omrisap/300K-numina-math-ds · Datasets at Hugging Face

problem stringlengths 10 5.02k	final_answer stringlengths 0 202
Given that the graph of the function $y=f(x)$ is symmetric to curve $C$ about the $y$-axis, and after shifting curve $C$ 1 unit to the left, we obtain the graph of the function $y=\log_{2}(-x-a)$. If $f(3)=1$, then the real number $a=$ ．	2
Let us call a ticket with a number from 000000 to 999999 excellent if the difference between some two neighboring digits of its number is 5. Find the number of excellent tickets.	409510
Let the curve $y=x^{n+1}$ ($n\in\mathbb{N}^{+}$) have a tangent line at the point $(1,1)$ that intersects the $x$-axis at the point with the $x$-coordinate $x_{n}$. Find the value of $\log _{2015}x_{1}+\log _{2015}x_{2}+\ldots+\log _{2015}x_{2014}$.	-1
If a person walks at 15 km/hr instead of 10 km/hr, he would have walked 20 km more. What is the actual distance traveled by him?	40
$PQRS$ is a rectangle whose area is 20 square units. Points $T$ and $U$ are positioned on $PQ$ and $RS$ respectively, dividing side $PQ$ into a ratio of 1:4 and $RS$ into a ratio of 4:1. Determine the area of trapezoid $QTUS$.	10
A normal lemon tree produces some lemons per year. Jim has specially engineered lemon trees that produce 50% more lemons per year. He has a grove that is 50 trees by 30 trees. He produces 675,000 lemons in 5 years. How many lemons does a normal lemon tree produce per year?	60
Three trains A, B, and C travel the same distance without stoppages at average speeds of 80 km/h, 100 km/h, and 120 km/h, respectively. When including stoppages, they cover the same distance at average speeds of 60 km/h, 75 km/h, and 90 km/h, respectively. The stoppage durations vary for each train. For train A, stoppages last for x minutes per hour, for train B, stoppages last for y minutes per hour, and for train C, stoppages last for z minutes per hour. Find the values of x, y, and z for each train.	15
Mathematicians found that when studying the reciprocals of the numbers $15$, $12$, and $10, it was discovered that $\frac{1}{12}-\frac{1}{15}=\frac{1}{10}-\frac{1}{12}$. Therefore, they named three numbers with this property as harmonic numbers, such as $6$, $3$, and $2$. Now, given a set of harmonic numbers: $x$, $5$, $3$ $(x>5)$, what is the value of $x$?	15
A cube with an edge length of 5 units has the same volume as a square-based pyramid whose base edge length is 10 units and height is $h$ units. What is the value of $h$?	3.75
The moon has a surface area that is 1/5 that of Earth. The surface area of the Earth is 200 square acres. The land on the moon is worth 6 times that of the land on the Earth. If the total value of all the land on the earth is 80 billion dollars, what is the total value in billions of all the land on the moon?	96
Given a set of positive real numbers $\left(x_{1}, x_{2}, \cdots, x_{n}\right)$ that satisfy the inequalities $$ \begin{array}{l} x_{1}^{2}+x_{2}^{2}+\cdots+x_{k}^{2}<\frac{1}{2}\left(x_{1}+x_{2}+\cdots+x_{k}\right), \\ x_{1}+x_{2}+\cdots+x_{k}<\frac{1}{2}\left(x_{1}^{3}+x_{2}^{3}+\cdots+x_{k}^{3}\right), \end{array} $$ find the minimum value of \(k\) that meets these conditions.	516
We are going to use 2 of our 3 number cards 1, 2, and 6 to create a two-digit number. Find the smallest possible multiple of 3.	12
The function $f(x)$ satisfies \[b^2 f(a) = a^2 f(b)\] for all real numbers $a$ and $b.$ If $f(2) \neq 0,$ find \[\frac{f(6) - f(3)}{f(2)}.\]	6.75
Working at a constant rate, P can finish a job in 4 hours. Q, also working at a constant rate, can finish the same job in 20 hours. They work together for 3 hours. How many more minutes will it take P to finish the job, working alone at his constant rate?	24
James decides to make a bathtub full of jello. For every pound of water, you need some tablespoons of jello mix. The bathtub can hold 6 cubic feet of water. Each cubic foot of water is 7.5 gallons. A gallon of water weighs 8 pounds. A tablespoon of jello mix costs $0.50. He spent $270 to fill his tub. How many tablespoons of jello mix are needed for every pound of water?	1.5
