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Three coplanar circles with distinct radii intersect as shown. What is the maximum number of points on the circles that a line passing through all three circles can touch?
[asy]import graph;
draw(Circle((-9,9),15));
draw(Circle((0,-9),10));
draw(Circle((9,9),5));
[/asy] | tiger_math |
The probability that event A occurs is 0.4, and the probability that events A and B both occur is 0.25. If the probability that either event A or event B occurs is 0.6, what is the probability that event B will occur?
Answer Choices: (A) 0.05 (B) 0.15 (C) 0.45 (D) 0.5 (E) 0.55 | tiger_lab_math |
Find the maximum value of
\[\begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 + \sin \theta & 6 \\ 7 + \cos \theta & 8 & 9 \end{vmatrix},\]as $\theta$ ranges over all angles. | tiger_math |
Consider the function
$$F(x)=\frac{x^2+ax+1}{x^2+bx+1},$$
where $a$ and $b$ are constants. Determine the values of $a$ and $b$ such that $F(x)$ is even, odd, or neither. | nvidia_math |
What is the greatest common factor of $1729$ and $1848$? | tiger_math |
What are the coordinates of the minimum point (vertex) of the parabola represented by the equation \( y = 2x^2 + 8x + 9 \)? | automath |
Find the greatest common factor of $39$ and $91$. | tiger_math |
Two identical machines have the ability to produce both nuts and bolts. However, it takes 1 second to produce a bolt but 2 seconds to produce a nut. What is the fastest the two machines, working together, can produce 2,000 nuts and 2,000 bolts?
Answer Choices: (A) 1,250 seconds (B) 1,500 seconds (C) 1,750 seconds (D) 2... | tiger_lab_math |
One hundred and fifty people were surveyed. Of these, $105$ indicated they liked Mozart and $100$ indicated they liked Bach. What is the minimum number of people surveyed who could have said they liked both Mozart and Bach? | tiger_math |
Apples are sold at the farmer's market at a rate of $\$4$ per five pounds. How much would it cost to purchase 20 pounds of apples? | tiger_math |
What is the value of \(\frac{1}{i}\) in terms of real and imaginary components? | automath |
In $\triangle ABC$, we have $AC=BC=7$ and $AB=2$. Suppose that $D$ is a point on line $AB$ such that $B$ lies between $A$ and $D$ and $CD=8$. What is $BD$? | nvidia_math |
Justin can run 2 blocks in 1.5 minutes. If he is 8 blocks from home, in how many minutes can he run home? Let's program in Python in the response. | tiger_lab_math |
A palindrome is an integer that reads the same forward and backward, such as 3663. What percent of the palindromes between 100 and 999 contain at least one 5? | tiger_math |
In Henry's collection of music CDs, he has 3 more country CDs than rock CDs but twice as many rock CDs as classical CDs. If he has 23 country CDs, how many classical CDs make up his collection? Please respond by writing a program in Python. | tiger_lab_math |
Compute $19^8 \div 19^5$. | tiger_math |
Given a double series of non-negative terms \( \sum_{i=1}^{\infty} \sum_{j=1}^{\infty} a_{ij} \), where each \( a_{ij} \geq 0 \), what is the value of the sum when you interchange the order of summation to obtain \( \sum_{j=1}^{\infty} \sum_{i=1}^{\infty} a_{ij} \)? Is it true that these two sums are equal, and if so, ... | automath |
A bag contains 5 red, 4 green, and 3 blue balls. What is the probability of randomly selecting a red ball and a green ball in any order, without replacement? | nvidia_math |
Suppose you have six candies of different colors: red, blue, green, yellow, orange, and purple. In how many different ways can you select three candies if you can select more than one candy of the same color? | tiger_lab_math |
What is the number of non-isomorphic groups of order 48, and what are their corresponding group structures and isomorphism classes? | tiger_lab_math |
Find the number of functions $f(n),$ taking the integers to the integers, such that
\[f(a \cdot b) + f(a+b) = f(a) f(b) - 1\]for all integers $a$ and $b.$ | nvidia_math |
Find the ones digit of the largest power of $2$ that divides into $(2^5)!$. | tiger_math |
In a city, there are 480 buildings, 60% of which are residential buildings. How many buildings are not residential? | nvidia_math |
What is the exact number of elements in the set of maps from the set \( \{1, 2, 3\} \) to the set \( \{1, 2\} \)? | automath |
A train running at the speed of 60 km/hr crosses a pole in 12 sec. What is the length of the train?
