problem stringlengths 14 1.99k | prompt stringlengths 14 1.99k | question stringlengths 14 1.99k | answer stringlengths 1 30 | ground_truth stringlengths 1 30 | solution stringlengths 0 8.23k | source stringclasses 6
values |
|---|---|---|---|---|---|---|
Lightning McQueen, the race car, cost 140000$. Mater only cost 10 percent of that, and Sally McQueen cost triple what Mater costs. How much does Sally McQueen cost? | Lightning McQueen, the race car, cost 140000$. Mater only cost 10 percent of that, and Sally McQueen cost triple what Mater costs. How much does Sally McQueen cost? | Lightning McQueen, the race car, cost 140000$. Mater only cost 10 percent of that, and Sally McQueen cost triple what Mater costs. How much does Sally McQueen cost? | 42000 | 42000 | Mater:140000(.10)=14000$
Sally:14000(3)=42000$
#### 42000 | gsm8k |
Markus is twice the age of his son, and Markus's son is twice the age of Markus's grandson. If the sum of the ages of Markus, his son, and his grandson is 140 years, then how many years old is Markus's grandson? | Markus is twice the age of his son, and Markus's son is twice the age of Markus's grandson. If the sum of the ages of Markus, his son, and his grandson is 140 years, then how many years old is Markus's grandson? | Markus is twice the age of his son, and Markus's son is twice the age of Markus's grandson. If the sum of the ages of Markus, his son, and his grandson is 140 years, then how many years old is Markus's grandson? | 20 | 20 | Let "x" be the age of Markus's grandson.
If Markus's son is twice the age of Markus's grandson, then Markus's son is 2*x.
If Markus is twice the age of his son, then Markus is 2*2*x.
Therefore, if the sum of the ages of Markus, his son, and his grandson is 140 years, then x+(2*x)+(2*2*x)=140 years.
Simplifying the equa... | gsm8k |
Going into the final game, Duke is so close to breaking the school's record for most points scored in a basketball season. He only needs 17 more points to tie the record. By the end of the game, Duke breaks the record by 5 points. The old record was 257 points. In the final game Duke made 5 free throws (worth one point... | Going into the final game, Duke is so close to breaking the school's record for most points scored in a basketball season. He only needs 17 more points to tie the record. By the end of the game, Duke breaks the record by 5 points. The old record was 257 points. In the final game Duke made 5 free throws (worth one point... | Going into the final game, Duke is so close to breaking the school's record for most points scored in a basketball season. He only needs 17 more points to tie the record. By the end of the game, Duke breaks the record by 5 points. The old record was 257 points. In the final game Duke made 5 free throws (worth one point... | 1 | 1 | Duke scored 22 points in the final game because 17 + 5 = <<17+5=22>>22
He scored 5 points of free-throws, because 5 x 1 = <<5*1=5>>5
He scored 8 points of regular baskets because 4 x 2 = <<4*2=8>>8
He made 9 points of three-pointers because 22 - 5 - 8 = <<22-5-8=9>>9
He scored 3 three-pointers in the final game because... | gsm8k |
As shown in the figure below, a regular dodecahedron (the polyhedron consisting of $12$ congruent regular pentagonal faces) floats in space with two horizontal faces. Note that there is a ring of five slanted faces adjacent to the top face, and a ring of five slanted faces adjacent to the bottom face. How many ways are... | As shown in the figure below, a regular dodecahedron (the polyhedron consisting of $12$ congruent regular pentagonal faces) floats in space with two horizontal faces. Note that there is a ring of five slanted faces adjacent to the top face, and a ring of five slanted faces adjacent to the bottom face. How many ways are... | As shown in the figure below, a regular dodecahedron (the polyhedron consisting of $12$ congruent regular pentagonal faces) floats in space with two horizontal faces. Note that there is a ring of five slanted faces adjacent to the top face, and a ring of five slanted faces adjacent to the bottom face. How many ways are... | 810 | 810 | To solve this problem, we need to consider the number of ways to move from the top face to the bottom face via a sequence of adjacent faces, visiting each face at most once, and without moving from the bottom ring to the top ring. We will use casework based on the number of moves made on the top ring of five slanted fa... | deepscaler |
Find the remainder when $x^4 +x + 2$ is divided by $x-3$. | Find the remainder when $x^4 +x + 2$ is divided by $x-3$. | Find the remainder when $x^4 +x + 2$ is divided by $x-3$. | 86 | 86 | To find the remainder when $x^4 + x + 2$ is divided by $x - 3$, we can use two approaches: polynomial division and the remainder theorem.
**Polynomial Division Approach:**
1. Divide $x^4$ by $x$ to get $x^3$. Multiply $x^3$ by $x - 3$ to get $x^4 - 3x^3$.
2. Subtract $x^4 - 3x^3$ from $x^4 + 0x^3 + 0x^2 + x + 2$ to e... | math |
The epic poem currently contains 24 lines. If a person adds 3 lines to the poem every month, in how many months will the poem contain 90 lines? | The epic poem currently contains 24 lines. If a person adds 3 lines to the poem every month, in how many months will the poem contain 90 lines? | The epic poem currently contains 24 lines. If a person adds 3 lines to the poem every month, in how many months will the poem contain 90 lines? | 22 | 22 | If the epic poem currently contains 24 lines, the total number of lines added for it to reach 90 is 90-24 = 66 lines.
If a person adds 3 lines to the poem every month, it will take 66/3= 22 months for the poem to contain 90 lines.
#### 22 | gsm8k |
I have 5 red plates and 4 blue plates. If I randomly select two plates to serve dinner on, what is the probability that they're both the same color? | I have 5 red plates and 4 blue plates. If I randomly select two plates to serve dinner on, what is the probability that they're both the same color? | I have 5 red plates and 4 blue plates. If I randomly select two plates to serve dinner on, what is the probability that they're both the same color? | \frac{4}{9} | \frac{4}{9} | To solve this problem, we need to calculate the probability that two plates selected at random are of the same color. We can break this down into a few steps:
1. **Calculate the total number of ways to select 2 plates out of 9**:
The total number of ways to select 2 plates out of 9 (regardless of color) can be calc... | math |
The average temperature in Orlando in a particular week was 60 degrees. If the temperature on each of the first 3 days in that week was 40, and the temperature for Thursday and Friday was 80 degrees each, calculate the total temperature of the remaining days of that week. | The average temperature in Orlando in a particular week was 60 degrees. If the temperature on each of the first 3 days in that week was 40, and the temperature for Thursday and Friday was 80 degrees each, calculate the total temperature of the remaining days of that week. | The average temperature in Orlando in a particular week was 60 degrees. If the temperature on each of the first 3 days in that week was 40, and the temperature for Thursday and Friday was 80 degrees each, calculate the total temperature of the remaining days of that week. | 140 | 140 | If the average temperature of Orlando for the week was 60, then the total temperature for the week was 7*60 = <<7*60=420>>420 degrees
For the first three days of that week, the temperature was 40 degrees each day, totaling 3*40 = <<3*40=120>>120 degrees in the three days.
The temperature for Thursday and Friday was 80 ... | gsm8k |
For how many different values of integer $n$, one can find $n$ different lines in the plane such that each line intersects with exactly 2004 of other lines? | For how many different values of integer $n$, one can find $n$ different lines in the plane such that each line intersects with exactly 2004 of other lines? | For how many different values of integer $n$, one can find $n$ different lines in the plane such that each line intersects with exactly 2004 of other lines? | 12 | 12 | deepscaler | |
Two different natural numbers are selected from the set $\{1, 2, 3, \ldots, 8\}$. What is the probability that the greatest common factor (GCF) of these two numbers is one? Express your answer as a common fraction. | Two different natural numbers are selected from the set $\{1, 2, 3, \ldots, 8\}$. What is the probability that the greatest common factor (GCF) of these two numbers is one? Express your answer as a common fraction. | Two different natural numbers are selected from the set $\{1, 2, 3, \ldots, 8\}$. What is the probability that the greatest common factor (GCF) of these two numbers is one? Express your answer as a common fraction. | \frac{3}{4} | \frac{3}{4} | deepscaler | |
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://artofproblemsolving.com/wiki/index.php/Point) $... | 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://artofproblemsolving.com/wiki/index.php/Point) $... | 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://artofproblemsolving.com/wiki/index.php/Point) $... | 737 | 737 |
1. **Identify the triangle and its properties**:
Triangle $ABC$ is a right triangle with $CA = 30$, $CB = 16$, and right angle at $C$. Using the Pythagorean theorem, we find the hypotenuse $AB$:
\[
AB = \sqrt{CA^2 + CB^2} = \sqrt{30^2 + 16^2} = \sqrt{900 + 256} = \sqrt{1156} = 34.
\]
2. **Understanding t... | amc_aime |
Solve the inequality
$$
\sqrt{3 x-7}-\sqrt{3 x^{2}-13 x+13} \geqslant 3 x^{2}-16 x+20
$$
In the answer, indicate the sum of all integer values \( x \) that satisfy the inequality. | Solve the inequality
$$
\sqrt{3 x-7}-\sqrt{3 x^{2}-13 x+13} \geqslant 3 x^{2}-16 x+20
$$
In the answer, indicate the sum of all integer values \( x \) that satisfy the inequality. | Solve the inequality
$$
\sqrt{3 x-7}-\sqrt{3 x^{2}-13 x+13} \geqslant 3 x^{2}-16 x+20
$$
In the answer, indicate the sum of all integer values \( x \) that satisfy the inequality. | 3 | 3 |
Consider the inequality:
\[ \sqrt{3x - 7} - \sqrt{3x^2 - 13x + 13} \geq 3x^2 - 16x + 20 \]
We will start by making substitutions to simplify the inequality. Let
\[ v = \sqrt{3x - 7} \]
\[ u = \sqrt{3x^2 - 13x + 13} \]
Then, the inequality can be rewritten as:
\[ \sqrt{3x - 7} - \sqrt{3x^2 - 13x + 13} \geq 3x^2 - ... | olympiads |
John is walking east at a speed of 3 miles per hour, while Bob is also walking east, but at a speed of 5 miles per hour. If Bob is now 1 mile west of John, how many minutes will it take for Bob to catch up to John?
$\text {(A) } 30 \qquad \text {(B) } 50 \qquad \text {(C) } 60 \qquad \text {(D) } 90 \qquad \text {(E) ... | John is walking east at a speed of 3 miles per hour, while Bob is also walking east, but at a speed of 5 miles per hour. If Bob is now 1 mile west of John, how many minutes will it take for Bob to catch up to John?
$\text {(A) } 30 \qquad \text {(B) } 50 \qquad \text {(C) } 60 \qquad \text {(D) } 90 \qquad \text {(E) ... | John is walking east at a speed of 3 miles per hour, while Bob is also walking east, but at a speed of 5 miles per hour. If Bob is now 1 mile west of John, how many minutes will it take for Bob to catch up to John?
$\text {(A) } 30 \qquad \text {(B) } 50 \qquad \text {(C) } 60 \qquad \text {(D) } 90 \qquad \text {(E) ... | \text{(A) } 30 | \text{(A) } 30 | 1. **Determine the relative speed of Bob with respect to John**: Since both are moving east, we subtract John's speed from Bob's speed to find the rate at which the distance between them is closing.
\[
\text{Relative speed} = 5 \text{ mph (Bob's speed)} - 3 \text{ mph (John's speed)} = 2 \text{ mph}
\]
2. **C... | amc_aime |
Yvan and Zoé play the following game. Let \( n \in \mathbb{N} \). The integers from 1 to \( n \) are written on \( n \) cards arranged in order. Yvan removes one card. Then, Zoé removes 2 consecutive cards. Next, Yvan removes 3 consecutive cards. Finally, Zoé removes 4 consecutive cards.
