps:
1. When $2\leqslant x\leqsla... | null | null |
In the Democratic Republic of Irun, 5 people are voting in an election among 5 candidates. If each person votes for a single candidate at random, what is the expected number of candidates that will be voted for? | Given:
- 5 people are voting in an election among 5 candidates.
- Each person votes for a single candidate at random.
We need to find the expected number of candidates that will receive at least one vote.
1. **Calculate the Probability of a Candidate Receiving No Votes:**
For a chosen candidate to receive no vote... | null | null |
The sum of two nonzero real numbers \(a\) and \(b\) equals \(3\) times their product plus \(1\). Find the sum of the square of the reciprocals of these two numbers.
\(\textbf{(A)}\ \frac{14}{k^2}\qquad\textbf{(B)}\ \frac{13}{k^2}\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ \frac{9ab + 1}{a^2b^2}\qquad\textbf{(E)}\ \frac{4a... | 1. Set the variables \(a\) and \(b\).
2. Establish the mathematically framed problem given in terms of these variables:
\[
a+b = 3ab + 1
\]
3. To find \(\frac{1}{a^2} + \frac{1}{b^2}\), first express in terms of a single variable ratio:
\[
\frac{1}{a^2} + \frac{1}{b^2} = \frac{a^2 + b^2}{a^2b^2}
\]
... | null | null |
Find all triples $(a, b, c)$ of positive integers for which $\frac{32a + 3b + 48c}{4abc}$ is also an integer. | To find all triples \((a, b, c)\) of positive integers for which \(\frac{32a + 3b + 48c}{4abc}\) is an integer, we start by setting the given expression equal to an integer \(k\):
1. **Set up the equation:**
\[
\frac{32a + 3b + 48c}{4abc} = k
\]
This implies:
\[
32a + 3b + 48c = 4kabc
\]
2. **Ana... | null | null |
Vasya and Tolya exchanged badges. Before the exchange, Vasya had 5 more badges than Tolya. After Vasya exchanged 24% of his badges for 20% of Tolya's badges, Vasya had one badge fewer than Tolya. How many badges did each boy have before the exchange? | We need to determine the number of badges (znachki) Vasya and Tolya had before the exchange. Let's denote the number of badges Vasya had before the exchange as \( V \) and the number Tolya had as \( T \).
Given:
- \( V \) is 5 badges more than \( T \), hence:
\[
V = T + 5
\]
- After the exchange, Vasya exchange... | null | null |
Given the functions $f(x) = \log_a(1+x)$ and $g(x) = \log_a(1-x)$ where $(a>0 \text{ and } a \neq 1)$, let $h(x) = f(x) - g(x)$.
(1) Find the domain of function $h(x)$, determine the parity of $h(x)$, and explain the reason;
(2) If $f(3) = 2$, find the set of $x$ for which $h(x) < 0$ holds;
(3) If $x \in \left[0, ... | (1) $h(x) = f(x) - g(x) = \log_a(1+x) - \log_a(1-x)$,
we have $1+x>0$ and $1-x>0$; solving gives $-1<x<1$;
The domain is $(-1, 1)$.
Also, since $h(-x) = \log_a \frac{1-x}{1+x} = -\log_a \frac{1+x}{1-x} = -h(x)$,
Therefore, the function $h(x)$ is an odd function.
(2) Since $f(3) = 2$, we solve to get $a=2$. ... | null | null |
At a loading dock, each worker on the night crew loaded 1/2 as many boxes as each worker on the day crew. The night crew has 4/5 as many workers as the day crew. What fraction of all the boxes loaded by the two crews did the day crew load? | Let's denote the number of boxes loaded by each worker on the day crew as D and the number of workers on the day crew as W.
The night crew loaded 1/2 as many boxes per worker, so each night crew worker loaded D/2 boxes. The night crew has 4/5 as many workers as the day crew, so the number of workers on the night crew... | null | null |
Among the following propositions, the number of incorrect statements is ( )
① Two lines perpendicular to the same line are parallel to each other;
② Two planes parallel to the same plane are parallel to each other;
③ If $a$ and $b$ are two lines, and $a \parallel b$, then $a$ is parallel to any plane that passes... | For statement ①, in space two lines perpendicular to the same line are not necessarily parallel to each other. Therefore, statement ① is incorrect.
Regarding statement ②, the assertion that two planes parallel to the same plane are parallel to each other is correct. That is, if $\alpha \parallel \gamma$ and $\beta \pa... | null | null |
Given that the regression equation of variable $y$ with respect to variable $x$ is $\hat{y}=b\ln x+0.24$, and a set of data is shown in the table below:<br/>
| $x$ | $e$ | $e^{3}$ | $e^{4}$ | $e^{6}$ | $e^{7}$ |
|-------|-----|---------|---------|---------|---------|
| $y$ | $1$ | $2$ | $3$ | $4$ | $5$... | To solve the problem, we start by understanding the given regression equation $\hat{y}=b\ln x+0.24$. We are told to find the value of $y$ when $x=e^{10}$, using the given data points.
Step 1: Convert the regression equation into a simpler form by letting $t=\ln x$. This gives us $\hat{y}=bt+0.24$.
Step 2: Calculate t... | null | null |
If x is a certain percent greater than 80, then x equals 88.8. What is the percentage by which x is greater than 80? | To find the percentage by which x is greater than 80, we can use the following formula:
Percentage Increase = [(New Value - Original Value) / Original Value] * 100
In this case, the new value (x) is 88.8 and the original value is 80.
Percentage Increase = [(88.8 - 80) / 80] * 100
Percentage Increase = [8.8 / 80] * 1... | null | null |
Let $F$ be the focus of the parabola ${C}_{1}:{y}^{2}=2x$, point $P$ lies on the parabola, and point $Q$ lies on the directrix $l$, satisfying $PQ$ <span class="special-symbol">∥</span> $x$-axis. If $|PQ|=|QF|$, then $|PF|=\left(\ \ \right)$
A: $2$
B: $2\sqrt{3}$
C: $3$
D: $3\sqrt{3}$ | Given the problem, we aim to find the length $|PF|$ under the conditions provided. Let's break down the solution step by step.