The Wholesome Bakery baked 5 loaves of bread on Wednesday, 7 loaves of bread on Thursday, some loaves of bread on Friday, 14 loaves of bread on Saturday, and 19 loaves of bread on Sunday. If this pattern continues, they will bake 25 loaves of bread on Monday. How many loaves of bread did they bake on Friday?	10
Every bag of Sweetsies (a fruit candy) contains the same number of pieces. The Sweetsies in one bag can't be divided equally among $8$ kids, because after each kid gets the same (whole) number of pieces, $5$ pieces are left over. If the Sweetsies in four bags are divided equally among $8$ kids, what is the smallest number of pieces that could possibly be left over?	4
the arithmetic mean and standard deviation of a certain normal distribution are 15.5 and 1.5 , respectively . what value is exactly 2 standard deviations less than the mean ?	12.5
Given a rectangular parallelepiped \( A B C D A_1 B_1 C_1 D_1 \) with dimensions \( A B = 4 \), \( A D = A A_1 = 14 \). Point \( M \) is the midpoint of edge \( C C_1 \). Find the area of the section of the parallelepiped by the plane passing through points \( A_1, D \), and \( M \).	42
Given the function $f(x)=2\sin x\cos x+\cos 2x$. - (I) Find the smallest positive period of $f(x)$ and the intervals of monotonic increase; - (II) Find the maximum and minimum values of $f(x)$ in the interval $\left[0, \frac{\pi}{2}\right]$.	-1
(1) Calculate: $\int_{-2}^{2}{\sqrt{4-{x}^{2}}}dx=$\_\_\_\_\_\_\_\_\_\_\_\_. (2) If $f(x)+\int_{0}^{1}{f(x)dx=2x}$, then $f(x)=$\_\_\_\_\_\_\_\_\_\_\_\_. (3) Arranging $3$ volunteers to complete $4$ tasks, with each person completing at least one task and each task being completed by one person, the total number of different arrangements is \_\_\_\_\_\_\_\_\_\_. (4) Let the function $f(x)=x\ln({e}^{x}+1)- \frac{1}{2}{x}^{2}+3, x\in[-t,t](t > 0)$, if the maximum value of the function $f(x)$ is $M$ and the minimum value is $m$, then $M+m=$\_\_\_\_\_\_\_\_\_\_\_\_\_.	6
It was Trevor's job to collect fresh eggs from the family's 4 chickens every morning. He got some eggs from Gertrude, 3 eggs from Blanche, 2 eggs from Nancy, and 2 eggs from Martha. On the way, he dropped 2 eggs. Trevor had 9 eggs left. How many eggs did he get from Gertrude?	4
A man is 18 years older than his son. In two years, his age will be twice the age of his son. What is the present age of his son?	16
Define the sequence $b_1, b_2, b_3, \ldots$ by $b_n = \sum\limits_{k=1}^n \cos{k}$, where $k$ is measured in radians. Find the index of the 50th term for which $b_n < 0$.	314
What is the coefficient of $x^6$ in the expansion of $(1 - 3x^3)^6$?	135
For an odd function $f(x)$ with the domain of all real numbers $\mathbb{R}$, find the value of $f(-2) + f(2)$.	0
Tom decides to take some dance lessons that cost $10 each, but he gets two of them for free. He pays $80. How many dance lessons did he take in total?	10
Of the votes cast on a certain proposal, some more were in favor of the proposal than were against it. The number of votes against the proposal was 40 percent of the total vote. The total number of votes cast was approximately 350. How many more votes were in favor of the proposal than were against it?	70
Find the number of positive integers $n$ for which (i) $n \leq 1991$ ; (ii) 6 is a factor of $(n^2 + 3n +2)$ .	1328
What is the largest perfect square factor of 1760?	4
Tennis rackets can be packaged in cartons holding 2 rackets each or in cartons holding 3 rackets each. Yesterday's packing slip showed that a certain number of cartons were used to pack a total of 100 rackets, and 24 cartons of 3 rackets size were used. How many cartons were used in total?	38
In a car dealership with some cars, 60% of the cars are hybrids, and 40% of the hybrids contain only one headlight. There are 216 hybrids with full headlights. How many cars are there in the dealership?	600
For an ellipse centered at point (2, -1) with a semi-major axis of 5 units and a semi-minor axis of 3 units, calculate the distance between the foci.	8
If \( x - y = 10 \) and \( x + y = 8 \), what is the value of \( y \)?	-1
Solve for $x$ in the equation $24 - 4 \times 2 = 3 + x$.	13
In a bag, there are 3 red balls and 2 white balls. If two balls are drawn successively without replacement, let A be the event that the first ball drawn is red, and B be the event that the second ball drawn is red. Find $p(B\|A)$.	1