What is the smallest value of ... | Yvan and Zoé play the following game. Let \( n \in \mathbb{N} \). The integers from 1 to \( n \) are written on \( n \) cards arranged in order. Yvan removes one card. Then, Zoé removes 2 consecutive cards. Next, Yvan removes 3 consecutive cards. Finally, Zoé removes 4 consecutive cards.
What is the smallest value of ... | Yvan and Zoé play the following game. Let \( n \in \mathbb{N} \). The integers from 1 to \( n \) are written on \( n \) cards arranged in order. Yvan removes one card. Then, Zoé removes 2 consecutive cards. Next, Yvan removes 3 consecutive cards. Finally, Zoé removes 4 consecutive cards.
What is the smallest value of ... | 14 | 14 | deepscaler | |
A restaurant buffet has 36 different dishes available to try. The restaurant features mango salsa on three of its dishes, fresh mangoes in a sixth of its dishes, and mango jelly in one dish. Oliver despises mangoes and won't eat them, but can pick them out of two of the dishes with fresh mango that he would be willing ... | A restaurant buffet has 36 different dishes available to try. The restaurant features mango salsa on three of its dishes, fresh mangoes in a sixth of its dishes, and mango jelly in one dish. Oliver despises mangoes and won't eat them, but can pick them out of two of the dishes with fresh mango that he would be willing ... | A restaurant buffet has 36 different dishes available to try. The restaurant features mango salsa on three of its dishes, fresh mangoes in a sixth of its dishes, and mango jelly in one dish. Oliver despises mangoes and won't eat them, but can pick them out of two of the dishes with fresh mango that he would be willing ... | 28 | 28 | There are fresh mangos in 36 / 6 = <<36/6=6>>6 dishes.
Thus, there are 3 + 6 + 1 = <<3+6+1=10>>10 dishes with mango on the buffet.
Oliver can pick the mango out of 2 of the dishes, so there are 10 - 2 = <<10-2=8>>8 dishes he won't eat.
Thus, there are 36 - 8 = <<36-8=28>>28 dishes left for Oliver on the buffet.
#### 28 | gsm8k |
To make pizza dough, Luca mixes 50 mL of milk for every 250 mL of flour. How many mL of milk does he mix with 750 mL of flour? | To make pizza dough, Luca mixes 50 mL of milk for every 250 mL of flour. How many mL of milk does he mix with 750 mL of flour? | To make pizza dough, Luca mixes 50 mL of milk for every 250 mL of flour. How many mL of milk does he mix with 750 mL of flour? | 150 | 150 | To solve the problem, we start by understanding the ratio of milk to flour that Luca uses for his pizza dough, which is 50 mL of milk for every 250 mL of flour.
Given that Luca wants to use 750 mL of flour, we need to find out how many times the base ratio (250 mL of flour) fits into 750 mL of flour. This is done by ... | math |
Stacy was 50 inches tall last year. If she grew 6 inches more than her brother who grew 1 inch last year, how tall is Stacy now? | Stacy was 50 inches tall last year. If she grew 6 inches more than her brother who grew 1 inch last year, how tall is Stacy now? | Stacy was 50 inches tall last year. If she grew 6 inches more than her brother who grew 1 inch last year, how tall is Stacy now? | 57 | 57 | Stacy grew 6+1=<<6+1=7>>7 inches.
Stacy is 50+7=<<50+7=57>>57 inches tall.
#### 57 | gsm8k |
Assume integers \( u \) and \( v \) satisfy \( 0 < v < u \), and let \( A \) be \((u, v)\). Points are defined as follows: \( B \) is the reflection of \( A \) over the line \( y = x \), \( C \) is the reflection of \( B \) over the \( y \)-axis, \( D \) is the reflection of \( C \) over the \( x \)-axis, and \( E \) i... | Assume integers \( u \) and \( v \) satisfy \( 0 < v < u \), and let \( A \) be \((u, v)\). Points are defined as follows: \( B \) is the reflection of \( A \) over the line \( y = x \), \( C \) is the reflection of \( B \) over the \( y \)-axis, \( D \) is the reflection of \( C \) over the \( x \)-axis, and \( E \) i... | Assume integers \( u \) and \( v \) satisfy \( 0 < v < u \), and let \( A \) be \((u, v)\). Points are defined as follows: \( B \) is the reflection of \( A \) over the line \( y = x \), \( C \) is the reflection of \( B \) over the \( y \)-axis, \( D \) is the reflection of \( C \) over the \( x \)-axis, and \( E \) i... | 21 | 21 |
Given that the integers \( u \) and \( v \) satisfy \( 0 < v < u \), and the coordinates of point \( A \) is given by \( (u, v) \). We perform a series of reflections to determine the coordinates of points \( B, C, D, E \) as follows:
1. Point \( B \) is obtained by reflecting \( A \) across the line \( y = x \). Thi... | olympiads |
If $A=2+i$, $O=-4$, $P=-i$, and $S=2+4i$, find $A-O+P+S$. | If $A=2+i$, $O=-4$, $P=-i$, and $S=2+4i$, find $A-O+P+S$. | If $A=2+i$, $O=-4$, $P=-i$, and $S=2+4i$, find $A-O+P+S$. | 8 + 4i | 8 + 4i | To solve $A-O+P+S$, we break down the calculation into real and imaginary parts separately.
Given:
- $A = 2 + i$
- $O = -4$ (Note: $O$ is purely real, so its imaginary part is $0$)
- $P = -i$ (Note: $P$ is purely imaginary, so its real part is $0$)
- $S = 2 + 4i$
We calculate the sum of the real parts and the sum of ... | math |
A certain university needs $40L$ of helium gas to make balloon decorations for its centennial celebration. The chemistry club voluntarily took on this task. The club's equipment can produce a maximum of $8L$ of helium gas per day. According to the plan, the club must complete the production within 30 days. Upon receivi... | A certain university needs $40L$ of helium gas to make balloon decorations for its centennial celebration. The chemistry club voluntarily took on this task. The club's equipment can produce a maximum of $8L$ of helium gas per day. According to the plan, the club must complete the production within 30 days. Upon receivi... | A certain university needs $40L$ of helium gas to make balloon decorations for its centennial celebration. The chemistry club voluntarily took on this task. The club's equipment can produce a maximum of $8L$ of helium gas per day. According to the plan, the club must complete the production within 30 days. Upon receivi... | 4640 | 4640 | deepscaler | |
The quadratic polynomials \( f(x) \) and \( g(x) \) are such that
\[
\begin{aligned}
& f(2) \\
& g(2)
\end{aligned}=\begin{aligned}
& f(3) \\
& g(3)
\end{aligned}=2
\]
Find \( f(1) \), given that \( g(1) = 2 \), \( f(5) = 7 \), and \( g(5) = 2 \). | The quadratic polynomials \( f(x) \) and \( g(x) \) are such that
\[
\begin{aligned}
& f(2) \\
& g(2)
\end{aligned}=\begin{aligned}
& f(3) \\
& g(3)
\end{aligned}=2
\]
Find \( f(1) \), given that \( g(1) = 2 \), \( f(5) = 7 \), and \( g(5) = 2 \). | The quadratic polynomials \( f(x) \) and \( g(x) \) are such that
\[
\begin{aligned}
& f(2) \\
& g(2)
\end{aligned}=\begin{aligned}
& f(3) \\
& g(3)
\end{aligned}=2
\]
Find \( f(1) \), given that \( g(1) = 2 \), \( f(5) = 7 \), and \( g(5) = 2 \). | 5 | 5 |
1. We consider the quadratic polynomials \( f(x) \) and \( g(x) \) defined by the given equations:
\[
\begin{aligned}
& f(2) = f(3) = 2 \\
& g(2) = g(3) = 2
\end{aligned}
\]
2. Define a new polynomial \( h(x) \) as follows:
\[
h(x) = 2g(x) - f(x)
\]
Given the conditions, notice that:
... | olympiads |
The school plans to schedule six leaders to be on duty from May 1st to May 3rd, with each leader on duty for one day and two leaders scheduled each day. Given that Leader A cannot be on duty on May 2nd and Leader B cannot be on duty on May 3rd, determine the number of different ways to arrange the duty schedule. | The school plans to schedule six leaders to be on duty from May 1st to May 3rd, with each leader on duty for one day and two leaders scheduled each day. Given that Leader A cannot be on duty on May 2nd and Leader B cannot be on duty on May 3rd, determine the number of different ways to arrange the duty schedule. | The school plans to schedule six leaders to be on duty from May 1st to May 3rd, with each leader on duty for one day and two leaders scheduled each day. Given that Leader A cannot be on duty on May 2nd and Leader B cannot be on duty on May 3rd, determine the number of different ways to arrange the duty schedule. | 42 | 42 | deepscaler | |
Given the sets $A=\{x|x^2 - mx + m^2 - 19 = 0\}$, $B=\{x|x^2 - 5x + 6 = 0\}$, and $C=\{2, -4\}$. If $A \cap B \neq \emptyset$ and $A \cap C = \emptyset$, find the value of the real number $m$. | Given the sets $A=\{x|x^2 - mx + m^2 - 19 = 0\}$, $B=\{x|x^2 - 5x + 6 = 0\}$, and $C=\{2, -4\}$. If $A \cap B \neq \emptyset$ and $A \cap C = \emptyset$, find the value of the real number $m$. | Given the sets $A=\{x|x^2 - mx + m^2 - 19 = 0\}$, $B=\{x|x^2 - 5x + 6 = 0\}$, and $C=\{2, -4\}$. If $A \cap B \neq \emptyset$ and $A \cap C = \emptyset$, find the value of the real number $m$. | -2 | -2 | deepscaler | |
Mary sees three breeding balls with 8 snakes each and 6 additional pairs of snakes. How many snakes did she see total? | Mary sees three breeding balls with 8 snakes each and 6 additional pairs of snakes. How many snakes did she see total? | Mary sees three breeding balls with 8 snakes each and 6 additional pairs of snakes. How many snakes did she see total? | 36 | 36 | First find how many snakes were in all the breeding balls: 3 balls * 8 snakes/ball = <<3*8=24>>24 snakes
Then find how many snakes are in all the pairs: 2 snakes/pair * 6 pairs = <<2*6=12>>12 snakes
Then add the number of snakes in both types of groups to find the total number of snakes: 24 snakes + 12 snakes = <<24+12... | gsm8k |
The body moves in a straight line with acceleration \(a=6t-4\). At \(t=0\), the initial position \(s_{0}=0\) and the initial velocity \(v_{0}=4\). Find the velocity and the distance traveled as functions of time. | The body moves in a straight line with acceleration \(a=6t-4\). At \(t=0\), the initial position \(s_{0}=0\) and the initial velocity \(v_{0}=4\). Find the velocity and the distance traveled as functions of time. | The body moves in a straight line with acceleration \(a=6t-4\). At \(t=0\), the initial position \(s_{0}=0\) and the initial velocity \(v_{0}=4\). Find the velocity and the distance traveled as functions of time. | s(t) = t^3 - 2t^2 + 4t | s(t) = t^3 - 2t^2 + 4t |
1. The given acceleration function is \( a(t) = 6t - 4 \).
2. To find the velocity function \( v(t) \), we integrate the acceleration function with respect to time \( t \).
\[
v(t) = \int a(t) \, dt = \int (6t - 4) \, dt
\]
3. Performing the integration:
\[
v(t) = \int 6t \, dt - \int 4 \, dt
\]
\[
v(t) = 6 \int ... | olympiads |
Emma is planning a dinner party, so she went to a shop to buy the products she needs. She bought 8 kg of cheese and 7 kg of vegetables. One kilogram of cheese costs $4 and one kilogram of vegetable costs is $2 more expensive. How much did she pay for her shopping? | Emma is planning a dinner party, so she went to a shop to buy the products she needs. She bought 8 kg of cheese and 7 kg of vegetables. One kilogram of cheese costs $4 and one kilogram of vegetable costs is $2 more expensive. How much did she pay for her shopping? | Emma is planning a dinner party, so she went to a shop to buy the products she needs. She bought 8 kg of cheese and 7 kg of vegetables. One kilogram of cheese costs $4 and one kilogram of vegetable costs is $2 more expensive. How much did she pay for her shopping? | 74 | 74 | Emma payed 8 kg * $4/kg = $<<8*4=32>>32 for the cheese.