1. **Understanding the Parabola and Its Elements**: The parabola ${C}_{1}:{y}^{2}=2x$ has its focus $F$ and a directrix $l$. For any point $P$ on the parabola and a corresponding point $Q$ on ... | null | null |
Given a function $f(x) = x^2e^{-ax}$, where $a > 0$ (e is the base of the natural logarithm, $e \approx 2.71828$...),
(I) Determine the intervals of monotonicity for $f(x)$;
(II) Find the maximum value of $f(x)$ on the interval $[1, 2]$. | (I) First, we find the derivative of $f(x)$ with respect to $x$:
$$ f'(x) = e^{-ax}(-ax^2 + 2x). $$
To determine the intervals of monotonicity, we set $f'(x) > 0$ since $e^{-ax} > 0$ for all $x$:
$$ -ax^2 + 2x > 0 \implies x(2 - ax) > 0. $$
Solving this inequality, we get:
$$ 0 < x < \frac{2}{a}. $$
Thus, $f(x)$ is d... | null | null |
If f ( f ( n ) ) + f ( n ) = 2 n + 3 , f ( 0 ) = 1, what is the value of f(2012)? | Let's denote f(n) as g for simplicity. Then we have:
f(g) + g = 2n + 3
Since f(0) = 1, let's find f(f(0)):
f(f(0)) + f(0) = 2*0 + 3
f(1) + 1 = 3
f(1) = 2
Now we have two values of the function: f(0) = 1 and f(1) = 2.
Let's try to find a pattern or a rule for f(n) that satisfies the given functional equation. We ca... | null | null |
In an aerial map, triangle XYZ is formed by the intersections of Pine Road, Oak Road, and a utility line. Pine Road runs north-south and Oak Road runs east-west. The triangle is further divided by Elm Street running from point X to midpoint M of YZ. YM measures 2 miles and MX measures 3 miles. If the entire distance of... | 1. **Calculate the dimensions of full triangle XYZ:**
- Since point M is the midpoint of YZ, MY = MZ = 2.5 miles.
- The full base YZ of triangle XYZ along Oak Road is given as 5 miles.
2. **Calculate the area of triangle XYZ:**
- The area of triangle XYZ can be calculated with base YZ and height XM (which run... | null | null |
Positive integers $k, m, n$ satisfy the equation $m^2 + n = k^2 + k$ . Show that $m \le n$ . | 1. Start with the given equation:
\[
m^2 + n = k^2 + k
\]
Rearrange the equation to isolate terms involving \(m\) and \(k\):
\[
m^2 - k^2 = k - n
\]
This can be factored using the difference of squares:
\[
(m - k)(m + k) = k - n
\]
2. Assume for contradiction that \(m > n\). This impli... | null | null |
Given an arithmetic sequence \(\{a_n\}\) where \(a_2=6\) and \(a_5=15\), let \(b_n=a_{2n}\). Find the sum of the first 5 terms of the sequence \(\{b_n\}\).
A: \(30\)
B: \(45\)
C: \(90\)
D: \(186\) | Let the common difference of the arithmetic sequence \(\{a_n\}\) be \(d\) and the first term be \(a_1\). From the given information, we have the system of equations:
\[\begin{cases}
a_1 + d = 6, \\
a_1 + 4d = 15.
\end{cases}\]
Solving this system yields:
\[\begin{cases}
a_1 = 3, \\
d = 3.
\end{cases}\]
Thus, the nth t... | null | null |
Class (1), Grade 3 students lineup in three rows for morning exercise, with an equal number of students in each row. Xiaohong is in the middle row. Counting from left to right, she is the 6th student; counting from right to left, she is the 7th student. How many students are there in total in the class? | 1. Let us denote the total number of people in one row as \( n \).
2. According to the problem statement:
- 小红 (Xiao Hong) is located in the middle row.
- Xiao Hong is the 6th person from the left.
- Xiao Hong is the 7th person from the right.
3. The total number of people in the middle row can be deduced as... | null | null |
Prove that there are infinitely many pairs of perfect squares $(a, b)$ such that: (1) $a$ and $b$ have the same number of digits; (2) the concatenation of $a$ and $b$ forms a new perfect square. | :
To prove the existence of an infinite number of pairs of perfect squares $(a, b)$ that satisfy:
1. $a$ and $b$ have the same number of digits.
2. $\overline{a b}$ is also a perfect square.
We construct the sequences $a_n$ and $b_n$ as follows:
\[ a_n = \left(5 \cdot 10^{n-1} - 1\right)^2, \]
\[ b_n = \left(10^n - 1... | null | null |
Let $\triangle AOB$ be a right-angled triangle with $m\angle AOB = 90^{\circ}$. Let $P$ and $Q$ be the midpoints of legs $OA$ and $OB$, respectively. Given that $AP=25$ and $BQ=15$, find $AB$. | Let $OP = x$ and $OQ = y$. Thus, using Pythagoras theorem in triangles $AP$ and $BQ$:
$$
25^2 = (2x)^2 + y^2 \quad \text{and} \quad 15^2 = x^2 + (2y)^2.
$$
This results in:
$$
625 = 4x^2 + y^2 \quad \text{and} \quad 225 = x^2 + 4y^2.
$$
Adding these two equations, we get:
$$
5x^2 + 5y^2 = 850 \implies x^2 + y^2 = 170.
... | null | null |
Given the function $f(x)=(ax^{2}+bx+c)e^{x}(a > 0)$, its derivative $y=f′(x)$ has two zeros at $-3$ and $0$, and the minimum value of $f(x)$ is $-1$. Find the value of $a+b+c$.
A: $3$
B: $1$
C: $0$
D: $-1$ | The derivative of the function is $f′(x)=(2ax+b)e^{x}+(ax^{2}+bx+c)e^{x}=[ax^{2}+(2a+b)x+b+c]e^{x}$. Let $g(x)=ax^{2}+(2a+b)x+b+c$. Since $e^{x} > 0$, the zeros of $y=f′(x)$ are the same as the zeros of $g(x)=ax^{2}+(2a+b)x+b+c$, and $f′(x)$ and $g(x)$ have the same sign. Given that $a > 0$, when $x < -3$ or $x > 0$, $... | null | null |
Let \(A, B, R, M, \) and \(L\) be positive real numbers such that
\begin{align*}
\log_{10} (ABL) + \log_{10} (ABM) &= 3, \\
\log_{10} (BML) + \log_{10} (BMR) &= 4, \\
\log_{10} (RAB) + \log_{10} (RBL) &= 5.