Answer Choices: (A) 120 m (B) 200 m (C) 115 m (D) 110 m (E) 150 m | tiger_lab_math |
What is the remainder when $3^{205}$ is divided by 5? | tiger_math |
When rolling a fair 8-sided die, what is the probability of a 3 or 5 being rolled? | nvidia_math |
Find the value of $x$ that satisfies the equation $123_{x}+134_{x}=267_{x}$. | tiger_math |
Three circles of radius 1 are externally tangent to each other and internally tangent to a larger circle. What is the area of the region inside the large circle but outside the three small circles? Express your answer in simplest radical form. | tiger_math |
What is the value of the sum \( \sum_{n=1}^{\infty} (-1)^{n+1} \frac{1}{2n-1} \) when evaluated at \( x = 1 \)? | automath |
The average weight of 10 girls increases by 5 kg when a new girl comes in place of one of them weighing 50 kg. What might be the weight of the new girl?
Answer Choices: (A) 90 kg (B) 100 kg (C) 95 kg (D) 85 kg (E) 92 kg | tiger_lab_math |
From a standard deck of 52 playing cards, three cards are drawn at random without replacement. What is the probability of drawing exactly one ace, one king, and one jack, regardless of the order in which they are drawn? Calculate the exact probability as a fraction in simplest terms. | automath |
Triangle $ABC$ is an obtuse, isosceles triangle. Angle $A$ measures $x$ degrees, where $x$ is a positive integer. What is number of degrees in the measure of the largest interior angle of triangle $ABC$? | tiger_math |
In $\Delta ABC$, $AB=5$, $BC=8$, and $AC=7$. Point $D$ is the midpoint of $\overline{BC}$. Find the length of the median $\overline{AD}$. Express your answer as a decimal to the nearest hundredth. | nvidia_math |
For positive integers $n,$ let $\tau (n)$ denote the number of positive integer divisors of $n,$ including 1 and $n.$ For example, $\tau (1)=1$ and $\tau(6) =4.$ Define $S(n)$ by $S(n)=\tau(1)+ \tau(2) + \cdots + \tau(n).$ Let $a$ denote the number of positive integers $n \leq 2010$ with $S(n)$ odd, and let $b$ denote ... | tiger_math |
If $h(x) = x^3$ and $f(x) = 3x + 2$, what is the value of $f(h(-1))$? | tiger_math |
Solve the following system of linear equations using LU decomposition method:
3x + 2y + z = 10
x - 2y + 4z = 7
2x + y - z = 2
Note: The answer should be provided in the form of x, y, and z values. | tiger_lab_math |
Working alone at its constant rate, machine K took 3 hours to produce 1/8 of the units produced last Friday. Then machine M started working and the two machines, working simultaneously at their respective constant rates, took 9 hours to produce the rest of the units produced last Friday. How many hours would it have ta... | tiger_lab_math |
A person spends 1/3rd of the money with him on clothes, 1/5th of the remaining on food and 1/4th of the remaining on travel. Now, he is left with Rs 600. How much did he have with him in the beginning?
Answer Choices: (A) s 200 (B) s 1500 (C) s 300 (D) s 450 (E) s 550 | tiger_lab_math |
Find the vertex of the parabola $y = 2x^2 + 4x + 1$. | nvidia_math |
Ms. Aria has a chicken farm with 30 chickens. Each chicken on the farm eats 60 kilograms of feed monthly. She also has a turkey farm with 5 times the number of chickens, with each turkey eating 1/2 times as much feed as a chicken does. How many kilograms of feed does Ms. Aria use to feed all of her birds in a year? | nvidia_math |
If \( n \) is a positive integer, what is the value of \( n^2 + n \) when \( n = 100 \)? Is this value even or odd? | automath |
If $x^{2y}= 8$ and $x = 4$, what is the value of $y$? Express your answer as a common fraction. | tiger_math |
Bella, Monica, and Rachel are friends who like jewelry. Bella has 10 earrings, which is 25% of Monica's earrings, and Monica has twice as many earrings as Rachel. How many earrings do all of the friends have? | tiger_lab_math |
Evaluate the integral $$I = \int_0^{\frac{\pi}{6}} \sec^3 \theta \, d\theta.$$ What is the exact value of \(I\)? | automath |
Rationalize the denominator: $\sqrt{\frac{5}{12}}$ | nvidia_math |
What is the exact value of the definite integral \(\int_{-1}^{0} \sqrt{t^2 + t^4} \, dt\)? | automath |
What is the probability that player A wins when two players alternately flip a coin, starting with player A, and the first to flip a head wins? | automath |
What is the decimal (base 10) equivalent of the number \( (2010221021)_3 \)? | automath |
Let $$\mathcal{X}$$ be the set of all nonempty subsets of the set $$\{1,2,3,\ldots,10\}$$. Define the relation $$\mathcal{R}$$ on $$\mathcal{X}$$ by $$A \mathcal{R} B$$ if and only if the smallest element of $$A$$ is equal to the smallest element of $$B$$.