The sum of two numbers is $20$ and their difference is $4$. Multiply the larger number by $3$ and then find the product of this result and the smaller number.	288
the perimeter of an equilateral triangle is 60 . if one of the sides of the equilateral triangle is the side of an isosceles triangle of perimeter 65 , then how long is the base of isosceles triangle ?	25
Given $f(x)=(1+x)+(1+x)^{2}+(1+x)^{3}+\ldots+(1+x)^{10}=a_{0}+a_{1}x+a_{2}x^{2}+\ldots+a_{10}x^{10}$, find the value of $a_{2}$.	165
What is the sum of the greatest common factor and the lowest common multiple of 36 and 56?	508
Find \(89^{-1} \pmod{90}\), as a residue modulo 90. (Give an answer between 0 and 89, inclusive.)	89
A train running at a certain speed crosses an electric pole in 12 seconds. It crosses a 320 m long platform in approximately 44 seconds. What is the speed of the train in kmph?	36
If the true discount on a sum due some years hence at 14% per annum is Rs. 168, the sum due is Rs. 768. How many years hence is the sum due?	2
Calculate the total number of pieces needed to create a 10-row triangle, where each piece comprises unit rods (which form the sides of the triangle) and connectors (which form the vertices of the triangle). A two-row triangle uses 9 unit rods and 6 connectors.	231
Shirley sold 20 boxes of Do-Si-Dos. She needs to deliver a certain number of cases of 4 boxes, plus extra boxes. How many cases does she need to deliver?	5
Find the sum of all positive integers such that their expression in base $4$ digits is the reverse of their expression in base $9$ digits, and they form a palindrome in base $10$. Express your answer in base $10$.	21
For $-49 \le x \le 49,$ find the maximum value of $\sqrt{49 + x} + \sqrt{49 - x}.$	14
Given vectors $\overrightarrow{a}=(2,1)$ and $\overrightarrow{b}=(2,x)$, if the projection vector of $\overrightarrow{b}$ in the direction of $\overrightarrow{a}$ is $\overrightarrow{a}$, then the value of $x$ is ______.	1
what is the maximum number of pieces of birthday cake of size 2 ” by 2 ” that can be cut from a cake 20 ” by 20 ” ?	100
Excluding stoppages, the speed of a bus is some km/hr, and including stoppages, it is 45 km/hr. The bus stops for 10 minutes per hour. What is the speed of the bus excluding stoppages?	54
Find $2537 + 240 \times 3 \div 60 - 347$.	2202
Form a four-digit number with no repeated digits using the numbers 1, 2, 3, and 4, where exactly one even digit is sandwiched between two odd digits. The number of such four-digit numbers is ______.	8
What is the surface area of a rectangular prism with edge lengths of 2, 3, and 4?	52
Coins denomination 1, 2, 10, 20 and 50 are available. How many ways are there of paying 100?	784
A man swims downstream 30 km and upstream 20 km taking 5 hours each time. What is the speed of the man in still water?	5
A reduction of 40% in the price of bananas would enable a man to obtain 67 more for Rs. 40. What is the reduced price per dozen?	2.87
When Betty makes cheesecake, she sweetens it with a ratio of one part sugar to four parts cream cheese, and she flavors it with one teaspoon of vanilla for every two cups of cream cheese. For every one teaspoon of vanilla, she uses two eggs. She used two cups of sugar in her latest cheesecake. How many eggs did she use?	8
Let $0 \leq a, b, c \leq 1.$ Find the maximum value of \[ \sqrt[3]{abc} + \sqrt[3]{(1-a)(1-b)(1-c)}. \]	1
There are numbers written on a board from 1 to 2021. Denis wants to choose 1010 of them so that the sum of any two selected numbers is not equal to 2021 or 2022. How many ways are there to do this?	511566
Determine the value of $$\lim_{\Delta x \to 0} \frac{f(1 - 2\Delta x) - f(1)}{\Delta x}$$ for the function $f(x) = 2\ln(3x) + 8x$.	-20
Complex numbers $u,$ $v,$ and $w$ are zeros of a polynomial $Q(z) = z^3 + 2z^2 + sz + t,$ and $\|u\|^2 + \|v\|^2 + \|w\|^2 = 350.$ The points corresponding to $u,$ $v,$ and $w$ in the complex plane are the vertices of an isosceles right triangle with hypotenuse $k.$ Find $k^2.$	525
Find the numerical value of $k$ such that \[\frac{9}{x + y} = \frac{k}{x + z} = \frac{15}{z - y}.\]	24