\end{align*}
Compute the value of the product \(ABRML\). | First, we express the equations using properties of logarithms:
\begin{align*}
\log_{10} (AB^2 ML) &= 3, \\
\log_{10} (BM^2 LR) &= 4, \\
\log_{10} (RAB^2 L) &= 5.
\end{align*}
This translates to:
\begin{align*}
AB^2 ML &= 10^3, \\
BM^2 LR &= 10^4, \\
RAB^2 L &= 10^5.
\end{align*}
Multiplying these equations results in:... | null | null |
The foci of the ellipse $\frac{x^2}{100} + \frac{y^2}{64} = 1$ are $F_1$ and $F_2$. A point $P$ on the ellipse satisfies $\angle F_1PF_2 = 60^\circ$. Calculate the area of the triangle $\triangle F_1PF_2$. | The given ellipse equation is $\frac{x^2}{100} + \frac{y^2}{64} = 1$.
By the definition of an ellipse, the sum of the distances from the foci to any point on the ellipse is constant, equal to the length of the major axis. For this ellipse, the length of the major axis is 20, thus:
$$|PF_1| + |PF_2| = 20$$
This result... | null | null |
Find the greatest common divisor of 13926, 20031, and 47058. | Applying the Euclidean algorithm and using the property $\text{gcd}(a, b, c) = \text{gcd}(\gcd(a, b), c)$, we begin by finding $\text{gcd}(13926, 20031)$:
1. Compute $\text{gcd}(13926, 20031)$:
$$ \text{gcd}(13926, 20031) = \text{gcd}(13926, 20031 - 13926) = \text{gcd}(13926, 6105). $$
2. Continue with the Euclidean... | null | null |
In how many ways can 100 be written as the sum of three positive distinct perfect squares if the order of the three perfect squares does not matter? | We need to find combinations of three distinct positive perfect squares that sum to 100.
1. **Check Larger Squares**:
- $100 - 81 = 19$ (since $9^2 = 81$). $19$ cannot be expressed as a sum of two distinct positive perfect squares.
- $100 - 64 = 36$ (since $8^2 = 64$). $36 = 36 + 0$, but zero is not a positive s... | null | null |
In the diagram, \(\triangle ABC\) is isosceles with \(AB = BC\). If \(\angle BAC = t^\circ\), what is the measure of \(\angle ABC\), in degrees? | To find the measure of \( \angle ABC \) in terms of \( t \), follow these steps:
1. **Identify Given Information:**
- The triangle \( \triangle ABC \) is isosceles with \( AB = BC \).
- Given \( \angle BAC = t^\circ \).
2. **Determine Congruent Angles:**
- Since \( \triangle ABC \) is isosceles with \( AB... | null | null |
There are 4 people of different heights standing in order of increasing height. The difference between the first person and the second person, and also between the second person and the third person is some inches. The difference between the third person and the fourth person is 6 inches and the average height is 76. H... | Let's denote the height of the first person as \( h \). Since the difference between the first and second person is 2 inches, the height of the second person would be \( h + 2 \) inches. Similarly, the height of the third person would be \( h + 2 + 2 = h + 4 \) inches. The difference between the third and fourth person... | null | null |
$a$ and $b$ are natural numbers. It is known that $a^{2} + b^{2}$ is divisible by $ab$. Prove that $a = b$. | Given $a$ and $b$ are natural numbers, and it's known that $a^{2} + b^{2}$ divides $ab$. We need to prove that $a = b$.
1. **Let $d$ be the GCD of $a$ and $b$**:
\[
d = \gcd(a, b)
\]
So, we can express $a$ and $b$ in terms of $d$:
\[
a = du \quad \text{and} \quad b = dv
\]
where $u$ and $v$ ... | null | null |
Define the operation "$⊗$" by $x⊗y= \frac {x^{2}-y^{2}}{xy}$ for $x, y \in \mathbb{R}, xy \neq 0$. When $x > 0$ and $y > 0$, the minimum value of $x⊗y+(2y)⊗x$ is \_\_\_\_\_\_ . | To solve this problem, we need to analyze the expression provided, using the operation $⊗$ that has been defined. Let's rewrite the expression for $x⊗y+(2y)⊗x$ using this operation:
Given that $x⊗y= \frac {x^{2}-y^{2}}{xy}$, we then have:
\[
x⊗y+(2y)⊗x= \frac {x^{2}-y^{2}}{xy} + \frac {4y^{2}-x^{2}}{2y \cdot x}
\]
C... | null | null |
Given that (x+1)<sup>10</sup> = a<sub>0</sub> + a<sub>1</sub>(x-1) + a<sub>2</sub>(x-1)<sup>2</sup> + ... + a<sub>10</sub>(x-1)<sup>10</sup>, find the value of a<sub>6</sub>. | Since we know that (x+1)<sup>10</sup> = [(x-1) + 2]<sup>10</sup> = a<sub>0</sub> + a<sub>1</sub>(x-1) + a<sub>2</sub>(x-1)<sup>2</sup> + ... + a<sub>10</sub>(x-1)<sup>10</sup>,
We can use the binomial theorem's general term formula to find the value of a<sub>6</sub>.
Hence, a<sub>6</sub> = $C_{10}^{4}$ • 2<sup>4</sup... | null | null |
Prove that in every convex hexagon of area $S$ one can draw a diagonal that cuts off a triangle of area not exceeding $\frac{1}{6}S.$ | 1. **Consider a convex hexagon \( ABCDEF \) with area \( S \).** We need to prove that there exists a diagonal that cuts off a triangle with area not exceeding \( \frac{1}{6}S \).
2. **Divide the hexagon into six triangles by drawing diagonals from one vertex.** Without loss of generality, let's choose vertex \( A \) ... | null | null |
Given a positive integer \( k \). If the sequence of subsets \( I_{1} \supseteq I_{2} \supseteq \cdots \supseteq I_{k} \) of the integer set \( \mathbf{Z} \) satisfies that for \( i=1,2, \cdots, k, \) \( 168 \in I_{i} \), and for any \( x, y \in I_{i} \), we have \( x-y \in I_{i} \) (where \( x \) and \( y \) can be eq... | :
Given a positive integer $k$. Consider a sequence of integer subsets \(I_{1} \supseteq I_{2} \supseteq \cdots \supseteq I_{k}\) such that for $i = 1, 2, \ldots, k$, and $168 \in I_i$, and for any $x, y \in I_i$, we have $x - y \in I_i$ (where $x$ and $y$ can be the same). This sequence \((I_{1}, I_{2}, \cdots, I_{k... | null | null |
Given $a=\log_{0.5}3$, $b=2^{0.5}$, $c=0.5^{0.3}$, then the relationship among $a$, $b$, and $c$ is ( )
A: $b＞a＞c$
B: $b＞c＞a$
C: $a＞b＞c$
D: $c＞b＞a$ | Since $a=\log_{0.5}3<\log_{0.5}1=0$,
$b=2^{0.5}>2^{0}=1$,
and $0<c=0.5^{0.3}<0.5^{0}=1$,
therefore, $b＞c＞a$.
