problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
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Cameron writes down the smallest positive multiple of 30 that is a perfect square, the smallest positive multiple of 30 that is a perfect cube, and all the multiples of 30 between them. How many integers are in Cameron's list? | 871 |
After eating half of the number of fruits he had, Martin remained with twice as many oranges as limes. If he has 50 oranges now, how many fruits did he initially have? | Since the number of oranges that Martin has now is twice the number of limes, there are 50/2 = <<50/2=25>>25 limes.
The number of fruits she has now is 50 oranges+25 limes= <<50+25=75>>75
If she had eaten half of the number of fruits that she had, initially Martin had 2*75 = <<2*75=150>>150 fruits
#### 150 |
Mark collects money for the homeless. He visits 20 households a day for 5 days and half of those households give him a pair of 20s. How much did he collect? | He got money from 20/2=<<20/2=10>>10 households per day
So that means he got money from 10*5=<<10*5=50>>50 households
Each of those houses gave 20*2=$<<20*2=40>>40
So he raised 40*50=$<<40*50=2000>>2000
#### 2000 |
Given the hyperbola $\frac {x^{2}}{4}-y^{2}$=1, a line $l$ with a slope angle of $\frac {π}{4}$ passes through the right focus $F\_2$ and intersects the right branch of the hyperbola at points $M$ and $N$. The midpoint of the line segment $MN$ is $P$. Determine the vertical coordinate of point $P$. | \frac{\sqrt {5}}{3} |
ABCD is a square. BDEF is a rhombus with A, E, and F collinear. Find ∠ADE. | 15 |
Compute $\sin 225^\circ$. | -\frac{\sqrt{2}}{2} |
Two numbers are such that their difference, their sum, and their product are to one another as $1:7:24$. The product of the two numbers is: | 48 |
Given $$(5x- \frac {1}{ \sqrt {x}})^{n}$$, the sum of the binomial coefficients in its expansion is 64. Find the constant term in the expansion. | 375 |
It is known that 9 cups of tea cost less than 10 rubles, and 10 cups of tea cost more than 11 rubles. How much does one cup of tea cost? | 111 |
What is the value of $\log_{10}{16} + 3\log_{5}{25} + 4\log_{10}{2} + \log_{10}{64} - \log_{10}{8}$? | 9.311 |
What is the last two digits of the decimal representation of $9^{8^{7^{\cdot^{\cdot^{\cdot^{2}}}}}}$ ? | 21 |
What is the number halfway between $\frac{1}{12}$ and $\frac{1}{10}$? | \frac{11}{120} |
Given the function $f(x)=\frac{cos2x+a}{sinx}$, if $|f(x)|\leqslant 3$ holds for any $x\in \left(0,\pi \right)$, then the set of possible values for $a$ is ______. | \{-1\} |
Let the first term of a geometric sequence be $\frac{3}{4}$, and let the second term be $15$. What is the smallest $n$ for which the $n$th term of the sequence is divisible by one million? | 7 |
Find the number of ordered 17-tuples $(a_1, a_2, a_3, \dots, a_{17})$ of integers, such that the square of any number in the 17-tuple is equal to the sum of the other 16 numbers. | 12378 |
Raul had $87 to spare so he decided to go to the bookshop. Raul bought 8 comics, each of which cost $4. How much money does Raul have left? | He spent 8 comics × $4/comic = $<<8*4=32>>32 on comics.
Raul has $87 - $32 = $<<87-32=55>>55 left.
#### 55 |
In the spelling bee, Max has 5 points, Dulce has 3 points, and Val has twice the combined points of Max and Dulce. If they are on the same team and their opponents' team has a total of 40 points, how many points are their team behind? | Max and Dulce's total combined points are 5 + 3 = <<5+3=8>>8.
So, Val got 8 x 2 = <<8*2=16>>16 points.
And the total points of their team is 8 + 16 = <<8+16=24>>24.
Therefore, their team is 40 - 24 = <<40-24=16>>16 points behind.
