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Return your final response within \boxed{}. The operation $\otimes$ is defined for all nonzero numbers by $a\otimes b =\frac{a^{2}}{b}$. Determine $[(1\otimes 2)\otimes 3]-[1\otimes (2\otimes 3)]$.
$\text{(A)}\ -\frac{2}{3}\qquad\text{(B)}\ -\frac{1}{4}\qquad\text{(C)}\ 0\qquad\text{(D)}\ \frac{1}{4}\qquad\text{(E)}\ \... | Okay, let me try to figure out this problem. So, we have this operation defined as a⊗b = a²/b. And we need to compute [(1⊗2)⊗3] - [1⊗(2⊗3)]. Then choose the correct answer from the options given. Alright, let's break it down step by step.
First, I need to remember that the operation ⊗ is not associative, right? Becaus... |
To determine the value of \([(1 \otimes 2) \otimes 3] - [1 \otimes (2 \otimes 3)]\) where the operation \(\otimes\) is defined by \(a \otimes b = \frac{a^2}{b}\), we proceed as follows:
First, compute \(1 \otimes 2\):
\[
1 \otimes 2 = \frac{1^2}{2} = \frac{1}{2}
\]
Next, use this result to compute \((1 \otimes 2) \o... | 0 | Okay, let me try to work through this problem step by step. The question is about an operation defined as a⊗b = a²/b for all nonzero numbers. We need to find [(1⊗2)⊗3] - [1⊗(2⊗3)] and choose the correct answer from the options given.
First, I need to make sure I understand the operation correctly. So, whenever I see ... |
Return your final response within \boxed{}. Doug constructs a square window using $8$ equal-size panes of glass, as shown. The ratio of the height to width for each pane is $5 : 2$, and the borders around and between the panes are $2$ inches wide. In inches, what is the side length of the square window?
[asy] fill((0,0... | Okay, let me try to figure out this problem. So Doug has a square window made up of 8 equal-sized panes of glass. Each pane has a height to width ratio of 5:2. There are borders around and between the panes that are 2 inches wide. The question is asking for the side length of the square window in inches. The answer cho... |
Doug constructs a square window using 8 equal-size panes of glass, with a height to width ratio of 5:2 for each pane. The borders around and between the panes are 2 inches wide. We need to determine the side length of the square window.
1. **Arrangement of Panes**: The panes are arranged in 2 rows and 4 columns. Thi... | 1 | Okay, let me try to figure out this problem. So Doug is building a square window with 8 equal-size panes of glass. Each pane has a height to width ratio of 5:2, and the borders around and between the panes are 2 inches wide. We need to find the side length of the square window. The answer choices are from 26 to 34 inch... |
Return your final response within \boxed{}. $P(x)$ is a polynomial of degree $3n$ such that
\begin{eqnarray*} P(0) = P(3) = \cdots &=& P(3n) = 2, \\ P(1) = P(4) = \cdots &=& P(3n-2) = 1, \\ P(2) = P(5) = \cdots &=& P(3n-1) = 0, \quad\text{ and }\\ && P(3n+1) = 730.\end{eqnarray*}
Determine $n$. | Okay, let's see. I need to find the value of n given this polynomial P(x) of degree 3n that satisfies certain conditions at points 0, 1, 2, 3, ..., 3n, and also P(3n+1) = 730. Hmm, this seems a bit involved, but let me break it down step by step.
First, let me try to understand the problem. The polynomial P(x) has deg... |
To determine the value of \( n \) given the polynomial \( P(x) \) of degree \( 3n \) that satisfies the conditions \( P(0) = P(3) = \cdots = P(3n) = 2 \), \( P(1) = P(4) = \cdots = P(3n-2) = 1 \), \( P(2) = P(5) = \cdots = P(3n-1) = 0 \), and \( P(3n+1) = 730 \), we need to consider the polynomial interpolation and t... | 2 | Okay, let me try to work through this problem. So, we have a polynomial P(x) of degree 3n. It has some specific values at certain points. Let me write down the given information again to make sure I have it right.
First, P(0) = P(3) = P(6) = ... = P(3n) = 2. So that's all the multiples of 3 up to 3n. Then, P(1) = P(4)... |
Return your final response within \boxed{}. Let $f$ be the function defined by $f(x)=ax^2-\sqrt{2}$ for some positive $a$. If $f(f(\sqrt{2}))=-\sqrt{2}$ then $a=$
$\text{(A) } \frac{2-\sqrt{2}}{2}\quad \text{(B) } \frac{1}{2}\quad \text{(C) } 2-\sqrt{2}\quad \text{(D) } \frac{\sqrt{2}}{2}\quad \text{(E) } \frac{2+\sqrt... | Okay, let's try to figure out this problem step by step. The function is given as f(x) = a x² - √2, and we need to find the positive value of a such that f(f(√2)) = -√2. The answer choices are given, so maybe once we solve for a, we can match it with one of the options. Let me start by breaking down what's needed here.... |
Given the function \( f(x) = ax^2 - \sqrt{2} \) where \( a \) is a positive constant, we need to find \( a \) such that \( f(f(\sqrt{2})) = -\sqrt{2} \).
First, compute \( f(\sqrt{2}) \):
\[
f(\sqrt{2}) = a(\sqrt{2})^2 - \sqrt{2} = 2a - \sqrt{2}
\]
Next, substitute \( f(\sqrt{2}) \) into the function \( f \):
\[
f(... | 3 | Okay, let me try to solve this problem step by step. The problem is about finding the value of 'a' in the function f(x) = ax² - sqrt(2) such that when you apply the function twice to sqrt(2), you get -sqrt(2). The answer choices are given, so I need to figure out which one is correct.
First, let me make sure I underst... |
Return your final response within \boxed{}. The total in-store price for an appliance is $\textdollar 99.99$. A television commercial advertises the same product for three easy payments of $\textdollar 29.98$ and a one-time shipping and handling charge of $\textdollar 9.98$. How many cents are saved by buying the appli... | Okay, let's try to figure out this problem step by step. So, the question is about comparing the total cost of buying an appliance in-store versus through a TV ad, and then finding out how many cents are saved by choosing the TV advertiser. The answer choices are from 6 to 10 cents, so the difference can't be too large... |
The in-store price for the appliance is $99.99. The television advertisement offers three easy payments of $29.98 each and a one-time shipping and handling charge of $9.98.
First, we calculate the total cost from the television advertiser:
- Three payments of $29.98:
\[
29.98 \times 3 = 89.94
\]
- Adding the... | 4 | Okay, let me try to figure out this problem. So, the question is about comparing the in-store price of an appliance to the price advertised on television. The in-store price is $99.99, and the TV ad says you can buy it for three easy payments of $29.98 plus a one-time shipping and handling charge of $9.98. The question... |
Return your final response within \boxed{}. Points $A,B,C,D,E$ and $F$ lie, in that order, on $\overline{AF}$, dividing it into five segments, each of length 1. Point $G$ is not on line $AF$. Point $H$ lies on $\overline{GD}$, and point $J$ lies on $\overline{GF}$. The line segments $\overline{HC}, \overline{JE},$ and ... | Okay, let's try to tackle this geometry problem step by step. First, I need to visualize the setup. There are points A, B, C, D, E, and F lying on the line AF in that order, dividing it into five segments each of length 1. So, the total length from A to F is 5 units. Let me draw a rough sketch in my mind: A----B----C--... |
Points \(A, B, C, D, E,\) and \(F\) lie on \(\overline{AF}\) in that order, dividing it into five segments each of length 1. We place these points on a coordinate system with \(A\) at \((0, 0)\), \(B\) at \((1, 0)\), \(C\) at \((2, 0)\), \(D\) at \((3, 0)\), \(E\) at \((4, 0)\), and \(F\) at \((5, 0)\). Point \(G\) i... | 5 | Okay, let me try to work through this geometry problem. Hmm, so we have points A, B, C, D, E, F on line AF, each dividing it into five equal segments of length 1. So AF is a line segment divided into five parts, each 1 unit long. Then point G is not on AF, which means it's somewhere off the line. Points H is on GD, and... |
Return your final response within \boxed{}. During the softball season, Judy had $35$ hits. Among her hits were $1$ home run, $1$ triple and $5$ doubles. The rest of her hits were single. What percent of her hits were single?
$\text{(A)}\ 28\% \qquad \text{(B)}\ 35\% \qquad \text{(C)}\ 70\% \qquad \text{(D)}\ 75\% \... | Okay, let's see. The problem is about Judy's hits during the softball season. She had a total of 35 hits. Among those, 1 was a home run, 1 was a triple, and 5 were doubles. The rest were singles. We need to find out what percent of her hits were singles. The options are given from A to E.
First, let me break down the ... |
Judy had a total of 35 hits, with 1 home run, 1 triple, and 5 doubles. To find the number of singles, we subtract these extra-base hits from the total:
\[
\text{Singles} = 35 - (1 + 1 + 5) = 35 - 7 = 28
\]
Next, we calculate the percentage of hits that were singles:
\[
\text{Percentage} = \left( \frac{28}{35} \rig... | 6 | Okay, let me try to figure out this problem. So, Judy had 35 hits in total during the softball season. Among those hits, there were 1 home run, 1 triple, and 5 doubles. The rest of her hits were singles. The question is asking what percent of her hits were singles. The answer choices are given from A to E, with percent... |
Return your final response within \boxed{}. The graph, $G$ of $y=\log_{10}x$ is rotated $90^{\circ}$ counter-clockwise about the origin to obtain a new graph $G'$. Which of the following is an equation for $G'$?
(A) $y=\log_{10}\left(\frac{x+90}{9}\right)$ (B) $y=\log_{x}10$ (C) $y=\frac{1}{x+1}$ (D) $y=10^{-x}$ (E) $y... | Okay, let's tackle this problem step by step. So, the question is about rotating the graph of y = log₁₀x 90 degrees counter-clockwise about the origin and finding the equation of the new graph G'. The options are given from A to E. Hmm, let's see.
First, I need to recall how rotation affects a graph. Rotating a graph ... |
To determine the equation of the graph \( G' \) obtained by rotating the graph \( G \) of \( y = \log_{10}x \) 90 degrees counter-clockwise about the origin, we need to apply the rotation transformation to each point on \( G \).
1. **Rotation Transformation**: A point \((x, y)\) on the original graph \( G \) is rota... | 7 | Okay, let me try to figure out how to solve this problem. So, the question is about rotating the graph of y = log₁₀x by 90 degrees counter-clockwise around the origin and then determining the equation of the new graph, G'. The options are given, and I need to pick the correct one.
First, I remember that when you rotat... |
Return your final response within \boxed{}. Jose, Thuy, and Kareem each start with the number 10. Jose subtracts 1 from the number 10, doubles his answer, and then adds 2. Thuy doubles the number 10, subtracts 1 from her answer, and then adds 2. Kareem subtracts 1 from the number 10, adds 2 to his number, and then d... | Okay, so I need to figure out who ends up with the largest final answer among Jose, Thuy, and Kareem. They all start with 10, but each of them does different operations. Let me take it step by step for each person.
First, let's break down what each person does. Maybe I'll write down each step for all three so I can co... |
Jose, Thuy, and Kareem each start with the number 10 and perform different operations. Let's break down their steps and compute their final results:
**Jose:**
1. Subtract 1 from 10: \(10 - 1 = 9\)
2. Double the result: \(9 \times 2 = 18\)
3. Add 2: \(18 + 2 = 20\)
**Thuy:**
1. Double the number 10: \(10 \times 2 = ... | 8 | Okay, let me try to figure out this problem step by step. So, Jose, Thuy, and Kareem each start with the number 10. Then they perform different operations, and we need to see who ends up with the largest final answer. The options are from A to E, with some being tied. Let me break down each person's operations one by o... |
Return your final response within \boxed{}. A set of consecutive positive integers beginning with $1$ is written on a blackboard. One number is erased. The average (arithmetic mean) of the remaining numbers is $35\frac{7}{17}$. What number was erased?
$\textbf{(A)}\ 6\qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 8 \qq... | Okay, let me try to figure out this problem. So, we have a set of consecutive positive integers starting from 1, and one number is erased. The average of the remaining numbers is 35 and 7/17. We need to find out which number was erased. The options are from 6 to 9, or cannot be determined. Hmm.
First, let's parse the ... |
Given a set of consecutive positive integers starting from 1, one number is erased, and the average of the remaining numbers is \(35 \frac{7}{17}\). We need to determine which number was erased.
1. **Convert the mixed number to an improper fraction**:
\[
35 \frac{7}{17} = \frac{602}{17}
\]
2. **Set up the ... | 9 | Okay, let me try to figure out this problem. So, we have a set of consecutive positive integers starting from 1, and one number is erased. The average of the remaining numbers is 35 and 7/17. We need to find which number was erased. The options are from A to E, with E saying it can't be determined. Alright, let's break... |
Return your final response within \boxed{}. A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths $15$ and $25$ meters. What fraction of the yard is occupied by the fl... | Okay, let me try to figure out this problem. So, we have a rectangular yard with two flower beds that are congruent isosceles right triangles. The rest of the yard is a trapezoid with parallel sides of 15 and 25 meters. We need to find the fraction of the yard occupied by the flower beds. Hmm, okay.
First, let me visu... |
The problem involves determining the fraction of a rectangular yard occupied by two congruent isosceles right triangle flower beds. The remaining part of the yard is a trapezoid with parallel sides of 15 meters and 25 meters.
1. **Visualizing the Yard and Flower Beds**:
- The yard is a rectangle with length 25 me... | 10 | Okay, let me try to figure out this problem. So, there's a rectangular yard with two flower beds that are congruent isosceles right triangles. The rest of the yard is a trapezoid with parallel sides of lengths 15 and 25 meters. We need to find the fraction of the yard occupied by the flower beds. The answer choices are... |
Return your final response within \boxed{}. What is the sum of all possible values of $t$ between $0$ and $360$ such that the triangle in the coordinate plane whose vertices are \[(\cos 40^\circ,\sin 40^\circ), (\cos 60^\circ,\sin 60^\circ), \text{ and } (\cos t^\circ,\sin t^\circ)\]
is isosceles?
$\textbf{(A)} \: 100... | Okay, let's see. I need to find the sum of all possible values of t between 0 and 360 degrees such that the triangle formed by the points (cos40°, sin40°), (cos60°, sin60°), and (cos t°, sin t°) is isosceles. The answer choices are given, so maybe after solving, I can check which one matches.
First, let me recall that... |
To determine the sum of all possible values of \( t \) between \( 0 \) and \( 360 \) such that the triangle formed by the points \((\cos 40^\circ, \sin 40^\circ)\), \((\cos 60^\circ, \sin 60^\circ)\), and \((\cos t^\circ, \sin t^\circ)\) is isosceles, we analyze the central angles between these points on the unit cir... | 11 | Okay, let me try to figure out this problem. We have three points on the unit circle: (cos40°, sin40°), (cos60°, sin60°), and (cost°, sint°). We need to find all possible values of t between 0° and 360° such that the triangle formed by these three points is isosceles. Then, sum all those t values and choose the correct... |
Return your final response within \boxed{}. Given: $x > 0, y > 0, x > y$ and $z\ne 0$. The inequality which is not always correct is:
$\textbf{(A)}\ x + z > y + z \qquad\textbf{(B)}\ x - z > y - z \qquad\textbf{(C)}\ xz > yz$
$\textbf{(D)}\ \frac {x}{z^2} > \frac {y}{z^2} \qquad\textbf{(E)}\ xz^2 > yz^2$ | Okay, let's see. The problem is asking which inequality is not always correct given that x and y are positive, x is greater than y, and z is not zero. The options are from A to E. I need to check each one to see if there's a scenario where the inequality doesn't hold. Let's go step by step.
