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Calculate the sum of $\log_{2002} (11^2) + \log_{2002} (13^2) + \log_{2002} (14^2)$, where $f(n) = \log_{2002} n^2$ for all positive integers $n$. | tiger_math |
Farmer Brown planted 300 cabbages in his main field. Over the course of the season, due to pests and bad weather, he lost 56 cabbages. He decided to plant 78 more cabbages in a new, smaller field that was more protected. At the end of the season, he harvested all the remaining cabbages from both fields. How many cabbag... | tiger_math |
What is the correlation coefficient between the mathematical aptitude and the final exam scores of a group of 50 students if the regression equation of aptitude (x) and exam scores (y) is y= 15x + 90? | tiger_lab_math |
What is the Discrete-Time Fourier Transform (DTFT) of the signal \( y[n] = \sin(\Omega_0 n + \phi) \) in terms of \( \Omega \), using the property that a time shift corresponds to multiplying by \( e^{i \Omega \phi} \) in the frequency domain? Specifically, express the DTFT for \( -\pi < \Omega < \pi \) and determine t... | automath |
In a certain competition, contestants are awarded points for each problem they solve correctly. The points are awarded according to the formula $p = 10a - 2b + 5c$, where $p$ is the total number of points, $a$ is the number of problems solved correctly in the algebra section, $b$ is the number of problems solved incorr... | nvidia_math |
What is the smallest number greater than 100,000 that is divisible by 8, 15, and 21, and is closest to 110,000? | automath |
Consider the equation $x^2 + 5x - 11 = 0$. Let the roots of this equation be $r_1$ and $r_2$. Find the value of $|r_1r_2 + r_1 + r_2|$. | nvidia_math |
A deck of 52 cards is shuffled, and then two cards are drawn randomly. What is the probability that the first card drawn is an ace, and the second card drawn is a king? | nvidia_math |
a man buys an article for $ 100 . and sells it for $ 120 . find the gain percent ? Please write a program to solve it | tiger_lab_math |
John decides to get gym memberships so he can get in shape. The gym close to his house is close but doesn't have everything he wants so he gets two different gym memberships. The cheap one costs some amount per month and has a sign-up fee of $50. The second gym is 3 times more expensive and it has a sign-up fee of 4 mo... | tiger_math |
A water tank is in the shape of a cone with a height of 10 meters and a base radius of 4 meters. The tank is filled with water, and a spherical buoy with a radius of 1 meter is floating on the surface. What is the volume of water displaced by the buoy, in cubic meters? Express your answer in terms of $\pi$. | nvidia_math |
What is the value of \( f(0.5) \) for the approximation of the blancmange function defined by
\[
f(x) = \sum_{n=0}^{10} f_n(x) \quad \text{where} \quad f_n(x) = \frac{1}{2^n} \left| 2^n x - \left[ 2^n x + \frac{1}{2} \right] \right| ?
\]
Calculate this sum and express your answer as a decimal rounded to three decim... | automath |
Given an infinite ray with direction vector \((0, -1, -1)\) that passes through the point \((0, 0, 10)\), and an infinite plane with a normal vector \((0, 0, 1)\) that passes through the point \((0, 0, 5)\), find the point of intersection between the ray and the plane. What are the coordinates of the intersection point... | automath |
Given a triangular region with vertices at the points \((-1, 1)\), \((0, 0)\), and \((1, 1)\), find a suitable change of variables that transforms this triangular region into a square or rectangular region in the \(uv\)-plane, and compute the Jacobian of this transformation. What is the area of the resulting square or ... | automath |
Given the vectors $\mathbf{u} = \begin{pmatrix} 3 \\ -2 \\ 1 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 1 \\ 3 \\ -2 \end{pmatrix},$ find the vector $\mathbf{w}$ such that $2 \mathbf{u} + 3 \mathbf{w} = \mathbf{v}.$ | nvidia_math |
Exactly 36% of the numbers in set R are even multiples of 3. If 40% of the even integers in set R are not multiples of 3, what percent of the numbers in set R are not even integers?