#### 16 |
How many three-digit numbers remain if we exclude all three-digit numbers in which all digits are the same or the middle digit is different from the two identical end digits? | 810 |
Determine the share of gold in the currency structure of the National Welfare Fund (NWF) as of December 1, 2022, using one of the following methods:
First Method:
a) Find the total amount of NWF funds allocated in gold as of December 1, 2022:
\[GOLD_{22} = 1388.01 - 41.89 - 2.77 - 478.48 - 309.72 - 0.24 = 554.91 \, (\text{billion rubles})\]
b) Determine the share of gold in the currency structure of NWF funds as of December 1, 2022:
\[\alpha_{22}^{GOLD} = \frac{554.91}{1388.01} \approx 39.98\% \]
c) Calculate by how many percentage points and in which direction the share of gold in the currency structure of NWF funds changed over the review period:
\[\Delta \alpha^{GOLD} = \alpha_{22}^{GOLD} - \alpha_{21}^{GOLD} = 39.98 - 31.8 = 8.18 \approx 8.2 \, (\text{p.p.})\]
Second Method:
a) Determine the share of euro in the currency structure of NWF funds as of December 1, 2022:
\[\alpha_{22}^{EUR} = \frac{41.89}{1388.01} \approx 3.02\% \]
b) Determine the share of gold in the currency structure of NWF funds as of December 1, 2022:
\[\alpha_{22}^{GOLD} = 100 - 3.02 - 0.2 - 34.47 - 22.31 - 0.02 = 39.98\%\]
c) Calculate by how many percentage points and in which direction the share of gold in the currency structure of NWF funds changed over the review period:
\[\Delta \alpha^{GOLD} = \alpha_{22}^{GOLD} - \alpha_{21}^{GOLD} = 39.98 - 31.8 = 8.18 \approx 8.2 \, (\text{p.p.})\] | 8.2 |
Kay has 14 siblings. Kay is 32 years old. The youngest sibling is 5 less than half Kay's age. The oldest sibling is four times as old as the youngest sibling. How old is the oldest sibling? | Kay 32 years old
Youngest:32/2-5=<<32/2-5=11>>11 years old
Oldest:11(4)=44 years old
#### 44 |
Let \( O \) be an interior point of \( \triangle ABC \). Extend \( AO \) to meet \( BC \) at \( D \). Similarly, extend \( BO \) and \( CO \) to meet \( CA \) and \( AB \) respectively at \( E \) and \( F \). Given \( AO = 30 \), \( FO = 20 \), \( BO = 60 \), \( DO = 10 \) and \( CO = 20 \), find \( EO \). | 20 |
Let $A B C D$ be a square of side length 5. A circle passing through $A$ is tangent to segment $C D$ at $T$ and meets $A B$ and $A D$ again at $X \neq A$ and $Y \neq A$, respectively. Given that $X Y=6$, compute $A T$. | \sqrt{30} |
A sequence of numbers is arranged in the following pattern: \(1, 2, 3, 2, 3, 4, 3, 4, 5, 4, 5, 6, \cdots\). Starting from the first number on the left, find the sum of the first 99 numbers. | 1782 |
The equation of the line joining the complex numbers $-2 + 3i$ and $1 + i$ can be expressed in the form
\[az + b \overline{z} = 10\]for some complex numbers $a$ and $b$. Find the product $ab$. | 13 |
Given that the equation $2kx+2m=6-2x+nk$ has a solution independent of $k$, the value of $4m+2n$ is ______. | 12 |
Let \(x\), \(y\), and \(z\) be complex numbers such that:
\[
xy + 3y = -9, \\
yz + 3z = -9, \\
zx + 3x = -9.
\]
Find all possible values of \(xyz\). | 27 |
36 liters of diesel fuel is worth €18. The tank of this pickup truck can hold 64 liters. How much does a full tank of diesel fuel cost? | A liter of diesel costs 18/ 36 = <<18/36=0.5>>0.5 €.
64 liters of diesel fuel cost 0.5 X 64 = <<64*0.5=32>>32 €
#### 32 |
Express .$\overline{28}$ as a common fraction. | \frac{28}{99} |
Find the sum: $(-39) + (-37) + \cdots + (-1)$. | -400 |
For a positive integer $n$ , define $n?=1^n\cdot2^{n-1}\cdot3^{n-2}\cdots\left(n-1\right)^2\cdot n^1$ . Find the positive integer $k$ for which $7?9?=5?k?$ .