Starting with option A: x +... |
The inequality that is not always correct is determined by analyzing each option under the given conditions \( x > 0 \), \( y > 0 \), \( x > y \), and \( z \ne 0 \):
- **Option A**: \( x + z > y + z \). Subtracting \( z \) from both sides simplifies to \( x > y \), which is always true.
- **Option B**: \( x - z > ... | 12 | Okay, let me try to figure out which inequality isn't always correct given the conditions x > 0, y > 0, x > y, and z ≠ 0. Hmm, let's look at each option one by one.
Starting with option A: x + z > y + z. Since x > y, if I add the same number z to both sides, the inequality should still hold. So adding z doesn't change... |
Return your final response within \boxed{}. The harmonic mean of two numbers can be calculated as twice their product divided by their sum. The harmonic mean of $1$ and $2016$ is closest to which integer?
$\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 45 \qquad \textbf{(C)}\ 504 \qquad \textbf{(D)}\ 1008 \qquad \textbf{(E)}\ 20... | Okay, let's see. The problem is about finding the harmonic mean of 1 and 2016 and determining which integer it's closest to among the given options. The harmonic mean formula is given as twice their product divided by their sum. Alright, let me recall what harmonic mean is.
So, harmonic mean of two numbers a and b is... |
The harmonic mean of two numbers \(a\) and \(b\) is calculated as \(\frac{2ab}{a + b}\). For the numbers 1 and 2016, we need to compute this harmonic mean and determine the closest integer.
1. **Calculate the numerator**: \(2 \times 1 \times 2016 = 4032\).
2. **Calculate the denominator**: \(1 + 2016 = 2017\).
3. **... | 13 | Okay, let me try to figure out this problem. So, the question is about the harmonic mean of 1 and 2016, and we need to find which integer it's closest to. The options are from A to E: 2, 45, 504, 1008, and 2015. Alright, let's start by recalling what harmonic mean is. The problem says the harmonic mean of two numbers i... |
Return your final response within \boxed{}. The first three terms of an arithmetic progression are $x - 1, x + 1, 2x + 3$, in the order shown. The value of $x$ is:
$\textbf{(A)}\ -2\qquad\textbf{(B)}\ 0\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ \text{undetermined}$ | Okay, so I have this problem here about an arithmetic progression. The first three terms are given as x - 1, x + 1, and 2x + 3. I need to find the value of x from the options provided. Let me try to work through this step by step.
First, I remember that in an arithmetic progression (AP), the difference between consecu... |
Given the first three terms of an arithmetic progression (AP) as \( x - 1 \), \( x + 1 \), and \( 2x + 3 \), we need to find the value of \( x \).
In an AP, the difference between consecutive terms is constant. Therefore, the difference between the second term and the first term should be equal to the difference bet... | 14 | Okay, let me try to solve this arithmetic progression problem. The question says that the first three terms of an arithmetic progression are x - 1, x + 1, and 2x + 3. We need to find the value of x from the given options. The options are A) -2, B) 0, C) 2, D) 4, and E) undetermined.
First, I remember that in an arithm... |
Return your final response within \boxed{}. Let a geometric progression with n terms have first term one, common ratio $r$ and sum $s$, where $r$ and $s$ are not zero. The sum of the geometric progression formed by replacing each term of the original progression by its reciprocal is
$\textbf{(A) }\frac{1}{s}\qquad \tex... | Alright, let me try to figure out this problem. So, we have a geometric progression (GP) with n terms. The first term is 1, common ratio is r, and the sum is s. Then, we're supposed to find the sum of another GP formed by taking the reciprocals of each term from the original progression. The options are given, and I ne... |
Given a geometric progression (GP) with \( n \) terms, where the first term is 1, the common ratio is \( r \), and the sum is \( s \). We need to find the sum of the GP formed by taking the reciprocals of each term of the original progression.
1. **Original GP Sum**:
The sum \( s \) of the original GP is given by... | 15 | Okay, let me try to work through this problem step by step. So, we have a geometric progression with n terms, first term 1, common ratio r, and sum s. We need to find the sum of the geometric progression formed by taking the reciprocal of each term of the original progression. The answer choices are given, so I need to... |
Return your final response within \boxed{}. Alicia had two containers. The first was $\tfrac{5}{6}$ full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was $\tfrac{3}{4}$ full of water. What is the ratio of the volume o... | Okay, let's see. Alicia has two containers. The first one is 5/6 full of water, and the second is empty. Then she pours all the water from the first into the second, and the second becomes 3/4 full. We need to find the ratio of the volume of the first container to the second. Hmm.
Alright, let's start by defining vari... |
Alicia poured \(\frac{5}{6}\) of the first container's volume (\(V_1\)) into the second container, making it \(\frac{3}{4}\) full of the second container's volume (\(V_2\)). This relationship is expressed as:
\[
\frac{5}{6}V_1 = \frac{3}{4}V_2
\]
Solving for the ratio \(\frac{V_1}{V_2}\):
\[
\frac{V_1}{V_2} = \fra... | 16 | Okay, let me try to figure out this problem. So, Alicia has two containers. The first one is 5/6 full of water, and the second one is empty. She pours all the water from the first container into the second one, and then the second container is 3/4 full. The question is asking for the ratio of the volume of the first co... |
Return your final response within \boxed{}. Misha rolls a standard, fair six-sided die until she rolls $1-2-3$ in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is $\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | Okay, let's tackle this problem step by step. So, Misha is rolling a standard six-sided die until she gets the sequence 1-2-3 in that order on three consecutive rolls. We need to find the probability that the number of rolls she makes is odd, and then compute m + n where the probability is expressed as a reduced fracti... |
To solve the problem, we need to determine the probability that Misha will roll the die an odd number of times before rolling the sequence 1-2-3. We model this using states and recursive equations.
### States:
- **S0**: No progress (starting state or last roll not part of 1-2-3)
- **S1**: Last roll was 1
- **S2**: L... | 17 | Okay, let me try to figure out this probability problem. So Misha is rolling a standard six-sided die until she gets the sequence 1-2-3 in that exact order on three consecutive rolls. We need to find the probability that she will roll the die an odd number of times. The answer is supposed to be m/n where m and n are co... |
Return your final response within \boxed{}. An architect is building a structure that will place vertical pillars at the vertices of regular hexagon $ABCDEF$, which is lying horizontally on the ground. The six pillars will hold up a flat solar panel that will not be parallel to the ground. The heights of pillars at $A$... | Alright, let's tackle this problem step by step. So, we have a regular hexagon ABCDEF with pillars at each vertex. The heights at A, B, and C are given as 12, 9, and 10 meters respectively. We need to find the height at E. The solar panel is flat but not parallel to the ground, which means it's a plane tilted in some w... |
Given a regular hexagon \(ABCDEF\) with pillars at vertices \(A\), \(B\), and \(C\) having heights 12, 9, and 10 meters respectively, we need to determine the height of the pillar at \(E\). The solar panel forms a plane passing through the tops of these pillars.
1. **Coordinate Setup**:
- Place the hexagon center... | 18 | Okay, let me try to figure out this problem. So, there's a regular hexagon ABCDEF lying on the ground, and pillars are placed at each vertex. The solar panel is flat but not parallel to the ground, and the heights at A, B, and C are given as 12, 9, and 10 meters. We need to find the height at E. The options are given, ... |
Return your final response within \boxed{}. Diameter $AB$ of a circle has length a $2$-digit integer (base ten). Reversing the digits gives the length of the perpendicular chord $CD$. The distance from their intersection point $H$ to the center $O$ is a positive rational number. Determine the length of $AB$. | Okay, let's try to tackle this problem step by step. The problem says that diameter AB of a circle is a two-digit integer, and reversing its digits gives the length of a perpendicular chord CD. The distance from their intersection point H to the center O is a positive rational number. We need to find the length of AB.
... |
The length of diameter \( AB \) is a two-digit integer, and reversing its digits gives the length of the perpendicular chord \( CD \). The distance from their intersection point \( H \) to the center \( O \) is a positive rational number. We need to determine the length of \( AB \).
1. **Setup and Equations**:
- ... | 19 | Okay, let me try to work through this problem step by step. First, let me make sure I understand what's being asked. We have a circle with diameter AB, and AB is a two-digit integer. If we reverse the digits of AB, that gives the length of a perpendicular chord CD. The intersection point of AB and CD is H, and the dist... |
Return your final response within \boxed{}. Find the smallest positive number from the numbers below.
$\textbf{(A)} \ 10-3\sqrt{11} \qquad \textbf{(B)} \ 3\sqrt{11}-10 \qquad \textbf{(C)}\ 18-5\sqrt{13}\qquad \textbf{(D)}\ 51-10\sqrt{26}\qquad \textbf{(E)}\ 10\sqrt{26}-51$ | Okay, so I need to find the smallest positive number among the given options. The options are all expressions involving square roots, so I guess I have to evaluate each one numerically to see which is the smallest positive value. Let me write down the options again to make sure I have them right:
A) 10 - 3√11
B) 3√11... |
To find the smallest positive number among the given options, we need to evaluate each expression and check if it is positive.
**Option A: \(10 - 3\sqrt{11}\)**
- Calculate \(3\sqrt{11}\):
\[
3\sqrt{11} \approx 3 \times 3.3166 \approx 9.9498
\]
- Subtract from 10:
\[
10 - 9.9498 \approx 0.0502
\]
- This... | 20 | Okay, so I need to find the smallest positive number among the options given. Let me look at each one and figure out their numerical values. Since these are all expressions involving square roots, maybe I can approximate them to see which is the smallest.
First, let's list out the options:
A) 10 - 3√11
B) 3√11 - 10... |
Return your final response within \boxed{}. The points $(2,-3)$, $(4,3)$, and $(5, k/2)$ are on the same straight line. The value(s) of $k$ is (are):
$\textbf{(A)}\ 12\qquad \textbf{(B)}\ -12\qquad \textbf{(C)}\ \pm 12\qquad \textbf{(D)}\ {12}\text{ or }{6}\qquad \textbf{(E)}\ {6}\text{ or }{6\frac{2}{3}}$ | Okay, so I need to find the value of k such that the three points (2, -3), (4, 3), and (5, k/2) lie on the same straight line. Hmm, let me think. Since they are on the same line, the slope between any two pairs of points should be the same. That's the key idea here. Let me recall, the slope formula between two points (... |
To determine the value of \( k \) such that the points \((2, -3)\), \((4, 3)\), and \((5, k/2)\) are collinear, we need to ensure that the slopes between any two pairs of points are equal.
1. **Calculate the slope between \((2, -3)\) and \((4, 3)\):**
\[
\text{slope} = \frac{3 - (-3)}{4 - 2} = \frac{6}{2} = 3
... | 21 | Okay, let me try to figure out this problem. We have three points: (2, -3), (4, 3), and (5, k/2). These points are supposed to be on the same straight line. We need to find the value(s) of k. The answer choices are given, so maybe after solving, I can check which one matches.
First, I remember that if three points are... |
Return your final response within \boxed{}. Suppose $q_1,q_2,...$ is an infinite sequence of integers satisfying the following two conditions:
(a) $m - n$ divides $q_m - q_n$ for $m>n \geq 0$
(b) There is a polynomial $P$ such that $|q_n|<P(n)$ for all $n$.
Prove that there is a polynomial $Q$ such that $q_n = Q(n)$ fo... | Okay, so I need to prove that if there's an infinite sequence of integers q₁, q₂, q₃,... that satisfies two conditions: (a) for any m > n ≥ 0, the difference m - n divides q_m - q_n, and (b) there exists a polynomial P such that the absolute value of q_n is less than P(n) for all n, then there must be some polynomial Q... |
Given an infinite sequence of integers \( q_1, q_2, \ldots \) satisfying the conditions:
(a) \( m - n \) divides \( q_m - q_n \) for \( m > n \geq 0 \)
(b) There exists a polynomial \( P \) such that \( |q_n| < P(n) \) for all \( n \),
we need to prove that there is a polynomial \( Q \) such that \( q_n = Q(n) \) fo... | 22 | Okay, let me try to work through this problem. So, we have an infinite sequence of integers q₁, q₂, ..., and it satisfies two conditions. First, for any m and n where m > n ≥ 0, the difference m - n divides q_m - q_n. Second, there's a polynomial P such that |q_n| < P(n) for all n. We need to prove that there's a polyn... |
Return your final response within \boxed{}. In a certain year the price of gasoline rose by $20\%$ during January, fell by $20\%$ during February, rose by $25\%$ during March, and fell by $x\%$ during April. The price of gasoline at the end of April was the same as it had been at the beginning of January. To the neares... | Okay, so I need to figure out the value of x, the percentage that gasoline prices fell in April, such that the price at the end of April is the same as it was at the beginning of January. The problem gives me a series of percentage changes each month: a 20% increase in January, a 20% decrease in February, a 25% increas... |
Let the initial price of gasoline be \( P \).
1. **January**: The price increases by 20%, so the new price is:
\[
P \times 1.20
\]
2. **February**: The price falls by 20%, so the new price is:
\[
P \times 1.20 \times 0.80 = P \times 0.96
\]
3. **March**: The price increases by 25%, so the new pri... | 23 | Okay, let me try to figure out this problem. So, the question is about the price of gasoline changing over the months of January, February, March, and April, and we need to find the percentage decrease in April that brings the price back to the original at the end of April. The answer choices are given, so maybe I can ... |
Return your final response within \boxed{}. Quadrilateral $ABCD$ satisfies $\angle ABC = \angle ACD = 90^{\circ}, AC=20,$ and $CD=30.$ Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point $E,$ and $AE=5.$ What is the area of quadrilateral $ABCD?$
$\textbf{(A) } 330 \qquad \textbf{(B) } 340 \qquad \textbf{(C... | Alright, let me try to figure out how to solve this geometry problem. Okay, so we have quadrilateral ABCD with two right angles at ABC and ACD. The diagonal AC is 20 units long, and CD is 30 units. The diagonals AC and BD intersect at point E, and AE is 5 units. We need to find the area of quadrilateral ABCD. The answe... |
Given quadrilateral \(ABCD\) with \(\angle ABC = 90^\circ\) and \(\angle ACD = 90^\circ\), \(AC = 20\), \(CD = 30\), and diagonals \(\overline{AC}\) and \(\overline{BD}\) intersecting at point \(E\) with \(AE = 5\). We need to find the area of quadrilateral \(ABCD\).