Hence, the correct option is: $\boxed{\text{B}}$.
This problem is solved by utilizing the monotonicity of exponential and logarithmic functions. It examines the comparison of the sizes of three numbers,... | null | null |
Find the integer $n$, $-90 \le n \le 90$, such that $\sin n^\circ = \sin 720^\circ$. | Since the sine function has a period of $360^\circ$, we start by reducing $720^\circ$:
\[
\sin 720^\circ = \sin (720^\circ - 2 \cdot 360^\circ) = \sin 0^\circ.
\]
Since $\sin 0^\circ = 0$, we can directly conclude:
\[
0 = \sin 0^\circ.
\]
Since $\sin n^\circ = \sin 0^\circ$, and the only angle within the range $-90 \le... | null | null |
Stan can type 50 words per minute. He needs to write a 5-page paper. Each page will contain 400 words. Each hour that he types he needs to drink a certain amount of water. He needs to drink 10 ounces of water while writing his paper. How many ounces of water does he need to drink per hour while typing? | First, we need to calculate the total number of words Stan needs to type for his 5-page paper. Since each page contains 400 words, the total number of words is:
5 pages * 400 words/page = 2000 words
Next, we need to determine how long it will take Stan to type 2000 words at a rate of 50 words per minute. To find the ... | null | null |
Circles of radius $4$ and $7$ are externally tangent to each other and are internally tangent to a circle of radius $13$. There is a circle of radius $13$ with a chord that is a common external tangent of the other two circles. Find the square of the length of this chord. | Let's label the circle with radius $4$ as $O_3$, the circle with radius $7$ as $O_6$, and the circle with radius $13$ as $O_9$. Let the endpoints of the tangent chord be $P$ and $Q$. Let $A_3$ and $A_6$ be the points of tangency on $O_3$ and $O_6$, and $A_9$ the projection of $O_9$ on $\overline{PQ}$.
Since $\overline... | null | null |
In the pattern below, the cat moves clockwise through the four squares, and the mouse moves counterclockwise through the eight exterior segments of the four squares.
[2003amc8prob23a.png](https://artofproblemsolving.com/wiki/index.php/File:2003amc8prob23a.png)
If the pattern is continued, where would the cat and m... | To solve this problem, we need to determine the positions of both the cat and the mouse after the 247th move. We will analyze their movements separately and then combine the results to find the correct answer.
#### Cat's Movement:
1. **Cycle Length:** The cat moves through four squares, completing a cycle every 4 move... | null | null |
A group of 5 elephants and 7 hippos ate 11 round and 20 cubic watermelons for breakfast. Another group of 8 elephants and 4 hippos ate 20 round and 8 cubic watermelons.
All elephants ate the same (whole number) amount of watermelons. Similarly, all hippos ate the same amount of watermelons. However, one type of animal... | 1. Let's denote:
- Number of watermelons each elephant eats as $e$
- Number of watermelons each hippo eats as $h$
2. Analyze the group of 5 elephants and 7 hippos:
- Total watermelons consumed: $11$ round + $20$ cubic = $31$
- Therefore, the system of equations becomes:
\[
5e + 7h = 31
\]
... | null | null |
Jenny has a tummy ache. Her brother Mike says that it is because Jenny ate 5 more than thrice the number of chocolate squares that he ate. If Mike ate 20 chocolate squares, how many did Jenny eat? | To find out how many chocolate squares Jenny ate, we follow these steps:
1. First, we calculate thrice the number of chocolate squares Mike ate: $20 \times 3 = 60$ squares.
2. Then, we add 5 to this number since Jenny ate 5 more than thrice what Mike ate: $60 + 5 = 65$ squares.
Therefore, Jenny ate $\boxed{65}$ choco... | null | null |
Let $ \mathcal{R}$ be the region consisting of the set of points in the coordinate plane that satisfy both $ |8 \minus{} x| \plus{} y \le 10$ and $ 3y \minus{} x \ge 15$ . When $ \mathcal{R}$ is revolved around the line whose equation is $ 3y \minus{} x \equal{} 15$ , the volume of the resulting solid is $ \fr... | 1. **Identify the region $\mathcal{R}$:**
- The inequality $|8 - x| + y \le 10$ can be split into two cases:
- Case 1: $x \leq 8 \implies 8 - x + y \le 10 \implies y \le x + 2$
- Case 2: $x > 8 \implies x - 8 + y \le 10 \implies y \le -x + 18$
- The inequality $3y - x \ge 15$ can be rewritten as $y \ge ... | null | null |
Calculate the product of $0.\overline{6}$ and $\frac{7}{3}$. | First, convert $0.\overline{6}$ to a fraction. Let $x = 0.\overline{6}$. Multiplying both sides by 10, we have $10x = 6.\overline{6}$. Subtracting the original equation from this, $10x - x = 6.\overline{6} - 0.\overline{6}$, we obtain:
\[ 9x = 6, \]
so,
\[ x = \frac{6}{9} = \frac{2}{3}. \]
Now, multiply $\frac{2}{3}$ b... | null | null |
Let $n \in \mathbb{N}_{+}$, the necessary and sufficient condition for the quadratic equation $x^2 - 4x + n = 0$ to have integer roots is $n = \_\_\_\_\_\_\_\_\_\_\_\_$. | To solve this, a quadratic equation $x^2 - 4x + n = 0$ has real roots if and only if $(-4)^2 - 4n \geq 0$, which simplifies to $n \leq 4$;
Since $n \in \mathbb{N}_{+}$, when $n=4$, the equation $x^2 - 4x + 4 = 0$ has an integer root of 2;
When $n=3$, the equation $x^2 - 4x + 3 = 0$ has integer roots of 1 and 3;
W... | null | null |
Let $a,$ $b,$ and $c$ be distinct real numbers. Simplify the expression
\[
\frac{(x + a)^4}{(a - b)(a - c)} + \frac{(x + b)^4}{(b - a)(b - c)} + \frac{(x + c)^4}{(c - a)(c - b)}.