*Proposed by Tristan Shin* | 10 |
Determine the length of $BC$ in an acute triangle $ABC$ with $\angle ABC = 45^{\circ}$ , $OG = 1$ and $OG \parallel BC$ . (As usual $O$ is the circumcenter and $G$ is the centroid.) | 12 |
The sum of 36 consecutive integers is $6^4$. What is their median? | 36 |
There are $n\geq 3$ cities in a country and between any two cities $A$ and $B$ , there is either a one way road from $A$ to $B$ , or a one way road from $B$ to $A$ (but never both). Assume the roads are built such that it is possible to get from any city to any other city through these roads, and define $d(A,B)$ to be the minimum number of roads you must go through to go from city $A$ to $B$ . Consider all possible ways to build the roads. Find the minimum possible average value of $d(A,B)$ over all possible ordered pairs of distinct cities in the country. | 3/2 |
Let $N$ denote the number of permutations of the $15$-character string $AAAABBBBBCCCCCC$ such that
None of the first four letters is an $A$.
None of the next five letters is a $B$.
None of the last six letters is a $C$.
Find the remainder when $N$ is divided by $1000$.
| 320 |
Let $\mathrm {Q}$ be the product of the roots of $z^8+z^6+z^4+z^3+z+1=0$ that have a positive imaginary part, and suppose that $\mathrm {Q}=s(\cos{\phi^{\circ}}+i\sin{\phi^{\circ}})$, where $0<s$ and $0\leq \phi <360$. Find $\phi$. | 180 |
If $\frac{1}{x} + \frac{1}{y} = 3$ and $\frac{1}{x} - \frac{1}{y} = -7$ what is the value of $x + y$? Express your answer as a common fraction. | -\frac{3}{10} |
There are three buckets, X, Y, and Z. The average weight of the watermelons in bucket X is 60 kg, the average weight of the watermelons in bucket Y is 70 kg. The average weight of the watermelons in the combined buckets X and Y is 64 kg, and the average weight of the watermelons in the combined buckets X and Z is 66 kg. Calculate the greatest possible integer value for the mean in kilograms of the watermelons in the combined buckets Y and Z. | 69 |
Calculate the following expression:
\[\left( 1 - \frac{1}{\cos 30^\circ} \right) \left( 1 + \frac{1}{\sin 60^\circ} \right) \left( 1 - \frac{1}{\sin 30^\circ} \right) \left( 1 + \frac{1}{\cos 60^\circ} \right).\] | -1 |
To make a living, Carl needs to drive a car for 2 hours every day. After he got promoted he needs to drive for 6 more hours every week. How many hours will Carl drive in two weeks? | Every week Carl needs to drive additional 6 hours, so in two weeks that would make 2 * 6 = <<2*6=12>>12 additional hours.
Two weeks of driving 2 hours every day, means 14 * 2 = <<14*2=28>>28 hours of driving.
So during two weeks, Carl will drive 28 + 12 = <<28+12=40>>40 hours.
#### 40 |
Tommy has 10 more sheets than Jimmy does. If Jimmy has 32 sheets, how many more sheets will Jimmy have than Tommy if his friend Ashton gives him 40 sheets. | If Jimmy has 32 sheets, and Tommy has 10 more sheets than Jimmy does, Tommy has 10+32 = <<32+10=42>>42 sheets.
When Ashton gives Jimmy 40 sheets, his total number of sheets increases to 32+ 40 = <<32+40=72>>72
The number of sheets that Jimmy now have than Tommy is 72-42 = <<72-42=30>>30
#### 30 |
Reggie's father gave him $48. Reggie bought 5 books, each of which cost $2. How much money does Reggie have left? | The cost of the books is 5 × $2 = $<<5*2=10>>10.
Reggie has $48 − $10 = $38 left.