Answer Choices: (A) 76% (B) 60% (C) 50% (D) 40% (E) 24% | tiger_lab_math |
How many ways are there to put 4 distinguishable balls into 2 distinguishable boxes? | tiger_math |
Bill is hoarding toilet paper in fear of another pandemic. He lives with his wife and two kids. Their bathroom usage patterns vary depending on the day of the week.
On weekdays, Bill goes to the bathroom three times a day and uses 5 squares of toilet paper each time, his wife goes to the bathroom four times a day and... | tiger_math |
Evaluate the following limit:
$$\lim_{\epsilon \to 0^+} \lim_{K \to \infty} \frac{(1-\epsilon)^K}{1+(1-\epsilon)^K} \cdot \frac{ \sum_{i=1}^{\frac{K-1}{2}} \binom{K}{i} \left[ \left( \frac{2\epsilon-\epsilon^2}{(1-\epsilon)^2} \right)^i - \left( \frac{\epsilon}{1-\epsilon} \right)^i \right] }{(1-\epsilon)^K \left[ 1 +... | automath |
Given \( K \) numbers that sum to \( S \), what is the minimum possible value of the sum of their squares? Provide your answer in terms of \( K \) and \( S \), and indicate if the numbers need to be integers. | automath |
Given an enumeration of the rational numbers \( r_n \) and the open dense set
\[
O = \bigcup_{n=1}^\infty \left( r_n - \frac{\epsilon}{2^{n + 2}}, r_n + \frac{\epsilon}{2^{n + 2}} \right),
\]
where \( \epsilon > 0 \) is fixed, can you construct a specific computable irrational number that is not contained in the set... | automath |
Lisa was asked to multiply 29 by 43 and round the result to the nearest hundred. Similar to Van, she rounded 29 to X and 43 to 40 before multiplying. She got an answer of 1200, which is incorrect. The correct answer is 1200. What is the value of X? | tiger_math |
Consider the cubic polynomial
\[ax^3+bx^2+cx+d,\]
where $a$, $b$, $c$ and $d$ are real numbers. We know that the linear binomials
$(x-1)$, $(x-2)$ and $(x-3)$ are factors of this polynomial and
that the constant term is equal to $6$. Find the value of $a+b+c+d$. | nvidia_math |
Find the largest number that is not a sum of two distinct primes. | nvidia_math |
The science club has 25 members: 10 boys and 15 girls. A 5-person committee is chosen at random. What is the probability that the committee has at least 1 boy and at least 1 girl? | nvidia_math |
In a right triangle PQR, where angle Q is 90 degrees and cos P is 5/8, what is the value of sin R? | tiger_math |
Tickets at a concert cost $8 for section A and $4.25 for section B. In total, 4500 tickets were sold worth $30,000. How many section A tickets were sold? | tiger_math |
Given a homogeneous thin stick of mass \( m \) and length \( l \) suspended from a ceiling, which can rotate freely about a pivot point \( P \), and a small ball of mass \( M \) attached to the end of the stick, if another small ball of mass \( M \) moves with velocity \( u \) and collides inelastically with the first ... | automath |
What is the molecular weight of 6 moles of Calcium hydride? | tiger_math |
In pentagon $ABCDE$, $AB = 10$, $BC = 14$, $CD = 17$, $DE = 20$, and $EA = 23$. Points $F$, $G$, $H$, $I$, and $J$ are on sides $AB$, $BC$, $CD$, $DE$, and $EA$, respectively, and divide their respective sides into segments whose lengths have a ratio of $1:2$. The ratio of the area of pentagon $FGHIJ$ to the area of pe... | nvidia_math |
If $(a - b)^2 = 50$ and $ab = 8$, what is $(a + b)^2$? | nvidia_math |
Mrs. Wilsborough saved $500 to buy concert tickets for her family. She bought 2 VIP tickets at $100 each and 3 regular tickets at $50 each. How much of her savings does Mrs. Wilsborough have after she buys the tickets? Let's write a Python program to solve it. | tiger_lab_math |
Consider a trapezoid where the lengths of the two bases are 6 and \( x \), and the height is \( 5 \). If the upper triangles formed by the height of the trapezoid are similar with a proportionality ratio of \( \frac{y}{x} = \frac{2}{6} \), express the area of the shaded trapezoidal region as a function of \( x \), and ... | automath |
The sum of the squares of two consecutive positive integers is 425. What is the value of the smaller integer? | nvidia_math |
Dacid obtained 96, 95, 82, 87 and 92 marks (out of 100) in English, Mathematics, Physics, Chemistry and Biology. What are his average marks?