\] | Let
\[p(x) = \frac{(x + a)^4}{(a - b)(a - c)} + \frac{(x + b)^4}{(b - a)(b - c)} + \frac{(x + c)^4}{(c - a)(c - b)}.\]
Then evaluating $p(-a)$:
\begin{align*}
p(-a) &= \frac{(-a+a)^4}{(a-b)(a-c)} + \frac{(-a+b)^4}{(b-a)(b-c)} + \frac{(-a+c)^4}{(c-a)(c-b)} \\
&= \frac{(b-a)^4}{(b-a)(b-c)} + \frac{(c-a)^4}{(c-a)(c-b)} \\... | null | null |
Extend a pattern consisting of 5x5 square tiles with 13 black and 12 white tiles by adding a border of white tiles around the original square. What is the new ratio of black tiles to white tiles in the extended pattern?
A) $\frac{13}{12}$
B) $\frac{13}{36}$
C) $\frac{36}{13}$
D) $\frac{12}{13}$ | 1. **Identify Dimensions and Tile Counts**: The original pattern consists of a square with 13 black tiles and 12 white tiles arranged in a 5x5 square format.
2. **Understanding the Extension**: A border of white tiles is added all around the original square, which increases the square size to 7x7.
3. **Calculate Numb... | null | null |
Calculate the product of $\frac{5}{4} \times \frac{6}{5} \times \frac{7}{6} \times \dots \times \frac{1004}{1003}$.
A) 250
B) 251
C) 252
D) 253
E) 254 | 1. **Identify the Sequence**: This product is a series of fractions where each fraction's numerator is one greater than its predecessor's denominator:
\[
\frac{5}{4} \times \frac{6}{5} \times \frac{7}{6} \times \dots \times \frac{1004}{1003}
\]
2. **Cancellation of Terms**: Each term's numerator, except fo... | null | null |
The graph of the function $f(x)=|\cos x|$ $(x\geqslant 0)$ intersects with a line passing through the origin at exactly four points. Let $\theta$ be the largest $x$-coordinate among these four intersection points. Then, $\dfrac {(1+\theta^{2})\sin 2\theta}{\theta}=$ \_\_\_\_\_\_. | Since the graph of the function $f(x)=|\cos x|$ $(x\geqslant 0)$ intersects with a line passing through the origin at exactly four points,
it follows that the line is tangent to the graph of $y=|\cos x|$ $(x\geqslant 0)$ within the interval $\left( \dfrac {3\pi}{2},2\pi\right)$.
In the interval $\left( \dfrac {3\pi}{... | null | null |
Prove that for every $0<\alpha<1$, there exist integers $1 \leq a_{1}<a_{2}<\cdots<a_{n} \leq 2^{n-1}$ such that
$$
\left[\alpha a_{1}\right] \leqslant\left[\alpha^{2} a_{2}\right] \leqslant \ldots \leqslant\left[\alpha^{n} a_{n}\right]
$$ | We aim to show that for any \(0 < \alpha < 1\), there exist integers \(1 \leq a_1 < a_2 < \cdots < a_n \leq 2^{n-1}\) such that:
\[
\left\lfloor \alpha a_1 \right\rfloor \leq \left\lfloor \alpha^2 a_2 \right\rfloor \leq \cdots \leq \left\lfloor \alpha^n a_n \right\rfloor
\]
Let's choose the sequence \(a_1 = 1, a_2 = ... | null | null |
A large rectangle is partitioned into four rectangles by two segments parallel to its sides. The areas of three of the resulting rectangles are shown. What is the area of the fourth rectangle?
$\textbf{(A)}\ 10 \qquad\textbf{(B)}\ 15 \qquad\textbf{(C)}\ 20 \qquad\textbf{(D)}\ 21 \qquad\textbf{(E)}\ 25$ | Let's denote the large rectangle as $ABCD$, with $AB$ and $CD$ being the length and $BC$ and $AD$ being the width. Suppose two segments parallel to $AB$ and $BC$ partition $ABCD$ into four smaller rectangles: $PQRS$, $PQTU$, $UVWX$, and $RSWX$. Given the areas of three of these rectangles, we need to find the area of t... | null | null |
In the figure, $AB$ is the diameter of semicircle $O$, and $C$ and $D$ are two points on the semicircle. A tangent to semicircle $O$ is drawn through $B$ and intersects $CD$ at $P$. The line $PO$ intersects lines $CA$ and $AD$ at points $E$ and $F$ respectively. Prove that $OE=OF$. | Given the semicircle with diameter \( AB \), points \( C \) and \( D \) on the semicircle, and \( P \) as the intersection of the extension of \( CD \) with the tangent line at \( B \), we aim to prove that \( OE = OF \).
#### Detailed Proof Method One:
1. **Construct Auxiliary Lines**:
Draw \( PG \) intersecting t... | null | null |
Consider polynomials \( Q(x) \) of degree at most 3, where each coefficient is an element of \(\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}\). How many such polynomials satisfy \( Q(-1) = 1 \)? | Let \( Q(x) = ax^3 + bx^2 + cx + d \). The condition \( Q(-1) = 1 \) translates to:
\[ -a + b - c + d = 1. \]
We can redefine the variables as \( a' = 9 - a \) and \( c' = 9 - c \), keeping \( b \) and \( d \) the same, which transforms the equation to:
\[ a' + b + c' + d = 19. \]
Using the stars and bars method to fin... | null | null |
In a race of six athletes, Andrei fell behind Boris, with two athletes finishing between them. Victor finished after Dmitry but before Gennady. Dmitry overtook Boris but still finished after Yevgeny. What place did each athlete finish in? | 1. **Define the variables**:
- Let’s use first letters of names to denote the athletes:
- $A$ for Андрей
- $B$ for Борис
- $D$ for Дмитрий
- $E$ for Евгений
- $V$ for Виктор
- $G$ for Геннадий
2. **Analyze the conditions**:
- The first condition states: Андрей отстал от Бориса и меж... | null | null |
Let \( A B C \) be a triangle. If \(\sin A = \frac{36}{\alpha}\), \(\sin B = \frac{12}{13}\) and \(\sin C = \frac{\beta}{y}\), find the value of \(\beta\), where \(\beta\) and \(y\) are in the lowest terms. | Given that:
- $\sin A = \frac{36}{\alpha}$,
- $\sin B = \frac{12}{13}$,
- $\sin C = \frac{\beta}{y}$,
we need to find the value of $\beta$, where $\beta$ and $y$ are in the lowest terms.
