instruction_seed stringlengths 13 12.7k | response_seed stringlengths 13 12.7k |
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Find the length of side $MN$ in triangle $MNO$ below.
[asy]
unitsize(1inch);
pair A,B,C;
A = (0,0);
B = (sqrt(2),0);
C = (0,sqrt(2));
draw (A--B--C--A,linewidth(0.9));
draw(rightanglemark(B,A,C,3));
label("$M$",A,SW);
label("$N$",B,SE);
label("$O$",C,N);
label("$45^\circ$",(0,0.25),E);
label("$30$",(A+C)/2,NW);
[/asy... | Find the length of side $MN$ in triangle $MNO$ below.
[asy]
unitsize(1inch);
pair A,B,C;
A = (0,0);
B = (sqrt(2),0);
C = (0,sqrt(2));
draw (A--B--C--A,linewidth(0.9));
draw(rightanglemark(B,A,C,3));
label("$M$",A,SW);
label("$N$",B,SE);
label("$O$",C,N);
label("$45^\circ$",(0,0.25),E);
label("$30$",(A+C)/2,NW);
[/asy... |
Find the value of $m$ such that the graph of the equation $x^2 + 4x - 5y = 2m$ passes through the point of intersection of the lines $x - 2y = 3$ and $2x + 3y = 5$. | Find the value of $m$ such that the graph of the equation $x^2 + 4x - 5y = 2m$ passes through the point of intersection of the lines $x - 2y = 3$ and $2x + 3y = 5$. |
A cylindrical container holds 1200 cm^3 of juice. The juice is to be repackaged into smaller cylindrical cans, each with half the radius of the original container. How many cans can be filled with the juice, assuming each can is filled to the same height as the original container? | A cylindrical container holds 1200 cm^3 of juice. The juice is to be repackaged into smaller cylindrical cans, each with half the radius of the original container. How many cans can be filled with the juice, assuming each can is filled to the same height as the original container? |
In a group of 25 people, 15 are wearing glasses and 10 are not. If we need to choose a committee of 5 people, with at least 2 wearing glasses and at least 1 not wearing glasses, how many different committees can we form? | In a group of 25 people, 15 are wearing glasses and 10 are not. If we need to choose a committee of 5 people, with at least 2 wearing glasses and at least 1 not wearing glasses, how many different committees can we form? |
A right circular cone has base radius $r$ and height $h$. Find the largest possible value of
\[\frac{r + h}{l},\]
where $l$ is the slant height of the cone. | A right circular cone has base radius $r$ and height $h$. Find the largest possible value of
\[\frac{r + h}{l},\]
where $l$ is the slant height of the cone. |
The equations $2x+7=3$ and $bx-10=-2$ have the same solution $x$. What is the value of $b$? | The equations $2x+7=3$ and $bx-10=-2$ have the same solution $x$. What is the value of $b$? |
Find the larger of the two distinct solutions to the equation:
$$x^2 - 12x + 32 = 0$$
Can you solve it using the quadratic formula? | Find the larger of the two distinct solutions to the equation:
$$x^2 - 12x + 32 = 0$$
Can you solve it using the quadratic formula? |
The 6-digit number $42\,185\square$ is a multiple of 9. Which digit is represented by $\square$? | The 6-digit number $42\,185\square$ is a multiple of 9. Which digit is represented by $\square$? |
Find the vector $\mathbf{x}$ such that $2\mathbf{x} + \begin{pmatrix} 3 \\ 2 \\ -1 \end{pmatrix} = \begin{pmatrix} 7 \\ 0 \\ 5 \end{pmatrix}.$ | Find the vector $\mathbf{x}$ such that $2\mathbf{x} + \begin{pmatrix} 3 \\ 2 \\ -1 \end{pmatrix} = \begin{pmatrix} 7 \\ 0 \\ 5 \end{pmatrix}.$ |
The base five numbers $12_5$ and $21_5$ are multiplied and their product is expressed in base ten. What is the remainder when this product is divided by 7? | The base five numbers $12_5$ and $21_5$ are multiplied and their product is expressed in base ten. What is the remainder when this product is divided by 7? |
At a local grocery store, Brand A laundry detergent is sold in 10-kg boxes for $\$15$each, while Brand B laundry detergent is sold in 8-kg boxes for$\$12$ each. However, an analysis reveals that Brand A detergent is only 80% as effective as Brand B detergent. What is the ratio of the unit price of Brand A detergent to ... | At a local grocery store, Brand A laundry detergent is sold in 10-kg boxes for $\$15$each, while Brand B laundry detergent is sold in 8-kg boxes for$\$12$ each. However, an analysis reveals that Brand A detergent is only 80% as effective as Brand B detergent. What is the ratio of the unit price of Brand A detergent to ... |
Find the number of terms in the sequence $1, 4, 9, 16, \ldots, 2008$, where each term is a perfect square. | Find the number of terms in the sequence $1, 4, 9, 16, \ldots, 2008$, where each term is a perfect square. |
What is the smallest number of sides a convex polygon can have such that the sum of its interior angles is greater than 2000 degrees? | What is the smallest number of sides a convex polygon can have such that the sum of its interior angles is greater than 2000 degrees? |
Consider a set of $n$ complex numbers, each with magnitude $1$. These numbers are equally spaced around the unit circle, with the first number at position $(1,0)$. We want to assign a sign ($+1$ or $-1$) to each number so that the real part of the sum of the signed numbers is maximized.