Given two vectors $\overrightarrow{a} = (1 - \sin\theta, 1)$ and $\overrightarrow{b} = \left(\frac{1}{2}, 1 + \sin\theta\right)$ (where $\theta$ is an acute angle), and $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$, find the value of $\tan\theta$.	1
Two trains are moving in opposite directions with speeds of 210 km/hr and 90 km/hr respectively. One train has a length of 1.10 km and they cross each other in 24 seconds. What is the length of the other train?	899.92
Daniel is building a rectangular playground from 480 feet of fencing. The fencing must cover three sides of the playground (the fourth side is bordered by Daniel's house). What is the maximum area of this playground?	28800
How many distinct four-digit positive integers have only even digits?	500
Johnny makes $8.25 per hour at his work. If he earns $16.5, how many hours did he work?	2
A quadrilateral is divided into 1000 triangles. What is the maximum number of distinct points that can be the vertices of these triangles?	1002
The numbers $x$ and $y$ are inversely proportional. When the sum of $x$ and $y$ is 50, $x$ is three times $y$. What is the value of $y$ when $x=-12$?	-39.0625
Patsy is gearing up for this weekend’s graduation. She needs to have 6 appetizers per each of her 30 guests. She’s making 3 dozen deviled eggs, 2 dozen pigs in a blanket and 2 dozen kebabs. How many more dozen appetizers does she need to make?	8
Solve for $x$: $\dfrac{1}{3} + \dfrac{1}{x} = \dfrac{2}{3}$.	3
If $f(x)$ is a function whose domain is $[-12, 12]$, and $p(x) = f\left(\frac{x}{3}\right)$, then the domain of $p(x)$ is an interval of what width?	72
George now has a palette of 10 colors. He wants to choose 3 colors to paint his room, but he insists that blue must be one of the colors. How many ways can he do this?	36
A number is interestingif it is a $6$ -digit integer that contains no zeros, its first $3$ digits are strictly increasing, and its last $3$ digits are non-increasing. What is the average of all interesting numbers?	308253
The zoo keeps 35 female (a) animals. Males outnumber females by 7. How many (a) animals are there in all?	77
Find the sum of all integers $k$ such that $\binom{29}{4} + \binom{29}{5} = \binom{30}{k}$.	30
In the Bank of Shower, a bored customer lays $n$ coins in a row. Then, each second, the customer performs ``The Process." In The Process, all coins with exactly one neighboring coin heads-up before The Process are placed heads-up (in its initial location), and all other coins are placed tails-up. The customer stops once all coins are tails-up. Define the function $f$ as follows: If there exists some initial arrangement of the coins so that the customer never stops, then $f(n) = 0$ . Otherwise, $f(n)$ is the average number of seconds until the customer stops over all initial configurations. It is given that whenever $n = 2^k-1$ for some positive integer $k$ , $f(n) > 0$ . Let $N$ be the smallest positive integer so that \[ M = 2^N \cdot \left(f(2^2-1) + f(2^3-1) + f(2^4-1) + \cdots + f(2^{10}-1)\right) \]is a positive integer. If $M = \overline{b_kb_{k-1}\cdots b_0}$ in base two, compute $N + b_0 + b_1 + \cdots + b_k$ . Proposed by Edward Wan and Brandon Wang	8
Lisa, Robert and Claire have taken photos on their school trip. Lisa has taken some times as many photos as Claire and Robert has taken 20 more photos than Claire. Claire has taken 10 photos. How many photos has Lisa taken?	3
The operation $\star$ is defined as $a \star b = a^2 \div b$. For how many integer values of $x$ will the value of $12 \star x$ be a positive integer?	15
if n is an integer , f ( n ) = f ( n - 1 ) - n and f ( 4 ) = 12 . what is the value of f ( 6 ) ?	1
At Billy's Restaurant, a group with 2 adults and some children came in to eat. If each meal cost 3 dollars, the bill was $21. How many children were in the group?	5
Tim has 22 books. Mike has 20 books. How many books do they have together ?	42
A 220 m long train running at the speed of 120 km/hr crosses another train running in opposite direction at the speed of 80 km/hr. The length of the other train is 280.04 m. How long does it take for the trains to cross each other?	9.00
What is the remainder when \(5^{303}\) is divided by \(11\)?	4
Given that $ab + bc + cd + da = 42$ and $b + d = 6$, find the value of $a + c$.	7