1. **Find $\alpha$:**
- Assume $\alpha = 45$.
- Then $\sin A = \frac{36}{45} = \frac{4}{5}$.
- Since $\sin A = \frac{4}{5}... | null | null |
Car A travels at a constant speed. Car B travels at 1.5 times the speed of Car A for the same distance. If the speed and time of Car A and Car B's motion are to be shown on a graph, which graph best represents the distance versus time for both cars? Assume the cars begin their journeys at the same time.
A. Two lines, w... | 1. **Understanding the Problem**: Car A travels at a constant speed, and Car B travels at 1.5 times Car A's speed. Both travel the same distance, and we need a graph showing distance vs. time.
2. **Defining Variables**:
- Let the speed of Car A be $v$, and the speed of Car B be $1.5v$.
- Let the time taken by Ca... | null | null |
Point P is on the curve $\rho=10\sin\theta$, and point Q is on the curve $\rho\sin\theta=10$. Find the minimum distance between points P and Q. | Given the curves $\rho\sin\theta=10$ and $\rho=10\sin\theta$ can be respectively rewritten as:
- $y=10$
- $x^2+y^2=10y$, which simplifies to a circle with center at $(0,5)$ and radius $5$.
The line $y=10$ is tangent to the circle centered at $(0,5)$ with a radius of $5$.
It is evident that the minimum distance betwe... | null | null |
I have 6 shirts, 6 pairs of pants, and 6 hats. Each item comes in the same 6 colors (so that I have one of each item of each color). I refuse to wear an outfit in which all 3 items are the same color. How many choices for outfits do I have? | To solve this problem, we start by calculating the total number of outfit combinations without any restrictions. Since there are 6 choices for each item (shirts, pants, and hats), and each item's choice is independent of the others, we multiply the number of choices for each item together:
\[6 \text{ shirts} \times 6 ... | null | null |
Prove that the rebus KUSY + UKSY = UKSUS has no solutions. (In the rebus, identical letters represent identical digits and different letters represent different digits). | 1. **Analyze the Structure of the Problem:**
- We are given the algebraic addition problem involving four-digit numbers, where:
\[
\text{KUSY} + \text{UKSY} = \text{UKSYUS}
\]
- We need to prove that this equation has no solutions, given the constraints that each letter represents a unique digit.
... | null | null |
Given \( f(x) = |1 - 2x|, x \in [0, 1] \), the number of solutions to the equation
\[ f(f(f(x))) = \frac{1}{2} x \]
is . | 1. First, let's define the function clearly:
\[
f(x) = |1 - 2x|, \quad x \in [0, 1].
\]
2. Next, let's determine how the function \( f(x) \) behaves over the interval \([0, 1]\). Since \( x \) is in this range, we have two cases to consider based on the definition of the absolute value:
- For \( 0 \... | null | null |
Given the function $f(x) = \frac {a\ln x}{x+1}+ \frac {b}{x}$, the equation of the tangent line to the curve $y=f(x)$ at the point $(1,f(1))$ is $x+2y-3=0$.
(Ⅰ) Find the values of $a$ and $b$;
(Ⅱ) Prove that when $x > 0$ and $x \neq 1$, $f(x) > \frac {\ln x}{x-1}$. | (Ⅰ) To find the slope of the tangent line at $x=1$, we first find the derivative of $f(x)$:
$$f'(x) = \frac {a\left( \frac {x+1}{x}-\ln x\right)}{(x+1)^{2}} - \frac {b}{x^{2}}.$$
Since the slope of the line $x+2y-3=0$ is $-\frac{1}{2}$ and it passes through the point $(1,f(1))$, which implies that $f(1) = 1$, we have t... | null | null |
The length of the major axis of the ellipse $x^{2}+9y^{2}=9$ is __________. | Solution: The equation of the ellipse $x^{2}+9y^{2}=9$ can be rewritten as $\frac{x^{2}}{9}+y^{2}=1$,
which means $a=3$, $b=1$,
thus, the length of the major axis is $2a=6$.
Therefore, the answer is $\boxed{6}$.
By converting the equation of the ellipse into its standard form, we find $a=3$, which leads us to t... | null | null |
Find all prime numbers \( p \) such that \( p^2 + 14 \) is also a prime number. | 1. **Check for Small Values of \( p \)**:
- Let us start by checking if \( p = 3 \) is a solution.
- Compute \( p^2 + 14 \) when \( p = 3 \):
\[
p^2 + 14 = 3^2 + 14 = 9 + 14 = 23
\]
- Since 23 is a prime number, \( p = 3 \) satisfies the condition.
2. **Analyze General Case \( p \ne... | null | null |
Let $S$ be a set of $n\ge2$ positive integers. Prove that there exist at least $n^2$ integers that can be written in the form $x+yz$ with $x,y,z\in S$ .
*Proposed by Gerhard Woeginger, Austria* | 1. Let \( S = \{a_1, a_2, \ldots, a_n\} \) with \( a_1 < a_2 < \cdots < a_n \). We need to show that there are at least \( n^2 \) integers that can be written in the form \( x + yz \) with \( x, y, z \in S \).
2. Consider the function \( \phi: S \times S \to \mathbb{Z}^+ \) defined by \( \phi(x, y) = x + y \cdot a_n \... | null | null |
John has to replace the ball bearings for machines he works with. He has 10 machines and they take 30 ball bearings each. It normally costs $1 per ball bearing but right now there is a sale where they are only $.75. Also since he is buying in bulk he gets a further 20% discount. How much money did he save by buying... | John needs to calculate the total number of ball bearings required for all his machines. Given that each machine requires 30 ball bearings and he has 10 machines, the total number of ball bearings needed is calculated as follows:
\[ \text{Total ball bearings} = 30 \times 10 = 300 \]
The normal cost for these ball bear... | null | null |
Given the sequence $\{a\_n\}$, where $a\_1=-60$ and $a\_{n+1}=a\_n+3$, find the value of $|a\_1|+|a\_2|+|a\_3|+...+|a\_{30}|$
A: 445
B: 765
C: 1080
D: 3105 | This problem involves comprehensive knowledge of arithmetic sequences. Since $a\_{n+1}=a\_n+3$, it follows that $a\_{n+1}-a\_n=3$. Therefore, $\{a\_n\}$ is an arithmetic sequence with first term $-60$ and common difference $3$. We can derive the nth term formula and then find the sum of the absolute values of the first... | null | null |
In a family, 19 people eat only vegetarian, 9 people eat only non-veg, and some people eat both veg and non-veg. There are 31 people who eat veg in the family. How many people eat both veg and non-veg? | To find out how many people eat both veg and non-veg, we can use the information given about the total number of people who eat veg and the number of people who eat only vegetarian.