The cost of one kilogram of vegetable is $4 + $2 = $<<4+2=6>>6.
So Emma payed 7 kg * $6/kg = $<<7*6=42>>42 for vegetables.
In total Emma payed $32 + $42 = $<<32+42=74>>74 for her shopping.
#### 74 | gsm8k |
Let $\alpha$ and $\beta$ be conjugate complex numbers such that $\frac{\alpha}{\beta^2}$ is a real number and $|\alpha - \beta| = 2 \sqrt{3}.$ Find $|\alpha|.$ | Let $\alpha$ and $\beta$ be conjugate complex numbers such that $\frac{\alpha}{\beta^2}$ is a real number and $|\alpha - \beta| = 2 \sqrt{3}.$ Find $|\alpha|.$ | Let $\alpha$ and $\beta$ be conjugate complex numbers such that $\frac{\alpha}{\beta^2}$ is a real number and $|\alpha - \beta| = 2 \sqrt{3}.$ Find $|\alpha|.$ | 2 | 2 | Given $\alpha$ and $\beta$ are conjugate complex numbers, we have $\alpha = x + yi$ and $\beta = x - yi$. The condition $\frac{\alpha}{\beta^2}$ being a real number and $|\alpha - \beta| = 2 \sqrt{3}$ guides us through the solution.
1. **Determining $|y|$ from $|\alpha - \beta|$:**
From the given $|\alpha - \beta|... | math |
Find maximum value of number $a$ such that for any arrangement of numbers $1,2,\ldots ,10$ on a circle, we can find three consecutive numbers such their sum bigger or equal than $a$ . | Find maximum value of number $a$ such that for any arrangement of numbers $1,2,\ldots ,10$ on a circle, we can find three consecutive numbers such their sum bigger or equal than $a$ . | Find maximum value of number $a$ such that for any arrangement of numbers $1,2,\ldots ,10$ on a circle, we can find three consecutive numbers such their sum bigger or equal than $a$ . | 18 | 18 | 1. Define \( x_1, x_2, \ldots, x_{10} \) to be the numbers on the circle consecutively, with \( x_{11} = x_1 \) and \( x_{12} = x_2 \). Let \( y_k = x_k + x_{k+1} + x_{k+2} \) for \( 1 \le k \le 10 \).
2. Note that the sum of all numbers on the circle is:
\[
\sum_{i=1}^{10} x_i = 1 + 2 + \cdots + 10 = 55
\]
... | aops_forum |
Vasya, Petya, and Kolya are in the same class. Vasya always lies in response to any question, Petya alternates between lying and telling the truth, and Kolya lies in response to every third question but tells the truth otherwise. One day, each of them was asked six consecutive times how many students are in their class... | Vasya, Petya, and Kolya are in the same class. Vasya always lies in response to any question, Petya alternates between lying and telling the truth, and Kolya lies in response to every third question but tells the truth otherwise. One day, each of them was asked six consecutive times how many students are in their class... | Vasya, Petya, and Kolya are in the same class. Vasya always lies in response to any question, Petya alternates between lying and telling the truth, and Kolya lies in response to every third question but tells the truth otherwise. One day, each of them was asked six consecutive times how many students are in their class... | 27 | 27 | deepscaler | |
Chuck breeds dogs. He has 3 pregnant dogs. They each give birth to 4 puppies. Each puppy needs 2 shots and each shot costs $5. How much did the shots cost? | Chuck breeds dogs. He has 3 pregnant dogs. They each give birth to 4 puppies. Each puppy needs 2 shots and each shot costs $5. How much did the shots cost? | Chuck breeds dogs. He has 3 pregnant dogs. They each give birth to 4 puppies. Each puppy needs 2 shots and each shot costs $5. How much did the shots cost? | 120 | 120 | He has 3*4=<<3*4=12>>12 puppies
So they need 12*2=<<12*2=24>>24 shots
That means the vaccines cost 24*5=$<<24*5=120>>120
#### 120 | gsm8k |
Miriam is trying to exercise more and figures if she counts her exercises it will be encouraging to see her numbers go up. On Monday she does 5 push-ups. On Tuesday she does 7 push-ups. On Wednesday she does twice as many push-ups as the day before. On Thursday she does half the number of total pushups she already did ... | Miriam is trying to exercise more and figures if she counts her exercises it will be encouraging to see her numbers go up. On Monday she does 5 push-ups. On Tuesday she does 7 push-ups. On Wednesday she does twice as many push-ups as the day before. On Thursday she does half the number of total pushups she already did ... | Miriam is trying to exercise more and figures if she counts her exercises it will be encouraging to see her numbers go up. On Monday she does 5 push-ups. On Tuesday she does 7 push-ups. On Wednesday she does twice as many push-ups as the day before. On Thursday she does half the number of total pushups she already did ... | 39 | 39 | On Monday Miriam does 5 push-ups + 7 push-ups on Tuesday = <<5+7=12>>12 push-ups on both days.
On Wednesday she does twice as many push-ups as she did on Tuesday, 7 x 2 = <<7*2=14>>14 push-ups on Wednesday.
Over the first 3 days, she did a total of 14 + 12 = <<14+12=26>>26 pushups
On Thursday Miriam did 1/2 the same nu... | gsm8k |
In triangle $ABC$ , points $M$ and $N$ are on segments $AB$ and $AC$ respectively such that $AM = MC$ and $AN = NB$ . Let $P$ be the point such that $PB$ and $PC$ are tangent to the circumcircle of $ABC$ . Given that the perimeters of $PMN$ and $BCNM$ are $21$ and $29$ respectively, and that ... | In triangle $ABC$ , points $M$ and $N$ are on segments $AB$ and $AC$ respectively such that $AM = MC$ and $AN = NB$ . Let $P$ be the point such that $PB$ and $PC$ are tangent to the circumcircle of $ABC$ . Given that the perimeters of $PMN$ and $BCNM$ are $21$ and $29$ respectively, and that ... | In triangle $ABC$ , points $M$ and $N$ are on segments $AB$ and $AC$ respectively such that $AM = MC$ and $AN = NB$ . Let $P$ be the point such that $PB$ and $PC$ are tangent to the circumcircle of $ABC$ . Given that the perimeters of $PMN$ and $BCNM$ are $21$ and $29$ respectively, and that ... | \frac{200}{21} | \frac{200}{21} | 1. **Identify the given conditions and setup the problem:**
- Points \( M \) and \( N \) are on segments \( AB \) and \( AC \) respectively such that \( AM = MC \) and \( AN = NB \).
- Point \( P \) is such that \( PB \) and \( PC \) are tangent to the circumcircle of \( \triangle ABC \).
- The perimeters of \... | aops_forum |
What is the area of the set of points $P(x ; y)$ in the right-angled coordinate system that satisfy the condition $|x+y|+|x-y| \leq 4?$
| What is the area of the set of points $P(x ; y)$ in the right-angled coordinate system that satisfy the condition $|x+y|+|x-y| \leq 4?$
| What is the area of the set of points $P(x ; y)$ in the right-angled coordinate system that satisfy the condition $|x+y|+|x-y| \leq 4?$
| 16 | 16 |
To determine the area of the set of points \( P(x, y) \) in the coordinate system satisfying \( |x+y| + |x-y| \leq 4 \), we will proceed as follows:
1. **Identify Sign Changes of Absolute Values**:
- The expression \( |x-y| \) changes sign when \( x = y \).
- The expression \( |x+y| \) changes sign when \( y ... | olympiads |
Find distinct positive integers $n_1<n_2<\dots<n_7$ with the least possible sum, such that their product $n_1 \times n_2 \times \dots \times n_7$ is divisible by $2016$ . | Find distinct positive integers $n_1<n_2<\dots<n_7$ with the least possible sum, such that their product $n_1 \times n_2 \times \dots \times n_7$ is divisible by $2016$ . | Find distinct positive integers $n_1<n_2<\dots<n_7$ with the least possible sum, such that their product $n_1 \times n_2 \times \dots \times n_7$ is divisible by $2016$ . | 31 | 31 | To solve the problem, we need to find distinct positive integers \( n_1 < n_2 < \dots < n_7 \) such that their product \( n_1 \times n_2 \times \dots \times n_7 \) is divisible by \( 2016 \). We also want the sum of these integers to be as small as possible.
1. **Factorize 2016**:
\[
2016 = 2^5 \times 3^2 \times... | aops_forum |
Let $\mathbf{A} =\begin{pmatrix} -1 & 2 \\ 3 & 4 \end{pmatrix}.$ Then there exist scalars $p$ and $q$ such that
\[\mathbf{A}^6 = p \mathbf{A} + q \mathbf{I}.\]Enter the ordered pair $(p,q).$ | Let $\mathbf{A} =\begin{pmatrix} -1 & 2 \\ 3 & 4 \end{pmatrix}.$ Then there exist scalars $p$ and $q$ such that
\[\mathbf{A}^6 = p \mathbf{A} + q \mathbf{I}.\]Enter the ordered pair $(p,q).$ | Let $\mathbf{A} =\begin{pmatrix} -1 & 2 \\ 3 & 4 \end{pmatrix}.$ Then there exist scalars $p$ and $q$ such that
\[\mathbf{A}^6 = p \mathbf{A} + q \mathbf{I}.\]Enter the ordered pair $(p,q).$ | (2223,4510) | (2223,4510) | To solve for the scalars $p$ and $q$ such that $\mathbf{A}^6 = p \mathbf{A} + q \mathbf{I}$, where $\mathbf{A} =\begin{pmatrix} -1 & 2 \\ 3 & 4 \end{pmatrix}$, we start by calculating $\mathbf{A}^2$:
\begin{align*}
\mathbf{A}^2 &= \begin{pmatrix} -1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} -1 & 2 \\ 3 & 4 \... | math |
Ivan is twice as old as Peter was when Ivan was as old as Peter is now. When Peter is as old as Ivan is now, the sum of their ages will be 54 years. How old is Peter? | Ivan is twice as old as Peter was when Ivan was as old as Peter is now. When Peter is as old as Ivan is now, the sum of their ages will be 54 years. How old is Peter? | Ivan is twice as old as Peter was when Ivan was as old as Peter is now. When Peter is as old as Ivan is now, the sum of their ages will be 54 years. How old is Peter? | 18 | 18 |
1. Define the variables for the ages:
- Let \( y \) be the age of Peter in the past when Ivan was his current age.
- Let \( x \) be the current age of Peter.
- Therefore, Ivan's current age is \( 2y \) since it's twice the age Peter was in the past.