Two trains of different lengths are running towards each other on parallel lines at 42 kmph and 30 kmph respectively. The trains will be clear of each other in 12.998960083193344 seconds from the moment they meet. If the second train is 160 m long, how long is the first train?	99.98
During the first week of performances of a certain play, some tickets were sold, all at reduced price. During the remaining weeks of performances, 5 times as many tickets were sold at full price as were sold at reduced price. The total number of tickets sold was 25200, and 16500 of them were sold at full price. How many tickets were sold at reduced price during the first week?	8700
Prove that there exist infinitely many positive integers \( n \) such that the largest prime divisor of \( n^4 + n^2 + 1 \) is equal to the largest prime divisor of \( (n+1)^4 + (n+1)^2 + 1 \).
Given that the graph of the function $y=f(x)$ is symmetric to the graph of the function $y=2^{x+2}$ with respect to the line $y=-x$, find the value of $f(-2)$.	1
John has 20% more boxes than Jules. Jules has 5 more boxes than Joseph. Joseph has 80% fewer boxes than Stan. If Stan has 100 boxes, how many boxes does John have?	30
Suppose the sum of seven consecutive even numbers is 400. What is the smallest of these seven numbers?	406
The closest approximation of (a number × 0.004) / 0.03 is 9.237333333333334. What is the number?	69.28
Lisa, Robert, and Claire have taken photos on their school trip. Lisa has taken some multiple of the number of photos Claire has taken, and Robert has taken 16 more photos than Claire. Claire has taken 8 photos. How many photos has Lisa taken?	24
Given the hyperbola $C$: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ with an eccentricity of $\sqrt{3}$, its real axis is $AB$, and a line parallel to $AB$ intersects hyperbola $C$ at points $M$ and $N$. Calculate the product of the slopes of lines $AM$ and $AN$.	-2
A middle school's 7th-grade literary and arts representative team took a bus to a county middle school 21 kilometers away from their school to participate in a competition. When they reached place A, the driver received a phone call from the school: "Hello, is this Mr. Zhang?" "Yes, it's me." "Please return immediately to pick up the 8th-grade sports representative team and take them to the county middle school, and let the 7th-grade team walk there." After resting for 10 minutes, the 7th-grade team started walking, and the bus did not stop on its return journey. When the bus arrived at the county middle school with the 8th-grade team, the 7th-grade team also arrived at the same time. It is known that the walking speed is 4 kilometers per hour, and the bus speed is 60 kilometers per hour. Find the distance from the school to place A and the distance walked by the 7th-grade representative team.	2
A grain storage warehouse has a total of 30 bins. Some hold 20 tons of grain each, and the rest hold a certain amount of grain each. The warehouse has a capacity of 510 tons, and there are 12 20-ton bins. How many tons of grain do the remaining bins hold each?	15
Kylie makes 10 beaded necklaces on Monday and 2 beaded necklaces on Tuesday. Then Kylie makes 5 beaded bracelets and 7 beaded earrings on Wednesday. A certain number of beads are needed to make one beaded necklace. 10 beads are needed to make one beaded bracelet. 5 beads are needed to make one beaded earring. Kylie uses 325 beads in total to make her jewelry. How many beads are needed to make one beaded necklace?	20
When you play a car game, you have to choose one character from three characters, and then one car from three cars. How many cases are there?	9
Sara made a complete list of the prime numbers between 10 and 50. What is the sum of the smallest prime number and the largest prime number on her list?	58
Paige was helping her mom plant flowers, and they decided to plant in three different sections in their garden. In the first section, they planted 470 seeds; in the second section, they planted 320 seeds; and in the third section, they planted 210 seeds. If they put 10 seeds in each flower bed for the first and second sections and 8 seeds in each flower bed for the third section, how many flower beds did they have in total?	105