Answer Choices: (A) 67 (B) 26 (C) 87 (D) 26 (E) 90 | tiger_lab_math |
In triangle $CAT$, we have $\angle{ACT}=\angle{ATC}$ and $\angle{CAT}=36^\circ$. If $\overline{TR}$ bisects $\angle{ATC}$, then how many degrees is $\angle{CRT}$? [asy]
/* AMC8 2000 #13 Problem */
draw((0,0)--(.5,1.75)--(1,0)--cycle);
draw((1,0)--(.15,.5));
label("$R$", (.15,.5), W);
label("$C$", (0,0), SW);
label("$T$... | nvidia_math |
The number of students who wish to go on a skiing trip is twice as many as those who wish to go on a scavenger hunting trip. If there are 4000 students who wish to go on the scavenger hunting trip, how many students are there altogether? | tiger_lab_math |
If $\sec x + \tan x = \frac{7}{3},$ then find $\sec x - \tan x.$ | tiger_math |
Determine the value of $b$ such that the lines described by the equations $x - 3y = 5$ and $y = bx + 2$ are perpendicular. | tiger_math |
In Yardley it snowed 0.12 inch in the morning and 0.5 inch in the afternoon. The total amount of snowfall was _____ . | tiger_math |
Lucas is in charge of organizing a dinner party at a triangular-shaped table where the lengths of the sides are all different. He has bought 60 inches of decorative trim to outline the table's edges. If the side lengths of the triangle must be whole numbers, what is the largest possible area that the table can have, ba... | tiger_math |
In $\triangle ABC$, we have $\angle A = 30^\circ$ and $\sin C = 0.5$. Find $\sec B$. | nvidia_math |
A man swims downstream 72 km and upstream 45 km taking 9 hours each time; what is the speed of the current ?
Answer Choices: (A) 1 kmph (B) 3.2 kmph (C) 1.5 kmph (D) 2 kmph (E) 2.5 kmph | tiger_lab_math |
The school is planning an event on the football field which is shaped like a rectangle. If the length of the rectangle is 60 meters more than its width and the perimeter of the field is 400 meters, what is the area of the football field? | tiger_math |
The line $y = b-x$ with $0 < b < 4$ intersects the $y$-axis at $P$ and the line $x=4$ at $S$. If the ratio of the area of triangle $QRS$ to the area of triangle $QOP$ is 9:25, what is the value of $b$? Express the answer as a decimal to the nearest tenth.
[asy]
draw((0,-3)--(0,5.5),Arrows);
draw((4,-3.5)--(4,5),Arrows... | tiger_lab_math |
Given a synthetic dataset of 1000 patients where the probability of having a heart attack in the next year is 0.5, and a logistic regression model predicting heart attacks based on a biomarker, if the model predicts that 343 out of 511 individuals with heart attacks are correctly identified, what is the precision of th... | automath |
Consider the transformation defined by \(\phi: M \to N\) where \(M\) is given by coordinates \((x, y)\) and \(N\) is given by coordinates \((u, v)\) with the equations \(u = x^2 + y^2\) and \(v = 2xy\).