2. According to the problem, when Peter becomes \( 2y \) ye... | olympiads |
In equilateral $\triangle ABC$ let points $D$ and $E$ trisect $\overline{BC}$. Then $\sin(\angle DAE)$ can be expressed in the form $\frac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is an integer that is not divisible by the square of any prime. Find $a+b+c$. | In equilateral $\triangle ABC$ let points $D$ and $E$ trisect $\overline{BC}$. Then $\sin(\angle DAE)$ can be expressed in the form $\frac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is an integer that is not divisible by the square of any prime. Find $a+b+c$. | In equilateral $\triangle ABC$ let points $D$ and $E$ trisect $\overline{BC}$. Then $\sin(\angle DAE)$ can be expressed in the form $\frac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is an integer that is not divisible by the square of any prime. Find $a+b+c$. | 20 | 20 | We find that, as before, $AE = \sqrt{7}$, and also the area of $\Delta DAE$ is 1/3 the area of $\Delta ABC$. Thus, using the area formula, $1/2 \cdot 7 \cdot \sin(\angle EAD) = 3\sqrt{3}/4$, and $\sin(\angle EAD) = \dfrac{3\sqrt{3}}{14}$. Therefore, $a + b + c = \boxed{020}.$ | deepscaler |
There are 3 boxes. If the weights of pairs of boxes are measured, their combined weights are 83 kg, 85 kg, and 86 kg, respectively. What is the weight of the lightest box? | There are 3 boxes. If the weights of pairs of boxes are measured, their combined weights are 83 kg, 85 kg, and 86 kg, respectively. What is the weight of the lightest box? | There are 3 boxes. If the weights of pairs of boxes are measured, their combined weights are 83 kg, 85 kg, and 86 kg, respectively. What is the weight of the lightest box? | 41 | 41 | :
1. We are given three boxes with their respective pairwise sums of weights:
\[
\begin{array}{c}
83 \, \text{kg (denoted as the sum of the middle and the smallest box)}, \\
85 \, \text{kg (denoted as the sum of the largest and the smallest box)}, \\
86 \, \text{kg (denoted as the sum of the largest and... | olympiads |
Given the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ with $(a>b>0)$ passing through the point $(2,1)$, the set of all points on these ellipses that satisfy $|y|>1$ is shaded in one of the diagrams below:
(A)
(B)
(C)
(D)
Answer: ( $\quad$ ) | Given the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ with $(a>b>0)$ passing through the point $(2,1)$, the set of all points on these ellipses that satisfy $|y|>1$ is shaded in one of the diagrams below:
(A)
(B)
(C)
(D)
Answer: ( $\quad$ ) | Given the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ with $(a>b>0)$ passing through the point $(2,1)$, the set of all points on these ellipses that satisfy $|y|>1$ is shaded in one of the diagrams below:
(A)
(B)
(C)
(D)
Answer: ( $\quad$ ) | C | C |
1. We begin by considering the equation of the ellipse given by:
\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad \text{where} \quad a > b > 0
\]
2. We are informed that the point \((2,1)\) lies on this ellipse. Substituting \(x = 2\) and \(y = 1\) into the ellipse equation, we obtain:
\[
\frac{2^2}{a^2}... | olympiads |
What is the area enclosed by the region defined by the equation $x^2+y^2+6x+8y=0$? | What is the area enclosed by the region defined by the equation $x^2+y^2+6x+8y=0$? | What is the area enclosed by the region defined by the equation $x^2+y^2+6x+8y=0$? | 25\pi | 25\pi | deepscaler | |
At a birthday party, 30% of the guests are married, 50% are single, and the rest are children. If there are 1000 guests, how many more married people are there than children? | At a birthday party, 30% of the guests are married, 50% are single, and the rest are children. If there are 1000 guests, how many more married people are there than children? | At a birthday party, 30% of the guests are married, 50% are single, and the rest are children. If there are 1000 guests, how many more married people are there than children? | 100 | 100 | There are 1000 x 30/100 = <<1000*30/100=300>>300 people who are married.
There are 1000 x 50/100 = <<1000*50/100=500>>500 people who are single.
So, there are a total of 300 + 500 = <<300+500=800>>800 that are either married or single.
This means, 1000 - 800 = <<1000-800=200>>200 are children.
Therefore, there are 300 ... | gsm8k |
$A$ and $B$ together can do a job in $2$ days; $B$ and $C$ can do it in four days; and $A$ and $C$ in $2\frac{2}{5}$ days.
The number of days required for A to do the job alone is:
$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 2.8$ | $A$ and $B$ together can do a job in $2$ days; $B$ and $C$ can do it in four days; and $A$ and $C$ in $2\frac{2}{5}$ days.
The number of days required for A to do the job alone is:
$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 2.8$ | $A$ and $B$ together can do a job in $2$ days; $B$ and $C$ can do it in four days; and $A$ and $C$ in $2\frac{2}{5}$ days.
The number of days required for A to do the job alone is:
$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 2.8$ | \textbf{(B)}\ 3 | \textbf{(B)}\ 3 | 1. **Define the rates of work**: Let $r_A$, $r_B$, and $r_C$ be the rates at which $A$, $B$, and $C$ can complete the job per day, respectively. The units are $\frac{\text{job}}{\text{day}}$.
2. **Set up the equations based on the given information**:
- $A$ and $B$ together can complete the job in 2 days:
\[
... | amc_aime |
Triangle $ABC$ has an inradius of $5$ and a circumradius of $16$. If $2\cos{B} = \cos{A} + \cos{C}$, then the area of triangle $ABC$ can be expressed as $\frac{a\sqrt{b}}{c}$, where $a, b,$ and $c$ are positive integers such that $a$ and $c$ are relatively prime and $b$ is not divisible by the square of any prime. Comp... | Triangle $ABC$ has an inradius of $5$ and a circumradius of $16$. If $2\cos{B} = \cos{A} + \cos{C}$, then the area of triangle $ABC$ can be expressed as $\frac{a\sqrt{b}}{c}$, where $a, b,$ and $c$ are positive integers such that $a$ and $c$ are relatively prime and $b$ is not divisible by the square of any prime. Comp... | Triangle $ABC$ has an inradius of $5$ and a circumradius of $16$. If $2\cos{B} = \cos{A} + \cos{C}$, then the area of triangle $ABC$ can be expressed as $\frac{a\sqrt{b}}{c}$, where $a, b,$ and $c$ are positive integers such that $a$ and $c$ are relatively prime and $b$ is not divisible by the square of any prime. Comp... | 141 | 141 | To solve for the area of triangle $ABC$ given the inradius ($r$) is $5$ and the circumradius ($R$) is $16$, and the condition $2\cos{B} = \cos{A} + \cos{C}$, we proceed as follows:
1. **Using the identity for a triangle's angles**: We start with the identity $\cos A + \cos B + \cos C = 1+\frac{r}{R}$. Substituting the... | math |
Given \( f^{(1)}(x) = f(x) \),
\( f^{(n)}(x) = \left( f^{(n-1)}(x) \right)' \) for \( n \geq 2 \).
If \( f(x) = \frac{x^2}{1 - x^2} \), find \( f^{(n)}(0) = \square \). | Given \( f^{(1)}(x) = f(x) \),
\( f^{(n)}(x) = \left( f^{(n-1)}(x) \right)' \) for \( n \geq 2 \).
If \( f(x) = \frac{x^2}{1 - x^2} \), find \( f^{(n)}(0) = \square \). | Given \( f^{(1)}(x) = f(x) \),
\( f^{(n)}(x) = \left( f^{(n-1)}(x) \right)' \) for \( n \geq 2 \).
If \( f(x) = \frac{x^2}{1 - x^2} \), find \( f^{(n)}(0) = \square \). | \frac{1 + (-1)^n}{2} \cdot n! | \frac{1 + (-1)^n}{2} \cdot n! | To find \( f^{(n)}(0) \), we begin by analyzing the given function \( f(x) = \frac{x^2}{1-x^2} \). Let's first expand \( f(x) \) and then compute its derivatives.
1. **Rewrite \( f(x) \):**
\[
f(x) = \frac{x^2}{1 - x^2}
\]
We can rewrite this using partial fractions or another form that is more managea... | olympiads |
Oliver collects trading cards. He has twice as many Monster Club cards as Alien Baseball cards. His Battle Gremlins card collection is the largest at 48 cards, three times the size of his Alien Baseball card collection. How many Monster Club cards does Oliver have? | Oliver collects trading cards. He has twice as many Monster Club cards as Alien Baseball cards. His Battle Gremlins card collection is the largest at 48 cards, three times the size of his Alien Baseball card collection. How many Monster Club cards does Oliver have? | Oliver collects trading cards. He has twice as many Monster Club cards as Alien Baseball cards. His Battle Gremlins card collection is the largest at 48 cards, three times the size of his Alien Baseball card collection. How many Monster Club cards does Oliver have? | 32 | 32 | Oliver’s Alien Baseball card collection has 48 / 3 = <<48/3=16>>16 cards.
Oliver has twice as many Monster Club cards, so he has 16 * 2 = <<16*2=32>>32 Monster Club cards.
#### 32 | gsm8k |
An 8-by-8 square is divided into 64 unit squares in the usual way. Each unit square is colored black or white. The number of black unit squares is even. We can take two adjacent unit squares (forming a 1-by-2 or 2-by-1 rectangle), and flip their colors: black becomes white and white becomes black. We call this oper... | An 8-by-8 square is divided into 64 unit squares in the usual way. Each unit square is colored black or white. The number of black unit squares is even. We can take two adjacent unit squares (forming a 1-by-2 or 2-by-1 rectangle), and flip their colors: black becomes white and white becomes black. We call this oper... | An 8-by-8 square is divided into 64 unit squares in the usual way. Each unit square is colored black or white. The number of black unit squares is even. We can take two adjacent unit squares (forming a 1-by-2 or 2-by-1 rectangle), and flip their colors: black becomes white and white becomes black. We call this oper... | 32 | 32 | deepscaler | |
Calculate the definite integral:
$$
\int_{\frac{\pi}{2}}^{2 \operatorname{arctg} 2} \frac{d x}{\sin ^{2} x(1-\cos x)}
$$ |
Calculate the definite integral:
$$
\int_{\frac{\pi}{2}}^{2 \operatorname{arctg} 2} \frac{d x}{\sin ^{2} x(1-\cos x)}
$$ |
Calculate the definite integral:
$$
\int_{\frac{\pi}{2}}^{2 \operatorname{arctg} 2} \frac{d x}{\sin ^{2} x(1-\cos x)}
$$ | 55/96 | 55/96 | deepscaler | |
Find the complex number $z$ such that
\[|z - 1| = |z + 3| = |z - i|.\] | Find the complex number $z$ such that
\[|z - 1| = |z + 3| = |z - i|.\] | Find the complex number $z$ such that
\[|z - 1| = |z + 3| = |z - i|.\] | -1 - i | -1 - i | To find the complex number $z$ that satisfies $|z - 1| = |z + 3| = |z - i|$, we let $z = a + bi$, where $a$ and $b$ are real numbers. This gives us three equations based on the absolute value conditions:
1. $|(a - 1) + bi| = |(a + 3) + bi|$
2. $|(a - 1) + bi| = |a + (b - 1)i|$
First, we simplify these equations using... | math |
The sequence of polynomials $\left( P_{n}(X)\right)_{n\in Z_{>0}}$ is defined as follows: $P_{1}(X)=2X$ $P_{2}(X)=2(X^2+1)$ $P_{n+2}(X)=2X\cdot P_{n+1}(X)-(X^2-1)P_{n}(X)$ , for all positive integers $n$ .
Find all $n$ for which $X^2+1\mid P_{n}(X)$ | The sequence of polynomials $\left( P_{n}(X)\right)_{n\in Z_{>0}}$ is defined as follows: $P_{1}(X)=2X$ $P_{2}(X)=2(X^2+1)$ $P_{n+2}(X)=2X\cdot P_{n+1}(X)-(X^2-1)P_{n}(X)$ , for all positive integers $n$ .
Find all $n$ for which $X^2+1\mid P_{n}(X)$ | The sequence of polynomials $\left( P_{n}(X)\right)_{n\in Z_{>0}}$ is defined as follows: $P_{1}(X)=2X$ $P_{2}(X)=2(X^2+1)$ $P_{n+2}(X)=2X\cdot P_{n+1}(X)-(X^2-1)P_{n}(X)$ , for all positive integers $n$ .