1. **Coordinates Setup**:
- Place \(C\) at the... | 24 | Okay, let me try to solve this problem step by step. So, we have quadrilateral ABCD with right angles at B and C. Wait, hold on, the problem says angle ABC is 90 degrees, and angle ACD is 90 degrees. So, point B is connected to A and C, forming a right angle, and point C is connected to A and D, forming another right a... |
Return your final response within \boxed{}. The angle bisector of the acute angle formed at the origin by the graphs of the lines $y = x$ and $y=3x$ has equation $y=kx.$ What is $k?$
$\textbf{(A)} \ \frac{1+\sqrt{5}}{2} \qquad \textbf{(B)} \ \frac{1+\sqrt{7}}{2} \qquad \textbf{(C)} \ \frac{2+\sqrt{3}}{2} \qquad \textbf... | Okay, so I need to find the equation of the angle bisector of the acute angle formed at the origin by the lines y = x and y = 3x. The bisector is given as y = kx, and I need to determine the value of k. The answer choices are all different expressions involving square roots, so I guess it's not a simple fraction. Let m... |
To find the equation of the angle bisector of the acute angle formed at the origin by the lines \( y = x \) and \( y = 3x \), we start by recognizing that the angle bisector will pass through the origin and have a positive slope \( k \).
We use the formula for the angle bisector between two lines \( L_1: a_1x + b_1y... | 25 | Okay, let me try to figure out this problem. So, the question is about finding the slope of the angle bisector of the acute angle formed at the origin by the lines y = x and y = 3x. The answer choices are given, and I need to determine which one is correct. Hmm, angle bisectors... I remember that an angle bisector spli... |
Return your final response within \boxed{}. A set $S$ of points in the $xy$-plane is symmetric about the origin, both coordinate axes, and the line $y=x$. If $(2,3)$ is in $S$, what is the smallest number of points in $S$?
$\mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 8\qquad ... | Okay, let's see. The problem says that a set S is symmetric about the origin, both coordinate axes, and the line y = x. If the point (2,3) is in S, what's the smallest number of points in S? The options are 1, 2, 4, 8, or 16. Hmm.
First, I need to recall what symmetry about each of these elements means. So, symmetry a... |
Given a set \( S \) of points in the \( xy \)-plane that is symmetric about the origin, both coordinate axes, and the line \( y = x \), we need to determine the smallest number of points in \( S \) if the point \( (2, 3) \) is in \( S \).
1. **Symmetry about the x-axis**: Reflecting \( (2, 3) \) over the x-axis give... | 26 | Okay, let me try to figure out this problem. So, we have a set S of points in the xy-plane that's symmetric about the origin, both coordinate axes, and the line y=x. And they tell us that the point (2,3) is in S. We need to find the smallest number of points in S. The options are from 1 to 16. Hmm, symmetries can somet... |
Return your final response within \boxed{}. Five positive consecutive integers starting with $a$ have average $b$. What is the average of $5$ consecutive integers that start with $b$?
$\textbf{(A)}\ a+3\qquad\textbf{(B)}\ a+4\qquad\textbf{(C)}\ a+5\qquad\textbf{(D)}\ a+6\qquad\textbf{(E)}\ a+7$ | Okay, let's tackle this problem step by step. So, we have five positive consecutive integers starting with 'a', and their average is 'b'. Then, we need to find the average of five consecutive integers starting with 'b'. The answer choices are from A to E, with options like a+3, a+4, up to a+7.
First, let me make sure... |
The problem involves finding the average of five consecutive integers starting with \( b \), where \( b \) is the average of five consecutive integers starting with \( a \).
1. **Calculate the average \( b \) of the first five consecutive integers starting with \( a \):**
- The five consecutive integers are \( a,... | 27 | Okay, let me try to work through this problem step by step. So, the problem says that there are five positive consecutive integers starting with 'a', and their average is 'b'. Then, we need to find the average of five consecutive integers that start with 'b'. The options are given from A to E, with possible answers inv... |
Return your final response within \boxed{}. Which of the following is the negation of the statement: For all $x$ of a certain set, $x^2>0$?
$\textbf{(A)}\ \text{For all x}, x^2 < 0\qquad \textbf{(B)}\ \text{For all x}, x^2 \le 0\qquad \textbf{(C)}\ \text{For no x}, x^2>0\qquad \\ \textbf{(D)}\ \text{For some x}, x^2>0\... | Okay, so I need to figure out the negation of the statement "For all x of a certain set, x² > 0." Let's break this down step by step.
First, the original statement is a universal quantification. It's saying that every element x in the set satisfies the condition x² being greater than zero. The logical form of this is... |
The original statement is a universal quantification: "For all \( x \), \( x^2 > 0 \)." The negation of a universal statement \( \forall x, P(x) \) is an existential statement \( \exists x, \neg P(x) \). Here, \( P(x) \) is \( x^2 > 0 \), so its negation \( \neg P(x) \) is \( x^2 \le 0 \).
Analyzing the options:
- ... | 28 | Okay, let me try to figure out the negation of the statement: "For all x of a certain set, x² > 0." Hmm, I remember that negating a universal statement requires transforming it into an existential statement. So, if the original statement is "For all x, P(x)", the negation should be "There exists an x such that not P(x)... |
Return your final response within \boxed{}. What is the greatest power of $2$ that is a factor of $10^{1002} - 4^{501}$?
$\textbf{(A) } 2^{1002} \qquad\textbf{(B) } 2^{1003} \qquad\textbf{(C) } 2^{1004} \qquad\textbf{(D) } 2^{1005} \qquad\textbf{(E) }2^{1006}$ | Okay, let's try to figure out the greatest power of 2 that divides \(10^{1002} - 4^{501}\). Hmm, so we need to find the highest power of 2 that is a factor of this expression. The answer choices go up to \(2^{1006}\), which is quite high. Let me break this down step by step.
First, let's rewrite the terms to see if th... |
To determine the greatest power of \(2\) that is a factor of \(10^{1002} - 4^{501}\), we start by expressing the terms in terms of powers of 2 and 5:
\[
10^{1002} = (2 \times 5)^{1002} = 2^{1002} \times 5^{1002}
\]
\[
4^{501} = (2^2)^{501} = 2^{1002}
\]
Factoring out \(2^{1002}\) from both terms, we get:
\[
10^{100... | 29 | Okay, let me try to figure out this problem. We need to find the greatest power of 2 that divides the number 10^1002 - 4^501. The answer choices are from 2^1002 up to 2^1006, so the exponents are getting really big here. Hmm, this seems like a problem that might involve some number theory, maybe looking at the factors ... |
Return your final response within \boxed{}. Consider all 1000-element subsets of the set $\{1, 2, 3, ... , 2015\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. | Okay, so I have this problem here: Consider all 1000-element subsets of the set {1, 2, 3, ..., 2015}. From each such subset, choose the least element. Then we need to find the arithmetic mean of all these least elements, express it as a reduced fraction p/q, and then find p + q. Hmm, that seems a bit complex, but let's... |
To find the arithmetic mean of the least elements of all 1000-element subsets of the set \(\{1, 2, 3, \ldots, 2015\}\), we start by considering the expectation of the minimum element in a random 1000-element subset.
The key insight is to use the known formula for the expectation of the minimum of a random \(n\)-ele... | 30 | Okay, let me try to figure out this problem. So, the question is about finding the arithmetic mean of the least elements of all 1000-element subsets of the set {1, 2, 3, ..., 2015}. The answer needs to be in the form p/q where p and q are coprime, and then find p + q. Hmm, that seems a bit complex, but let's break it d... |
Return your final response within \boxed{}. A man on his way to dinner short after $6: 00$ p.m. observes that the hands of his watch form an angle of $110^{\circ}$. Returning before $7: 00$ p.m. he notices that again the hands of his watch form an angle of $110^{\circ}$. The number of minutes that he has been away is: ... | Okay, so I need to figure out how many minutes the man was away based on the times when the angle between the clock hands was 110 degrees. Let's start by recalling how clock angles work.
First, I remember that the formula to calculate the angle between the hour and minute hands is:
θ = |30H - 5.5M|
where θ is the a... |
To determine the number of minutes the man was away, we need to find the times between 6:00 p.m. and 7:00 p.m. when the angle between the clock hands is \(110^\circ\). The formula for the angle between the hour and minute hands is:
\[
\theta = |30H - 5.5M|
\]
where \(H\) is the hour and \(M\) is the minutes past th... | 31 | Okay, let me try to figure out this problem. So, the problem is about a watch where the minute and hour hands form a 110-degree angle both before and after dinner. We need to find the number of minutes the man was away. The answer choices are given, so maybe after working through it, I can check which one matches.
Fir... |
Return your final response within \boxed{}. A 3x3x3 cube is made of $27$ normal dice. Each die's opposite sides sum to $7$. What is the smallest possible sum of all of the values visible on the $6$ faces of the large cube?
$\text{(A)}\ 60 \qquad \text{(B)}\ 72 \qquad \text{(C)}\ 84 \qquad \text{(D)}\ 90 \qquad \text{(E... | Alright, so I've got this problem here about a 3x3x3 cube made up of 27 dice. The question is asking for the smallest possible sum of all the visible faces on the six faces of the large cube. Each die has opposite sides that sum to 7. The answer choices are 60, 72, 84, 90, and 96. Let me try to figure this out step by ... |
To determine the smallest possible sum of all visible values on the 6 faces of a 3x3x3 cube made of 27 dice, we need to consider the optimal orientation of each die to minimize the sum of visible faces. Each die's opposite sides sum to 7.
1. **Categorizing Dice Positions**:
- **Corner Dice**: 8 corners, each with... | 32 | Okay, let me try to figure out this problem. So, we have a 3x3x3 cube made up of 27 dice. Each die has opposite faces adding up to 7. The question is asking for the smallest possible sum of all the visible faces on the large cube. The options are from 60 to 96, with 60 being the smallest. Hmm, let me break this down st... |
Return your final response within \boxed{}. In the multiplication problem below $A$, $B$, $C$, $D$ are different digits. What is $A+B$?
\[\begin{array}{cccc}& A & B & A\\ \times & & C & D\\ \hline C & D & C & D\\ \end{array}\]
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf... | Okay, so I need to solve this multiplication problem where A, B, C, D are different digits. The problem is set up like a vertical multiplication:
A B A
× C D
----------
C D C D
And the question is asking for A + B, with the answer choices given from 1 to 9. Let me try to figure this out step by step.
First, let ... |
Given the multiplication problem where \( A \), \( B \), \( C \), and \( D \) are different digits:
\[
\begin{array}{cccc}
& A & B & A \\
\times & & C & D \\
\hline
C & D & C & D \\
\end{array}
\]
We need to determine the value of \( A + B \).
1. **Analyzing the structure**:
- The three-digit number \( ABA \) ... | 33 | Okay, let me try to solve this problem step by step. So, we have this multiplication problem where each letter (A, B, C, D) represents a different digit. The problem looks like:
```
A B A
× C D
--------
C D C D
```
And we need to find A + B. The answer choices are 1, 2, 3, 4, or 9. Hmm, interesting. Let me start ... |
Return your final response within \boxed{}. Let $\, a =(m^{m+1} + n^{n+1})/(m^m + n^n), \,$ where $\,m\,$ and $\,n\,$ are positive integers. Prove that $\,a^m + a^n \geq m^m + n^n$.
[You may wish to analyze the ratio $\,(a^N - N^N)/(a-N),$ for real $\, a \geq 0 \,$ and integer $\, N \geq 1$.] | Okay, so I need to prove that for positive integers m and n, if a is defined as (m^{m+1} + n^{n+1})/(m^m + n^n), then a^m + a^n is greater than or equal to m^m + n^n. Hmm, let's start by understanding what a is. It looks like a weighted average of m and n, but with exponents involved.
First, let me compute a. The num... |
Given \( a = \frac{m^{m+1} + n^{n+1}}{m^m + n^n} \) where \( m \) and \( n \) are positive integers, we need to prove that \( a^m + a^n \geq m^m + n^n \).
### Key Steps and Analysis:
1. **Definition of \( a \)**:
- \( a \) is a weighted average of \( m \) and \( n \) with weights proportional to \( m^m \) and \(... | 34 | Okay, let me try to work through this problem. We need to prove that \( a^m + a^n \geq m^m + n^n \), where \( a = \frac{m^{m+1} + n^{n+1}}{m^m + n^n} \). Hmm, so first, let's understand what \( a \) is. It's the average of \( m^{m+1} \) and \( n^{n+1} \), divided by the sum of \( m^m \) and \( n^n \). Wait, maybe not e... |
Return your final response within \boxed{}. The sum of two natural numbers is $17{,}402$. One of the two numbers is divisible by $10$. If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers?
$\textbf{(A)} ~10{,}272\qquad\textbf{(B)} ~11{,}700\qquad\textbf{... | Okay, let's tackle this problem step by step. So, we have two natural numbers whose sum is 17,402. One of them is divisible by 10, which means it ends with a 0 in the units place. The other number is obtained by erasing the units digit of the first number. We need to find the difference between these two numbers. The o... |
Let's denote the two natural numbers as \( x \) and \( y \), where \( x \) is divisible by 10. According to the problem, erasing the units digit of \( x \) gives \( y \). This implies \( x = 10y \). The sum of the two numbers is 17,402, so we can set up the equation:
\[
x + y = 17,402
\]
Substituting \( x = 10y \) ... | 35 | Okay, let me try to work through this problem step by step. So, we have two natural numbers that add up to 17,402. One of them is divisible by 10, which means it ends with a 0. If we erase the units digit of that number, we get the other number. We need to find the difference between these two numbers. The options are ... |
Return your final response within \boxed{}. What is the value of \[2^{\left(0^{\left(1^9\right)}\right)}+\left(\left(2^0\right)^1\right)^9?\]
$\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 4$ | Okay, let me try to figure out this problem step by step. The question is asking for the value of this expression:
\[2^{\left(0^{\left(1^9\right)}\right)}+\left(\left(2^0\right)^1\right)^9\]
And the options are from A to E, with possible answers 0, 1, 2, 3, or 4. Hmm, let's break this down into parts.
First, I not... |
The given expression is \[2^{\left(0^{\left(1^9\right)}\right)}+\left(\left(2^0\right)^1\right)^9.\]
**First Term Analysis:**
1. Compute the innermost exponent: \(1^9 = 1\).
2. Substitute into the next exponent: \(0^{1} = 0\).
3. Substitute into the outer term: \(2^{0} = 1\).
**Second Term Analysis:**
1. Compute th... | 36 | Okay, let me try to figure out this problem step by step. The question is asking for the value of the expression:
\[2^{\left(0^{\left(1^9\right)}\right)}+\left(\left(2^0\right)^1\right)^9\]
And the options are A) 0, B) 1, C) 2, D) 3, E) 4. Alright, I need to break this down into two parts since there's a plus sign in... |
Return your final response within \boxed{}. Which of the following describes the graph of the equation $(x+y)^2=x^2+y^2$?
$\textbf{(A) } \text{the\,empty\,set}\qquad \textbf{(B) } \textrm{one\,point}\qquad \textbf{(C) } \textrm{two\,lines} \qquad \textbf{(D) } \textrm{a\,circle} \qquad \textbf{(E) } \textrm{the\,entire... | Okay, let me try to figure out what the graph of the equation \((x + y)^2 = x^2 + y^2\) looks like. Hmm, so the question gives me multiple choices: empty set, one point, two lines, a circle, or the entire plane. I need to figure out which one it is.