How many equivalence classes are formed by this relation $$\m... | automath |
Let $S$ be the set of all nonzero real numbers. Let $f : S \to S$ be a function such that
\[f(x) + f(y) = f(x + yf(x - y))\]for all $x,$ $y \in S$ such that $x - y \neq 0.$
Let $n$ be the number of possible values of $f(5),$ and let $s$ be the sum of all possible values of $f(5).$ Find $n \times s.$ | tiger_math |
Let $f(x)$ be a function such that $f(0) = 1$ and
\[f(xy) = f \left( \frac{x^2 + y^2}{2} \right) + (x - y)^2\]for all real numbers $x$ and $y.$ Find $f(x).$ | nvidia_math |
For the graph of a certain quadratic $y = ax^2 + bx + c$, the vertex of the parabola is $(3,7)$ and one of the $x$-intercepts is $(-2,0)$. What is the $x$-coordinate of the other $x$-intercept? | nvidia_math |
I need to learn vocabulary words for my Spanish exam. There are 500 words and the exam grade is the percentage of these words that I recall correctly. Assuming I will recall correctly the words I learn and assuming my guesses will yield $50\%$ of the points for the words I do not know, what is the least number of words... | tiger_math |
Gracie and Joe are choosing numbers on the complex plane. Joe chooses the point $2-i$. Gracie chooses $-3+2i$. How far apart are Gracie and Joe's points? | tiger_math |
Consider the function $$f(x) = \frac{x^{4}}{4} - 2x^{3} + \frac{11}{2}x^{2} - 6x + 2$$ defined for all $$x \in \mathbb{R}$$.
1. Find the critical points of $$f$$.
2. Determine whether each critical point is a local maximum, local minimum, or neither.
3. Identify the global maximum and global minimum values of $$f$$ o... | automath |
Given that $x + y = 4$ and $x^2y + xy^2 = 12$, find $x^4 + y^4$. | nvidia_math |
In 2023, the sum of the year's digits is $7$. What is the first year after 2023 in which the sum of the digits is 7 again? | tiger_math |
Suppose we have a chaotic pendulum with an initial angle of 0.2 radians and initial velocity of 0.5 radians per second. If the length of the pendulum is 1 meter and the gravitational acceleration is 9.81 m/s^2, find the Poincaré map after 10 oscillations of the pendulum. | tiger_lab_math |
The polynomial $x^3 - 2004 x^2 + mx + n$ has integer coefficients and three distinct positive zeros. Exactly one of these is an integer, and it is twice the sum of the other two. How many values of $n$ are possible? | tiger_math |
Given a large network whose degree distribution appears to follow a power law, how can you estimate its scaling exponent \( \gamma \) using linear regression on its complementary cumulative distribution function (CCDF), and what is the expected approximation for \( \gamma \) if the CCDF shows the behavior consistent wi... | automath |
Using the Lorenz system, what is the smallest value of the parameter 'r' for which the system exhibits chaotic behavior? | tiger_lab_math |
The length of the shortest trip from $A$ to $B$ along the edges of the cube shown is the length of 3 edges. How many different 3-edge trips are there from $A$ to $B$ if we cannot move along the dotted edges?