1. Compute the Jacobian matrix of the transformation \(\phi\).
2. At which points does the transformation fail to b... | automath |
Given the function \( f(x, y) = \frac{1}{x^2 - y} \), determine the critical points of the function. What are the coordinates of these critical points, and how can you classify them as local maxima, local minima, or saddle points? | automath |
A pet shop has 2 puppies and some kittens. A puppy costs $20, and a kitten costs $15. If the stock is worth $100, how many kittens does the pet shop have? | nvidia_math |
In the infinite sequence 50, 60, 70, 80, 90, . . . , where each term is 10 greater than the previous term, the 100th term is
Answer Choices: (A) 1330 (B) 1807 (C) 1910 (D) 1040 (E) 2060 | tiger_lab_math |
Let F(m,n) =45*m+36*n,where m and n are integers (+ve or -ve) What is the minimum positive value for f(m,n) (This may be achived for various values of m and n)?
Answer Choices: (A) 5 (B) 6 (C) 7 (D) 8 (E) 9 | tiger_lab_math |
A toy car track is made up of three sections: a straight section, a circular section, and a spiral section. The straight section is 15 meters long, the circular section has a diameter of 8 meters, and the spiral section is 3 meters high. If the total length of the track is 45 meters, what is the radius of the spiral se... | nvidia_math |
A lady grows cabbage in her garden that is in the shape of a square. Each cabbage takes 1 square foot of area in her garden. This year, she has increased her output by 211 cabbages when compared to last year. The shape of the area used for growing the cabbage has remained a square in both these years. How many cabbages... | tiger_lab_math |
A train speeds past a pole in 15 sec and a platform 100 m long in 25 sec, its length is?
Answer Choices: (A) 238 (B) 150 (C) 988 (D) 177 (E) 171 | tiger_lab_math |
What is the sum of the coordinates of the midpoint of the segment with endpoints $(8, 5)$ and $(2, -1)$? Let's write a Python program. | tiger_lab_math |
To 15 lts of water containing 20% alcohol, we add 5 lts of pure water. What is % alcohol.
Answer Choices: (A) 25% (B) 35% (C) 15% (D) 18% (E) 19% Please respond by writing a program in Python. | tiger_lab_math |
Let \[f(x) =
\begin{cases}
2x^2 - 3&\text{if } x\le X, \\
ax + 4 &\text{if } x>2.
\end{cases}
\]Find $a$ if the graph of $y=f(x)$ is continuous (which means the graph can be drawn without lifting your pencil from the paper). The answer is 2. What is the value of unknown variable X? | tiger_math |
Find the greatest common factor of the three numbers: $1! + 2! + 3!$, $2! + 3! + 4!$, and $3! + 4! + 5!$. | nvidia_math |
Given two mutually exclusive propositions \( \alpha \) and \( \beta \) where \( P(\alpha) = 0.3 \) and \( P(\beta) = 0.5 \), what is the probability \( P(\alpha \vee \beta) \)? | automath |
Problem:
A deck of 52 cards is shuffled and a card is drawn at random. What is the probability that the card is not a face card (King, Queen, Jack) and its suit is not hearts? Express your answer as a common fraction. | nvidia_math |
A person got Rs.48 more when he invested a certain sum at compound interest instead of simple interest for two years at 8% p.a. Find the sum?