First, maybe I should start by expanding the left side of the equatio... |
The equation given is \((x + y)^2 = x^2 + y^2\). To analyze this, we start by expanding the left-hand side:
\[
(x + y)^2 = x^2 + 2xy + y^2.
\]
Substituting this back into the equation, we have:
\[
x^2 + 2xy + y^2 = x^2 + y^2.
\]
Subtracting \(x^2 + y^2\) from both sides simplifies to:
\[
2xy = 0.
\]
Dividing bo... | 37 | Okay, let me try to figure out this problem. The equation given is (x + y)^2 = x^2 + y^2. And I need to determine which of the options A to E describes the graph of this equation. The options are empty set, one point, two lines, a circle, or the entire plane. Hmm, let's start by expanding the left side of the equation ... |
Return your final response within \boxed{}. If $\texttt{a}$ and $\texttt{b}$ are digits for which
$\begin{array}{ccc}& 2 & a\\ \times & b & 3\\ \hline & 6 & 9\\ 9 & 2\\ \hline 9 & 8 & 9\end{array}$
then $\texttt{a+b =}$
$\mathrm{(A)\ } 3 \qquad \mathrm{(B) \ }4 \qquad \mathrm{(C) \ } 7 \qquad \mathrm{(D) \ } 9 \qquad... | Okay, so I have this multiplication problem here, and I need to figure out the values of a and b, which are digits. Then, add them together to get the answer. Let me try to break this down step by step.
First, let me write down the problem as it's given:
```
2 a
× b 3
------
6 9
9 2
------
9 8 9
```
Hmm, okay. T... |
Given the multiplication problem:
\[
\begin{array}{ccc}
& 2 & a \\
\times & b & 3 \\
\hline
& 6 & 9 \\
9 & 2 \\
\hline
9 & 8 & 9 \\
\end{array}
\]
We need to determine the values of \(a\) and \(b\) (digits) and then find \(a + b\).
1. **First Partial Product (69):** This is the result of multiplying the two-digi... | 38 | Okay, let me try to figure out this problem. So, we have a multiplication problem here where a two-digit number, 2a, multiplied by another two-digit number, b3, gives a three-digit number, 989. The digits a and b are unknown, and we need to find their sum. The answer choices are given, so that might help in checking.
... |
Return your final response within \boxed{}. At $2: 15$ o'clock, the hour and minute hands of a clock form an angle of:
$\textbf{(A)}\ 30^{\circ} \qquad\textbf{(B)}\ 5^{\circ} \qquad\textbf{(C)}\ 22\frac {1}{2}^{\circ} \qquad\textbf{(D)}\ 7\frac {1}{2} ^{\circ} \qquad\textbf{(E)}\ 28^{\circ}$ | Okay, so I need to figure out the angle between the hour and minute hands at 2:15. Hmm, let's think. I remember that clocks can be a bit tricky because both hands are moving. The minute hand moves around the clock as the minutes pass, but the hour hand also moves, not just jumping from hour to hour. So, at 2:15, the mi... |
At 2:15, the minute hand is at \(15 \times 6 = 90^\circ\) (since it moves \(6^\circ\) per minute). The hour hand starts at \(2 \times 30 = 60^\circ\) and moves \(15 \times 0.5 = 7.5^\circ\) (as it moves \(0.5^\circ\) per minute). Thus, the hour hand is at \(60 + 7.5 = 67.5^\circ\). The angle between them is \(90 - 67... | 39 | Okay, let me try to figure out this clock angle problem. So, the question is: At 2:15 o'clock, the hour and minute hands of a clock form an angle of which option? The options are A through E with different angles. Hmm, I need to remember how to calculate the angle between the hour and minute hands at a given time.
Fir... |
Return your final response within \boxed{}. In [right triangle](https://artofproblemsolving.com/wiki/index.php/Right_triangle) $ABC$ with [right angle](https://artofproblemsolving.com/wiki/index.php/Right_angle) $C$, $CA = 30$ and $CB = 16$. Its legs $CA$ and $CB$ are extended beyond $A$ and $B$. [Points](https://art... | Okay, let's try to figure out this problem. So, we have a right triangle ABC with a right angle at C. The legs CA and CB are 30 and 16 units long, respectively. The legs are extended beyond A and B, and there are two circles with centers O₁ and O₂ in the exterior of the triangle. Both circles have the same radius, and ... |
In right triangle \(ABC\) with right angle at \(C\), the legs \(CA = 30\) and \(CB = 16\). The hypotenuse \(AB\) is calculated using the Pythagorean theorem:
\[
AB = \sqrt{30^2 + 16^2} = \sqrt{900 + 256} = \sqrt{1156} = 34
\]
The circles with centers \(O_1\) and \(O_2\) are tangent to the hypotenuse \(AB\) and the ... | 40 | Okay, let me try to work through this problem step by step. First, let me make sure I understand the problem correctly. We have a right triangle ABC with a right angle at C. The legs are CA = 30 and CB = 16. The legs are extended beyond A and B. Then, there are two circles with centers O1 and O2, both with equal radii.... |
Return your final response within \boxed{}. A finite [sequence](https://artofproblemsolving.com/wiki/index.php/Sequence) of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term a... | Okay, let's try to tackle this problem. So, we have a finite sequence of three-digit numbers. Each term in the sequence has the property that the tens and units digits become the hundreds and tens digits of the next term. Also, the last term's tens and units digits loop back to become the first term's hundreds and tens... |
The sequence forms a cyclic structure where each term's tens and units digits become the next term's hundreds and tens digits. Denote each term as \( T_i = 100a_i + 10b_i + c_i \). The cyclic property implies:
- \( a_{i+1} = b_i \) and \( b_{i+1} = c_i \) for \( 1 \leq i \leq n-1 \)
- \( a_1 = b_n \) and \( b_1 = c_... | 41 | Okay, let me try to work through this problem step by step. First, let me make sure I understand the problem correctly.
We have a finite sequence of three-digit integers. The property is that each term's tens and units digits become the hundreds and tens digits of the next term. Additionally, the last term's tens and ... |
Return your final response within \boxed{}. What time was it $2011$ minutes after midnight on January 1, 2011?
$\textbf{(A)}\ \text{January 1 at 9:31PM}$
$\textbf{(B)}\ \text{January 1 at 11:51PM}$
$\textbf{(C)}\ \text{January 2 at 3:11AM}$
$\textbf{(D)}\ \text{January 2 at 9:31AM}$
$\textbf{(E)}\ \text{January 2 at 6... | Okay, so I need to figure out what time it was 2011 minutes after midnight on January 1, 2011. The options are from A to E, with different dates and times. Let me start by breaking down the problem step by step.
First, midnight on January 1 is 12:00 AM. Adding minutes to that time should give me the correct time later... |
To determine the time 2011 minutes after midnight on January 1, 2011, we first convert minutes into hours and days:
1. **Convert 2011 minutes to hours**:
\[
2011 \div 60 = 33 \text{ hours and } 31 \text{ minutes.}
\]
This means 2011 minutes is equivalent to 33 hours and 31 minutes.
2. **Break down 3... | 42 | Okay, let me try to figure out what time it was 2011 minutes after midnight on January 1, 2011. Hmm, so I need to convert 2011 minutes into hours and minutes to add to midnight. Let me start by breaking down the problem step by step.
First, I remember that one hour has 60 minutes. So if I divide 2011 by 60, I can find... |
Return your final response within \boxed{}. A basketball player made $5$ baskets during a game. Each basket was worth either $2$ or $3$ points. How many different numbers could represent the total points scored by the player?
$\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \... | Okay, let's try to figure out this basketball problem. So, the player made 5 baskets, each worth either 2 or 3 points. We need to find out how many different total points they could have scored. The options are from A to E, with possible answers 2, 3, 4, 5, or 6. Hmm, let's break this down step by step.
First, each ba... |
The basketball player made 5 baskets, each worth either 2 or 3 points. We need to determine how many different total points could be scored.
1. Let \( x \) be the number of 2-pointers and \( y \) be the number of 3-pointers. We know \( x + y = 5 \).
2. The total points can be expressed as \( 2x + 3y \).
3. Substitut... | 43 | Okay, let me try to figure out this problem. So, the question is about a basketball player who made 5 baskets, each worth either 2 or 3 points. We need to find how many different total points could be scored. The options are from 2 to 6, so the answer is probably somewhere in that range. Let me break it down step by st... |
Return your final response within \boxed{}. Square $EFGH$ has one vertex on each side of square $ABCD$. Point $E$ is on $AB$ with $AE=7\cdot EB$. What is the ratio of the area of $EFGH$ to the area of $ABCD$?
$\text{(A)}\,\frac{49}{64} \qquad\text{(B)}\,\frac{25}{32} \qquad\text{(C)}\,\frac78 \qquad\text{(D)}... | Okay, let's see. The problem is about two squares: a larger square ABCD and a smaller square EFGH. Each vertex of EFGH is on a different side of ABCD. Specifically, point E is on side AB, and they tell us that AE is 7 times EB. We need to find the ratio of the area of EFGH to the area of ABCD. The answer choices are gi... |
Square \(EFGH\) has one vertex on each side of square \(ABCD\). Point \(E\) is on \(AB\) with \(AE = 7 \cdot EB\). We need to find the ratio of the area of \(EFGH\) to the area of \(ABCD\).
1. **Assign Coordinates:**
- Let the side length of square \(ABCD\) be \(s\).
- Coordinates of \(A(0, 0)\), \(B(s, 0)\), ... | 44 | Okay, let me try to figure out this geometry problem. So, we have square ABCD, and square EFGH is inside it with each vertex on a different side of ABCD. The problem states that point E is on side AB such that AE is 7 times EB. We need to find the ratio of the area of EFGH to the area of ABCD. The answer choices are gi... |
Return your final response within \boxed{}. A house worth $ $9000$ is sold by Mr. A to Mr. B at a $10$ % loss. Mr. B sells the house back to Mr. A at a $10$ % gain.
The result of the two transactions is:
$\textbf{(A)}\ \text{Mr. A breaks even} \qquad \textbf{(B)}\ \text{Mr. B gains }$900 \qquad \textbf{(C)}\ \text{Mr... | Okay, let's see. The problem is about Mr. A selling a house to Mr. B at a 10% loss, and then Mr. B selling it back to Mr. A at a 10% gain. We need to figure out the result of these two transactions for both parties. The options given are A through E, with possible outcomes like breaking even, gaining or losing specific... |
Mr. A initially sells the house worth $9000 to Mr. B at a 10% loss.
**First transaction:**
- Loss amount: \(10\% \times 9000 = 900\)
- Selling price: \(9000 - 900 = 8100\)
Mr. A receives $8100, incurring a $900 loss.
Mr. B then sells the house back to Mr. A at a 10% gain.
**Second transaction:**
- Gain... | 45 | Okay, let me try to work through this problem step by step. So, we have a house that's worth $9000. Mr. A sells it to Mr. B with a 10% loss. Then Mr. B sells it back to Mr. A with a 10% gain. We need to figure out the result of these two transactions. The options are about whether A breaks even, B gains $900, A loses $... |
Return your final response within \boxed{}. Chandler wants to buy a $500$ dollar mountain bike. For his birthday, his grandparents
send him $50$ dollars, his aunt sends him $35$ dollars and his cousin gives him $15$ dollars. He earns
$16$ dollars per week for his paper route. He will use all of his birthday money and a... | Okay, let's see. Chandler wants to buy a mountain bike that costs $500. He received some money from his grandparents, aunt, and cousin for his birthday. Plus, he earns money from his paper route every week. The question is asking how many weeks he needs to work to save up enough for the bike, using all his birthday mon... |
Chandler receives a total of $50 + $35 + $15 = $100 from his birthday gifts. The remaining amount he needs to buy the $500 mountain bike is $500 - $100 = $400. He earns $16 per week from his paper route, so the number of weeks required to save $400 is calculated by dividing $400 by $16:
\[
\frac{400}{16} = 25
\]
Th... | 46 | Okay, let me try to figure out how many weeks Chandler needs to save up to buy the mountain bike. The problem says he wants a $500 bike. His grandparents gave him $50, his aunt $35, and his cousin $15. So first, I should add up all the money he already received from his birthday. Let me check that.
Starting with the g... |
Return your final response within \boxed{}. The harmonic mean of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. What is the harmonic mean of 1, 2, and 4?
$\textbf{(A) }\frac{3}{7}\qquad\textbf{(B) }\frac{7}{12}\qquad\textbf{(C) }\frac{12}{7}\qquad\textbf{(D) }\frac{7}{4}\q... | Okay, let's see. The problem is asking for the harmonic mean of the numbers 1, 2, and 4. The answer choices are given, so once I figure it out, I can check which one matches. Hmm, harmonic mean... I remember it's different from the arithmetic mean. Let me recall the definition.
The harmonic mean of a set of non-zero n... |
The harmonic mean of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. To find the harmonic mean of 1, 2, and 4, we follow these steps:
1. **Find the reciprocals of the numbers**:
- The reciprocal of 1 is \(1\).
- The reciprocal of 2 is \(\frac{1}{2}\).
- The reci... | 47 | Okay, let me try to figure out how to find the harmonic mean of the numbers 1, 2, and 4. Hmm, the problem says the harmonic mean is the reciprocal of the average of the reciprocals of the numbers. So first, I need to remember the formula for harmonic mean. Let me recall... I think harmonic mean is calculated by taking ... |
Return your final response within \boxed{}. Julie is preparing a speech for her class. Her speech must last between one-half hour and three-quarters of an hour. The ideal rate of speech is 150 words per minute. If Julie speaks at the ideal rate, which of the following number of words would be an appropriate length f... | Okay, so Julie needs to prepare a speech that's between half an hour and three-quarters of an hour long. The ideal speaking rate is 150 words per minute. The question is asking which number of words from the options given would be appropriate for her speech. Let me break this down step by step.
First, I need to figure... |
Julie's speech must last between 30 minutes (0.5 hours) and 45 minutes (0.75 hours). At an ideal rate of 150 words per minute, the required word count range is calculated as follows:
- **Minimum words**: \(150 \, \text{words/min} \times 30 \, \text{min} = 4500 \, \text{words}\)
- **Maximum words**: \(150 \, \text... | 48 | Okay, let me try to figure out this problem. So, Julie needs to prepare a speech that's between half an hour and three-quarters of an hour. The ideal rate is 150 words per minute. The question is asking which of the given options (A to E) is the appropriate number of words for her speech.
First, I need to understand ... |
Return your final response within \boxed{}. If the following instructions are carried out by a computer, which value of $X$ will be printed because of instruction $5$?
1. START $X$ AT $3$ AND $S$ AT $0$.
2. INCREASE THE VALUE OF $X$ BY $2$.
3. INCREASE THE VALUE OF $S$ BY THE VALUE OF $X$.
4. IF $S$ IS AT LEAST ... | Okay, let me try to figure out this problem. So, the question is about a computer program with instructions 1 through 6, and we need to find the value of X that gets printed when instruction 5 is executed. The options are given from (A) 19 up to (E) 201. Alright, let's break this down step by step.
First, let's parse ... |
The problem involves determining the value of \( X \) printed by a computer program after executing a series of instructions. Here is the systematic breakdown:
1. **Initialization**: Start with \( X = 3 \) and \( S = 0 \).