[asy]
size(4cm,4cm);
pair a1, b1, c1, d1;
a1=(1,1);
b1=(0,1);
c1=(1.6,1.4);
d1=(1,0);
pair e1, f1, g1, h1;
... | tiger_math |
The fifth and ninth terms of a geometric sequence are 11 and 44, respectively. What is the 12th term of this progression? | nvidia_math |
Christian is twice as old as Brian. In eight more years, Brian will be 40 years old. How old will Christian be in eight years? | tiger_lab_math |
Forest and his friends have prepared a birthday cake for their friend Juelz having 240 cake pieces. After singing the birthday song, they ate 60% of the cake's pieces, and later, Juelz divided the remaining pieces among his three sisters. What is the number of pieces of cake each sister received? | nvidia_math |
Sure, here's a numerical problem for a math student to solve:
The manager of a restaurant desires to motivate their server to provide excellent service to all customers. The server can either provide high-quality service or low-quality service to a customer, but the manager cannot directly observe the service provided... | tiger_lab_math |
The number 720 is the product of three consecutive positive integers and is also the product of four consecutive integers. What is the sum of those seven integers? | nvidia_math |
A circle is graphed in the Cartesian plane. The circle has a radius of 4 units and is tangent to both the x-axis and y-axis. The center of the circle is in the first quadrant. What are the coordinates of the center of the circle? | nvidia_math |
Suppose you have three identical prisms with congruent equilateral triangles as the end-polygons. Suppose you attach them by the rectangular faces so they are perfectly aligned. There will be some large faces created by two or more co-planar faces of the individual prisms: count each such large face as one. Given that,... | tiger_lab_math |
What are the possible values of \( n \) for the number of sheets in a covering of a Moebius band, and why are only odd values feasible? | automath |
Alex has three main appliances in her home: a washing machine, a clothes dryer, and an air conditioner, each with different energy consumption rates. The amount she pays for the energy used by the washing machine is four times the amount she pays for the power used by the clothes dryer. If the air conditioner uses powe... | nvidia_math |
In a deck of 12 playing cards consisting of two suits of 6 cards each, where each suit has cards numbered 1 to 6, what is the probability that when drawing 4 cards from this deck, at least one pair of cards (having the same value) is drawn? | automath |
A teacher wants to grade his students' tests using a weighted average of their scores on several subjects: mathematics, language, science, and social studies. The weights of the subjects are in the ratio 3:2:1:4, and the total score is calculated by multiplying the subject scores with their respective weights and addin... | nvidia_math |
Given the equation
\[
\frac{jx}{1+jy} = \frac{3x+j4}{x+3y}
\]
where \( x \) and \( y \) are real numbers, determine all pairs \( (x, y) \) that satisfy this equation. | automath |
Evelyn studied for 6 hours each day on Saturday, Sunday, and Tuesday. She studied for 4 hours on both Monday and Wednesday. How many hours would Evelyn study each day if she studied the same number of hours each day? | nvidia_math |
Aaron is gathering can lids to take to the recycling center. He uses 3 equal-sized boxes of canned tomatoes and adds the lids he gets to the 14 can lids he already has. He is now taking 53 can lids to the recycling center. How many cans lids did he get from each box? | tiger_lab_math |
What is the value of \( 8^7 \mod 187 \)? | automath |
The price of a T.V. set worth Rs. 60000 is to be paid in 20 installments of Rs. 1000 each. If the rate of interest be 6% per annum, and the first installment be paid at the time of purchase, then the value of the last installment covering the interest as well will be?
Answer Choices: (A) 22678 (B) 59000 (C) 26788 (D) 1... | tiger_lab_math |
A box contains 8 red marbles and $m$ blue marbles, where $m$ is an unknown positive integer. A marble is drawn at random from the box. If a red marble is drawn, the player wins 3 dollars, but if a blue marble is drawn, the player loses 1 dollar. If the expected amount won for playing the game is 1 dollar, then what is ... | nvidia_math |
In right triangle $MNO$, $\tan{M}=\frac{5}{4}$, $ON=10$, and $\angle O = 90^\circ$. Find $MN$. Express your answer in simplest radical form. | tiger_math |
Given the matrix \( A = \begin{bmatrix} 4 & -2 \\ -2 & 1 \end{bmatrix} \), find the multiplicative inverse \( A^{-1} \) of matrix \( A \). | automath |
14, 23, 32, 41, 50, 59, …
In the sequence above, each term is 9 more than the previous term. What is the 41st term Q of the sequence?
Answer Choices: (A) 360 (B) 365 (C) 369 (D) 374 (E) 383 | tiger_lab_math |
Problem:
A bag contains 10 balls, each with a different color. Alice picks two balls without replacement. What is the probability that the second ball has a color that is exactly two shades darker or lighter than the first ball?