Answer Choices: (A) 7500 (B) 2776 (C) 2877 (D) 2667 (E) 2671 | tiger_lab_math |
Find the number of integers between 100 and 200 that are not divisible by 2, 3, or 5. | nvidia_math |
A 72.0 kg man hoists himself into a tree using a nylon rope that he has tied around his waist. He pulls downward on the free end of the rope with a force of 358 N. Neglecting friction between the rope and the branch, what is the man's upward acceleration? | automath |
Of the 60 students in the drama club, 36 take mathematics, 27 take physics and 20 students take both mathematics and physics. How many drama club students take neither mathematics nor physics? | tiger_math |
There are constants $\alpha$ and $\beta$ such that $\frac{x-\alpha}{x+\beta} = \frac{x^2-80x+1551}{x^2+57x-2970}$. What is $\alpha+\beta$? | nvidia_math |
Given an array of positive integers, how many subarrays have a sum that lies within the range [L, R]? Provide an algorithm that achieves this in O(N log N) time complexity. | automath |
Eula Jones invests a total of $6000 in two accounts, with a portion at 4.5% interest and the remainder at 6%. If she invests \( d \) dollars at 4.5%, what expression calculates the total amount of interest earned from the investment at 6% after one year? Provide the correct expression. | automath |
Given three points in the plane: \( A(0, 0) \), \( B(2, 4) \), and \( C(4, 0) \), find the convex combination of these points that corresponds to the weights \( \alpha_1 = \frac{1}{4} \), \( \alpha_2 = \frac{1}{2} \), and \( \alpha_3 = \frac{1}{4} \). What are the coordinates of the resulting point? | automath |
Given that 0C04328 is a multiple of 88, find the smallest possible digit C that makes the number a multiple of 88. | nvidia_math |
The expression $\cos x + \cos 3x + \cos 7x + \cos 9x$ can be written in the equivalent form
\[a \cos bx \cos cx \cos dx\]for some positive integers $a,$ $b,$ $c,$ and $d.$ Find $a + b + c + d.$ | tiger_math |
What value of \( x \) makes the denominator of the function \( g(x) = \frac{x+2}{x-1} \) equal to zero? | automath |
A rectangular prism has a base with length 8 and width 4, and a height of 15. A pyramid is placed on top of the prism, with the base of the pyramid coinciding with the top face of the prism. The apex of the pyramid is directly above the center of the base and has a distance of 17 from the base. Find the total volume of... | tiger_math |
What is the simplified form of $\left( \frac{1}{2k} \right)^{-2} \cdot (-k)^3$? | tiger_math |
Given a topological space X consisting of two circles intersecting at exactly two points, compute the total number of ways to assign colors (black or white) to each point of intersection such that no two adjacent points have the same color. | tiger_lab_math |
If there are 50 students in Kylie's class and in a specific test, ten students scored 90 marks, 15 scored ten marks less than the first ten students, and the remaining students scored 60 marks each, what is the average mark for the entire class? | tiger_math |
Given the equation
$$908.80 = 250\left(1+\frac{0.24}{m}\right)^{6m},$$
find the value of \( m \) such that \( m \) is approximately equal to 0.99. What numerical method can be used to verify the solution, and what is the value of \( m \) to four decimal places? | automath |
a vessel of capacity 2 litre has 30 % of alcohol and another vessel of capacity 6 litre had 45 % alcohol . the total liquid of 8 litre was poured out in a vessel of capacity 10 litre and thus the rest part of the vessel was filled with the water . what is the new concentration of mixture ? | tiger_math |
Find all solutions to
\[x^2 + 4x + 4x \sqrt{x + 3} = 13.\]Enter all the solutions, separated by commas. Let's write a Python program to solve it. | tiger_lab_math |
A ball is dropped from a height of 1 yard and rebounds to \(\frac{2}{3}\) of its previous height with each bounce. What is the height of the ball after its 10th rebound, and what is the total vertical distance traveled by the ball after it touches the ground for the 10th time? | automath |
The expression $x(x + y)$ is expanded, where $x = (a + b)^5$ and $y = (c + d)^7$. What is the coefficient of $a^2b^3c^2d^4$ in the expansion? | nvidia_math |
In the same diagram, let $T$ be the midpoint of $\overline{PQ}$. If the area of $\triangle QTS$ is $2$, what is the area of the trapezoid $PQRS$? | tiger_math |
Let $V$ be the vector space of all $3\times3$ matrices, and let $W$ be the subspace of all matrices of the form $$
\begin{pmatrix}a&b&c\\ d&0&-a\\ e&-d&0\end{pmatrix}
$$Find the dimension of $W$. | nvidia_math |
Let $p(x) = x^3 + ax^2 + bx + c$ be a polynomial with integer coefficients, and let $r,$ $s,$ and $t$ be its roots. Suppose that
\begin{align*}
\log_2(rst) + \log_3(rs) &= 2, \\
\log_3(rst) + \log_4(rt) &= 3, \\
\log_4(rst) + \log_5(st) &= 4.