2. **Loop Execution**:
- **Instruction 2**: Increase \( X \) by 2.
- **Instruction 3**: ... | 49 | Okay, let me try to work through this problem step by step. So, we have a computer program with instructions 1 through 6. The question is asking which value of X will be printed because of instruction 5. The options are from A to E, with numbers 19, 21, 23, 199, and 201. Let me break down each instruction to understand... |
Return your final response within \boxed{}. Which of the following is equivalent to "If P is true, then Q is false."? | Okay, let's tackle this logic problem. The question is asking which of the following is equivalent to "If P is true, then Q is false." Hmm, first, I need to recall how to translate conditional statements into logical expressions. So, the original statement is "If P, then not Q." In symbolic terms, that would be P → ¬Q,... |
The statement "If P is true, then Q is false" (symbolically \( P \rightarrow \neg Q \)) is logically equivalent to:
1. **Contrapositive**: "If Q is true, then P is false" (\( Q \rightarrow \neg P \)).
This follows because the contrapositive of \( P \rightarrow \neg Q \) is formed by reversing and negating the a... | 50 | Okay, let me try to figure out which of the given options is equivalent to "If P is true, then Q is false." Hmm, logical equivalences can be tricky, especially with implications. First, I need to recall some basic rules about if-then statements, also known as implications in logic.
An implication "If P, then Q" is lo... |
Return your final response within \boxed{}. Let $m$ be the least positive integer divisible by $17$ whose digits sum to $17$. Find $m$. | Okay, let's see. I need to find the smallest positive integer that's divisible by 17 and whose digits add up to 17. Hmm, so m has two conditions: divisible by 17 and digit sum of 17. Alright, let's break this down step by step.
First, understanding the problem: We need the smallest number (least positive integer) such... |
To find the least positive integer \( m \) divisible by 17 whose digits sum to 17, we need to check numbers systematically.
First, we check two-digit numbers with a digit sum of 17 (89 and 98), but neither is divisible by 17. Next, we consider three-digit numbers. The smallest three-digit numbers with a digit sum o... | 51 | Okay, let me try to solve this problem. The question is: Find the least positive integer divisible by 17 whose digits sum to 17. So, I need to find the smallest number that is a multiple of 17 and the sum of its digits is 17. Hmm, let's break this down.
First, the number has to be divisible by 17. That means when the ... |
Return your final response within \boxed{}. A shopper plans to purchase an item that has a listed price greater than $\textdollar 100$ and can use any one of the three coupons. Coupon A gives $15\%$ off the listed price, Coupon B gives $\textdollar 30$ off the listed price, and Coupon C gives $25\%$ off the amount by w... | Okay, let's try to figure out this coupon problem. So, we have an item that costs more than $100, and the shopper can use one of three coupons: A, B, or C. We need to find the smallest and largest prices where Coupon A is at least as good as B or C. Then subtract those two numbers to get the answer.
First, let's make ... |
To determine the smallest and largest prices for which Coupon A saves at least as many dollars as Coupon B or C, we analyze the savings for each coupon:
- **Coupon A**: 15% off the listed price. Savings: \(0.15P\)
- **Coupon B**: $30 off the listed price. Savings: $30
- **Coupon C**: 25% off the amount by which the ... | 52 | Okay, let me try to work through this problem step by step. So, the shopper is buying an item that costs more than $100. They can use one of three coupons: A, B, or C. Each coupon gives a discount, and we need to find the smallest and largest prices where Coupon A saves at least as much as Coupon B or C. Then, we have ... |
Return your final response within \boxed{}. In $\triangle ABC$, $\angle ABC=45^\circ$. Point $D$ is on $\overline{BC}$ so that $2\cdot BD=CD$ and $\angle DAB=15^\circ$. Find $\angle ACB.$
$\text{(A) }54^\circ \qquad \text{(B) }60^\circ \qquad \text{(C) }72^\circ \qquad \text{(D) }75^\circ \qquad \text{(E) }90^\circ$ | Okay, let me try to solve this geometry problem. Hmm, so we have triangle ABC where angle ABC is 45 degrees. Point D is on BC such that 2 times BD equals CD, and angle DAB is 15 degrees. We need to find angle ACB. The answer choices are 54°, 60°, 72°, 75°, and 90°.
First, let me draw a rough sketch of triangle ABC. L... |
Given triangle \( \triangle ABC \) with \( \angle ABC = 45^\circ \). Point \( D \) is on \( \overline{BC} \) such that \( 2 \cdot BD = CD \) and \( \angle DAB = 15^\circ \). We need to find \( \angle ACB \).
1. **Identify Ratios and Angles**:
- \( BD : CD = 1 : 2 \), so \( BD = x \) and \( CD = 2x \), making \( B... | 53 | Okay, let me try to solve this geometry problem. So, we have triangle ABC with angle ABC equal to 45 degrees. Point D is on BC such that BD is half of CD, meaning CD is twice BD. Also, angle DAB is 15 degrees. We need to find angle ACB, which is one of the options given: 54°, 60°, 72°, 75°, or 90°. Hmm, okay. Let me st... |
Return your final response within \boxed{}. Mary's top book shelf holds five books with the following widths, in centimeters: $6$, $\dfrac{1}{2}$, $1$, $2.5$, and $10$. What is the average book width, in centimeters?
$\mathrm{(A)}\ 1 \qquad \mathrm{(B)}\ 2 \qquad \mathrm{(C)}\ 3 \qquad \mathrm{(D)}\ 4 \qquad \mathrm{(E... | Okay, so Mary has five books on her top shelf, and their widths are given as 6 cm, 1/2 cm, 1 cm, 2.5 cm, and 10 cm. The question is asking for the average width of these books. The answer choices are A through E, with the numbers 1 to 5. Hmm, let me think about how to approach this.
First, I remember that the average ... |
To find the average width of Mary's books, we first convert all measurements to decimals for easier addition:
- \(6\) cm remains \(6.0\)
- \(\dfrac{1}{2}\) cm converts to \(0.5\)
- \(1\) cm remains \(1.0\)
- \(2.5\) cm remains \(2.5\)
- \(10\) cm remains \(10.0\)
Next, we sum these widths:
\[
6.0 + 0.5 + 1.0 + 2.5 +... | 54 | Okay, let me try to figure out how to solve this problem. So, Mary has a top bookshelf with five books, and their widths are given as 6, 1/2, 1, 2.5, and 10 centimeters. The question is asking for the average book width, and the answer choices are from 1 to 5. Alright, let me start by recalling what average means. The ... |
Return your final response within \boxed{}. The equation $2^{2x}-8\cdot 2^x+12=0$ is satisfied by:
$\text{(A) } log(3)\quad \text{(B) } \tfrac{1}{2}log(6)\quad \text{(C) } 1+log(\tfrac{3}{2})\quad \text{(D) } 1+\frac{log(3)}{log(2)}\quad \text{(E) none of these}$ | Okay, let's see. I need to solve the equation \(2^{2x} - 8 \cdot 2^x + 12 = 0\) and check which of the options A to E is correct. Hmm, the equation looks a bit complicated with exponents. Maybe I can simplify it by substituting a variable for \(2^x\). That often helps with exponential equations. Let me try that.
Let’s... |
To solve the equation \(2^{2x} - 8 \cdot 2^x + 12 = 0\), we start by substituting \(y = 2^x\). This transforms the equation into a quadratic equation:
\[
y^2 - 8y + 12 = 0
\]
Factoring the quadratic equation, we get:
\[
(y - 6)(y - 2) = 0
\]
This gives the solutions \(y = 6\) and \(y = 2\). Substituting back \(y ... | 55 | Okay, let me try to solve this equation step by step. The equation given is 2^(2x) - 8*2^x + 12 = 0. The question is asking which of the options A to E satisfy this equation. The options include logarithmic expressions, so I need to manipulate the equation to involve logarithms or solve for x in terms of logarithms.
F... |
Return your final response within \boxed{}. The sum of the greatest integer less than or equal to $x$ and the least integer greater than or equal to $x$ is $5$. The solution set for $x$ is
$\textbf{(A)}\ \Big\{\frac{5}{2}\Big\}\qquad \textbf{(B)}\ \big\{x\ |\ 2 \le x \le 3\big\}\qquad \textbf{(C)}\ \big\{x\ |\ 2\le x <... | Alright, let me try to figure out this problem step by step. So, the question is about finding all real numbers x such that the sum of the greatest integer less than or equal to x and the least integer greater than or equal to x is equal to 5. The answer choices are given from A to E, and I need to determine which one ... |
To solve the problem, we need to find all real numbers \( x \) such that the sum of the greatest integer less than or equal to \( x \) (floor function \( \lfloor x \rfloor \)) and the least integer greater than or equal to \( x \) (ceiling function \( \lceil x \rceil \)) is 5.
1. **Case 1: \( x \) is an integer** ... | 56 | Okay, let me try to figure out this problem. So, the question is about finding the solution set for x where the sum of the greatest integer less than or equal to x (which is the floor function, right?) and the least integer greater than or equal to x (which is the ceiling function) is equal to 5. The answer choices are... |
Return your final response within \boxed{}. A powderman set a fuse for a blast to take place in $30$ seconds. He ran away at a rate of $8$ yards per second. Sound travels at the rate of $1080$ feet per second. When the powderman heard the blast, he had run approximately:
$\textbf{(A)}\ \text{200 yd.}\qquad\textbf{(B)}\... | Okay, let's try to figure out this problem step by step. So, the powderman sets a fuse for 30 seconds and starts running away at 8 yards per second. Then, when he hears the blast, we need to find out how far he has run. The options are given, but I need to work through it myself first.
First, let me parse the informat... |
The powderman sets a fuse for 30 seconds and runs away at 8 yards per second. Sound travels at 1080 feet per second, which converts to 360 yards per second.
1. **Distance run by the powderman in 30 seconds**:
\[
8 \, \text{yards/second} \times 30 \, \text{seconds} = 240 \, \text{yards}
\]
2. **Let \( t_2... | 57 | Okay, let me try to figure out this problem. So, the powderman sets a fuse that will detonate the blast in 30 seconds. Then he starts running away at 8 yards per second. Sound travels at 1080 feet per second, and we need to find out how far he had run when he heard the blast. The answer choices are in yards, so I shoul... |
Return your final response within \boxed{}. A student council must select a two-person welcoming committee and a three-person planning committee from among its members. There are exactly $10$ ways to select a two-person team for the welcoming committee. It is possible for students to serve on both committees. In how ma... | Okay, let's see. The problem is about selecting committees from a student council. Specifically, we need to find out how many ways there are to select a three-person planning committee, given that there are exactly 10 ways to select a two-person welcoming committee. Both committees can have overlapping members, so the ... |
Given that there are exactly 10 ways to select a two-person welcoming committee, we need to determine the number of ways to select a three-person planning committee from the same student council members.
First, we use the combination formula to find the number of members \( n \) in the student council. The number of... | 58 | Okay, let me try to figure out this problem. So, the question is about a student council selecting a two-person welcoming committee and a three-person planning committee. It says there are exactly 10 ways to select the two-person committee, and we need to find out how many different ways the three-person planning commi... |
Return your final response within \boxed{}. Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be $2$ : $1$ ?
$\textbf{(A)}\ 2 \qquad\textbf{(B)} \ 4 \qquad\textbf{(C)} \ 5 \qquad\textbf{(D)} \ 6 \qq... | Okay, so I need to solve this age problem. Let me read it again carefully. The question says:
"Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be 2 : 1? The options are A through E: 2, 4, 5, 6, 8... |
Let \( P \) be Pete's current age and \( C \) be Claire's current age.
From the problem, we know:
1. Two years ago, Pete was three times as old as Claire:
\[
P - 2 = 3(C - 2)
\]
2. Four years ago, Pete was four times as old as Claire:
\[
P - 4 = 4(C - 4)
\]
Expanding and simplifying these equatio... | 59 | Okay, let me try to solve this problem step by step. So, the problem is about Pete and Claire's ages, and we need to find out in how many years the ratio of their ages will be 2:1. The answer choices are given from A to E, with numbers 2, 4, 5, 6, 8. Hmm, let's start by breaking down the information given.
First, the ... |
Return your final response within \boxed{}. If $\theta$ is a constant such that $0 < \theta < \pi$ and $x + \dfrac{1}{x} = 2\cos{\theta}$, then for each positive integer $n$, $x^n + \dfrac{1}{x^n}$ equals
$\textbf{(A)}\ 2\cos\theta\qquad \textbf{(B)}\ 2^n\cos\theta\qquad \textbf{(C)}\ 2\cos^n\theta\qquad \textbf{(D)}\ ... | Okay, let's see. The problem says that if x + 1/x = 2 cos θ, where θ is between 0 and π, then for any positive integer n, what is x^n + 1/x^n equal to? The options are given as D, which is 2 cos nθ, and others. Hmm, I need to figure out why it's D. Let me think step by step.
First, I recall that expressions like x + 1... |
Given the equation \( x + \dfrac{1}{x} = 2\cos{\theta} \) where \( 0 < \theta < \pi \), we need to determine the value of \( x^n + \dfrac{1}{x^n} \) for any positive integer \( n \).
First, we recognize that \( x \) can be represented as a complex exponential \( e^{i\theta} \) or \( e^{-i\theta} \) because \( e^{i\t... | 60 | Okay, let me try to figure out this problem. So, the question is: If θ is a constant between 0 and π, and x + 1/x equals 2 cos θ, then for each positive integer n, what does x^n + 1/x^n equal? The options are given from A to E. Hmm, okay. Let me start by understanding what's given and what's being asked.
First, we hav... |
Return your final response within \boxed{}. Positive integers $a$ and $b$ are such that the graphs of $y=ax+5$ and $y=3x+b$ intersect the $x$-axis at the same point. What is the sum of all possible $x$-coordinates of these points of intersection?
$\textbf{(A)}\ {-20}\qquad\textbf{(B)}\ {-18}\qquad\textbf{(C)}\ {-15}\qq... | Okay, so I need to solve this problem where positive integers a and b are such that the graphs of y = ax + 5 and y = 3x + b intersect the x-axis at the same point. Then, I have to find the sum of all possible x-coordinates of these intersection points. The answer choices are given from A to E. Let me try to break this ... |
To solve the problem, we need to find the sum of all possible \( x \)-coordinates where the graphs of \( y = ax + 5 \) and \( y = 3x + b \) intersect the \( x \)-axis, given that \( a \) and \( b \) are positive integers and the graphs intersect the \( x \)-axis at the same point.
1. **Determine the \( x \)-intercep... | 61 | Okay, let me try to work through this problem step by step. So, we have two lines: y = ax + 5 and y = 3x + b. The question says that their graphs intersect the x-axis at the same point. We need to find the sum of all possible x-coordinates of these points of intersection. The answer choices are given, so hopefully, aft... |
Return your final response within \boxed{}. (Titu Andreescu)
Prove that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd. | Okay, so I need to prove that for every positive integer n, there's an n-digit number divisible by 5^n, and all its digits are odd. Hmm, let's start by understanding the problem better.
First, let's break it down. For each n, we need to construct a number with exactly n digits (so the first digit can't be zero), each... |
To prove that for every positive integer \( n \) there exists an \( n \)-digit number divisible by \( 5^n \) with all digits odd, we use mathematical induction.