What is the largest possible va... | Consider a set of $n$ complex numbers, each with magnitude $1$. These numbers are equally spaced around the unit circle, with the first number at position $(1,0)$. We want to assign a sign ($+1$ or $-1$) to each number so that the real part of the sum of the signed numbers is maximized.
What is the largest possible va... |
Compute without using a calculator: $9!/8!$ | Compute without using a calculator: $9!/8!$ |
How many 4-digit positive integers exist that satisfy the following conditions: (A) Each of the first two digits must be 1, 4, or 5, and (B) the last two digits cannot be the same digit, and (C) each of the last two digits must be 5, 7, or 8? | How many 4-digit positive integers exist that satisfy the following conditions: (A) Each of the first two digits must be 1, 4, or 5, and (B) the last two digits cannot be the same digit, and (C) each of the last two digits must be 5, 7, or 8? |
Given that $a-b=5$ and $a^2+b^2=35$, find $a^3-b^3$. | Given that $a-b=5$ and $a^2+b^2=35$, find $a^3-b^3$. |
A polynomial $p(x) = x^3 + bx^2 + cx + d$ has real coefficients and satisfies $p(i) = p(-i) = 0$. Find the smallest possible value of $|b| + |c| + |d|$. | A polynomial $p(x) = x^3 + bx^2 + cx + d$ has real coefficients and satisfies $p(i) = p(-i) = 0$. Find the smallest possible value of $|b| + |c| + |d|$. |
If $x = 2$ and $y = 5$, then what is the value of $\frac{x^4+2y^2}{6}$ ? | If $x = 2$ and $y = 5$, then what is the value of $\frac{x^4+2y^2}{6}$ ? |
A function $g(x)$ has a graph that consists of four vertices forming a square of side length 10 units.
The function $h(x) = g(x + 1)$ also forms a square shape with its four vertices. What is the side length of the new square? | A function $g(x)$ has a graph that consists of four vertices forming a square of side length 10 units.
The function $h(x) = g(x + 1)$ also forms a square shape with its four vertices. What is the side length of the new square? |
Let $g(x) = \frac{2}{2^x + 1}.$ Find
\[g \left( \frac{1}{501} \right) + g \left( \frac{2}{501} \right) + g \left( \frac{3}{501} \right) + \dots + g \left( \frac{500}{501} \right).\] | Let $g(x) = \frac{2}{2^x + 1}.$ Find
\[g \left( \frac{1}{501} \right) + g \left( \frac{2}{501} \right) + g \left( \frac{3}{501} \right) + \dots + g \left( \frac{500}{501} \right).\] |
A water tank can be filled at a rate of 2 cubic meters per minute. However, due to a leak, water is being drained from the tank at a constant rate. If it takes 3 hours and 20 minutes to fill the tank completely, and the tank's capacity is 300 cubic meters, what is the rate at which water is being drained from the tank? | A water tank can be filled at a rate of 2 cubic meters per minute. However, due to a leak, water is being drained from the tank at a constant rate. If it takes 3 hours and 20 minutes to fill the tank completely, and the tank's capacity is 300 cubic meters, what is the rate at which water is being drained from the tank? |
How many integers from 1 through 999, inclusive, do not contain any of the digits 0, 6, 7, or 8? | How many integers from 1 through 999, inclusive, do not contain any of the digits 0, 6, 7, or 8? |
Let $f(x)=6x^2+x-1$. Prove that for all positive integers $n$ we have
$$\binom{n}{2} f(1) + \binom{n}{3} f(2) + \dots + \binom{n}{n} f(n-1) = f(n).$$ | Let $f(x)=6x^2+x-1$. Prove that for all positive integers $n$ we have
$$\binom{n}{2} f(1) + \binom{n}{3} f(2) + \dots + \binom{n}{n} f(n-1) = f(n).$$ |
Let $x_1,$ $x_2,$ $\dots,$ $x_{100}$ be real numbers such that $x_1 + x_2 + \dots + x_{100} = 1$ and
\[\frac{x_1}{1 - x_1} + \frac{x_2}{1 - x_2} + \dots + \frac{x_{100}}{1 - x_{100}} = 1.\]Find
\[\frac{x_1^2}{1 - x_1} + \frac{x_2^2}{1 - x_2} + \dots + \frac{x_{100}^2}{1 - x_{100}}.\] | Let $x_1,$ $x_2,$ $\dots,$ $x_{100}$ be real numbers such that $x_1 + x_2 + \dots + x_{100} = 1$ and
\[\frac{x_1}{1 - x_1} + \frac{x_2}{1 - x_2} + \dots + \frac{x_{100}}{1 - x_{100}} = 1.\]Find
\[\frac{x_1^2}{1 - x_1} + \frac{x_2^2}{1 - x_2} + \dots + \frac{x_{100}^2}{1 - x_{100}}.\] |
What are the possible degrees of the remainder when a rational function is divided by $x^3 + 2x^2 - 5x - 6$? Enter all the possible values, separated by commas. | What are the possible degrees of the remainder when a rational function is divided by $x^3 + 2x^2 - 5x - 6$? Enter all the possible values, separated by commas. |
Given two real numbers $1<a<b$ so that $\frac{1}{a} - \frac{1}{b} = \frac{1}{2}$ and $ab = 8$, what is $a$? | Given two real numbers $1<a<b$ so that $\frac{1}{a} - \frac{1}{b} = \frac{1}{2}$ and $ab = 8$, what is $a$? |