Given that the graph of the function $y=f(x)$ is symmetric to curve $C$ about the $y$-axis, and after shifting curve $C$ 1 unit to the left, we obtain the graph of the function $y=\log_{2}(-x-a)$. If $f(3)=1$, then the real number $a=$ ．

2

Let us call a ticket with a number from 000000 to 999999 excellent if the difference between some two neighboring digits of its number is 5. Find the number of excellent tickets.

409510

Let the curve $y=x^{n+1}$ ($n\in\mathbb{N}^{+}$) have a tangent line at the point $(1,1)$ that intersects the $x$-axis at the point with the $x$-coordinate $x_{n}$. Find the value of $\log _{2015}x_{1}+\log _{2015}x_{2}+\ldots+\log _{2015}x_{2014}$.

-1

If a person walks at 15 km/hr instead of 10 km/hr, he would have walked 20 km more. What is the actual distance traveled by him?

40

$PQRS$ is a rectangle whose area is 20 square units. Points $T$ and $U$ are positioned on $PQ$ and $RS$ respectively, dividing side $PQ$ into a ratio of 1:4 and $RS$ into a ratio of 4:1. Determine the area of trapezoid $QTUS$.

10

A normal lemon tree produces some lemons per year. Jim has specially engineered lemon trees that produce 50% more lemons per year. He has a grove that is 50 trees by 30 trees. He produces 675,000 lemons in 5 years. How many lemons does a normal lemon tree produce per year?

60

Three trains A, B, and C travel the same distance without stoppages at average speeds of 80 km/h, 100 km/h, and 120 km/h, respectively. When including stoppages, they cover the same distance at average speeds of 60 km/h, 75 km/h, and 90 km/h, respectively. The stoppage durations vary for each train. For train A, stoppages last for x minutes per hour, for train B, stoppages last for y minutes per hour, and for train C, stoppages last for z minutes per hour. Find the values of x, y, and z for each train.

15

Mathematicians found that when studying the reciprocals of the numbers $15$, $12$, and $10, it was discovered that $\frac{1}{12}-\frac{1}{15}=\frac{1}{10}-\frac{1}{12}$. Therefore, they named three numbers with this property as harmonic numbers, such as $6$, $3$, and $2$. Now, given a set of harmonic numbers: $x$, $5$, $3$ $(x>5)$, what is the value of $x$?

15

A cube with an edge length of 5 units has the same volume as a square-based pyramid whose base edge length is 10 units and height is $h$ units. What is the value of $h$?

3.75

The moon has a surface area that is 1/5 that of Earth. The surface area of the Earth is 200 square acres. The land on the moon is worth 6 times that of the land on the Earth. If the total value of all the land on the earth is 80 billion dollars, what is the total value in billions of all the land on the moon?

96

Given a set of positive real numbers $\left(x_{1}, x_{2}, \cdots, x_{n}\right)$ that satisfy the inequalities $$ \begin{array}{l} x_{1}^{2}+x_{2}^{2}+\cdots+x_{k}^{2}<\frac{1}{2}\left(x_{1}+x_{2}+\cdots+x_{k}\right), \\ x_{1}+x_{2}+\cdots+x_{k}<\frac{1}{2}\left(x_{1}^{3}+x_{2}^{3}+\cdots+x_{k}^{3}\right), \end{array} $$ find the minimum value of $k$ that meets these conditions.

516

We are going to use 2 of our 3 number cards 1, 2, and 6 to create a two-digit number. Find the smallest possible multiple of 3.

12

The function $f(x)$ satisfies \[b^2 f(a) = a^2 f(b)\] for all real numbers $a$ and $b.$ If $f(2) \neq 0,$ find \[\frac{f(6) - f(3)}{f(2)}.\]

6.75

Working at a constant rate, P can finish a job in 4 hours. Q, also working at a constant rate, can finish the same job in 20 hours. They work together for 3 hours. How many more minutes will it take P to finish the job, working alone at his constant rate?