(Note: assume that the colors are arranged in a circular manner, so the darkest ball is tw... | nvidia_math |
Find the number of permutations $(a_1, a_2, a_3, a_4, a_5, a_6)$ of $(1,2,3,4,5,6)$ that satisfy
\[\frac{a_1 + 1}{2} \cdot \frac{a_2 + 2}{3} \cdot \frac{a_3 + 3}{4} \cdot \frac{a_4 + 4}{5} \cdot \frac{a_5 + 5}{6} \cdot \frac{a_6 + 6}{7} > 6!.\] | tiger_math |
Let $a$ and $b$ be the solutions of the equation $2x^2+6x-14=0$. What is the value of $(2a-3)(4b-6)$? | nvidia_math |
Set J consists of the terms {a, b, c, d, e}, where e > d > c > b > a > 1.5. Which of the following operations would decrease the standard deviation of Set J?
Answer Choices: (A) Multiply each term by e/d (B) Divide each term by b/c (C) Multiply each term by −1 (D) Divide each term by a/e (E) Multiply each term by c/e | tiger_lab_math |
A ball is dropped from a height \( h \) and another ball is thrown upward from the ground at the same instant. If the two balls meet at a height of \( \frac{h}{2} \), what must be the initial speed of the second ball in terms of \( h \)? | automath |
Is it true that \(0^0 = 1\)? If yes, can you provide a rigorous justification for this definition in the context of combinatorics or set theory? If not, what are the reasons for considering \(0^0\) as undefined or indeterminate? | automath |
The dimensions of each book in an encyclopedia set measures 8 in x 4 in x 12 in. (l x w x h). Each shelf (i.e. each divider) has a surface area of 848 sq in., where the thickness is 1 in and the width is 16 in. Assuming there is enough space in between shelves, how many shelves are needed to fit all of the books?
Answe... | tiger_lab_math |
The function $f(x)$ satisfies
\[f(xy) = f(x)f(y)\]for all real numbers $x$ and $y.$ If $f(2) = 3$ and $f(3) = 4,$ find $f(6).$ | nvidia_math |
Let $S$ be the set of all real values of $x$ with $0 < x < \frac{\pi}{4}$ such that $\sin x$, $\cos x$, and $\tan x$ form the side lengths (in some order) of a right triangle. Compute the sum of $\sin^2 x$ over all $x$ in $S$. | tiger_math |
How many ways are there to partition a set containing 6 distinct elements into 3 non-empty subsets? | tiger_lab_math |
Simplify the following expression:
\[ (x^2+2x+1)(x^2-2x+1)-(x^2+1)^2+x^4 \] | nvidia_math |
What is the mass moment of inertia about the x-axis (or y-axis) of a homogeneous spherical cap with radius \( R = 5 \), height \( h = 2 \), and mass \( m = 4.28 \times 10^5 \), using the correct expression for the moment of inertia that accounts for the shift in the axis from the center of mass to the center of the sph... | automath |
Given two groups of people with the following statistics:
Group 1: Mean age = 35 years, Standard Deviation = 14 years, Number of people = 137
Group 2: Mean age = 31 years, Standard Deviation = 11 years, Number of people = 112
What is the combined standard deviation of the ages for both groups, using the means and... | automath |
A dietitian ate three-fourths of her lunch during her lunch break. If the total amount of food she had prepared for lunch had 40 calories, and the recommended calorie intake by the FDA is 25, how many more calories than the recommended amount did she eat? | tiger_lab_math |
In rectangle $ABCD$, side $AB$ measures $6$ units and side $BC$ measures $3$ units, as shown. Points $F$ and $G$ are on side $CD$ with segment $DF$ measuring $2$ units and segment $GC$ measuring $1$ unit, and lines $AF$ and $BG$ intersect at $E$. What is the area of triangle $AEB$? | tiger_math |
Done with her new problems, Wendy takes a break from math. Still without any fresh reading material, she feels a bit antsy. She starts to feel annoyed that Michael's loose papers clutter the family van. Several of them are ripped, and bits of paper litter the floor. Tired of trying to get Michael to clean up after hims... | tiger_math |
Suppose Alice and Bob are exchanging messages using the Diffie-Hellman Key Exchange method. Alice chooses a prime number $p = 19$ and a base $g = 2$. She also chooses a secret number $a = 3$ and computes $A = g^a \mod p$. Bob chooses a secret number $b = 5$ and computes $B = g^b \mod p$. What is the shared secret key t... | tiger_lab_math |
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