In a regular hexagon with side length 6 units, a smaller regular hexagon with side length 3 units is inscribed. What is the ratio of the area of the smaller hexagon to the area of the region between the two hexagons? Express your answer as a common fraction. | In a regular hexagon with side length 6 units, a smaller regular hexagon with side length 3 units is inscribed. What is the ratio of the area of the smaller hexagon to the area of the region between the two hexagons? Express your answer as a common fraction. |
The circle with center $(-1, 4)$ and radius $r$ has an area of $16\pi$ square units. What is the value of $r$? | The circle with center $(-1, 4)$ and radius $r$ has an area of $16\pi$ square units. What is the value of $r$? |
Find the product of the values of $x$ that satisfy the equation $|4x|+3=35$. | Find the product of the values of $x$ that satisfy the equation $|4x|+3=35$. |
Let $h(x)=e^{x^2}$. If $j(x)=h^{-1}(x)-5$, determine $j(e^{16})$. | Let $h(x)=e^{x^2}$. If $j(x)=h^{-1}(x)-5$, determine $j(e^{16})$. |
Compute the domain of the function $$f(x)=\frac{1}{\lfloor x^2-7x+13\rfloor}.$$ | Compute the domain of the function $$f(x)=\frac{1}{\lfloor x^2-7x+13\rfloor}.$$ |
A bookstore has a selection of 5 different fiction books and 4 different non-fiction books. How many different bundles of 3 books can be created, where each bundle must contain at least one fiction book and one non-fiction book? | A bookstore has a selection of 5 different fiction books and 4 different non-fiction books. How many different bundles of 3 books can be created, where each bundle must contain at least one fiction book and one non-fiction book? |
Let $A$ be a $3 \times 3$ matrix with integer entries such that $A^3 = I$, where $I$ is the $3 \times 3$ identity matrix. Find the smallest possible value of the sum of the absolute values of the entries of $A$. | Let $A$ be a $3 \times 3$ matrix with integer entries such that $A^3 = I$, where $I$ is the $3 \times 3$ identity matrix. Find the smallest possible value of the sum of the absolute values of the entries of $A$. |
How many different positive integers are there, whose fourth powers are 16-digit numbers? | How many different positive integers are there, whose fourth powers are 16-digit numbers? |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(x) + g(g(y)) = g(xy)\]for all $x,$ $y.$
Let $m$ be the number of possible values of $g(1),$ and let $t$ be the product of all possible values of $g(1).$ Find $m + t.$ | Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(x) + g(g(y)) = g(xy)\]for all $x,$ $y.$
Let $m$ be the number of possible values of $g(1),$ and let $t$ be the product of all possible values of $g(1).$ Find $m + t.$ |
Let $a$, $b$, and $c$ be complex numbers such that $|a|=|b|=|c|=1$ and $a+b+c = -1$. Find $abc$. | Let $a$, $b$, and $c$ be complex numbers such that $|a|=|b|=|c|=1$ and $a+b+c = -1$. Find $abc$. |
What is the smallest prime factor of the product of all positive integers less than or equal to 30? | What is the smallest prime factor of the product of all positive integers less than or equal to 30? |
A bus travels from city A to city B with multiple stops in between. The bus travels at a constant speed of 60 kph between stops, but it spends 5 minutes at each stop picking up passengers. If the total distance between city A and city B is 120 kilometers, and there are 6 stops along the way, what is the average speed o... | A bus travels from city A to city B with multiple stops in between. The bus travels at a constant speed of 60 kph between stops, but it spends 5 minutes at each stop picking up passengers. If the total distance between city A and city B is 120 kilometers, and there are 6 stops along the way, what is the average speed o... |
The slope of a line is equal to the product of its $x$- and $y$-intercepts. Find the angle between the line and the $x$-axis. | The slope of a line is equal to the product of its $x$- and $y$-intercepts. Find the angle between the line and the $x$-axis. |
What is the value of $x$ in the figure shown, where the three angles around the point are $100^{\circ}$, $2x^{\circ}$, and $2x^{\circ}$? | What is the value of $x$ in the figure shown, where the three angles around the point are $100^{\circ}$, $2x^{\circ}$, and $2x^{\circ}$? |
Simplify the expression:
$$\frac{1}{\sqrt{2}+1}\times\frac{\sqrt{3}+1}{\sqrt{2}+\sqrt{3}}\times\frac{\sqrt{2}+\sqrt{3}}{\sqrt{3}+1}\times\frac{\sqrt{2}+1}{1}$$
Can you simplify this expression and find its value? | Simplify the expression:
$$\frac{1}{\sqrt{2}+1}\times\frac{\sqrt{3}+1}{\sqrt{2}+\sqrt{3}}\times\frac{\sqrt{2}+\sqrt{3}}{\sqrt{3}+1}\times\frac{\sqrt{2}+1}{1}$$
Can you simplify this expression and find its value? |
The volume of a rectangular prism is given by the product of its length, width, and height. If the prism has a length of 6 meters, a width of 2 meters, and a height of 5 meters, what is the total volume in cubic meters of the prism if it is cut into three identical boxes? | The volume of a rectangular prism is given by the product of its length, width, and height. If the prism has a length of 6 meters, a width of 2 meters, and a height of 5 meters, what is the total volume in cubic meters of the prism if it is cut into three identical boxes? |
A regular hexagon is inscribed in a circle with radius 2 inches. What is the area of the hexagon?
(Note: This problem requires a different approach than the original problem, as it involves finding the area of a hexagon rather than the side length of a square.) | A regular hexagon is inscribed in a circle with radius 2 inches. What is the area of the hexagon?
(Note: This problem requires a different approach than the original problem, as it involves finding the area of a hexagon rather than the side length of a square.) |
Determine the number of unique quadrilaterals that can be formed by connecting four different vertices of a regular octahedron. (Two quadrilaterals are considered unique if they occupy different spatial positions.) | Determine the number of unique quadrilaterals that can be formed by connecting four different vertices of a regular octahedron. (Two quadrilaterals are considered unique if they occupy different spatial positions.) |
Determine the value of $x$ in the equation
$$\frac{x+2}{x-3}=\frac{5}{2}$$
such that when $x$ is rounded to the nearest tenth, the result is exactly halfway between two consecutive integers. | Determine the value of $x$ in the equation
$$\frac{x+2}{x-3}=\frac{5}{2}$$
such that when $x$ is rounded to the nearest tenth, the result is exactly halfway between two consecutive integers. |
The length of a car tire is 60 cm. If the tire completes one rotation in 2 seconds, how many kilometers will the car travel in 1 hour if it maintains a constant speed? (You may use a calculator on this problem; answer to the nearest tenth.) | The length of a car tire is 60 cm. If the tire completes one rotation in 2 seconds, how many kilometers will the car travel in 1 hour if it maintains a constant speed? (You may use a calculator on this problem; answer to the nearest tenth.) |
Express the value of the following as a common fraction:
$\left(1-\frac12\right)\cdot\left(1-\frac13\right)\cdot\left(1-\frac14\right) \dotsm \left(1-\frac1{n+1}\right) \dotsm \left(1-\frac1{100}\right)$ | Express the value of the following as a common fraction:
$\left(1-\frac12\right)\cdot\left(1-\frac13\right)\cdot\left(1-\frac14\right) \dotsm \left(1-\frac1{n+1}\right) \dotsm \left(1-\frac1{100}\right)$ |
What is the sum of all the even divisors of 240? | What is the sum of all the even divisors of 240? |
How many numbers are in the first $20$ rows of Pascal's Triangle (from the $0$th row to the $19$th row)? | How many numbers are in the first $20$ rows of Pascal's Triangle (from the $0$th row to the $19$th row)? |
In a city, the population grows by 10% every year. In 2010, the population was 100,000. Determine the year when the population will exceed 150,000. | In a city, the population grows by 10% every year. In 2010, the population was 100,000. Determine the year when the population will exceed 150,000. |
Find the value of $$\frac{(2^4 \cdot 3^3 \cdot 5^2 \cdot 7) \cdot (3^2 \cdot 5 \cdot 11)}{(2^3 \cdot 3^2 \cdot 5) \cdot (2^2 \cdot 3 \cdot 7 \cdot 13)}$$ | Find the value of $$\frac{(2^4 \cdot 3^3 \cdot 5^2 \cdot 7) \cdot (3^2 \cdot 5 \cdot 11)}{(2^3 \cdot 3^2 \cdot 5) \cdot (2^2 \cdot 3 \cdot 7 \cdot 13)}$$ |
In a physical education class, students line up in four rows to do exercises. One particular class has over 30 students, with three rows having the same number of students and one row having one more student than the other three rows. What is the smallest possible class size for this physical education class? | In a physical education class, students line up in four rows to do exercises. One particular class has over 30 students, with three rows having the same number of students and one row having one more student than the other three rows. What is the smallest possible class size for this physical education class? |
Find the area of the region enclosed by the curve $y=\sqrt{x^2-6x+9}$ and the line $y=x$.