**Base Case (\( n = 1 \)):**
The number \( 5 \) is a 1-digit number divisible by \( 5^1 = 5 \), and its digit is odd. Thus, the base case holds.
**Induct... | 62 | Okay, so I need to prove that for every positive integer n, there exists an n-digit number divisible by 5^n all of whose digits are odd. Hmm, let's start by understanding the problem.
First, an n-digit number with all odd digits. The odd digits are 1, 3, 5, 7, 9. So each digit in the number is one of these. Then, the... |
Return your final response within \boxed{}. Four circles, no two of which are congruent, have centers at $A$, $B$, $C$, and $D$, and points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\tfrac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\tfrac{5}{8}$ times the radius of c... | Okay, let's try to tackle this problem step by step. First, I need to understand what's being asked here. We have four circles with centers at A, B, C, and D. None of the circles are congruent, which means all their radii are different. Points P and Q lie on all four circles, so P and Q are common intersection points o... |
Given four circles with centers at \(A\), \(B\), \(C\), and \(D\), where points \(P\) and \(Q\) lie on all four circles, we need to find the sum of the distances from the midpoint \(R\) of \(\overline{PQ}\) to each center. The radii of circles \(A\) and \(C\) are \(\frac{5}{8}\) times the radii of circles \(B\) and \... | 63 | Okay, let me try to work through this problem step by step. First, let me make sure I understand the problem correctly.
We have four circles with centers at points A, B, C, D. All four circles pass through points P and Q. The radii of the circles are related: the radius of circle A is 5/8 times the radius of circle B,... |
Return your final response within \boxed{}. A figure is an equiangular parallelogram if and only if it is a
$\textbf{(A)}\ \text{rectangle}\qquad \textbf{(B)}\ \text{regular polygon}\qquad \textbf{(C)}\ \text{rhombus}\qquad \textbf{(D)}\ \text{square}\qquad \textbf{(E)}\ \text{trapezoid}$ | Okay, let's tackle this problem step by step. The question is asking: "A figure is an equiangular parallelogram if and only if it is a..." with options given. The answer choices are rectangle, regular polygon, rhombus, square, trapezoid. Hmm. So first, let's parse the terms here.
An equiangular parallelogram. Let's re... |
To determine the correct answer, we need to analyze the definitions and properties of the given options in relation to an equiangular parallelogram.
1. **Definitions and Properties**:
- A **parallelogram** is a quadrilateral with both pairs of opposite sides parallel.
- **Equiangular** means all angles are equ... | 64 | Okay, let me try to figure out this problem. The question is: "A figure is an equiangular parallelogram if and only if it is a..." and there are five options: A through E. The answer choices are rectangle, regular polygon, rhombus, square, or trapezoid.
First, I need to recall what an equiangular parallelogram is. A ... |
Return your final response within \boxed{}. Find the minimum value of $\sqrt{x^2+y^2}$ if $5x+12y=60$.
$\textbf{(A)}\ \frac{60}{13}\qquad \textbf{(B)}\ \frac{13}{5}\qquad \textbf{(C)}\ \frac{13}{12}\qquad \textbf{(D)}\ 1\qquad \textbf{(E)}\ 0$ | Okay, so I need to find the minimum value of √(x² + y²) given that 5x + 12y = 60. Hmm, let me think. The problem is asking for the minimum distance from the origin to the line 5x + 12y = 60, right? Because √(x² + y²) is the distance formula from the origin to the point (x, y). So, if I can find the shortest distance fr... |
To find the minimum value of \(\sqrt{x^2 + y^2}\) given that \(5x + 12y = 60\), we recognize that this is the distance from the origin to the line \(5x + 12y = 60\). The formula for the distance from a point \((x_0, y_0)\) to the line \(ax + by + c = 0\) is \(\frac{|ax_0 + by_0 + c|}{\sqrt{a^2 + b^2}}\).
Rewriting t... | 65 | Okay, let me try to figure out how to solve this problem. The question is asking for the minimum value of the square root of (x squared plus y squared) given the equation 5x + 12y = 60. The answer choices are from A to E, with B, C, D, and E being some fractions or whole numbers. Hmm, I remember that the square root of... |
Return your final response within \boxed{}. On average, for every 4 sports cars sold at the local dealership, 7 sedans are sold. The dealership predicts that it will sell 28 sports cars next month. How many sedans does it expect to sell?
$\textbf{(A)}\ 7\qquad\textbf{(B)}\ 32\qquad\textbf{(C)}\ 35\qquad\textbf{(D)}\ 4... | Okay, so I need to solve this problem where the dealership sells sports cars and sedans in a ratio, and they predict selling a certain number of sports cars next month. The question is asking how many sedans they expect to sell. Let me break this down step by step.
First, the problem says that for every 4 sports cars ... |
The dealership's sales ratio is 4 sports cars to 7 sedans. For 28 sports cars, the scaling factor is \( \frac{28}{4} = 7 \). Multiplying the sedan part of the ratio by this factor:
\[
7 \times 7 = 49
\]
Verifying via proportion:
\[
\frac{4}{7} = \frac{28}{x} \implies 4x = 196 \implies x = \frac{196}{4} = 49
\]
Bo... | 66 | Okay, let me try to figure out this problem. So, the question is about the dealership selling sports cars and sedans. It says that for every 4 sports cars sold, 7 sedans are sold on average. The dealership predicts selling 28 sports cars next month, and they want to know how many sedans they expect to sell. The options... |
Return your final response within \boxed{}. Two fair dice, each with at least $6$ faces are rolled. On each face of each die is printed a distinct integer from $1$ to the number of faces on that die, inclusive. The probability of rolling a sum of $7$ is $\frac34$ of the probability of rolling a sum of $10,$ and the pro... | Okay, let me try to figure out this problem step by step. So, we have two fair dice, each with at least 6 faces. When rolled, the probability of getting a sum of 7 is three-fourths the probability of getting a sum of 10. Also, the probability of rolling a sum of 12 is 1/12. We need to find the least possible number of ... |
Given two fair dice with at least 6 faces each, we need to find the least possible number of faces on the two dice combined, considering the following conditions:
1. The probability of rolling a sum of 7 is \(\frac{3}{4}\) of the probability of rolling a sum of 10.
2. The probability of rolling a sum of 12 is \(\frac... | 67 | Okay, let me try to work through this problem step by step. So, we have two fair dice, each with at least 6 faces. Each die has distinct integers from 1 to the number of faces on that die. The probability of rolling a sum of 7 is 3/4 of the probability of rolling a sum of 10, and the probability of rolling a sum of 12 ... |
Return your final response within \boxed{}. Five towns are connected by a system of roads. There is exactly one road connecting each pair of towns. Find the number of ways there are to make all the roads one-way in such a way that it is still possible to get from any town to any other town using the roads (possibly pas... | Okay, so I need to figure out the number of ways to make all roads between five towns one-way such that you can still get from any town to any other town. Hmm, let's start by understanding the problem better.
First, there are five towns, and each pair is connected by a single road. Since each pair has a road, this is ... |
To determine the number of ways to make all roads between five towns one-way such that it is possible to travel from any town to any other town, we need to count the number of strongly connected tournaments on five vertices. A tournament is a complete oriented graph where every pair of vertices is connected by a sing... | 68 | Okay, so I have this problem here about five towns connected by roads, and each pair of towns has exactly one road connecting them. The question is asking for the number of ways to make all the roads one-way such that it's still possible to get from any town to any other town using the roads. Hmm, let's try to break th... |
Return your final response within \boxed{}. The number $2.5252525\ldots$ can be written as a fraction.
When reduced to lowest terms the sum of the numerator and denominator of this fraction is:
$\textbf{(A) }7\qquad \textbf{(B) }29\qquad \textbf{(C) }141\qquad \textbf{(D) }349\qquad \textbf{(E) }\text{none of these}$ | Okay, let's see. I need to convert the repeating decimal 2.5252525... into a fraction and then find the sum of the numerator and denominator when it's in its simplest form. The options are given, but I should figure it out step by step.
First, I remember that repeating decimals can often be converted into fractions by... |
Let \( x = 2.5252525\ldots \). The repeating part is two digits long, so we multiply both sides by 100:
\[
100x = 252.5252525\ldots
\]
Subtracting the original equation from this:
\[
100x - x = 252.5252525\ldots - 2.5252525\ldots
\]
This simplifies to:
\[
99x = 250
\]
Solving for \( x \):
\[
x = \frac{250}{99}... | 69 | Okay, let me try to figure out how to convert the repeating decimal 2.5252525... into a fraction and then find the sum of the numerator and denominator when it's in its simplest form. Hmm, repeating decimals can be tricky at first, but I remember there's a standard method for converting them to fractions. Let me recall... |
Return your final response within \boxed{}. For all non-zero numbers $x$ and $y$ such that $x = 1/y$, $\left(x-\frac{1}{x}\right)\left(y+\frac{1}{y}\right)$ equals
$\textbf{(A) }2x^2\qquad \textbf{(B) }2y^2\qquad \textbf{(C) }x^2+y^2\qquad \textbf{(D) }x^2-y^2\qquad \textbf{(E) }y^2-x^2$ | Okay, so I have this problem here. Let me read it again: For all non-zero numbers x and y such that x equals 1 over y, the expression (x minus 1/x) times (y plus 1/y) equals one of the given options. The options are A through E, with various expressions involving x squared and y squared. Alright, so my goal is to simpl... |
Given the problem, we need to simplify the expression \(\left(x - \frac{1}{x}\right)\left(y + \frac{1}{y}\right)\) for non-zero numbers \(x\) and \(y\) such that \(x = \frac{1}{y}\).
First, we substitute \(y = \frac{1}{x}\) into the expression. This gives us:
\[
\left(x - \frac{1}{x}\right)\left(\frac{1}{x} + x\righ... | 70 | Okay, let me try to work through this problem step by step. The question says: For all non-zero numbers x and y such that x = 1/y, what does (x - 1/x)(y + 1/y) equal? The options are from A to E, with B, C, D, E being 2y², x² + y², x² - y², and y² - x² respectively. Hmm, okay.
First, I need to substitute the given con... |
Return your final response within \boxed{}. The values of $k$ for which the equation $2x^2-kx+x+8=0$ will have real and equal roots are:
$\textbf{(A)}\ 9\text{ and }-7\qquad\textbf{(B)}\ \text{only }-7\qquad\textbf{(C)}\ \text{9 and 7}\\ \textbf{(D)}\ -9\text{ and }-7\qquad\textbf{(E)}\ \text{only 9}$ | Okay, let me try to solve this problem. The question is about finding the values of k for which the quadratic equation 2x² - kx + x + 8 = 0 has real and equal roots. The options are given from A to E, and I need to choose the correct one.
First, I remember that for a quadratic equation ax² + bx + c = 0 to have real a... |
To determine the values of \( k \) for which the quadratic equation \( 2x^2 - kx + x + 8 = 0 \) has real and equal roots, we need to ensure the discriminant is zero.
First, we combine the like terms in the equation:
\[ 2x^2 - kx + x + 8 = 0 \]
\[ 2x^2 + (-k + 1)x + 8 = 0 \]
This is in the standard form \( ax^2 + b... | 71 | Okay, let me try to figure out this problem. So, the question is about finding the values of k such that the quadratic equation 2x² - kx + x + 8 = 0 has real and equal roots. The options are given from A to E. Hmm, real and equal roots... That reminds me of the discriminant. Right, for a quadratic equation ax² + bx + c... |
Return your final response within \boxed{}. How many perfect cubes lie between $2^8+1$ and $2^{18}+1$, inclusive?
$\textbf{(A) }4\qquad\textbf{(B) }9\qquad\textbf{(C) }10\qquad\textbf{(D) }57\qquad \textbf{(E) }58$ | Alright, let me try to figure out how many perfect cubes lie between 2^8 + 1 and 2^18 + 1, inclusive. Hmm, okay. First, I need to understand the problem properly. So, we have two numbers: the lower bound is 2^8 + 1, and the upper bound is 2^18 + 1. We need to count how many perfect cubes are within this range, includin... |
To determine how many perfect cubes lie between \(2^8 + 1\) and \(2^{18} + 1\), inclusive, we start by calculating the values of these bounds.
- \(2^8 = 256\), so \(2^8 + 1 = 257\).
- \(2^{18} = 262,144\), so \(2^{18} + 1 = 262,145\).
Next, we need to find the smallest and largest integers \(n\) such that \(n^3\) ... | 72 | Okay, let me try to figure out how many perfect cubes are between 2^8 + 1 and 2^18 + 1, inclusive. Hmm, first, I need to understand the problem clearly. The question is asking for the count of perfect cubes that lie within that range. So, perfect cubes are numbers like 1, 8, 27, 64, etc., which are n³ for integer n. Th... |
Return your final response within \boxed{}. A line that passes through the origin intersects both the line $x = 1$ and the line $y=1+ \frac{\sqrt{3}}{3} x$. The three lines create an equilateral triangle. What is the perimeter of the triangle?
$\textbf{(A)}\ 2\sqrt{6} \qquad\textbf{(B)} \ 2+2\sqrt{3} \qquad\textbf{(C)}... | Okay, let's try to figure out this problem. So, we have a line that passes through the origin, and it intersects two other lines: x = 1 and y = 1 + (√3/3)x. These three lines form an equilateral triangle. We need to find the perimeter of this triangle. The answer choices are given, so maybe that can help check our work... |
Given a line passing through the origin that intersects both the vertical line \( x = 1 \) and the line \( y = 1 + \frac{\sqrt{3}}{3}x \), we need to determine the perimeter of the equilateral triangle formed by these three lines.
1. **Equations of the Lines**:
- The line passing through the origin is \( y = mx \... | 73 | Okay, let me try to figure out this problem. So, we have a line that passes through the origin and intersects both the line x=1 and the line y=1 + (√3/3)x. These three lines form an equilateral triangle. We need to find the perimeter of this triangle. The answer choices are given, so maybe I can use that to check my wo... |
Return your final response within \boxed{}. Alicia earns 20 dollars per hour, of which $1.45\%$ is deducted to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes?
$\mathrm{(A) \ } 0.0029 \qquad \mathrm{(B) \ } 0.029 \qquad \mathrm{(C) \ } 0.29 \qquad \mathrm{(D) \ } 2.9 \qquad \math... | Okay, let's see. The problem is about Alicia's wages and calculating how many cents per hour go to local taxes. Hmm. Let me start by breaking down the information given.
First, Alicia earns $20 per hour. The tax rate is 1.45%, which is deducted from her wages. The question is asking for how many cents per hour are use... |
Alicia earns $20 per hour, and 1.45% of her wages are deducted for local taxes. To find how many cents per hour are used to pay local taxes, we start by calculating the tax amount in dollars:
1. Convert 1.45% to a decimal: \(1.45\% = 0.0145\).
2. Multiply Alicia's hourly wage by the tax rate:
\[
20 \times 0.0... | 74 | Okay, let me try to figure out this problem. So Alicia earns $20 per hour, and 1.45% of that is deducted for local taxes. The question is asking how many cents per hour she is paying in taxes. The answer choices are from 0.0029 to 29 cents. Hmm, let's break this down step by step.