#### 38 |
Brad's car broke down on an isolated road. At the time, Brad was traveling with Jim and Marcus. So, the three decided to push the car back into town, which was 10 miles away. For the first three miles, Brad steered as Jim and Marcus pushed at a speed of 6 miles per hour. Then, for the next 3 miles, Jim steered, as Brad and Marcus pushed at a speed of 3 miles per hour. For the last four miles, Marcus steered as Brad and Jim pushed at a speed of 8 miles per hour. How long did it take, in hours, to push the car back to town? | Three miles at 6 miles per hour is 3/6=1/2 hours.
Then, 3 miles at 3 miles per hour is 3/3=<<3/3=1>>1 hour.
The last 4 miles at 8 miles per hour is 4/8=1/2 hour.
In total, it took them 1/2 + 1 + 1/2 = <<1/2+1+1/2=2>>2 hours to push the car back to town.
#### 2 |
Given that in triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Angle $B$ is obtuse. Let the area of $\triangle ABC$ be $S$. If $4bS=a(b^{2}+c^{2}-a^{2})$, then the maximum value of $\sin A + \sin C$ is ____. | \frac{9}{8} |
Farmer Randy has 1700 acres of cotton he needs to have planted in 5 days. With a crew of 2 tractors working for 2 days and then a crew of 7 tractors working for another 3 days, how many acres of cotton per day does each tractor need to plant to meet their planting deadline? | With 2 tractor teams working for 2 days, this is like 2*2=<<2*2=4>>4 days worth of work by a single tractor, which they call tractor days.
With 5 tractor teams working for 4 days, this is 7*3=21 tractor days.
In total, the effort of all the tractor teams is 4+21=<<4+21=25>>25 tractor days.
If they need to plant 1700 acres with their 25 tractor days, then each tractor team needs to plant 1700/25 = <<1700/25=68>>68 acres of cotton per day.
#### 68 |
Four chess players - Ivanov, Petrov, Vasiliev, and Kuznetsov - played a round-robin tournament (each played one game against each of the others). A victory awards 1 point, a draw awards 0.5 points to each player. It was found that the player in first place scored 3 points, and the player in last place scored 0.5 points. How many possible distributions of points are there among the named chess players, if some of them could have scored the same number of points? (For example, the cases where Ivanov has 3 points and Petrov has 0.5 points, and where Petrov has 3 points and Ivanov has 0.5 points, are considered different!) | 36 |
Point \( M \) lies on side \( BC \) of parallelogram \( ABCD \) with a \(45^{\circ}\) angle at vertex \( A \), such that \(\angle AMD = 90^{\circ}\) and the ratio \( BM : MC = 2 : 3 \). Find the ratio of the adjacent sides of the parallelogram. | \frac{2\sqrt{2}}{5} |
Recall that the conjugate of the complex number $w = a + bi$, where $a$ and $b$ are real numbers and $i = \sqrt{-1}$, is the complex number $\overline{w} = a - bi$. For any complex number $z$, let $f(z) = 4i\overline{z}$. The polynomial $P(z) = z^4 + 4z^3 + 3z^2 + 2z + 1$ has four complex roots: $z_1$, $z_2$, $z_3$, and $z_4$. Let $Q(z) = z^4 + Az^3 + Bz^2 + Cz + D$ be the polynomial whose roots are $f(z_1)$, $f(z_2)$, $f(z_3)$, and $f(z_4)$, where the coefficients $A,$ $B,$ $C,$ and $D$ are complex numbers. What is $B + D?$ | 208 |
Given the variance of a sample is $$s^{2}= \frac {1}{20}[(x_{1}-3)^{2}+(x_{2}-3)^{2}+\ldots+(x_{n}-3)^{2}]$$, then the sum of this set of data equals \_\_\_\_\_\_. | 60 |
A fair six-sided die with faces numbered 1, 2, 3, 4, 5, and 6 is rolled twice. Let $a$ and $b$ denote the outcomes of the first and second rolls, respectively.
(1) Find the probability that the line $ax + by + 5 = 0$ is tangent to the circle $x^2 + y^2 = 1$.