The final answer is:
(Step 1)
The curve $y=\sqrt{x^2-6x+9}$ can be rewritten as $y=|x-3|$, which is a V-shaped graph with its vertex at $(3, 0)$.
The line $y=x$ is a straight line passing through the origin.
(Step 2)
To find ... | Find the area of the region enclosed by the curve $y=\sqrt{x^2-6x+9}$ and the line $y=x$.
The final answer is:
(Step 1)
The curve $y=\sqrt{x^2-6x+9}$ can be rewritten as $y=|x-3|$, which is a V-shaped graph with its vertex at $(3, 0)$.
The line $y=x$ is a straight line passing through the origin.
(Step 2)
To find ... |
Given the function $f(x) = \frac{1 + x^2}{x^4 + x^2 + 1},$ what is the maximum value that $f(x)$ can achieve? | Given the function $f(x) = \frac{1 + x^2}{x^4 + x^2 + 1},$ what is the maximum value that $f(x)$ can achieve? |
In a small town, there are five different types of houses: A, B, C, D, and E. The town has 120 houses in total, with 10 houses of type A, 20 houses of type B, 30 houses of type C, 15 houses of type D, and 45 houses of type E. The property tax on these houses is calculated based on their type, with the tax being $1000 f... | In a small town, there are five different types of houses: A, B, C, D, and E. The town has 120 houses in total, with 10 houses of type A, 20 houses of type B, 30 houses of type C, 15 houses of type D, and 45 houses of type E. The property tax on these houses is calculated based on their type, with the tax being $1000 f... |
The vertices of triangle $ABC$ are $A(3,4)$, $B(5,6)$, and $C(7,8)$. The triangle is then reflected over the $x$-axis to form triangle $A'B'C'$ and then reflected over the line $y=x$ to form triangle $A''B''C''$. What is the ratio of the area of triangle $A''B''C''$ to the area of triangle $ABC$? | The vertices of triangle $ABC$ are $A(3,4)$, $B(5,6)$, and $C(7,8)$. The triangle is then reflected over the $x$-axis to form triangle $A'B'C'$ and then reflected over the line $y=x$ to form triangle $A''B''C''$. What is the ratio of the area of triangle $A''B''C''$ to the area of triangle $ABC$? |
A right cone with a height of 8 inches has a radius of 3 inches. What is the volume of the cone, in cubic inches? Express your answer in terms of $\pi$. | A right cone with a height of 8 inches has a radius of 3 inches. What is the volume of the cone, in cubic inches? Express your answer in terms of $\pi$. |
Factor the expression $(x^3 + 27)^2 - 72x^3$. | Factor the expression $(x^3 + 27)^2 - 72x^3$. |
Determine the smallest integer value of x such that x^3 + 3x^2 + 2x + 5 is divisible by 11. | Determine the smallest integer value of x such that x^3 + 3x^2 + 2x + 5 is divisible by 11. |
Let $a,$ $b,$ $c$ be distinct integers, and let $\omega$ be a complex number such that $\omega^3 = 1$ and $\omega \neq 1.$ Find the smallest possible value of
\[|a + b \omega + c \omega^2|.\] | Let $a,$ $b,$ $c$ be distinct integers, and let $\omega$ be a complex number such that $\omega^3 = 1$ and $\omega \neq 1.$ Find the smallest possible value of
\[|a + b \omega + c \omega^2|.\] |
A city's transportation department wants to estimate the number of people who use a particular bus route. They notice that on a given day, 15% of the passengers get off at a specific stop. If they count the number of passengers who get off at that stop and assume it is representative of the total ridership, how can the... | A city's transportation department wants to estimate the number of people who use a particular bus route. They notice that on a given day, 15% of the passengers get off at a specific stop. If they count the number of passengers who get off at that stop and assume it is representative of the total ridership, how can the... |
Simplify $(3+2i)(3-2i)(2+3i)$. (Your answer should be of the form $a+bi$.) | Simplify $(3+2i)(3-2i)(2+3i)$. (Your answer should be of the form $a+bi$.) |
Ana walks along the circumference of a circle with a radius of 4 km. From any point on her path, she can see exactly 2 km horizontally in all directions. What is the area of the region consisting of all points Ana can see during her walk, expressed in square kilometers and rounded to the nearest whole number? | Ana walks along the circumference of a circle with a radius of 4 km. From any point on her path, she can see exactly 2 km horizontally in all directions. What is the area of the region consisting of all points Ana can see during her walk, expressed in square kilometers and rounded to the nearest whole number? |