Find all $n$ for which $X^2+1\mid P_{n}(X)$ | n \equiv 2 \pmod{4} | n \equiv 2 \pmod{4} | 1. **Initial Definitions and Observations:**
- The sequence of polynomials \(\left( P_{n}(X)\right)_{n\in \mathbb{Z}_{>0}}\) is defined as follows:
\[
P_{1}(X) = 2X, \quad P_{2}(X) = 2(X^2 + 1)
\]
\[
P_{n+2}(X) = 2X \cdot P_{n+1}(X) - (X^2 - 1)P_{n}(X) \quad \text{for all positive integers }... | aops_forum |
Let \( 0 < a < 1 \), and let \( b = 1 - a \). The inequality \( \mathrm{e}^{x} \leq \frac{1+ax}{1-bx} \) holds for all \( x \in [0, 1] \). Find the maximum value of \( a \). | Let \( 0 < a < 1 \), and let \( b = 1 - a \). The inequality \( \mathrm{e}^{x} \leq \frac{1+ax}{1-bx} \) holds for all \( x \in [0, 1] \). Find the maximum value of \( a \). | Let \( 0 < a < 1 \), and let \( b = 1 - a \). The inequality \( \mathrm{e}^{x} \leq \frac{1+ax}{1-bx} \) holds for all \( x \in [0, 1] \). Find the maximum value of \( a \). | \frac{1}{2} | \frac{1}{2} | Given the inequality \( \mathrm{e}^x \leqslant \frac{1 + ax}{1 - bx} \) for \( x \in [0,1] \), where \( 0 < a < 1 \) and \( b = 1 - a \):
1. Consider the inequality for \( x \in (0,1] \):
\[
\mathrm{e}^x \leqslant \frac{1 + ax}{1 - bx}
\]
2. Rewrite the inequality in a different form:
\[
\mathrm{e}^x(... | olympiads |
In seven years, Talia will be 20 years old. Talia's mom is currently three times as old as Talia is today. In three years, Talia's father will be the same age as Talia's mom is today. Currently, how many years old is Talia's father? | In seven years, Talia will be 20 years old. Talia's mom is currently three times as old as Talia is today. In three years, Talia's father will be the same age as Talia's mom is today. Currently, how many years old is Talia's father? | In seven years, Talia will be 20 years old. Talia's mom is currently three times as old as Talia is today. In three years, Talia's father will be the same age as Talia's mom is today. Currently, how many years old is Talia's father? | 36 | 36 | In seven years, Talia will be 20 years old, and therefore Talia is currently 20-7=<<20-7=13>>13 years old.
Talia's mom is currently three times as old as Talia is today, or 3*13=<<3*13=39>>39 years old.
If in three years, Talia's father will be the same age as Talia's mom is today, then Talia's father is currently 39-3... | gsm8k |
In Century Park, there is a large meadow that grows a lot of weeds every day (assume the number of weeds that grow per minute is constant). Every morning at 8:00 AM, some workers go to remove the weeds (each worker removes weeds at the same rate). Once all the weeds are removed (the number of weeds is 0, with no good g... | In Century Park, there is a large meadow that grows a lot of weeds every day (assume the number of weeds that grow per minute is constant). Every morning at 8:00 AM, some workers go to remove the weeds (each worker removes weeds at the same rate). Once all the weeds are removed (the number of weeds is 0, with no good g... | In Century Park, there is a large meadow that grows a lot of weeds every day (assume the number of weeds that grow per minute is constant). Every morning at 8:00 AM, some workers go to remove the weeds (each worker removes weeds at the same rate). Once all the weeds are removed (the number of weeds is 0, with no good g... | 8:39 \, \text{AM} | 8:39 \, \text{AM} |
1. Let us assume that the grass grows at a rate of \(1\) unit per minute. This assumption simplifies the calculations and allows us to standardize the problem.
2. On the first day, the workers start at 8:00 AM and finish at 9:00 AM. This means they worked for \(60\) minutes. During this time, the workers were able to... | olympiads |
If \( 4^3 + 4^r + 4^4 \) is a perfect square and \( r \) is a positive integer, find the minimum value of \( r \). | If \( 4^3 + 4^r + 4^4 \) is a perfect square and \( r \) is a positive integer, find the minimum value of \( r \). | If \( 4^3 + 4^r + 4^4 \) is a perfect square and \( r \) is a positive integer, find the minimum value of \( r \). | 4 | 4 |
1. Given the expression \(4^3 + 4^r + 4^4\), we recognize that it needs to be a perfect square. To simplify the expression, we start by rewriting the powers of 4 in terms of powers of 2:
$$4^3 + 4^r + 4^4 = (2^2)^3 + (2^2)^r + (2^2)^4$$
Which simplifies to:
$$2^6 + 2^{2r} + 2^8$$
2. Factor out the greatest common po... | olympiads |
A circle is tangential to sides $AB$ and $AD$ of convex quadrilateral $ABCD$ at $G$ and $H$ respectively, and cuts diagonal $AC$ at $E$ and $F$ . What are the necessary and sufficient conditions such that there exists another circle which passes through $E$ and $F$ , and is tangential to $DA$ and ... | A circle is tangential to sides $AB$ and $AD$ of convex quadrilateral $ABCD$ at $G$ and $H$ respectively, and cuts diagonal $AC$ at $E$ and $F$ . What are the necessary and sufficient conditions such that there exists another circle which passes through $E$ and $F$ , and is tangential to $DA$ and ... | A circle is tangential to sides $AB$ and $AD$ of convex quadrilateral $ABCD$ at $G$ and $H$ respectively, and cuts diagonal $AC$ at $E$ and $F$ . What are the necessary and sufficient conditions such that there exists another circle which passes through $E$ and $F$ , and is tangential to $DA$ and ... | AB + CD = BC + DA | AB + CD = BC + DA | 1. **Correcting the Typo and Setting Up the Problem:**
The problem should state that a circle is tangential to sides \(AB\) and \(AD\) of convex quadrilateral \(ABCD\) at \(G\) and \(H\) respectively, and cuts diagonal \(AC\) at \(E\) and \(F\). We need to find the necessary and sufficient conditions for the existen... | aops_forum |
The points $A = (3,-4,2),$ $B = (5,-8,5),$ $C = (4,-3,0),$ and $D = (6,-7,3)$ in space form a flat quadrilateral. Find the area of this quadrilateral. | The points $A = (3,-4,2),$ $B = (5,-8,5),$ $C = (4,-3,0),$ and $D = (6,-7,3)$ in space form a flat quadrilateral. Find the area of this quadrilateral. | The points $A = (3,-4,2),$ $B = (5,-8,5),$ $C = (4,-3,0),$ and $D = (6,-7,3)$ in space form a flat quadrilateral. Find the area of this quadrilateral. | \sqrt{110} | \sqrt{110} | To find the area of the quadrilateral formed by points $A = (3,-4,2),$ $B = (5,-8,5),$ $C = (4,-3,0),$ and $D = (6,-7,3)$ in space, we first represent these points as vectors:
- $\mathbf{a} = \begin{pmatrix} 3 \\ -4 \\ 2 \end{pmatrix},$
- $\mathbf{b} = \begin{pmatrix} 5 \\ -8 \\ 5 \end{pmatrix},$
- $\mathbf{c} = \begin... | math |
The longest professional tennis match ever played lasted a total of $11$ hours and $5$ minutes. How many minutes was this?
$\textbf{(A) }605\qquad\textbf{(B) }655\qquad\textbf{(C) }665\qquad\textbf{(D) }1005\qquad \textbf{(E) }1105$ | The longest professional tennis match ever played lasted a total of $11$ hours and $5$ minutes. How many minutes was this?
$\textbf{(A) }605\qquad\textbf{(B) }655\qquad\textbf{(C) }665\qquad\textbf{(D) }1005\qquad \textbf{(E) }1105$ | The longest professional tennis match ever played lasted a total of $11$ hours and $5$ minutes. How many minutes was this?
$\textbf{(A) }605\qquad\textbf{(B) }655\qquad\textbf{(C) }665\qquad\textbf{(D) }1005\qquad \textbf{(E) }1105$ | 665 | 665 | To find the total duration of the tennis match in minutes, we need to convert the hours into minutes and then add the remaining minutes.
1. **Convert hours to minutes**:
- There are 60 minutes in one hour.
- Therefore, for 11 hours, the total minutes are:
\[
11 \text{ hours} \times 60 \text{ minutes/h... | amc_aime |
Find the remainder when $7145 + 7146 + 7147 + 7148 + 7149$ is divided by 8. | Find the remainder when $7145 + 7146 + 7147 + 7148 + 7149$ is divided by 8. | Find the remainder when $7145 + 7146 + 7147 + 7148 + 7149$ is divided by 8. | 7 | 7 | To find the remainder when $7145 + 7146 + 7147 + 7148 + 7149$ is divided by 8, we first reduce each number modulo 8. This process involves finding the remainder when each number is divided by 8. Let's do this step by step:
- $7145 \mod 8 = 1$ because when 7145 is divided by 8, the remainder is 1.
- $7146 \mod 8 = 2$ b... | math |
I take variable $b$, double it, and add four. I subtract $4b$ from this new expression, and divide the resulting difference by two. What is my final expression in simplest form? | I take variable $b$, double it, and add four. I subtract $4b$ from this new expression, and divide the resulting difference by two. What is my final expression in simplest form? | I take variable $b$, double it, and add four. I subtract $4b$ from this new expression, and divide the resulting difference by two. What is my final expression in simplest form? | 2 - b | 2 - b | deepscaler | |
The number of positive integers from 1 to 2002 that contain exactly one digit 0. | The number of positive integers from 1 to 2002 that contain exactly one digit 0. | The number of positive integers from 1 to 2002 that contain exactly one digit 0. | 414 | 414 | deepscaler | |
The school has 14 boys and 10 girls. If 4 boys and 3 girls drop out, how many boys and girls are left? | The school has 14 boys and 10 girls. If 4 boys and 3 girls drop out, how many boys and girls are left? | The school has 14 boys and 10 girls. If 4 boys and 3 girls drop out, how many boys and girls are left? | 17 | 17 | There are 14 boys - 4 boys = <<14-4=10>>10 boys left.
There are 10 girls - 3 girls = <<10-3=7>>7 girls left.
In total there are 10 boys + 7 girls = <<10+7=17>>17 boys and girls left.
#### 17 | gsm8k |
4 12-sided dice are rolled. What is the probability that the number of dice showing a two digit number is equal to the number of dice showing a one digit number? Express your answer as a common fraction. (Assume that the numbers on the 12 sides are the numbers from 1 to 12 expressed in decimal.) | 4 12-sided dice are rolled. What is the probability that the number of dice showing a two digit number is equal to the number of dice showing a one digit number? Express your answer as a common fraction. (Assume that the numbers on the 12 sides are the numbers from 1 to 12 expressed in decimal.) | 4 12-sided dice are rolled. What is the probability that the number of dice showing a two digit number is equal to the number of dice showing a one digit number? Express your answer as a common fraction. (Assume that the numbers on the 12 sides are the numbers from 1 to 12 expressed in decimal.) | \dfrac{27}{128} | \dfrac{27}{128} | deepscaler | |
Let $N$ be the number of ordered triples $(a,b,c) \in \{1, \ldots, 2016\}^{3}$ such that $a^{2} + b^{2} + c^{2} \equiv 0 \pmod{2017}$ . What are the last three digits of $N$ ? | Let $N$ be the number of ordered triples $(a,b,c) \in \{1, \ldots, 2016\}^{3}$ such that $a^{2} + b^{2} + c^{2} \equiv 0 \pmod{2017}$ . What are the last three digits of $N$ ? | Let $N$ be the number of ordered triples $(a,b,c) \in \{1, \ldots, 2016\}^{3}$ such that $a^{2} + b^{2} + c^{2} \equiv 0 \pmod{2017}$ . What are the last three digits of $N$ ? | 000 | 000 | 1. **Understanding the Problem:**
We need to find the number of ordered triples \((a, b, c)\) such that \(a^2 + b^2 + c^2 \equiv 0 \pmod{2017}\). Here, \(a, b, c\) are elements of the set \(\{1, 2, \ldots, 2016\}\).