(2) Find the probability that the segments with lengths $a$, $b$, and $5$ form an isosceles triangle. | \frac{7}{18} |
The parabola $y^2 = 12x$ and the circle $x^2 + y^2 - 4x - 6y = 0$ intersect at two points $C$ and $D$. Find the distance $CD$. | 3\sqrt{5} |
Given a pyramid-like structure with a rectangular base consisting of $4$ apples by $7$ apples, each apple above the first level resting in a pocket formed by four apples below, and the stack topped off with a single row of apples, determine the total number of apples in the stack. | 60 |
Marcy is the lunch monitor in an elementary school cafeteria. She gives 5 time-outs for running, 1 less than five times that number of time-outs for throwing food, and 1/3 the number of food-throwing time-outs for swearing. If each time-out is 5 minutes, how much time do the students spend in time-out total? | First multiply the number of running time-outs by 5: 5 time-outs * 5 = <<5*5=25>>25 time-outs
Then subtract 1 from that number to find the number of food-throwing time-outs: 25 time-outs - 1 = <<25-1=24>>24 time-outs
Then divide that number by 3 to find the number of swearing time-outs: 24 time-outs / 3 = <<24/3=8>>8 time-outs
Then add the number of each kind of time-out to find the total number: 8 time-outs + 24 time-outs + 5 time-outs = <<8+24+5=37>>37 time-outs
Then multiply that number by the length of each time-out to find the total time the students spend in time-out: 37 time-outs * 5 minutes/time-out = <<37*5=185>>185 minutes
#### 185 |
The graph of the rational function $\frac{2x^6+3x^5 - x^2 - 1}{q(x)}$ has a horizontal asymptote. What is the smallest possible degree of $q(x)$? | 6 |
I am playing a walking game with myself. On move 1, I do nothing, but on move $n$ where $2 \le n \le 25$, I take one step forward if $n$ is prime and two steps backwards if the number is composite. After all 25 moves, I stop and walk back to my original starting point. How many steps long is my walk back? | 21 |
Josh wants to build a square sandbox that is 3 ft long, 3 ft wide for his son. He can buy sand in 3 sq ft bags for $4.00 a bag. How much will it cost him to fill up the sandbox? | The sandbox is 3' long, 3' wide so the sq footage of the sandbox is 3*3 = <<3*3=9>>9 sq ft
He needs 9 sq ft of sand to fill the sandbox and he can buy it in 3 sq ft bags so he needs 9/3 = <<9/3=3>>3 bags of sand
The sand costs $4.00 a bag and he needs 3 bags so it will cost him 4*3 = $<<4*3=12.00>>12.00 in sand
#### 12 |
Find all $t$ such that $x-t$ is a factor of $6x^2+13x-5.$
Enter your answer as a list separated by commas. | -\frac{5}{2} |
In an office at various times during the day, the boss gives the secretary a letter to type, each time putting the letter on top of the pile in the secretary's inbox. When there is time, the secretary takes the top letter off the pile and types it. There are nine letters to be typed during the day, and the boss delivers them in the order $1, 2, 3, 4, 5, 6, 7, 8, 9$.
While leaving for lunch, the secretary tells a colleague that letter $8$ has already been typed but says nothing else about the morning's typing. The colleague wonders which of the nine letters remain to be typed after lunch and in what order they will be typed. Based on the above information, how many such after-lunch typing orders are possible? (That there are no letters left to be typed is one of the possibilities.) | 704 |
It took $4$ days for $75$ workers, all working together at the same rate, to build an embankment. If only $50$ workers had been available, how many total days would it have taken to build the embankment? | 6 |
Points $M$ and $N$ are located on side $AC$ of triangle $ABC$, and points $K$ and $L$ are on side $AB$, with $AM : MN : NC = 1 : 3 : 1$ and $AK = KL = LB$. It is known that the area of triangle $ABC$ is 1. Find the area of quadrilateral $KLNM$. | 7/15 |