Find the range of the function $f(x) = \arctan x + \arctanh x.$ All functions are in radians. | Find the range of the function $f(x) = \arctan x + \arctanh x.$ All functions are in radians. |
What is the value of the infinite sum $S = \sum_{n=1}^{\infty} \frac{n}{2^n}$? | What is the value of the infinite sum $S = \sum_{n=1}^{\infty} \frac{n}{2^n}$? |
Suppose the square matrix $M$ has columns $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c}$ and $N$ has columns $\mathbf{p},$ $\mathbf{q},$ $\mathbf{r}.$ In terms of $\det M$ and $\det N$, find $\det (M+N).$ | Suppose the square matrix $M$ has columns $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c}$ and $N$ has columns $\mathbf{p},$ $\mathbf{q},$ $\mathbf{r}.$ In terms of $\det M$ and $\det N$, find $\det (M+N).$ |
The line $y = 3x + d$ is tangent to the parabola $y^2 = 12x.$ Find $d.$ | The line $y = 3x + d$ is tangent to the parabola $y^2 = 12x.$ Find $d.$ |
Tom has been saving money for a new bike and has $120 in his savings account. He wants to buy a bike that costs $180. Tom's parents have agreed to give him an allowance of $5 per week for doing chores. However, every week Tom spends $2 on candy and snacks. How many weeks will it take Tom to save enough money to buy the... | Tom has been saving money for a new bike and has $120 in his savings account. He wants to buy a bike that costs $180. Tom's parents have agreed to give him an allowance of $5 per week for doing chores. However, every week Tom spends $2 on candy and snacks. How many weeks will it take Tom to save enough money to buy the... |
What is the base ten equivalent of $54321_6$? | What is the base ten equivalent of $54321_6$? |
In Biology class, a student discovers that a specific nutrient can be created by combining 0.15 liters of solution B with 0.30 liters of a buffer (yielding 0.45 liters of the nutrient). If the student aims to produce a total of 1.8 liters of this nutrient, how many liters of the buffer will be required? | In Biology class, a student discovers that a specific nutrient can be created by combining 0.15 liters of solution B with 0.30 liters of a buffer (yielding 0.45 liters of the nutrient). If the student aims to produce a total of 1.8 liters of this nutrient, how many liters of the buffer will be required? |
Expand $(3x^2 + 2x - 1)(4x^2 - 5x + 3)$. | Expand $(3x^2 + 2x - 1)(4x^2 - 5x + 3)$. |
What is the value of $3^{12} \div (3^2)^3$? | What is the value of $3^{12} \div (3^2)^3$? |
Let $\mathbf{A}$ and $\mathbf{B}$ be matrices such that
\[\mathbf{A} + \mathbf{B} = \mathbf{A} \mathbf{B}.\]If $\mathbf{A} \mathbf{B} = \begin{pmatrix} 20/3 & 4/3 \\ -8/3 & 8/3 \end{pmatrix},$ find $\mathbf{B} \mathbf{A}.$ | Let $\mathbf{A}$ and $\mathbf{B}$ be matrices such that
\[\mathbf{A} + \mathbf{B} = \mathbf{A} \mathbf{B}.\]If $\mathbf{A} \mathbf{B} = \begin{pmatrix} 20/3 & 4/3 \\ -8/3 & 8/3 \end{pmatrix},$ find $\mathbf{B} \mathbf{A}.$ |
It takes Mary 30 minutes to walk uphill 1 km from her home to school, but it takes her only 10 minutes to walk from school to home along the same route. What is her average speed, in km/hr, for the round trip? | It takes Mary 30 minutes to walk uphill 1 km from her home to school, but it takes her only 10 minutes to walk from school to home along the same route. What is her average speed, in km/hr, for the round trip? |
Forty lights in a shopping mall are to be connected so that each light has a direct connection to exactly two other lights. How many connections will be necessary? | Forty lights in a shopping mall are to be connected so that each light has a direct connection to exactly two other lights. How many connections will be necessary? |
Let $m$ be a positive integer. In the set of ordered pairs of positive integers $(x,y)$, where $\gcd(x,y)=m$, it is known that the pair with the greatest sum $x+y$ has $x=30m$ and $y=45m$. Find $m$. | Let $m$ be a positive integer. In the set of ordered pairs of positive integers $(x,y)$, where $\gcd(x,y)=m$, it is known that the pair with the greatest sum $x+y$ has $x=30m$ and $y=45m$. Find $m$. |
A store is having a sale on shirts, and everything is discounted by 20%. If the original price of a shirt was $40, how much will it cost after the discount?