2. **Simplifying the Problem:**
Since \(2017\) is a prime number, we can use properties of quadra... | aops_forum |
The positive integers $N$ and $N^2$ both end in the same sequence of four digits $abcd$ when written in base 10, where digit $a$ is not zero. Find the three-digit number $abc$ . | The positive integers $N$ and $N^2$ both end in the same sequence of four digits $abcd$ when written in base 10, where digit $a$ is not zero. Find the three-digit number $abc$ . | The positive integers $N$ and $N^2$ both end in the same sequence of four digits $abcd$ when written in base 10, where digit $a$ is not zero. Find the three-digit number $abc$ . | 937 | 937 | 1. Given that both \( N \) and \( N^2 \) end in the same sequence of four digits \( abcd \), we can express this condition mathematically as:
\[
N \equiv N^2 \pmod{10000}
\]
This implies:
\[
N^2 - N \equiv 0 \pmod{10000}
\]
Therefore:
\[
N(N-1) \equiv 0 \pmod{10000}
\]
2. To satisfy \(... | aops_forum |
Two players are playing a turn based game on a $n \times n$ chessboard. At the beginning, only the bottom left corner of the chessboard contains a piece. At each turn, the player moves the piece to either the square just above, or the square just right, or the diagonal square just right-top. If a player cannot make a... | Two players are playing a turn based game on a $n \times n$ chessboard. At the beginning, only the bottom left corner of the chessboard contains a piece. At each turn, the player moves the piece to either the square just above, or the square just right, or the diagonal square just right-top. If a player cannot make a... | Two players are playing a turn based game on a $n \times n$ chessboard. At the beginning, only the bottom left corner of the chessboard contains a piece. At each turn, the player moves the piece to either the square just above, or the square just right, or the diagonal square just right-top. If a player cannot make a... | 4 | 4 | 1. **Understanding the Game Mechanics**:
- The game is played on an \( n \times n \) chessboard.
- The piece starts at the bottom-left corner \((1,1)\).
- Players take turns moving the piece to either the square just above, the square just right, or the diagonal square just right-top.
- The player who canno... | aops_forum |
John draws a regular five pointed star in the sand, and at each of the 5 outward-pointing points and 5 inward-pointing points he places one of ten different sea shells. How many ways can he place the shells, if reflections and rotations of an arrangement are considered equivalent? | John draws a regular five pointed star in the sand, and at each of the 5 outward-pointing points and 5 inward-pointing points he places one of ten different sea shells. How many ways can he place the shells, if reflections and rotations of an arrangement are considered equivalent? | John draws a regular five pointed star in the sand, and at each of the 5 outward-pointing points and 5 inward-pointing points he places one of ten different sea shells. How many ways can he place the shells, if reflections and rotations of an arrangement are considered equivalent? | 362880 | 362880 | To solve this problem, we follow these steps:
1. **Count the total arrangements without considering symmetries**: Since there are 10 different sea shells and 10 positions where they can be placed, the total number of ways to arrange these shells without considering any symmetries is the number of permutations of 10 it... | math |
Two cars covered the same distance. The speed of the first car was constant and three times less than the initial speed of the second car. The second car traveled the first half of the journey without changing speed, then its speed was suddenly halved, then traveled with constant speed for another quarter of the journe... | Two cars covered the same distance. The speed of the first car was constant and three times less than the initial speed of the second car. The second car traveled the first half of the journey without changing speed, then its speed was suddenly halved, then traveled with constant speed for another quarter of the journe... | Two cars covered the same distance. The speed of the first car was constant and three times less than the initial speed of the second car. The second car traveled the first half of the journey without changing speed, then its speed was suddenly halved, then traveled with constant speed for another quarter of the journe... | 5/3 | 5/3 | deepscaler | |
In the Cartesian coordinate system $xOy$, it is known that the distance from any point on curve $C$ to point $M(0, \frac{1}{2})$ is equal to its distance to the line $y = -\frac{1}{2}$.
(Ⅰ) Find the equation of curve $C$;
(Ⅱ) Let $A_1(x_1, 0)$ and $A_2(x_2, 0)$ be two points on the x-axis ($x_1 + x_2 \neq 0$, $x_1x_... | In the Cartesian coordinate system $xOy$, it is known that the distance from any point on curve $C$ to point $M(0, \frac{1}{2})$ is equal to its distance to the line $y = -\frac{1}{2}$.
(Ⅰ) Find the equation of curve $C$;
(Ⅱ) Let $A_1(x_1, 0)$ and $A_2(x_2, 0)$ be two points on the x-axis ($x_1 + x_2 \neq 0$, $x_1x_... | In the Cartesian coordinate system $xOy$, it is known that the distance from any point on curve $C$ to point $M(0, \frac{1}{2})$ is equal to its distance to the line $y = -\frac{1}{2}$.
(Ⅰ) Find the equation of curve $C$;
(Ⅱ) Let $A_1(x_1, 0)$ and $A_2(x_2, 0)$ be two points on the x-axis ($x_1 + x_2 \neq 0$, $x_1x_... | \frac{6}{7} | \frac{6}{7} | deepscaler | |
Dan spent an hour doing 400 work tasks at $0.25 each. Then Dan spent an hour doing 5 work tasks at $2.00 each. How much more did Dan make doing the good work compared to the lower-paid work? | Dan spent an hour doing 400 work tasks at $0.25 each. Then Dan spent an hour doing 5 work tasks at $2.00 each. How much more did Dan make doing the good work compared to the lower-paid work? | Dan spent an hour doing 400 work tasks at $0.25 each. Then Dan spent an hour doing 5 work tasks at $2.00 each. How much more did Dan make doing the good work compared to the lower-paid work? | 90 | 90 | Dan spent an hour doing 400 tasks * $.25 = $100.
Dan spent an hour doing 5 tasks * $2.00 each = $<<5*2=10.00>>10.00.
Dan made $100 - $10 = $<<100-10=90.00>>90.00 per hour more doing the higher paid work.
#### 90 | gsm8k |
What is the largest three-digit integer whose digits are distinct and form a geometric sequence? | What is the largest three-digit integer whose digits are distinct and form a geometric sequence? | What is the largest three-digit integer whose digits are distinct and form a geometric sequence? | 964 | 964 | To find the largest three-digit integer with distinct digits forming a geometric sequence, we start by considering the hundreds digit. For the number to be as large as possible, we aim for the hundreds digit to be $9$, the largest digit possible. This approach ensures that any number we find will be larger than any num... | math |
Alex needs to be 54 inches tall to ride the newest roller coaster at the theme park. He is 48 inches tall this year. He hears a rumor that for every hour he hangs upside down, he can grow 1/12 of an inch. Normally he grows 1/3 of an inch per month. On average, how many hours does he need to hang upside down each month ... | Alex needs to be 54 inches tall to ride the newest roller coaster at the theme park. He is 48 inches tall this year. He hears a rumor that for every hour he hangs upside down, he can grow 1/12 of an inch. Normally he grows 1/3 of an inch per month. On average, how many hours does he need to hang upside down each month ... | Alex needs to be 54 inches tall to ride the newest roller coaster at the theme park. He is 48 inches tall this year. He hears a rumor that for every hour he hangs upside down, he can grow 1/12 of an inch. Normally he grows 1/3 of an inch per month. On average, how many hours does he need to hang upside down each month ... | 2 | 2 | He is 6 inches too short this year because 54 - 48 = <<54-48=6>>6
He will naturally grow 4 inches taller because 12 x (1/3) = <<12*(1/3)=4>>4
He needs to grow 2 inches from hanging because 6 - 4 = <<6-4=2>>2
He needs to hang for 24 hours because 2 / (1/12) = <<2/(1/12)=24>>24
He needs to hang for 2 hours a month becaus... | gsm8k |
The price of a home is $98 per square foot (sq ft). The house is 2,400 sq ft and the barn out back is 1,000 sq ft. How much is this property? | The price of a home is $98 per square foot (sq ft). The house is 2,400 sq ft and the barn out back is 1,000 sq ft. How much is this property? | The price of a home is $98 per square foot (sq ft). The house is 2,400 sq ft and the barn out back is 1,000 sq ft. How much is this property? | 333200 | 333200 | The house is 2,400 sq ft and the barn is 1,000 sq ft so it's 2400+1000 = <<2400+1000=3400>>3,400 sq ft
The price is $98 per sq ft and it's 3,400 sq ft big so the property costs 98*3400 = $<<98*3400=333200.00>>333,200.00
#### 333200 | gsm8k |
Linda makes $10.00 an hour babysitting. There is a $25.00 application fee for each college application she submits. If she is applying to 6 colleges, how many hours will she need to babysit to cover the application fees? | Linda makes $10.00 an hour babysitting. There is a $25.00 application fee for each college application she submits. If she is applying to 6 colleges, how many hours will she need to babysit to cover the application fees? | Linda makes $10.00 an hour babysitting. There is a $25.00 application fee for each college application she submits. If she is applying to 6 colleges, how many hours will she need to babysit to cover the application fees? | 15 | 15 | The application fee is $25.00 per college and she is applying to 6 colleges so that's 25*6 = $<<25*6=150.00>>150.00
She makes $10.00 an hour babysitting. Her application fees total $150.00 so she needs to work 150/10 = <<150/10=15>>15 hours to cover the cost
#### 15 | gsm8k |
Find the number of integers $n$ such that $$ 1+\left\lfloor\frac{100 n}{101}\right\rfloor=\left\lceil\frac{99 n}{100}\right\rceil $$ | Find the number of integers $n$ such that $$ 1+\left\lfloor\frac{100 n}{101}\right\rfloor=\left\lceil\frac{99 n}{100}\right\rceil $$ | Find the number of integers $n$ such that $$ 1+\left\lfloor\frac{100 n}{101}\right\rfloor=\left\lceil\frac{99 n}{100}\right\rceil $$ | 10100 | 10100 | Consider $f(n)=\left\lceil\frac{99 n}{100}\right\rceil-\left\lfloor\frac{100 n}{101}\right\rfloor$. Note that $f(n+10100)=\left\lceil\frac{99 n}{100}+99 \cdot 101\right\rceil-\left\lfloor\frac{100 n}{101}+100^{2}\right\rfloor=f(n)+99 \cdot 101-100^{2}=f(n)-1$. Thus, for each residue class $r$ modulo 10100, there is exa... | deepscaler |
Solve for the positive integer(s) \( n \) such that \( \phi\left(n^{2}\right) = 1000 \phi(n) \). | Solve for the positive integer(s) \( n \) such that \( \phi\left(n^{2}\right) = 1000 \phi(n) \). | Solve for the positive integer(s) \( n \) such that \( \phi\left(n^{2}\right) = 1000 \phi(n) \). | 1000 | 1000 | deepscaler | |
Let a sequence $\{u_n\}$ be defined by $u_1=5$ and the relationship $u_{n+1}-u_n=3+4(n-1), n=1,2,3\cdots.$If $u_n$ is expressed as a polynomial in $n$, the algebraic sum of its coefficients is:
$\text{(A) 3} \quad \text{(B) 4} \quad \text{(C) 5} \quad \text{(D) 6} \quad \text{(E) 11}$ | Let a sequence $\{u_n\}$ be defined by $u_1=5$ and the relationship $u_{n+1}-u_n=3+4(n-1), n=1,2,3\cdots.$If $u_n$ is expressed as a polynomial in $n$, the algebraic sum of its coefficients is:
$\text{(A) 3} \quad \text{(B) 4} \quad \text{(C) 5} \quad \text{(D) 6} \quad \text{(E) 11}$ | Let a sequence $\{u_n\}$ be defined by $u_1=5$ and the relationship $u_{n+1}-u_n=3+4(n-1), n=1,2,3\cdots.$If $u_n$ is expressed as a polynomial in $n$, the algebraic sum of its coefficients is:
$\text{(A) 3} \quad \text{(B) 4} \quad \text{(C) 5} \quad \text{(D) 6} \quad \text{(E) 11}$ | \text{(C) 5} | \text{(C) 5} | 1. **Identify the nature of the sequence**: Given the recurrence relation $u_{n+1} - u_n = 3 + 4(n-1)$, we can simplify this to $u_{n+1} - u_n = 4n - 1$. This indicates that the sequence $\{u_n\}$ is defined by a quadratic polynomial because the difference between consecutive terms is a linear function.