We know that there are 31 people who eat veg (this includes those who eat only veg and those who eat both veg and non-veg) and 19 people ... | null | null |
When XYZ Co discontinued an agency arrangement with John, management held his commission of Rs. 25000 for one month. Earlier, John had availed an advance agency fees from the company, but Robert forgot that. After one month, John asked for his fees and the accountant gave him Rs. 18500. The incentive amount given to Jo... | Let's denote the advance agency fees that John had availed as A.
According to the information provided, John's commission was Rs. 25000, and he was given an incentive amount of Rs. 1780. Therefore, the total amount due to John before considering the advance agency fees would be:
Total due = Commission + Incentive
Tot... | null | null |
Determine the prime number $p$, knowing that the number of its digits is even and the number is a palindrome, meaning that if you write its digits in reverse order, you still get $p$. | 1. Identify all two-digit palindromic numbers:
- Two-digit palindromic numbers are $11, 22, 33, 44, 55, 66, 77, 88, 99$.
2. Filter out prime numbers:
- The number $11$ is prime.
- All other numbers are multiples of $11$ and hence not prime.
3. Demonstrate that an even-digit palindromic number is divisible... | null | null |
The solution set of the inequality $|2x-5| \leqslant 7$ is _______. | **Analysis**
This question examines the method of solving absolute value inequalities, which is a basic problem.
**Solution**
Given the inequality $|2x-5| \leqslant 7$,
We have $-7 \leqslant 2x-5 \leqslant 7$,
Which leads to $-1 \leqslant x \leqslant 6$,
Therefore, the solution set of the original inequality is $... | null | null |
1. Use the Euclidean algorithm or the method of successive subtraction to find the greatest common divisor (GCD) of 228 and 1995.
2. Convert the number 104 to ternary (base 3). | 1. Using the Euclidean algorithm:
Since $1995 \div 228 = 8$ with a remainder of $171$,
$$
1995 = 228 \cdot 8 + 171
$$
Then divide $228$ by $171$,
$$
228 = 171 \cdot 1 + 57
$$
Next, divide $171$ by $57$,
$$
171 = 57 \cdot 3
$$
As there is no remainder in the last division, the GCD ... | null | null |
Natasha exercised for 30 minutes every day for a certain period. Esteban exercised for 10 minutes on each of nine days. Natasha and Esteban exercised for a total of 5 hours. How long did Natasha exercise daily? | Let's denote the number of days Natasha exercised as \( d \).
We know that Natasha exercised for 30 minutes every day, so the total time she exercised is \( 30d \) minutes.
Esteban exercised for 10 minutes on each of nine days, so the total time he exercised is \( 10 \times 9 = 90 \) minutes.
Together, Natasha and E... | null | null |
The altitudes of triangle \(ABC\) intersect at point \(O\). It is known that \(OC = AB\). Find the angle at vertex \(C\). | 1. Given that the altitudes of triangle \(ABC\) intersect at point \(O\), and \(OC = AB\). We need to find the angle at vertex \(C\).
2. Label \(D\) as the foot of the altitude dropped from vertex \(A\) to side \(BC\).
3. To solve this problem, we start by showing that triangles \(COD\) and \(ABD\) are congruent.
4.... | null | null |
The sale price of sarees listed for Rs. 400 after successive discounts of 10% and another discount is Rs. 342. What was the second discount percentage? | Let's calculate the sale price after the first discount of 10%.
First discount of 10% on Rs. 400:
10% of 400 = (10/100) * 400 = 40
So, the price after the first discount:
400 - 40 = Rs. 360
Now, we know that after the second discount, the sale price is Rs. 342.
Let's denote the second discount percentage as 'D'.
T... | null | null |
A chord intercepted on the circle $x^{2}+y^{2}=4$ by the line $\begin{cases} x=2-\frac{1}{2}t \\ y=-1+\frac{1}{2}t \end{cases} (t\text{ is the parameter})$ has a length of $\_\_\_\_\_\_\_.$ | Since the line $\begin{cases} x=2-\frac{1}{2}t \\ y=-1+\frac{1}{2}t \end{cases} (t\text{ is the parameter})$
We have the general equation of the line as $x+y-1=0$.
The distance between the center of the circle and the line is $d=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}$.
The length of the chord is given by $l=2\sqrt{4-... | null | null |
Given a square table with four legs $L_{1}, L_{2}, L_{3}, L_{4}$, each with a height of a positive integer $n$, determine how many ordered quadruples of non-negative integers $\left(k_{1}, k_{2}, k_{3}, k_{4}\right)$ exist such that when each leg $L_{i}$ is sawed off by a length of $k_{i}$ from the ground, $i = 1, 2, 3... | **Step 1: Understand the Problem**
We are given four legs of a table $L_1$, $L_2$, $L_3$, and $L_4$, each of an integer height $n$. We need to determine the number of ordered quadruples of non-negative integers $(k_1, k_2, k_3, k_4)$ such that after each leg is cut by $k_i$ (from the ground), the table remains stable.... | null | null |
In the vertices of a 17-sided polygon, different integers were written (one in each vertex). Then all the numbers were simultaneously replaced with new ones: each was replaced by the difference of the two numbers following it in a clockwise direction (subtracting the next one from the neighboring one). Could the produc... | ### Problem:
In the vertices of a 17-gon, distinct integers were written (one in each vertex). Then, all numbers were simultaneously replaced with new numbers: each was replaced with the difference of the two successive numbers following it in the clockwise direction (the predecessor subtracts the successor). Could the... | null | null |
Find the common ratio of the infinite geometric series: $$\frac{4}{7} + \frac{12}{7} + \frac{36}{7} + \dots$$ | To find the common ratio of the given series, we take the ratio of consecutive terms. Let's denote the first term as $a = \frac{4}{7}$ and calculate the ratio using the second term $b = \frac{12}{7}$:
$$ r = \frac{b}{a} = \frac{\frac{12}{7}}{\frac{4}{7}} = \frac{12}{7} \cdot \frac{7}{4} = \frac{12 \cdot 7}{7 \cdot 4} =... | null | null |
In triangle \( A B C \), points \( M \) and \( N \) are chosen on sides \( A B \) and \( B C \) respectively such that \( A M = 2 M B \) and \( B N = N C \). Segments \( A N \) and \( C M \) intersect at point \( P \). Find the area of quadrilateral \( M B N P \) if it is known that the area of triangle \( A B C \) is ... | 1. **Analyzing the Given Information**:
We are given a triangle \(ABC\) with points \(M\) on \(AB\) and \(N\) on \(BC\) such that:
\[
AM = 2MB \quad \text{and} \quad BN = NC
\]
The area of triangle \(ABC\) is 30.