(This problem requires a different approach than the original, as it involves calculating a percentage decrease rather than estimating a fraction.) | A store is having a sale on shirts, and everything is discounted by 20%. If the original price of a shirt was $40, how much will it cost after the discount?
(This problem requires a different approach than the original, as it involves calculating a percentage decrease rather than estimating a fraction.) |
There are two ${\bf positive}$ integers $c$ for which the equation $$5x^2+11x+c=0$$has rational solutions. What is the product of those two values of $c$? | There are two ${\bf positive}$ integers $c$ for which the equation $$5x^2+11x+c=0$$has rational solutions. What is the product of those two values of $c$? |
The number of games won by five softball teams are displayed in the graph. However, the names of the teams are missing. The following clues provide information about the teams:
1. The Tigers won more games than the Eagles.
2. The Patriots won more games than the Cubs, but fewer games than the Mounties.
3. The Cubs w... | The number of games won by five softball teams are displayed in the graph. However, the names of the teams are missing. The following clues provide information about the teams:
1. The Tigers won more games than the Eagles.
2. The Patriots won more games than the Cubs, but fewer games than the Mounties.
3. The Cubs w... |
Problem:
A regular polygon with an odd number of sides is inscribed in a circle. The length of the largest diagonal is $d$ and the radius of the circle is $r$. What is the largest possible ratio of $d$ to $r$?
(Note: I've tried to create a problem that requires a different approach to solve, while still being related ... | Problem:
A regular polygon with an odd number of sides is inscribed in a circle. The length of the largest diagonal is $d$ and the radius of the circle is $r$. What is the largest possible ratio of $d$ to $r$?
(Note: I've tried to create a problem that requires a different approach to solve, while still being related ... |
Consider the quadratic function $f(x) = x^2 + 6x + c$, where $c$ is a constant. Determine the value of $c$ such that the graph of $f(x)$ is tangent to the line $y = 2x + 3$. | Consider the quadratic function $f(x) = x^2 + 6x + c$, where $c$ is a constant. Determine the value of $c$ such that the graph of $f(x)$ is tangent to the line $y = 2x + 3$. |
How many integers between 100 and 999 have a prime digit (2, 3, 5, or 7) in their tens place? | How many integers between 100 and 999 have a prime digit (2, 3, 5, or 7) in their tens place? |
Let $p(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e$ be a polynomial with integer coefficients such that $p(17) = p(19) = 179$. What is the smallest possible value of $p(18)$? | Let $p(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e$ be a polynomial with integer coefficients such that $p(17) = p(19) = 179$. What is the smallest possible value of $p(18)$? |
Tom has answered 80% of his 30 multiple-choice questions correctly. After answering 10 more questions, he improves his percentage to 82%. How many of the last 10 questions did he answer correctly? | Tom has answered 80% of his 30 multiple-choice questions correctly. After answering 10 more questions, he improves his percentage to 82%. How many of the last 10 questions did he answer correctly? |
A rectangle has a perimeter of 64 inches and each side has an integer length. How many non-congruent rectangles meet these criteria? | A rectangle has a perimeter of 64 inches and each side has an integer length. How many non-congruent rectangles meet these criteria? |
Let's consider two sets $A$ and $B$, each containing a certain number of distinct elements. Suppose we have a function $f(x)$, such that the number of elements in $f(A)$ is exactly $5$, and the number of elements in $f(B)$ is exactly $15$. If $f(A \cap B)$ has at least $2$ elements, what is the maximum number of elemen... | Let's consider two sets $A$ and $B$, each containing a certain number of distinct elements. Suppose we have a function $f(x)$, such that the number of elements in $f(A)$ is exactly $5$, and the number of elements in $f(B)$ is exactly $15$. If $f(A \cap B)$ has at least $2$ elements, what is the maximum number of elemen... |
Let $f(x)$ be a quadratic function that satisfies $f(x) = f(1-x)$ for all real numbers $x$.