2. **Determine... | amc_aime |
Given points $A=(4,10)$ and $B=(10,8)$ lie on circle $\omega$ in the plane, and the tangent lines to $\omega$ at $A$ and $B$ intersect at a point on the $x$-axis, find the area of $\omega$. | Given points $A=(4,10)$ and $B=(10,8)$ lie on circle $\omega$ in the plane, and the tangent lines to $\omega$ at $A$ and $B$ intersect at a point on the $x$-axis, find the area of $\omega$. | Given points $A=(4,10)$ and $B=(10,8)$ lie on circle $\omega$ in the plane, and the tangent lines to $\omega$ at $A$ and $B$ intersect at a point on the $x$-axis, find the area of $\omega$. | \frac{100\pi}{9} | \frac{100\pi}{9} | deepscaler | |
Mason is trying to download a 880 MB game to his phone. After downloading 310 MB, his Internet connection slows to 3 MB/minute. How many more minutes will it take him to download the game? | Mason is trying to download a 880 MB game to his phone. After downloading 310 MB, his Internet connection slows to 3 MB/minute. How many more minutes will it take him to download the game? | Mason is trying to download a 880 MB game to his phone. After downloading 310 MB, his Internet connection slows to 3 MB/minute. How many more minutes will it take him to download the game? | 190 | 190 | First find how many MB are left to download: 880 MB - 310 MB = <<880-310=570>>570 MB
Then divide that number by the download speed to find the download time: 570 MB / 3 MB/minute = <<570/3=190>>190 minutes
#### 190 | gsm8k |
Brad has a cylindrical barrel with radius 10 inches and height 15 inches. He wants to fill it from a well, using a bucket in the shape of a hemisphere with a radius of 10 inches. How many trips must Brad make to the well in order to fill the barrel? | Brad has a cylindrical barrel with radius 10 inches and height 15 inches. He wants to fill it from a well, using a bucket in the shape of a hemisphere with a radius of 10 inches. How many trips must Brad make to the well in order to fill the barrel? | Brad has a cylindrical barrel with radius 10 inches and height 15 inches. He wants to fill it from a well, using a bucket in the shape of a hemisphere with a radius of 10 inches. How many trips must Brad make to the well in order to fill the barrel? | 3 | 3 | To solve the problem, we calculate the volumes of both the bucket and the barrel, and then determine how many buckets are needed to fill the barrel. Let's denote the radius $r$ as 10 inches for both the bucket and the barrel.
**Step 1: Calculate the volume of the bucket.**
The bucket is a hemisphere with radius $r$. ... | math |
In a box, there is a large mixed batch of six types of flowers. Vasya randomly takes flowers one by one from the box. As soon as he gathers 5 flowers of one type, he makes a bouquet and sells it. What is the minimum number of flowers he needs to take to guarantee selling 10 bouquets? | In a box, there is a large mixed batch of six types of flowers. Vasya randomly takes flowers one by one from the box. As soon as he gathers 5 flowers of one type, he makes a bouquet and sells it. What is the minimum number of flowers he needs to take to guarantee selling 10 bouquets? | In a box, there is a large mixed batch of six types of flowers. Vasya randomly takes flowers one by one from the box. As soon as he gathers 5 flowers of one type, he makes a bouquet and sells it. What is the minimum number of flowers he needs to take to guarantee selling 10 bouquets? | 70 | 70 | **Step 1:** Understand the task. We have a box containing a large number of mixed flowers of six different types, and Vasya is randomly picking flowers one by one. Once Vasya collects 5 flowers of one type, he makes a bouquet and sells it. We need to determine the minimum number of flowers Vasya has to pick to guarante... | olympiads |
Let \( A = \{1, 2, 3, \cdots, 4n+2\} \) and \( M = \{2n+1, 4n+3, 6n+5\} \). For any non-empty subset \( B \) of \( A \), \( B \) is called an \( M \)-free set if the sum of any two numbers in \( B \) does not belong to \( M \). If \( A = A_1 \cup A_2 \), \( A_1 \cap A_2 = \varnothing \), and both \( A_1 \) and \( A_2 \... | Let \( A = \{1, 2, 3, \cdots, 4n+2\} \) and \( M = \{2n+1, 4n+3, 6n+5\} \). For any non-empty subset \( B \) of \( A \), \( B \) is called an \( M \)-free set if the sum of any two numbers in \( B \) does not belong to \( M \). If \( A = A_1 \cup A_2 \), \( A_1 \cap A_2 = \varnothing \), and both \( A_1 \) and \( A_2 \... | Let \( A = \{1, 2, 3, \cdots, 4n+2\} \) and \( M = \{2n+1, 4n+3, 6n+5\} \). For any non-empty subset \( B \) of \( A \), \( B \) is called an \( M \)-free set if the sum of any two numbers in \( B \) does not belong to \( M \). If \( A = A_1 \cup A_2 \), \( A_1 \cap A_2 = \varnothing \), and both \( A_1 \) and \( A_2 \... | 2^{n+1} | 2^{n+1} |
1. **Identify Relationship and Set Definitions**:
- We are given the set \(A = \{1, 2, 3, \cdots, 4n+2\}\).
- The set \(M = \{2n + 1, 4n + 3, 6n + 5\}\) contains elements which we want to avoid as sums of pairs from \(B\).
2. **Define Relationship**:
- For \(m, p \in A\), if \(m + p \in M\), we say \(m\) ... | olympiads |
Find the sum of all positive integers $n$ such that there exists an integer $b$ with $|b| \neq 4$ such that the base -4 representation of $n$ is the same as the base $b$ representation of $n$. | Find the sum of all positive integers $n$ such that there exists an integer $b$ with $|b| \neq 4$ such that the base -4 representation of $n$ is the same as the base $b$ representation of $n$. | Find the sum of all positive integers $n$ such that there exists an integer $b$ with $|b| \neq 4$ such that the base -4 representation of $n$ is the same as the base $b$ representation of $n$. | 1026 | 1026 | All 1 digit numbers, $0,1,2,3$, are solutions when, say, $b=5$. (Of course, $d \in \{0,1,2,3\}$ works for any base $b$ of absolute value greater than $d$ but not equal to 4 .) Consider now positive integers $n=\left(a_{d} \ldots a_{1} a_{0}\right)_{4}$ with more than one digit, so $d \geq 1, a_{d} \neq 0$, and $0 \leq ... | deepscaler |
In a building, there are a hundred ladies on the first-floor studying. There are three times that many girls at a party being held on the second floor of the building. How many ladies are on the two floors in total? | In a building, there are a hundred ladies on the first-floor studying. There are three times that many girls at a party being held on the second floor of the building. How many ladies are on the two floors in total? | In a building, there are a hundred ladies on the first-floor studying. There are three times that many girls at a party being held on the second floor of the building. How many ladies are on the two floors in total? | 400 | 400 | If there are a hundred ladies in the living room and three times that many girls at the party, there are 3*100=<<3*100=300>>300 girls at the party.
In the building there are 100+300=<<100+300=400>>400 ladies.
#### 400 | gsm8k |
Van Helsing gets paid by the town to remove all the vampires and werewolves. He gets $5 per vampire and $10 per werewolf. He removes half the vampires and removes 8 werewolves, and earned $105. There were 4 times as many werewolves as vampires. What percentage of the werewolves did he remove? | Van Helsing gets paid by the town to remove all the vampires and werewolves. He gets $5 per vampire and $10 per werewolf. He removes half the vampires and removes 8 werewolves, and earned $105. There were 4 times as many werewolves as vampires. What percentage of the werewolves did he remove? | Van Helsing gets paid by the town to remove all the vampires and werewolves. He gets $5 per vampire and $10 per werewolf. He removes half the vampires and removes 8 werewolves, and earned $105. There were 4 times as many werewolves as vampires. What percentage of the werewolves did he remove? | 20 | 20 | He earns $80 from werewolf removal because 8 x 10 = <<8*10=80>>80
He earned $25 from vampire removal because 105 - 80 = <<105-80=25>>25
He removed 5 vampires from town because 25 / 5 = <<25/5=5>>5
There were 10 vampires because 5 / .5 = 10
There were 40 werewolves because 4 x 10 = <<4*10=40>>40
The proportion of werewo... | gsm8k |
Given that \(a, b, c\) are non-zero rational numbers and satisfy \(a b^{2}=\frac{c}{a}-b\), then \[\left(\frac{a^{2} b^{2}}{c^{2}}-\frac{2}{c}+\frac{1}{a^{2} b^{2}}+\frac{2 a b}{c^{2}}-\frac{2}{a b c}\right) \div\left(\frac{2}{a b}-\frac{2 a b}{c}\right) \div \frac{101}{c}=\] | Given that \(a, b, c\) are non-zero rational numbers and satisfy \(a b^{2}=\frac{c}{a}-b\), then \[\left(\frac{a^{2} b^{2}}{c^{2}}-\frac{2}{c}+\frac{1}{a^{2} b^{2}}+\frac{2 a b}{c^{2}}-\frac{2}{a b c}\right) \div\left(\frac{2}{a b}-\frac{2 a b}{c}\right) \div \frac{101}{c}=\] | Given that \(a, b, c\) are non-zero rational numbers and satisfy \(a b^{2}=\frac{c}{a}-b\), then \[\left(\frac{a^{2} b^{2}}{c^{2}}-\frac{2}{c}+\frac{1}{a^{2} b^{2}}+\frac{2 a b}{c^{2}}-\frac{2}{a b c}\right) \div\left(\frac{2}{a b}-\frac{2 a b}{c}\right) \div \frac{101}{c}=\] | -\frac{1}{202} | -\frac{1}{202} | Given \(a, b, c\) are non-zero rational numbers and satisfy the equation
\[
a b^{2} = \frac{c}{a} - b
\]
We are required to find the value of
\[
\left(\frac{a^{2} b^{2}}{c^{2}} - \frac{2}{c} + \frac{1}{a^{2} b^{2}} + \frac{2 a b}{c^{2}} - \frac{2}{a b c}\right) \div \left(\frac{2}{a b} - \frac{2 a b}{c}\right) \div ... | olympiads |
When Jack traveled to Canada, he had to wait 20 hours to get through customs, plus 14 days in coronavirus quarantine. How many hours total did Jack have to wait? | When Jack traveled to Canada, he had to wait 20 hours to get through customs, plus 14 days in coronavirus quarantine. How many hours total did Jack have to wait? | When Jack traveled to Canada, he had to wait 20 hours to get through customs, plus 14 days in coronavirus quarantine. How many hours total did Jack have to wait? | 356 | 356 | First convert the quarantine wait time to hours by multiplying the number of days by the number of hours in a day: 14 days * 24 hours/day = <<14*24=336>>336 hours
Then add the time Jack spent waiting in customs to the quarantine time: 336 hours + 20 hours = <<336+20=356>>356 hours
#### 356 | gsm8k |
Each edge of a regular tetrahedron is given a stripe. The choice of which edge to stripe is made at random. What is the probability that there is at least one triangle face with all its edges striped? | Each edge of a regular tetrahedron is given a stripe. The choice of which edge to stripe is made at random. What is the probability that there is at least one triangle face with all its edges striped? | Each edge of a regular tetrahedron is given a stripe. The choice of which edge to stripe is made at random. What is the probability that there is at least one triangle face with all its edges striped? | \frac{1695}{4096} | \frac{1695}{4096} | deepscaler | |
Find the product of all $x$ such that the expression $\frac{x^2+2x+1}{x^2+2x-3}$ is undefined. | Find the product of all $x$ such that the expression $\frac{x^2+2x+1}{x^2+2x-3}$ is undefined. | Find the product of all $x$ such that the expression $\frac{x^2+2x+1}{x^2+2x-3}$ is undefined. | -3 | -3 | To find the product of all $x$ such that the expression $\frac{x^2+2x+1}{x^2+2x-3}$ is undefined, we need to determine when the denominator equals zero. This is because a fraction is undefined when its denominator is zero. So, we solve the equation:
1. Set the denominator equal to zero: $x^2+2x-3=0$.
2. To understand... | math |
If $\frac{4^x}{2^{x+y}}=8$ and $\frac{9^{x+y}}{3^{5y}}=243$, $x$ and $y$ real numbers, then $xy$ equals:
$\text{(A) } \frac{12}{5} \quad \text{(B) } 4 \quad \text{(C) } 6 \quad \text{(D)} 12 \quad \text{(E) } -4$ | If $\frac{4^x}{2^{x+y}}=8$ and $\frac{9^{x+y}}{3^{5y}}=243$, $x$ and $y$ real numbers, then $xy$ equals:
$\text{(A) } \frac{12}{5} \quad \text{(B) } 4 \quad \text{(C) } 6 \quad \text{(D)} 12 \quad \text{(E) } -4$ | If $\frac{4^x}{2^{x+y}}=8$ and $\frac{9^{x+y}}{3^{5y}}=243$, $x$ and $y$ real numbers, then $xy$ equals:
$\text{(A) } \frac{12}{5} \quad \text{(B) } 4 \quad \text{(C) } 6 \quad \text{(D)} 12 \quad \text{(E) } -4$ | 4 | 4 | 1. **Simplify the first equation:**
\[
\frac{4^x}{2^{x+y}} = 8
\]
Since $4^x = (2^2)^x = 2^{2x}$, we can rewrite the equation as:
\[
\frac{2^{2x}}{2^{x+y}} = 8
\]
Simplifying the left side using the properties of exponents:
\[
2^{2x - (x+y)} = 8
\]
\[
2^{x-y} = 8
\]
Since $8... | amc_aime |
What is $(a^3+b^3)\div(a^2-ab+b^2+c)$ for $a=7$, $b=6$, and $c=1$? | What is $(a^3+b^3)\div(a^2-ab+b^2+c)$ for $a=7$, $b=6$, and $c=1$? | What is $(a^3+b^3)\div(a^2-ab+b^2+c)$ for $a=7$, $b=6$, and $c=1$? | \frac{559}{44} | \frac{559}{44} | deepscaler | |
Let $ABCD$ be a square with side length $6$ . Circles $X, Y$ , and $Z$ are congruent circles with centers inside the square such that $X$ is tangent to both sides $\overline{AB}$ and $\overline{AD}$ , $Y$ is tangent to both sides $\overline{AB}$ and $\overline{BC}$ , and $Z$ is tangent to side $\ove... | Let $ABCD$ be a square with side length $6$ . Circles $X, Y$ , and $Z$ are congruent circles with centers inside the square such that $X$ is tangent to both sides $\overline{AB}$ and $\overline{AD}$ , $Y$ is tangent to both sides $\overline{AB}$ and $\overline{BC}$ , and $Z$ is tangent to side $\ove... | Let $ABCD$ be a square with side length $6$ . Circles $X, Y$ , and $Z$ are congruent circles with centers inside the square such that $X$ is tangent to both sides $\overline{AB}$ and $\overline{AD}$ , $Y$ is tangent to both sides $\overline{AB}$ and $\overline{BC}$ , and $Z$ is tangent to side $\ove... | 195 | 195 | 1. Let us denote the centers of circles \(X, Y, Z\) as \(X', Y', Z'\) respectively, and let the radius of each circle be \(r\). Since circle \(X\) is tangent to both sides \(\overline{AB}\) and \(\overline{AD}\), the center \(X'\) is at a distance \(r\) from both these sides. Similarly, circle \(Y\) is tangent to both ... | aops_forum |
Each of the sides of five congruent rectangles is labeled with an integer. In rectangle A, $w = 4, x = 1, y = 6, z = 9$. In rectangle B, $w = 1, x = 0, y = 3, z = 6$. In rectangle C, $w = 3, x = 8, y = 5, z = 2$. In rectangle D, $w = 7, x = 5, y = 4, z = 8$. In rectangle E, $w = 9, x = 2, y = 7, z = 0$. These five rect... | Each of the sides of five congruent rectangles is labeled with an integer. In rectangle A, $w = 4, x = 1, y = 6, z = 9$. In rectangle B, $w = 1, x = 0, y = 3, z = 6$. In rectangle C, $w = 3, x = 8, y = 5, z = 2$. In rectangle D, $w = 7, x = 5, y = 4, z = 8$. In rectangle E, $w = 9, x = 2, y = 7, z = 0$. These five rect... | Each of the sides of five congruent rectangles is labeled with an integer. In rectangle A, $w = 4, x = 1, y = 6, z = 9$. In rectangle B, $w = 1, x = 0, y = 3, z = 6$. In rectangle C, $w = 3, x = 8, y = 5, z = 2$. In rectangle D, $w = 7, x = 5, y = 4, z = 8$. In rectangle E, $w = 9, x = 2, y = 7, z = 0$. These five rect... | E | E | 1. **Identify Unique Values**: We start by identifying the unique values among the $w$ and $y$ values of each rectangle, as these values will help us determine the placement of the rectangles. The pairs $(w, y)$ for each rectangle are:
- $A(4,6)$
- $B(1,3)$
- $C(3,5)$
- $D(7,4)$
- $E(9,7)$
2. **Determin... | amc_aime |
Johnny has 7 different colored marbles in his bag. In how many ways can he choose three different marbles from his bag to play a game? | Johnny has 7 different colored marbles in his bag. In how many ways can he choose three different marbles from his bag to play a game? | Johnny has 7 different colored marbles in his bag. In how many ways can he choose three different marbles from his bag to play a game? | 35 | 35 | To solve this problem, we can use the combination formula, which allows us to calculate the number of ways to choose $k$ items from a set of $n$ distinct items without regard to the order. The formula for a combination is given by $\binom{n}{k} = \frac{n!}{k!(n-k)!}$, where $n!$ denotes the factorial of $n$, which is t... | math |
The sum of the first few terms of a geometric sequence is 11, the sum of their squares is 341, and the sum of their cubes is 3641. Determine the terms of the sequence. | The sum of the first few terms of a geometric sequence is 11, the sum of their squares is 341, and the sum of their cubes is 3641. Determine the terms of the sequence. | The sum of the first few terms of a geometric sequence is 11, the sum of their squares is 341, and the sum of their cubes is 3641. Determine the terms of the sequence. | 16, -8, 4, -2, 1 | 16, -8, 4, -2, 1 |
1. Let us denote the first term of the geometric sequence by \( a \), its common ratio by \( q \), and the number of terms by \( n \).
2. Given three conditions for the sequence:
\[
S = a \left( \frac{q^n - 1}{q - 1} \right) = 11
\]
\[
S_2 = a^2 \left( \frac{q^{2n} - 1}{q^2 - 1} \right) = 341
\]
... | olympiads |
Four identical regular tetrahedrons are thrown simultaneously on a table. Calculate the probability that the product of the four numbers on the faces touching the table is divisible by 4. | Four identical regular tetrahedrons are thrown simultaneously on a table. Calculate the probability that the product of the four numbers on the faces touching the table is divisible by 4. | Four identical regular tetrahedrons are thrown simultaneously on a table. Calculate the probability that the product of the four numbers on the faces touching the table is divisible by 4. | \frac{13}{16} | \frac{13}{16} | deepscaler | |
Bill milked his cow and got 16 gallons of milk. He turned 1/4 into sour cream, 1/4 into butter, and kept the rest as whole milk. It takes 4 gallons of milk to make one gallon of butter and 2 gallons of milk to make 1 gallon of sour cream. If Bill sells butter for $5/gallon, sour cream for $6/gallon, and whole milk for ... | Bill milked his cow and got 16 gallons of milk. He turned 1/4 into sour cream, 1/4 into butter, and kept the rest as whole milk. It takes 4 gallons of milk to make one gallon of butter and 2 gallons of milk to make 1 gallon of sour cream. If Bill sells butter for $5/gallon, sour cream for $6/gallon, and whole milk for ... | Bill milked his cow and got 16 gallons of milk. He turned 1/4 into sour cream, 1/4 into butter, and kept the rest as whole milk. It takes 4 gallons of milk to make one gallon of butter and 2 gallons of milk to make 1 gallon of sour cream. If Bill sells butter for $5/gallon, sour cream for $6/gallon, and whole milk for ... | 41 | 41 | First find how much milk Bill turned into sour cream and butter: 16 gallons * 1/4 = <<16*1/4=4>>4 gallons
Then find how many gallons of butter he makes out of 4 gallons of milk: 4 gallons milk / 4 gallons milk/1 gallon butter = <<4/4/1=1>>1 gallon butter
Then find how many gallons of sour cream he makes out of 4 gallon... | gsm8k |
Find the range of the function \( f(x) = x - 1 + \sqrt{6x - x^2} \). | Find the range of the function \( f(x) = x - 1 + \sqrt{6x - x^2} \). | Find the range of the function \( f(x) = x - 1 + \sqrt{6x - x^2} \). | [-1, 2] | [-1, 2] |
Given the function \( f(x) = x - 1 + \sqrt{6x - x^2} \), we want to find the set of values that \( f(x) \) can take.
1. **Find the domain of the function:**
The domain of the function \( f(x) \) is determined by the expression under the square root, \( 6x - x^2 \). To ensure that \(\sqrt{6x - x^2}\) is defined, ... | olympiads |
Elias uses a bar of soap every month. If each bar of soap costs $4, how much does he spend on bars of soap in two years? | Elias uses a bar of soap every month. If each bar of soap costs $4, how much does he spend on bars of soap in two years? | Elias uses a bar of soap every month. If each bar of soap costs $4, how much does he spend on bars of soap in two years? | 96 | 96 | A year is 12 months, so Elias spends 4 * 12 = $<<4*12=48>>48 per year on bars of soap.
In two years, he spends 2 * 48 = $<<2*48=96>>96 on bars of soap.
#### 96 | gsm8k |
What is $2\frac{1}{4}$ divided by $\frac{3}{5}$? | What is $2\frac{1}{4}$ divided by $\frac{3}{5}$? | What is $2\frac{1}{4}$ divided by $\frac{3}{5}$? | 3 \frac{3}{4} | 3 \frac{3}{4} | deepscaler | |
How many positive multiples of 6 that are less than 150 have a units digit of 6? | How many positive multiples of 6 that are less than 150 have a units digit of 6? | How many positive multiples of 6 that are less than 150 have a units digit of 6? | 25 | 25 | deepscaler |
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