First, I need to understand what 1.45... |
Return your final response within \boxed{}. The numbers $-2, 4, 6, 9$ and $12$ are rearranged according to these rules:
1. The largest isn't first, but it is in one of the first three places.
2. The smallest isn't last, but it is in one of the last three places.
3. The median isn't first or l... | Alright, let's tackle this problem step by step. So, we have the numbers -2, 4, 6, 9, and 12. We need to rearrange them according to three rules and then find the average of the first and last numbers. Let me try to break down each rule and see how they constrain the possible arrangements.
First, let's list the number... |
The numbers \(-2, 4, 6, 9,\) and \(12\) need to be rearranged according to the given rules. Let's analyze each rule systematically:
1. **Largest number (12)** isn't first but must be in one of the first three positions. Thus, 12 can be in position 2 or 3.
2. **Smallest number (-2)** isn't last but must be in one of ... | 75 | Okay, let me try to solve this problem step by step. So, we have the numbers -2, 4, 6, 9, and 12, and we need to rearrange them according to three specific rules. Then, we have to find the average of the first and last numbers. The answer choices are given, so maybe after figuring out the correct arrangement, we can ju... |
Return your final response within \boxed{}. Mary thought of a positive two-digit number. She multiplied it by $3$ and added $11$. Then she switched the digits of the result, obtaining a number between $71$ and $75$, inclusive. What was Mary's number?
$\textbf{(A)}\ 11\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 13\qquad\... | Okay, so Mary thought of a positive two-digit number. Let me call this number N. The problem says she multiplied it by 3 and then added 11. Then she switched the digits of the result, and that switched number is between 71 and 75, inclusive. We need to find Mary's original number from the options given: 11, 12, 13, 14,... |
Mary thought of a positive two-digit number \( N \). She multiplied it by 3 and added 11. Then she switched the digits of the result, obtaining a number between 71 and 75, inclusive. We need to find Mary's original number.
1. Let \( N \) be the original two-digit number.
2. The operations performed on \( N \) are: \... | 76 | Okay, let me try to solve this problem step by step. So, Mary thought of a positive two-digit number. Let's call that number N. Then she multiplied it by 3 and added 11. After that, she switched the digits of the result, and the new number was between 71 and 75, inclusive. We need to find what her original number was. ... |
Return your final response within \boxed{}. Trapezoid $ABCD$ has $\overline{AB}\parallel\overline{CD},BC=CD=43$, and $\overline{AD}\perp\overline{BD}$. Let $O$ be the intersection of the diagonals $\overline{AC}$ and $\overline{BD}$, and let $P$ be the midpoint of $\overline{BD}$. Given that $OP=11$, the length of $AD$... | Okay, so I need to solve this trapezoid problem. Let's break down the given information step by step.
First, we have trapezoid ABCD with AB parallel to CD. The sides BC and CD are both 43 units long. Also, AD is perpendicular to BD. The diagonals AC and BD intersect at point O, and P is the midpoint of BD. We're told... |
Given trapezoid \(ABCD\) with \(\overline{AB} \parallel \overline{CD}\), \(BC = CD = 43\), and \(\overline{AD} \perp \overline{BD}\). The intersection of the diagonals \(AC\) and \(BD\) is \(O\), and \(P\) is the midpoint of \(\overline{BD}\) with \(OP = 11\). We need to find the length of \(AD\) in the form \(m\sqrt... | 77 | Okay, let me try to work through this geometry problem. Hmm, it's about a trapezoid with some specific properties. Let me start by recalling the given information and drawing a rough sketch in my mind.
We have trapezoid ABCD where AB is parallel to CD. So, AB and CD are the two bases. BC and CD are both 43 units long.... |
Return your final response within \boxed{}. The expression $\frac{1^{4y-1}}{5^{-1}+3^{-1}}$ is equal to:
$\textbf{(A)}\ \frac{4y-1}{8}\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ \frac{15}{2}\qquad\textbf{(D)}\ \frac{15}{8}\qquad\textbf{(E)}\ \frac{1}{8}$ | Okay, let's see. I need to figure out which of the options A through E is equal to the expression (1^{4y-1})/(5^{-1} + 3^{-1}). Hmm, first, I should probably simplify both the numerator and the denominator step by step.
Starting with the numerator: 1 raised to any exponent is still 1, right? Because 1 multiplied by i... |
The expression simplifies by recognizing that \(1^{4y-1} = 1\) for any \(y\). The denominator \(5^{-1} + 3^{-1}\) converts to \(\frac{1}{5} + \frac{1}{3}\). Finding a common denominator (15) gives \(\frac{3}{15} + \frac{5}{15} = \frac{8}{15}\). Taking the reciprocal, the expression becomes \(1 \div \frac{8}{15} = \fr... | 78 | Okay, let me try to work through this problem step by step. The expression given is (1^{4y - 1}) divided by (5^{-1} + 3^{-1}). The question is asking which of the options A to E it equals. Let's break this down.
First, let's look at the numerator: 1^{4y - 1}. Hmm. Any number raised to any power is still 1, right? Beca... |
Return your final response within \boxed{}. Carrie has a rectangular garden that measures $6$ feet by $8$ feet. She plants the entire garden with strawberry plants. Carrie is able to plant $4$ strawberry plants per square foot, and she harvests an average of $10$ strawberries per plant. How many strawberries can she ex... | Okay, let's try to figure out how many strawberries Carrie can expect to harvest from her garden. Hmm, let's start by understanding the problem step by step.
First, Carrie has a rectangular garden. The dimensions are 6 feet by 8 feet. So, the area of the garden must be calculated first because the number of plants dep... |
Carrie's rectangular garden has an area of \(6 \, \text{feet} \times 8 \, \text{feet} = 48 \, \text{square feet}\). She plants \(4 \, \text{strawberry plants per square foot}\), resulting in \(48 \times 4 = 192 \, \text{plants}\). Each plant yields an average of \(10 \, \text{strawberries}\), so the total harvest is ... | 79 | Okay, let me try to figure out how to solve this problem. So, Carrie has a rectangular garden that's 6 feet by 8 feet. She plants strawberries all over, and she can plant 4 plants per square foot. Each plant gives her an average of 10 strawberries. We need to find out how many strawberries she can expect to harvest. Th... |
Return your final response within \boxed{}. When the natural numbers $P$ and $P'$, with $P>P'$, are divided by the natural number $D$, the remainders are $R$ and $R'$, respectively. When $PP'$ and $RR'$ are divided by $D$, the remainders are $r$ and $r'$, respectively. Then:
$\text{(A) } r>r' \text{ always}\quad \text... | Alright, let me try to tackle this problem step by step. Okay, so we have two natural numbers P and P', where P is greater than P'. When both are divided by another natural number D, they leave remainders R and R' respectively. Then, we're looking at what happens when we divide the products PP' and RR' by D. The remain... |
When natural numbers \( P \) and \( P' \) (with \( P > P' \)) are divided by \( D \), they leave remainders \( R \) and \( R' \), respectively. By modular arithmetic, \( P \equiv R \pmod{D} \) and \( P' \equiv R' \pmod{D} \). Multiplying these congruences gives \( PP' \equiv RR' \pmod{D} \). This implies that the rem... | 80 | Okay, let me try to work through this problem step by step. So, we have two natural numbers, P and P', where P is greater than P'. When we divide both by a natural number D, they leave remainders R and R', respectively. Then, when we multiply P by P' to get PP', we divide that by D and get a remainder r. Similarly, whe... |
Return your final response within \boxed{}. The sides of a triangle have lengths $6.5$, $10$, and $s$, where $s$ is a whole number. What is the smallest possible value of $s$?
$\text{(A)}\ 3 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 7$ | Okay, so I need to find the smallest possible whole number value for the third side of a triangle where the other two sides are 6.5 and 10. The answer choices are from 3 to 7, so I guess the answer is one of those. Let me remember the triangle inequality theorem. I think it states that the sum of any two sides of a tri... |
To determine the smallest possible whole number value \( s \) for the third side of a triangle with sides 6.5 and 10, we apply the triangle inequality theorem. The theorem states that the sum of any two sides must be greater than the third side.
We need to check the following inequalities:
1. \( 6.5 + 10 > s \)
2. ... | 81 | Okay, let me try to figure out this problem. So, the question is about finding the smallest possible value of the third side of a triangle when two sides are given as 6.5 and 10, and the third side is a whole number. The answer choices are from 3 to 7, so I need to check which one is the smallest possible whole number ... |
Return your final response within \boxed{}. One proposal for new postage rates for a letter was $30$ cents for the first ounce and $22$ cents for each additional ounce (or fraction of an ounce). The postage for a letter weighing $4.5$ ounces was
$\text{(A)}\ \text{96 cents} \qquad \text{(B)}\ \text{1.07 dollars} \qqua... | Okay, let's see here. The problem is about calculating the postage cost for a letter that weighs 4.5 ounces. The options are given, and I need to figure out which one is correct. The pricing structure is 30 cents for the first ounce and 22 cents for each additional ounce or fraction of an ounce. Hmm, so fractions of an... |
The problem involves calculating the postage cost for a letter weighing 4.5 ounces with a pricing structure of 30 cents for the first ounce and 22 cents for each additional ounce or fraction thereof.
1. **Initial Analysis**: The first ounce costs 30 cents. The remaining weight after the first ounce is 4.5 - 1 = 3.5... | 82 | Okay, let me try to figure out this postage rate problem. So, the question is about calculating the postage for a letter that weighs 4.5 ounces. The proposal says 30 cents for the first ounce and 22 cents for each additional ounce or fraction. The options are given from A to E, with different amounts. Hmm, let's break ... |
Return your final response within \boxed{}. Let $(a_n)$ and $(b_n)$ be the sequences of real numbers such that
\[ (2 + i)^n = a_n + b_ni \]for all integers $n\geq 0$, where $i = \sqrt{-1}$. What is\[\sum_{n=0}^\infty\frac{a_nb_n}{7^n}\,?\]
$\textbf{(A) }\frac 38\qquad\textbf{(B) }\frac7{16}\qquad\textbf{(C) }\frac12\qq... | Okay, let me try to figure out this problem. So we have two sequences, a_n and b_n, defined by the equation (2 + i)^n = a_n + b_n i for all integers n ≥ 0. We need to find the sum from n=0 to infinity of (a_n b_n)/7^n. The answer choices are fractions, so maybe it simplifies to a nice fraction. Let's break it down step... |
Given the sequences of real numbers \((a_n)\) and \((b_n)\) defined by \((2 + i)^n = a_n + b_n i\) for all integers \(n \geq 0\), we need to find the sum \(\sum_{n=0}^\infty \frac{a_n b_n}{7^n}\).
First, we convert the complex number \(2 + i\) to polar form. The modulus is \(\sqrt{5}\) and the argument is \(\theta =... | 83 | Okay, let me try to work through this problem step by step. So, we have two sequences (a_n) and (b_n) such that (2 + i)^n = a_n + b_n i for all integers n ≥ 0. We need to find the sum S = Σ_{n=0}^∞ a_n b_n / 7^n. The answer choices are given, so maybe after working through it, I can check which one matches.
First, let... |
Return your final response within \boxed{}. An $11\times 11\times 11$ wooden cube is formed by gluing together $11^3$ unit cubes. What is the greatest number of unit cubes that can be seen from a single point?
$\text{(A) 328} \quad \text{(B) 329} \quad \text{(C) 330} \quad \text{(D) 331} \quad \text{(E) 332}$ | Okay, so I have this problem here: an 11×11×11 wooden cube made up of 11³ unit cubes. The question is asking for the greatest number of unit cubes that can be seen from a single point. The answer choices are from 328 to 332, so the answer is probably one of those. Let me think about how to approach this.
First, when y... |
To determine the greatest number of unit cubes that can be seen from a single point on an \(11 \times 11 \times 11\) wooden cube, we need to consider the visibility of the cubes from a corner viewpoint. When viewed from a corner, three adjacent faces of the cube are visible. However, some cubes are counted multiple t... | 84 | Okay, let me try to figure out this problem. So, we have an 11×11×11 cube made up of 11³ unit cubes. The question is asking for the greatest number of unit cubes that can be seen from a single point. The options are from 328 to 332, with 11×11×11 being 1331. Hmm, so the answer is pretty close to that, but not exactly. ... |
Return your final response within \boxed{}. If each of two intersecting lines intersects a hyperbola and neither line is tangent to the hyperbola,
then the possible number of points of intersection with the hyperbola is:
$\textbf{(A)}\ 2\qquad \textbf{(B)}\ 2\text{ or }3\qquad \textbf{(C)}\ 2\text{ or }4\qquad \textb... | Okay, so I need to figure out the possible number of points where two intersecting lines, neither of which is tangent to a hyperbola, can intersect the hyperbola. The options are given as A through E, with E being 2, 3, or 4. Let me start by recalling some properties about hyperbolas and lines intersecting them.
First... |
To determine the possible number of points of intersection between two intersecting lines and a hyperbola, where neither line is tangent to the hyperbola, we need to consider the following:
1. **Hyperbola and Line Intersections**: A line that is not tangent to a hyperbola can intersect it in 0, 1, or 2 points. Howev... | 85 | Okay, let me try to figure out this problem. So, the question is about two intersecting lines that both intersect a hyperbola, and neither line is tangent to the hyperbola. We need to determine the possible number of points where these lines intersect the hyperbola. The options are from A to E, with E being 2, 3, or 4.... |
Return your final response within \boxed{}. Are there integers $a$ and $b$ such that $a^5b+3$ and $ab^5+3$ are both perfect cubes of integers? | Okay, so I need to figure out if there are integers a and b such that both a^5b + 3 and ab^5 + 3 are perfect cubes. Hmm, interesting problem. Let me start by understanding what the question is asking. We have two expressions: a^5b + 3 and ab^5 + 3. Both of these need to be perfect cubes. That means there exist some int... |
After a thorough exploration, we analyze the problem using modular arithmetic and factorization techniques.
1. **Modulo 9 Analysis**: Perfect cubes modulo 9 can only be 0, 1, or 8. For \(a^5b + 3\) and \(ab^5 + 3\) to be cubes, they must align with these residues. Through detailed modular computations, we find cont... | 86 | Okay, so I need to figure out if there are integers a and b such that both a⁵b + 3 and ab⁵ + 3 are perfect cubes. Hmm, let's start by understanding the problem. We need to find integers a and b where each of these expressions results in a cube number.
First, let's recall that a perfect cube is an integer k such that ... |
Return your final response within \boxed{}. If $\frac {1}{x} - \frac {1}{y} = \frac {1}{z}$, then $z$ equals:
$\textbf{(A)}\ y - x\qquad \textbf{(B)}\ x - y\qquad \textbf{(C)}\ \frac {y - x}{xy}\qquad \textbf{(D)}\ \frac {xy}{y - x}\qquad \textbf{(E)}\ \frac {xy}{x - y}$ | Okay, let's see here. The problem is asking to solve for z given the equation 1/x minus 1/y equals 1/z. The answer choices are labeled from A to E, and I need to figure out which one is correct. Alright, let's start by writing down the equation again to make sure I have it right:
1/x - 1/y = 1/z
Hmm, so we need to so... |
To solve for \( z \) given the equation \( \frac{1}{x} - \frac{1}{y} = \frac{1}{z} \):
1. **Combine the fractions on the left side**:
\[
\frac{1}{x} - \frac{1}{y} = \frac{y - x}{xy}
\]
This is achieved by finding a common denominator \( xy \), resulting in \( \frac{y - x}{xy} \).