Suppose that the graph of $f(x)$ is tangent to the line $y=1$ at the point $(0,1)$ and it also passes through the point $\left(\frac{1}{2}, 4\right)$.
Find the expression of $f(x)$. | Let $f(x)$ be a quadratic function that satisfies $f(x) = f(1-x)$ for all real numbers $x$.
Suppose that the graph of $f(x)$ is tangent to the line $y=1$ at the point $(0,1)$ and it also passes through the point $\left(\frac{1}{2}, 4\right)$.
Find the expression of $f(x)$. |
Determine the number of real solutions to the equation $$\log_{\sqrt{x}}\left(\sin\left(\arctan\left(\frac{1}{\sqrt{x}}\right)\right)\right)=\frac{1}{2}.$$ | Determine the number of real solutions to the equation $$\log_{\sqrt{x}}\left(\sin\left(\arctan\left(\frac{1}{\sqrt{x}}\right)\right)\right)=\frac{1}{2}.$$ |
Simplify $(3x^2 - 2x + 1) + (x^2 + 5x - 2)(2x - 1)$ by combining like terms and performing the necessary calculations. | Simplify $(3x^2 - 2x + 1) + (x^2 + 5x - 2)(2x - 1)$ by combining like terms and performing the necessary calculations. |
A right circular cylinder with a height of 6 inches and a radius of 4 inches is cut into two equal parts along its axis. The upper part is filled with ice cream, and the lower part is left empty. What is the volume of the ice cream in terms of π? | A right circular cylinder with a height of 6 inches and a radius of 4 inches is cut into two equal parts along its axis. The upper part is filled with ice cream, and the lower part is left empty. What is the volume of the ice cream in terms of π? |
What is the range of the function $y=\dfrac{x^2+1}{x^2-1}$? (Express your answer using interval notation.) | What is the range of the function $y=\dfrac{x^2+1}{x^2-1}$? (Express your answer using interval notation.) |
Express $\frac{25}{121}$ as a decimal rounded to two decimal places. | Express $\frac{25}{121}$ as a decimal rounded to two decimal places. |
A group of 8 friends want to form a committee consisting of a president, a vice-president, and two teams of 3 members each, Team A and Team B. The teams will work on different projects and the president and vice-president will oversee both teams. How many ways can they choose the committee if the president and vice-pre... | A group of 8 friends want to form a committee consisting of a president, a vice-president, and two teams of 3 members each, Team A and Team B. The teams will work on different projects and the president and vice-president will oversee both teams. How many ways can they choose the committee if the president and vice-pre... |
What is the length of the altitude dropped from the vertex of a right triangle with legs of length 3 and 4 to its hypotenuse? | What is the length of the altitude dropped from the vertex of a right triangle with legs of length 3 and 4 to its hypotenuse? |
What is the length, in units, of the median to the hypotenuse of a right triangle with leg lengths of 25 and 60 units? | What is the length, in units, of the median to the hypotenuse of a right triangle with leg lengths of 25 and 60 units? |
A tree has 5 branches, and some of these branches have 3 sub-branches each, while others do not. The total number of branches and sub-branches is 17. However, due to disease, the gardener had to remove some of the branches and sub-branches. The removed branches and sub-branches were those that had the most sub-branches... | A tree has 5 branches, and some of these branches have 3 sub-branches each, while others do not. The total number of branches and sub-branches is 17. However, due to disease, the gardener had to remove some of the branches and sub-branches. The removed branches and sub-branches were those that had the most sub-branches... |
Let $f(x) = 2x - 3$ and $g(x) = x + 1$. What is the value of $f(1 + g(2))$? | Let $f(x) = 2x - 3$ and $g(x) = x + 1$. What is the value of $f(1 + g(2))$? |
Determine the values of $m$ and $c$ for which the lines with equations $y = mx + c$ and $y = x - 4$ are perpendicular. | Determine the values of $m$ and $c$ for which the lines with equations $y = mx + c$ and $y = x - 4$ are perpendicular. |
In a regular hexagon, the length of the longest diagonal is numerically equal to the area of one of its equilateral triangles. What is the length of a side of the hexagon, in units? Express your answer in simplest radical form. | In a regular hexagon, the length of the longest diagonal is numerically equal to the area of one of its equilateral triangles. What is the length of a side of the hexagon, in units? Express your answer in simplest radical form. |
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