Given lines $l_{1}$: $(3+a)x+4y=5-3a$ and $l_{2}$: $2x+(5+a)y=8$, find the value of $a$ such that the lines are parallel. | -5 |
If the line $ax+by-1=0$ ($a>0$, $b>0$) passes through the center of symmetry of the curve $y=1+\sin(\pi x)$ ($0<x<2$), find the smallest positive period for $y=\tan\left(\frac{(a+b)x}{2}\right)$. | 2\pi |
Given the function $f(x) = |x^2 + bx|$ ($b \in \mathbb{R}$), when $x \in [0, 1]$, find the minimum value of the maximum value of $f(x)$. | 3-2\sqrt{2} |
Given a triangle \(ABC\) with sides opposite to the angles \(A\), \(B\), and \(C\) being \(a\), \(b\), and \(c\) respectively, and it is known that \( \sqrt {3}\sin A-\cos (B+C)=1\) and \( \sin B+\sin C= \dfrac {8}{7}\sin A\) with \(a=7\):
(Ⅰ) Find the value of angle \(A\);
(Ⅱ) Calculate the area of \( \triangle ABC\). | \dfrac {15 \sqrt {3}}{4} |
If $\log_{10}{m}= b-\log_{10}{n}$, then $m=$ | \frac{10^{b}}{n} |
In triangle $ABC,$ $\angle B = 30^\circ,$ $AB = 150,$ and $AC = 50 \sqrt{3}.$ Find the sum of all possible values of $BC.$ | 150 \sqrt{3} |
Given the polar equation of curve C is $\rho - 6\cos\theta + 2\sin\theta + \frac{1}{\rho} = 0$, with the pole as the origin of the Cartesian coordinate system and the polar axis as the positive x-axis, establish a Cartesian coordinate system in the plane xOy. The line $l$ passes through point P(3, 3) with an inclination angle $\alpha = \frac{\pi}{3}$.
(1) Write the Cartesian equation of curve C and the parametric equation of line $l$;
(2) Suppose $l$ intersects curve C at points A and B, find the value of $|AB|$. | 2\sqrt{5} |
A certain product has a cost price of $40$ yuan per unit. When the selling price is $60$ yuan per unit, 300 units can be sold per week. It is now necessary to reduce the price for clearance. According to market research, for every $1$ yuan reduction in price, an additional 20 units can be sold per week. Answer the following questions under the premise of ensuring profitability:
1. If the price reduction per unit is $x$ yuan and the profit from selling the goods per week is $y$ yuan, write the function relationship between $y$ and $x$, and determine the range of values for the independent variable $x$.
2. How much should the price be reduced by to maximize the profit per week? What is the maximum profit? | 6125 |
The last 5 digits of $99 \times 10101 \times 111 \times 1001001$ are _____. | 88889 |
Find the constant $t$ such that \[(5x^2 - 6x + 7)(4x^2 +tx + 10) = 20x^4 -54x^3 +114x^2 -102x +70.\] | -6 |
Cornelia likes to travel. She visited already 42 different countries. 20 of them were in Europe and 10 in South America. From the rest of the countries, only half of them were in Asia. How many Asian countries has Cornelia visited? | Cornelia visited 42 countries - 20 countries - 10 countries = <<42-20-10=12>>12 countries which were outside of Europe and South America.
Half of those 12 countries were in Asia, so there were 12 countries / 2 = <<12/2=6>>6 Asian countries that Cornelia visited.
#### 6 |
In the Cartesian coordinate system $(xOy)$, the parametric equations of line $l$ are given by $\begin{cases}x=1+\frac{\sqrt{2}}{2}t\\y=\frac{\sqrt{2}}{2}t\end{cases}$ (where $t$ is the parameter), and in the polar coordinate system with the origin $O$ as the pole and the $x$-axis as the polar axis, the polar equation of curve $C$ is $\rho=4\sin\theta$.
(1) Find the Cartesian equation of line $l$ and the polar equation of curve $C$.
(2) Let $M$ be a moving point on curve $C$, and $P$ be the midpoint of $OM$. Find the minimum distance from point $P$ to line $l$. | \sqrt{2}-1 |
Find the angle of inclination of the tangent line to the curve $y= \frac {1}{2}x^{2}-2x$ at the point $(1,- \frac {3}{2})$. | \frac{3\pi}{4} |
Given the function $f(x)=4\cos x\sin \left(x+ \dfrac{\pi}{6} \right)$.