24

James decides to make a bathtub full of jello. For every pound of water, you need some tablespoons of jello mix. The bathtub can hold 6 cubic feet of water. Each cubic foot of water is 7.5 gallons. A gallon of water weighs 8 pounds. A tablespoon of jello mix costs $0.50. He spent $270 to fill his tub. How many tablespoons of jello mix are needed for every pound of water?

1.5

The Wholesome Bakery baked 5 loaves of bread on Wednesday, 7 loaves of bread on Thursday, some loaves of bread on Friday, 14 loaves of bread on Saturday, and 19 loaves of bread on Sunday. If this pattern continues, they will bake 25 loaves of bread on Monday. How many loaves of bread did they bake on Friday?

10

Every bag of Sweetsies (a fruit candy) contains the same number of pieces. The Sweetsies in one bag can't be divided equally among $8$ kids, because after each kid gets the same (whole) number of pieces, $5$ pieces are left over. If the Sweetsies in four bags are divided equally among $8$ kids, what is the smallest number of pieces that could possibly be left over?

4

the arithmetic mean and standard deviation of a certain normal distribution are 15.5 and 1.5 , respectively . what value is exactly 2 standard deviations less than the mean ?

12.5

Given a rectangular parallelepiped $ A B C D A_1 B_1 C_1 D_1 $ with dimensions $ A B = 4 $, $ A D = A A_1 = 14 $. Point $ M $ is the midpoint of edge $ C C_1 $. Find the area of the section of the parallelepiped by the plane passing through points $ A_1, D $, and $ M $.

42

Given the function $f(x)=2\sin x\cos x+\cos 2x$. - (I) Find the smallest positive period of $f(x)$ and the intervals of monotonic increase; - (II) Find the maximum and minimum values of $f(x)$ in the interval $\left[0, \frac{\pi}{2}\right]$.

-1

(1) Calculate: $\int_{-2}^{2}{\sqrt{4-{x}^{2}}}dx=$\_\_\_\_\_\_\_\_\_\_\_\_. (2) If $f(x)+\int_{0}^{1}{f(x)dx=2x}$, then $f(x)=$\_\_\_\_\_\_\_\_\_\_\_\_. (3) Arranging $3$ volunteers to complete $4$ tasks, with each person completing at least one task and each task being completed by one person, the total number of different arrangements is \_\_\_\_\_\_\_\_\_\_. (4) Let the function $f(x)=x\ln({e}^{x}+1)- \frac{1}{2}{x}^{2}+3, x\in[-t,t](t > 0)$, if the maximum value of the function $f(x)$ is $M$ and the minimum value is $m$, then $M+m=$\_\_\_\_\_\_\_\_\_\_\_\_\_.

6

It was Trevor's job to collect fresh eggs from the family's 4 chickens every morning. He got some eggs from Gertrude, 3 eggs from Blanche, 2 eggs from Nancy, and 2 eggs from Martha. On the way, he dropped 2 eggs. Trevor had 9 eggs left. How many eggs did he get from Gertrude?

4

A man is 18 years older than his son. In two years, his age will be twice the age of his son. What is the present age of his son?

16

Define the sequence $b_1, b_2, b_3, \ldots$ by $b_n = \sum\limits_{k=1}^n \cos{k}$, where $k$ is measured in radians. Find the index of the 50th term for which $b_n < 0$.

314

What is the coefficient of $x^6$ in the expansion of $(1 - 3x^3)^6$?

135

For an odd function $f(x)$ with the domain of all real numbers $\mathbb{R}$, find the value of $f(-2) + f(2)$.

0

Tom decides to take some dance lessons that cost $10 each, but he gets two of them for free. He pays $80. How many dance lessons did he take in total?

10

Of the votes cast on a certain proposal, some more were in favor of the proposal than were against it. The number of votes against the proposal was 40 percent of the total vote. The total number of votes cast was approximately 350. How many more votes were in favor of the proposal than were against it?

70

Find the number of positive integers $n$ for which (i) $n \leq 1991$ ; (ii) 6 is a factor of $(n^2 + 3n +2)$ .

1328

What is the largest perfect square factor of 1760?

4

Tennis rackets can be packaged in cartons holding 2 rackets each or in cartons holding 3 rackets each. Yesterday's packing slip showed that a certain number of cartons were used to pack a total of 100 rackets, and 24 cartons of 3 rackets size were used. How many cartons were used in total?

38

In a car dealership with some cars, 60% of the cars are hybrids, and 40% of the hybrids contain only one headlight. There are 216 hybrids with full headlights. How many cars are there in the dealership?