\end{align*}Compute the value of the product $rst.$ | nvidia_math |
Carly had 42 lollipops to share with her friends. Half of the lollipops were cherry, and the rest were equal amounts of watermelon, sour apple, and grape. How many lollipops were grape? Let's write a program. | tiger_lab_math |
A square and a triangle have equal perimeters. The lengths of the three sides of the triangle are $6.1$ cm, $8.2$ cm and $9.7$ cm. What is the area of the square in square centimeters? Let's program in Python in the response. | tiger_lab_math |
Given the demand function for bagels \( Q_d = 300 - 100P \) and the supply function \( Q_s = -7 + 2P \), determine the equilibrium price \( P \) and quantity \( Q \) of bagels. Then, calculate the consumer surplus at the equilibrium price. | automath |
An infinite geometric series has common ratio $\frac{-2}{5}$ and sum $30.$ What is the second term of the sequence? | tiger_math |
a man is 18 years older than his son . in two years , his age will be twice the age of his son . the present age of this son is ? Let's write a Python program to solve it. | tiger_lab_math |
A line through the points $(2, -9)$ and $(j, 17)$ is parallel to the line $2x + 3y = 21$. What is the value of $j$? | tiger_lab_math |
Calculate the average of first 12 even numbers is?
Answer Choices: (A) 15 (B) 14 (C) 12 (D) 11 (E) 13 | tiger_lab_math |
Increasing the original price of a certain item by 15 percent and then increasing the new price by 15 percent is equivalent to increasing the original price by what percent? | tiger_math |
A project manager needs to select a group of 3 people from a total of 3 men and 3 women. How many possible group combinations exist such that no group has all men or all women?
Answer Choices: (A) 18 (B) 24 (C) 74 (D) 82 (E) 48 | tiger_lab_math |
Given the identity \(\frac{n^3 + a^3}{n^3 + b^3} = \frac{n + a}{n + b}\) when \(a + b = n\), derive the conditions under which this identity holds true. Specifically, evaluate the following expression for integer values of \(n\), \(a\), and \(b\):
\[
\frac{124^3 + 43^3}{124^3 + 81^3}
\]
Is this equal to \(\frac{124 ... | automath |
The side length of square $C$ is 27 cm. The side length of square $D$ is 35 cm. What is the ratio of the area of square $C$ to the area of square $D$? Express your answer as a common fraction. | nvidia_math |
I chose a number and divide it by 4. Then I subtracted 18 from the result and got 7. What was the number I chose?
Answer Choices: (A) 600 (B) 100 (C) 800 (D) 900 (E) None | tiger_lab_math |
What is the total sum of the values of A, B, C, and D in the simplified function $y = Ax^2 + Bx + C$, where the original function $y = \frac{x^3 + 8x^2 + 21x + 18}{x + 2}$ is defined everywhere except at $x = D$? | tiger_math |
Laura loves to cook. One day she decided to make dinner for her family. She bought the necessary ingredients for this occasion: two salads, 2 kilograms of beef, 1 kilogram of potatoes, and two liters of juice. One salad costs $3, and one kilogram of beef is two times more expensive. One kilogram of potatoes costs one-t... | nvidia_math |
The point $A$ $(3,4)$ is reflected over the $x$-axis to $B$. Then $B$ is reflected over the line $y=x$ to $C$. What is the area of triangle $ABC$? | nvidia_math |
Let \( D \subset \mathbb{C} \) be an open set, and consider the infinite product of analytic functions \( f(z) = \prod_{n=1}^{\infty} f_n(z) \) that converges locally uniformly on \( D \setminus \{ z \in D : f(z) = 0 \} \). Assume that the logarithmic derivative of \( f \) satisfies the relation
\[
\frac{f'(z)}{f(z)}... | automath |
Given a particle moving in a straight line with a velocity \( v \) that is a function of its position \( s \), it is known that the acceleration \( a \) can be expressed as \( a = v \left( \frac{dv}{ds} \right) \). If the velocity of the particle is given by \( v(s) = 5s^2 + 3s + 2 \), determine the acceleration \( a \... | automath |
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