$(1)$ Find the smallest positive period of $f(x)$;
$(2)$ Find the maximum and minimum values of $f(x)$ in the interval $\left[- \dfrac{\pi}{6}, \dfrac{\pi}{4} \right]$. | -1 |
In the set of numbers 1, 2, 3, 4, 5, select an even number a and an odd number b to form a vector $\overrightarrow{a} = (a, b)$ with the origin as the starting point. From all the vectors obtained with the origin as the starting point, select any two vectors as adjacent sides to form a parallelogram. Let the total number of parallelograms formed be n, and among them, let the number of parallelograms with an area not exceeding 4 be m. Calculate the value of $\frac{m}{n}$. | \frac{1}{3} |
Gumball was counting his water intake for the previous week. He checked his list and saw that he had drank 60 liters of water for the week. He drank nine liters of water on Monday, Thursday and Saturday and 8 liters of water on Tuesday, Friday and Sunday. Unfortunately, no data was input on Wednesday. How many liters of water did he drink on Wednesday? | Gumball's total water intake on Monday, Thursday, and Saturday is 9 x 3 = <<9*3=27>>27 liters.
And his total water intake on Tuesday, Friday, and Sunday is 8 x 3 = <<8*3=24>>24 liters.
So his total water intake for 6 days is 24 + 27 = <<24+27=51>>51 liters.
Therefore Gumball's water intake on Wednesday is 60 - 51 = <<60-51=9>>9 liters.
#### 9 |
Calculate the definite integral:
$$
\int_{0}^{2} \frac{(4 \sqrt{2-x}-\sqrt{3 x+2}) d x}{(\sqrt{3 x+2}+4 \sqrt{2-x})(3 x+2)^{2}}
$$ | \frac{1}{32} \ln 5 |
Remove all positive integers that are divisible by 7, and arrange the remaining numbers in ascending order to form a sequence $\{a_n\}$. Calculate the value of $a_{100}$. | 116 |
Solve $\log_4 x + \log_2 x^2 = 10$. | 16 |
Solve for $r$: \[\frac{r-45}{2} = \frac{3-2r}{5}.\] | \frac{77}{3} |
Claudia has 12 coins, each of which is a 5-cent coin or a 10-cent coin. There are exactly 17 different values that can be obtained as combinations of one or more of her coins. How many 10-cent coins does Claudia have? | 6 |
If $P = 3012 \div 4$, $Q = P \div 2$, and $Y = P - Q$, then what is the value of $Y$? | 376.5 |
Find $d$, given that $\lfloor d\rfloor$ is a solution to \[3x^2 + 19x - 70 = 0\] and $\{d\} = d - \lfloor d\rfloor$ is a solution to \[4x^2 - 12x + 5 = 0.\] | -8.5 |
Let $q(x)=q^{1}(x)=2x^{2}+2x-1$, and let $q^{n}(x)=q(q^{n-1}(x))$ for $n>1$. How many negative real roots does $q^{2016}(x)$ have? | \frac{2017+1}{3} |
Two players play alternately on a $ 5 \times 5$ board. The first player always enters a $ 1$ into an empty square and the second player always enters a $ 0$ into an empty square. When the board is full, the sum of the numbers in each of the nine $ 3 \times 3$ squares is calculated and the first player's score is the largest such sum. What is the largest score the first player can make, regardless of the responses of the second player? | 6 |
Given four non-coplanar points \(A, B, C, D\) in space where the distances between any two points are distinct, consider a plane \(\alpha\) that satisfies the following properties: The distances from three of the points \(A, B, C, D\) to \(\alpha\) are equal, while the distance from the fourth point to \(\alpha\) is twice the distance of one of the three aforementioned points. Determine the number of such planes \(\alpha\). | 32 |
Kylie has 5 daisies. Her sister gave her another 9 daisies. Kylie then gave half her daisies to her mother. How many daisies does Kylie have left? | Kylie had 5 + 9 = <<5+9=14>>14 daisies.
She has 14/2 = <<14/2=7>>7 daisies left.