2. **Set the combined f... | 87 | Okay, let me try to solve this problem step by step. The question is: If 1/x minus 1/y equals 1/z, then what does z equal? The options are given from A to E. Hmm, let's see.
First, I need to rearrange the equation to solve for z. The original equation is 1/x - 1/y = 1/z. My goal is to isolate z on one side. So, I'll s... |
Return your final response within \boxed{}. Ashley, Betty, Carlos, Dick, and Elgin went shopping. Each had a whole number of dollars to spend, and together they had $56$ dollars. The absolute difference between the amounts Ashley and Betty had to spend was $19$ dollars. The absolute difference between the amounts Betty... | Okay, let's see. There are five people: Ashley, Betty, Carlos, Dick, and Elgin. Each has a whole number of dollars, and the total they all have together is 56. The problem gives the absolute differences between each pair in this order: Ashley and Betty is 19, Betty and Carlos is 7, Carlos and Dick is 5, Dick and Elgin ... |
Let \( A, B, C, D, \) and \( E \) represent the amounts of money Ashley, Betty, Carlos, Dick, and Elgin had, respectively. We know the following:
1. \( A + B + C + D + E = 56 \)
2. \( |A - B| = 19 \)
3. \( |B - C| = 7 \)
4. \( |C - D| = 5 \)
5. \( |D - E| = 4 \)
6. \( |E - A| = 11 \)
We need to find the value of \(... | 88 | Okay, let me try to solve this problem step by step. It seems like a system of equations problem with absolute values, and we need to find how much Elgin had. Let me start by listing out all the given information.
First, there are five people: Ashley (A), Betty (B), Carlos (C), Dick (D), and Elgin (E). Each has a whol... |
Return your final response within \boxed{}. A palindrome between $1000$ and $10,000$ is chosen at random. What is the probability that it is divisible by $7$?
$\textbf{(A)}\ \dfrac{1}{10} \qquad \textbf{(B)}\ \dfrac{1}{9} \qquad \textbf{(C)}\ \dfrac{1}{7} \qquad \textbf{(D)}\ \dfrac{1}{6} \qquad \textbf{(E)}\ \dfrac{1}... | Okay, so I need to find the probability that a randomly chosen palindrome between 1000 and 10,000 is divisible by 7. Hmm. Let me break this down step by step.
First, let's recall what a palindrome is. A palindrome is a number that reads the same forwards and backwards. Since we're dealing with four-digit numbers (bet... |
To find the probability that a randomly chosen palindrome between 1000 and 10,000 is divisible by 7, we start by noting that a four-digit palindrome has the form ABBA. This can be expressed numerically as \(1001A + 110B\), where \(A\) and \(B\) are digits with \(A\) ranging from 1 to 9 and \(B\) ranging from 0 to 9.
... | 89 | Okay, let me try to figure out this probability problem. So, we need to find the probability that a randomly chosen palindrome between 1000 and 10,000 is divisible by 7. The answer choices are fractions, so I guess the answer will be one of those. Let me start by understanding the problem step by step.
First, let's re... |
Return your final response within \boxed{}. Let $P$ be an interior point of circle $K$ other than the center of $K$. Form all chords of $K$ which pass through $P$, and determine their midpoints. The locus of these midpoints is
$\textbf{(A)} \text{ a circle with one point deleted} \qquad \\ \textbf{(B)} \text{ a circle... | Okay, let's tackle this geometry problem. So, we have a circle K with an interior point P that's not the center. We need to consider all the chords passing through P and find the locus of their midpoints. The answer choices suggest it could be a circle, a semicircle, or some variation with points deleted. Let me think ... |
To determine the locus of the midpoints of all chords passing through an interior point \( P \) (other than the center) of a circle \( K \), we start by setting up a coordinate system with the center \( O \) of the circle at the origin \((0,0)\) and the radius of the circle as \( R \). Let point \( P \) be at \((h, 0... | 90 | Okay, let me try to figure out this geometry problem. So, we have a circle K, and an interior point P that's not the center. We need to consider all chords of K that pass through P, find their midpoints, and determine the locus of these midpoints. The answer choices suggest it's a circle, semicircle, or something with ... |
Return your final response within \boxed{}. What is the greatest three-digit positive integer $n$ for which the sum of the first $n$ positive integers is $\underline{not}$ a divisor of the product of the first $n$ positive integers?
$\textbf{(A) } 995 \qquad\textbf{(B) } 996 \qquad\textbf{(C) } 997 \qquad\textbf{(D) } ... | Okay, so I need to find the greatest three-digit number n where the sum of the first n positive integers (which is n(n+1)/2) is not a divisor of the product of the first n positive integers (which is n!). The options are from 995 to 999, so it's one of those. Let me think step by step.
First, I remember that the sum o... |
To determine the greatest three-digit positive integer \( n \) such that the sum of the first \( n \) positive integers is not a divisor of the product of the first \( n \) positive integers, we need to check when \( \frac{n(n+1)}{2} \) does not divide \( n! \). This is equivalent to checking when \( n+1 \) does not ... | 91 | Okay, let me try to figure out this problem. So, the question is asking for the greatest three-digit positive integer n such that the sum of the first n positive integers is NOT a divisor of the product of the first n positive integers. The options are from 995 up to 999, so I need to check these numbers and see which ... |
Return your final response within \boxed{}. In $\triangle ABC$ with right angle at $C$, altitude $CH$ and median $CM$ trisect the right angle. If the area of $\triangle CHM$ is $K$, then the area of $\triangle ABC$ is
$\textbf{(A)}\ 6K\qquad\textbf{(B)}\ 4\sqrt3\ K\qquad\textbf{(C)}\ 3\sqrt3\ K\qquad\textbf{(D)}\ 3K... | Okay, let's tackle this geometry problem step by step. So, we have a right triangle ABC with a right angle at C. The altitude CH and the median CM trisect the right angle. The area of triangle CHM is K, and we need to find the area of triangle ABC in terms of K. The answer choices are given, so that might help later.
... |
Given a right triangle \( \triangle ABC \) with a right angle at \( C \), the altitude \( CH \) and the median \( CM \) trisect the right angle. The area of \( \triangle CHM \) is \( K \), and we need to find the area of \( \triangle ABC \).
1. **Trisection of the Right Angle**:
- The right angle at \( C \) is tr... | 92 | Okay, let me try to work through this geometry problem. So, we have a right triangle ABC with a right angle at C. The altitude CH and the median CM trisect the right angle. That means the right angle is split into three equal angles by CH and CM. Since the right angle is 90 degrees, each of these angles would be 30 deg... |
Return your final response within \boxed{}. Claire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of $2017$. She then discovers that she forgot to include one angle. What is the degree measure of the forgotten angle?
$\textbf{(A)}\ 37\qquad\textbf{(B)}\ 63\qquad\textbf{(C)}\ 11... | Okay, let me try to figure out this problem step by step. So, Claire was adding the degree measures of the interior angles of a convex polygon and got a sum of 2017. But then she realized she forgot one angle. We need to find the measure of that forgotten angle. The answer choices are 37, 63, 117, 143, and 163. Alright... |
Claire calculated the sum of the interior angles of a convex polygon as 2017 degrees, but she missed one angle. The formula for the sum of the interior angles of a convex polygon with \( n \) sides is \((n-2) \times 180\) degrees. Let \( x \) be the measure of the forgotten angle. The correct sum of the interior angl... | 93 | Okay, let me try to figure out this problem. So, Claire added the degree measures of the interior angles of a convex polygon and got a sum of 2017. But she forgot to include one angle, and we need to find the measure of that forgotten angle. The answer choices are 37, 63, 117, 143, and 163. Alright, let's start by reca... |
Return your final response within \boxed{}. Zara has a collection of $4$ marbles: an Aggie, a Bumblebee, a Steelie, and a Tiger. She wants to display them in a row on a shelf, but does not want to put the Steelie and the Tiger next to one another. In how many ways can she do this?
$\textbf{(A) }6 \qquad \textbf{(B) }8 ... | Okay, so Zara has four marbles: Aggie, Bumblebee, Steelie, and Tiger. She wants to arrange them in a row on a shelf. But there's a condition: she doesn't want the Steelie and the Tiger to be next to each other. I need to find out how many different ways she can do this. The answer choices are 6, 8, 12, 18, or 24. Let m... |
Zara has four marbles: Aggie (A), Bumblebee (B), Steelie (S), and Tiger (T). We need to find the number of ways to arrange these marbles such that Steelie and Tiger are not next to each other.
First, calculate the total number of permutations without any restrictions:
\[
4! = 24
\]
Next, calculate the number of per... | 94 | Okay, let me try to figure out how to solve this problem. So, Zara has four marbles: Aggie, Bumblebee, Steelie, and Tiger. She wants to display them in a row on a shelf, but she doesn't want the Steelie and the Tiger next to each other. The question is asking how many ways she can arrange them without those two being a... |
Return your final response within \boxed{}. Ray's car averages $40$ miles per gallon of gasoline, and Tom's car averages $10$ miles per gallon of gasoline. Ray and Tom each drive the same number of miles. What is the cars' combined rate of miles per gallon of gasoline?
$\textbf{(A)}\ 10 \qquad \textbf{(B)}\ 16 \qquad \... | Okay, let's see. The problem is about Ray and Tom's cars and their mileage. The question is asking for the combined rate of miles per gallon for both cars when they each drive the same number of miles. The answer choices are given, so I need to figure out which one is correct.
First, let me parse the information. Ray'... |
Ray's car averages 40 miles per gallon, and Tom's car averages 10 miles per gallon. Both drive the same number of miles. To find the combined rate of miles per gallon, we need to calculate the total miles driven and the total gallons used.
Let \( d \) be the distance each drives in miles.
- Gasoline used by Ray: \(... | 95 | Okay, let me try to figure out this problem. So, Ray's car gets 40 miles per gallon, and Tom's car gets 10 miles per gallon. Both of them drive the same number of miles. The question is asking for the combined rate of miles per gallon for both cars. Hmm, combined rate... I need to think about how to combine their fuel ... |
Return your final response within \boxed{}. Triangle $ABC$ has vertices $A = (3,0)$, $B = (0,3)$, and $C$, where $C$ is on the line $x + y = 7$. What is the area of $\triangle ABC$?
$\mathrm{(A)}\ 6\qquad \mathrm{(B)}\ 8\qquad \mathrm{(C)}\ 10\qquad \mathrm{(D)}\ 12\qquad \mathrm{(E)}\ 14$ | Okay, so I need to find the area of triangle ABC where points A and B are given, and point C lies on the line x + y = 7. Let me start by visualizing the problem.
First, let's recall the coordinates of the vertices. Point A is at (3, 0), and point B is at (0, 3). Point C is somewhere on the line x + y = 7. Since the l... |
To find the area of triangle \(ABC\) with vertices \(A = (3,0)\), \(B = (0,3)\), and \(C\) on the line \(x + y = 7\), we use the shoelace formula.
The coordinates of \(C\) satisfy \(x + y = 7\). Using the shoelace formula for the area of a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\):
... | 96 | Okay, let me try to figure out this problem. So, we have triangle ABC with vertices A at (3,0), B at (0,3), and point C somewhere on the line x + y = 7. We need to find the area of triangle ABC. The answer choices are from 6 to 14, so the area isn't going to be too big. Let me start by recalling how to calculate the ar... |
Return your final response within \boxed{}. A team won $40$ of its first $50$ games. How many of the remaining $40$ games must this team win so it will have won exactly $70 \%$ of its games for the season?
$\text{(A)}\ 20 \qquad \text{(B)}\ 23 \qquad \text{(C)}\ 28 \qquad \text{(D)}\ 30 \qquad \text{(E)}\ 35$ | Okay, let's see. The problem is about a team that has won 40 out of their first 50 games. They want to know how many more games they need to win out of the remaining 40 games to have a total of 70% wins for the entire season. The answer choices are given from A to E. Hmm, let me break this down step by step.
First, le... |
The team has won 40 out of its first 50 games and has 40 remaining games. The total number of games in the season is \(50 + 40 = 90\). To determine how many games they need to win to achieve exactly 70% of the season games, we calculate 70% of 90:
\[
0.7 \times 90 = 63
\]
They need a total of 63 wins. Since they al... | 97 | Okay, let me try to figure out this problem. So, the question is about a team that has already played 50 games and won 40 of them. They have 40 more games left in the season. The question is asking how many of those remaining 40 games they need to win in order to have a total of 70% wins for the entire season. The opti... |
Return your final response within \boxed{}. For each real number $x$, let $\textbf{[}x\textbf{]}$ be the largest integer not exceeding $x$
(i.e., the integer $n$ such that $n\le x<n+1$). Which of the following statements is (are) true?
$\textbf{I. [}x+1\textbf{]}=\textbf{[}x\textbf{]}+1\text{ for all }x \\ \textbf{II.... | Okay, let's tackle this problem step by step. It's about the floor function, which is denoted here as [x]. The floor function, as I remember, gives the greatest integer less than or equal to x. So, for example, [3.7] would be 3, and [-1.2] would be -2. The question has three statements, I, II, and III, and we need to d... |
To determine which statements about the floor function are true, we analyze each statement systematically:
**Statement I: \([x + 1] = [x] + 1\) for all \(x\)**
- For any real number \(x\), adding 1 to \(x\) increments the integer part of \(x\) by 1 while the fractional part remains the same.
- Testing with various... | 98 | Okay, let me try to work through this problem. So, the question is about the floor function, which is the greatest integer less than or equal to x. The floor function, denoted as [x], gives the largest integer not exceeding x. The statements I, II, and III are about properties of the floor function when applied to sums... |
Return your final response within \boxed{}. Positive integers $a$ and $b$ are each less than $6$. What is the smallest possible value for $2 \cdot a - a \cdot b$?
$\textbf{(A)}\ -20\qquad\textbf{{(B)}}\ -15\qquad\textbf{{(C)}}\ -10\qquad\textbf{{(D)}}\ 0\qquad\textbf{{(E)}}\ 2$ | Okay, so I have this problem here: Positive integers a and b are each less than 6. What is the smallest possible value for 2·a - a·b? The options are from A to E, with the answers given. Let me try to figure this out step by step.
First, let me make sure I understand the problem correctly. We need to find the smallest... |
To find the smallest possible value of \(2a - ab\) where \(a\) and \(b\) are positive integers less than 6, we start by rewriting the expression as \(a(2 - b)\).
We analyze the possible values of \(a\) and \(b\) which can be 1, 2, 3, 4, or 5. The term \(2 - b\) can take values 1, 0, -1, -2, or -3 depending on \(b\)... | 99 | Okay, let me try to figure out this problem. So, we need to find the smallest possible value for the expression 2a - ab, where a and b are positive integers each less than 6. The options given are from A to E, with the smallest one being -20. Hmm, but let me make sure I understand the problem correctly.
First, the que... |
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