2. **Using Ratios**:
Let's place coordinate points for easier calculation, assuming \(B\... | null | null |
Ms. Thompson sold two books at $\$4.50$ each. On the first book, she made a $25\%$ profit, and on the second book, she incurred a $25\%$ loss. Determine Ms. Thompson's overall financial outcome from the sale of these books.
$\textbf{(A)}\ \text{broke even}$
$\textbf{(B)}\ \text{lost }60\text{ cents}$
$\textbf{(C)}\ \te... | 1. **Analyzing the first book's sale:**
- Ms. Thompson made a $25\%$ profit on the first book. Therefore, she sold the book for 125% of its cost price.
- Let the cost price of the first book be $C_1$. Then the selling price is $4.50 = 1.25 \times C_1$.
- Solving for $C_1$, we have:
\[
C_1 = \frac{4.... | null | null |
Let $x$ be a positive real number. Find the maximum possible value of $$\frac{x^2+3-\sqrt{x^4+9}}{x}.$$ | Rationalizing the numerator, we proceed as follows:
\begin{align*}
\frac{x^2+3-\sqrt{x^4+9}}{x}\cdot\frac{x^2+3+\sqrt{x^4+9}}{x^2+3+\sqrt{x^4+9}} &= \frac{(x^2+3)^2 - (x^4+9)}{x(x^2+3+\sqrt{x^4+9})} \\
&= \frac{x^4 + 6x^2 + 9 - x^4 - 9}{x(x^2+3+\sqrt{x^4+9})} \\
&= \frac{6x^2}{x(x^2+3+\sqrt{x^4+9})} \\
&= \frac{6}{x+\... | null | null |
In the Cartesian coordinate system, given points $A(2,m)$ and $B(n,-3)$ are symmetric about the $x$-axis, then the value of $m+n$ is ( )
A: $-1$
B: $1$
C: $5$
D: $-5$ | To solve this problem, let's analyze the symmetry of points $A$ and $B$ about the $x$-axis in the Cartesian coordinate system.
Given points are $A(2,m)$ and $B(n,-3)$. For two points to be symmetric about the $x$-axis, their $x$-coordinates must be the same, and their $y$-coordinates must be equal in magnitude but op... | null | null |
Marcus scored 5 3-point goals, 10 2-point goals, 8 free throws (1-point each), and 2 4-point goals. If his team scored 110 points overall, what percentage of the team's total points did Marcus score? | First, let's calculate the total points Marcus scored:
5 3-point goals = 5 * 3 = 15 points
10 2-point goals = 10 * 2 = 20 points
8 free throws = 8 * 1 = 8 points
2 4-point goals = 2 * 4 = 8 points
Now, add all of Marcus's points together:
15 (from 3-point goals) + 20 (from 2-point goals) + 8 (from free throws) + 8 (... | null | null |
You have 30 dimes, 40 quarters, and 10 half dollars. What percent of the value of your coins is in quarters? | 1. **Calculate the total value of each type of coin**:
- Dimes: \(30 \times 10\) cents = 300 cents.
- Quarters: \(40 \times 25\) cents = 1000 cents.
- Half dollars: \(10 \times 50\) cents = 500 cents.
2. **Calculate the total value of all coins**:
\( \text{Total} = 300 + 1000 + 500 = 1800 \text{ cents} \).... | null | null |
There are 3 teachers and 3 students standing in a row for a photo. The total number of arrangements where no two students stand next to each other is ( )
A: 36
B: 72
C: 144
D: 288 | **Answer:** To solve for the arrangements where 3 students are not adjacent to each other, we can use the slot method.
First, arrange the 3 teachers, then there are 4 slots available for the students.
Therefore, there are $A_3^3 \cdot A_4^3 = 144$ arrangements.
Hence, the answer is: $\boxed{C}$
**Analysis:** T... | null | null |
A factory produces 4340 toys per week. The workers at this factory work a certain number of days a week and make the same number of toys every day. They produce 2170 toys each day. How many days a week do they work? | To find out how many days a week the workers work, we can divide the total number of toys produced per week by the number of toys produced each day.
Total toys per week = 4340
Toys produced each day = 2170
Number of days worked per week = Total toys per week / Toys produced each day
Number of days worked per week = 4... | null | null |
Determine the sum of all prime numbers $p$ for which there exists no integer solution in $x$ to the congruence $5(10x+2) \equiv 7 \pmod{p}$. | First, simplify the congruence $5(10x+2) \equiv 7 \pmod{p}$:
\[ 50x + 10 \equiv 7 \pmod{p} \]
\[ 50x \equiv -3 \pmod{p} \]
\[ 50x \equiv p - 3 \pmod{p} \]
This congruence has solutions if and only if 50 is invertible modulo $p$, which occurs if $\gcd(50, p) = 1$. The prime factors of 50 are 2 and 5. Therefore, if $p =... | null | null |
End of preview. Expand in Data Studio
NuminaMath Worked Solutions
Competition and school maths problems with step-by-step solutions.
| Rows | 41,100 |
| 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-numina-worked-41k", split="train")
import pandas as pd
df = pd.read_parquet("hf://datasets/rodriguescarson/adaption-math-numina-worked-41k/data.parquet")
Published by Carson Rodrigues (Hugging Face, Kaggle).
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