#### 7 |
Lucas, Emma, and Noah collected shells at the beach. Lucas found four times as many shells as Emma, and Emma found twice as many shells as Noah. Lucas decides to share some of his shells with Emma and Noah so that all three will have the same number of shells. What fraction of his shells should Lucas give to Emma? | \frac{5}{24} |
The altitudes of an acute-angled triangle \( ABC \) drawn from vertices \( B \) and \( C \) are 7 and 9, respectively, and the median \( AM \) is 8. Points \( P \) and \( Q \) are symmetric to point \( M \) with respect to sides \( AC \) and \( AB \), respectively. Find the perimeter of the quadrilateral \( APMQ \). | 32 |
In a certain ellipse, the endpoints of the major axis are $(-11,4)$ and $(9,4).$ Also, the ellipse passes through the point $(7,7).$ Find the area of the ellipse. | 50 \pi |
Kim has 4 dozen shirts. She lets her sister have 1/3 of them. How many shirts does she have left? | She had 4*12=<<4*12=48>>48 shirts to start with
She gives away 48/3=<<48/3=16>>16
So she has 48-16=<<48-16=32>>32 left
#### 32 |
If I roll 7 standard 6-sided dice and multiply the number on the face of each die, what is the probability that the result is a composite number and the sum of the numbers rolled is divisible by 3? | \frac{1}{3} |
Mary is paying her monthly garbage bill for a month with exactly four weeks. The garbage company charges Mary $10 per trash bin and $5 per recycling bin every week, and Mary has 2 trash bins and 1 recycling bin. They're giving her an 18% discount on the whole bill before fines for being elderly, but also charging her a $20 fine for putting inappropriate items in a recycling bin. How much is Mary's garbage bill? | First find how much Mary pays weekly for the trash bins: $10/trash bin * 2 trash bins = $<<10*2=20>>20
Then add this to the cost of the recycling bin: $20 + $5 = $<<20+5=25>>25
Then multiply the week cost by the number of weeks per month to find the monthly cost: $25/week * 4 weeks/month = $<<25*4=100>>100/month
Now calculate Mary's senior discount: 18% * $100 = $<<18*.01*100=18>>18
Now subtract the discount and add the fine to find the total monthly cost: $100 - $18 + $20 = $<<100-18+20=102>>102
#### 102 |
Find $\lfloor |-4.2| \rfloor + |\lfloor -4.2 \rfloor|$. | 9 |
Find the volume of the solid $T$ consisting of all points $(x, y, z)$ such that $|x| + |y| \leq 2$, $|x| + |z| \leq 2$, and $|y| + |z| \leq 2$. | \frac{32}{3} |
In $\triangle ABC,$ $AB=AC=30$ and $BC=28.$ Points $G, H,$ and $I$ are on sides $\overline{AB},$ $\overline{BC},$ and $\overline{AC},$ respectively, such that $\overline{GH}$ and $\overline{HI}$ are parallel to $\overline{AC}$ and $\overline{AB},$ respectively. What is the perimeter of parallelogram $AGHI$? | 60 |
If $x$ is real, compute the maximum integer value of
\[\frac{3x^2 + 9x + 17}{3x^2 + 9x + 7}.\] | 41 |
Given that in triangle $ABC$, $\overrightarrow{A B} \cdot \overrightarrow{B C} = 3 \overrightarrow{C A} \cdot \overrightarrow{A B}$, find the maximum value of $\frac{|\overrightarrow{A C}| + |\overrightarrow{A B}|}{|\overrightarrow{B C}|}$: | $\sqrt{3}$ |
Compute
\[
\left( 1 + \sin \frac {\pi}{12} \right) \left( 1 + \sin \frac {5\pi}{12} \right) \left( 1 + \sin \frac {7\pi}{12} \right) \left( 1 + \sin \frac {11\pi}{12} \right).
\] | \frac{1}{16} |
In the figure, polygons $A$, $E$, and $F$ are isosceles right triangles; $B$, $C$, and $D$ are squares with sides of length $1$; and $G$ is an equilateral triangle. The figure can be folded along its edges to form a polyhedron having the polygons as faces. The volume of this polyhedron is | 5/6 |
Given that the function $f(x)$ is an even function with a period of $2$, and when $x \in (0,1)$, $f(x) = 2^x - 1$, find the value of $f(\log_{2}{12})$. | -\frac{2}{3} |
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