600

For an ellipse centered at point (2, -1) with a semi-major axis of 5 units and a semi-minor axis of 3 units, calculate the distance between the foci.

8

If $ x - y = 10 $ and $ x + y = 8 $, what is the value of $ y $?

-1

Solve for $x$ in the equation $24 - 4 \times 2 = 3 + x$.

13

In a bag, there are 3 red balls and 2 white balls. If two balls are drawn successively without replacement, let A be the event that the first ball drawn is red, and B be the event that the second ball drawn is red. Find $p(B|A)$.

1

The sum of two numbers is $20$ and their difference is $4$. Multiply the larger number by $3$ and then find the product of this result and the smaller number.

288

the perimeter of an equilateral triangle is 60 . if one of the sides of the equilateral triangle is the side of an isosceles triangle of perimeter 65 , then how long is the base of isosceles triangle ?

25

Given $f(x)=(1+x)+(1+x)^{2}+(1+x)^{3}+\ldots+(1+x)^{10}=a_{0}+a_{1}x+a_{2}x^{2}+\ldots+a_{10}x^{10}$, find the value of $a_{2}$.

165

What is the sum of the greatest common factor and the lowest common multiple of 36 and 56?

508

Find $89^{-1} \pmod{90}$, as a residue modulo 90. (Give an answer between 0 and 89, inclusive.)

89

A train running at a certain speed crosses an electric pole in 12 seconds. It crosses a 320 m long platform in approximately 44 seconds. What is the speed of the train in kmph?

36

If the true discount on a sum due some years hence at 14% per annum is Rs. 168, the sum due is Rs. 768. How many years hence is the sum due?

2

Calculate the total number of pieces needed to create a 10-row triangle, where each piece comprises unit rods (which form the sides of the triangle) and connectors (which form the vertices of the triangle). A two-row triangle uses 9 unit rods and 6 connectors.

231

Shirley sold 20 boxes of Do-Si-Dos. She needs to deliver a certain number of cases of 4 boxes, plus extra boxes. How many cases does she need to deliver?

5

Find the sum of all positive integers such that their expression in base $4$ digits is the reverse of their expression in base $9$ digits, and they form a palindrome in base $10$. Express your answer in base $10$.

21

For $-49 \le x \le 49,$ find the maximum value of $\sqrt{49 + x} + \sqrt{49 - x}.$

14

Given vectors $\overrightarrow{a}=(2,1)$ and $\overrightarrow{b}=(2,x)$, if the projection vector of $\overrightarrow{b}$ in the direction of $\overrightarrow{a}$ is $\overrightarrow{a}$, then the value of $x$ is ______.

1

what is the maximum number of pieces of birthday cake of size 2 ” by 2 ” that can be cut from a cake 20 ” by 20 ” ?

100

Excluding stoppages, the speed of a bus is some km/hr, and including stoppages, it is 45 km/hr. The bus stops for 10 minutes per hour. What is the speed of the bus excluding stoppages?

54

Find $2537 + 240 \times 3 \div 60 - 347$.

2202

Form a four-digit number with no repeated digits using the numbers 1, 2, 3, and 4, where exactly one even digit is sandwiched between two odd digits. The number of such four-digit numbers is ______.

8

What is the surface area of a rectangular prism with edge lengths of 2, 3, and 4?

52

Coins denomination 1, 2, 10, 20 and 50 are available. How many ways are there of paying 100?

784

A man swims downstream 30 km and upstream 20 km taking 5 hours each time. What is the speed of the man in still water?

5

A reduction of 40% in the price of bananas would enable a man to obtain 67 more for Rs. 40. What is the reduced price per dozen?

2.87

When Betty makes cheesecake, she sweetens it with a ratio of one part sugar to four parts cream cheese, and she flavors it with one teaspoon of vanilla for every two cups of cream cheese. For every one teaspoon of vanilla, she uses two eggs. She used two cups of sugar in her latest cheesecake. How many eggs did she use?

8

Let $0 \leq a, b, c \leq 1.$ Find the maximum value of \[ \sqrt[3]{abc} + \sqrt[3]{(1-a)(1-b)(1-c)}. \]

1

There are numbers written on a board from 1 to 2021. Denis wants to choose 1010 of them so that the sum of any two selected numbers is not equal to 2021 or 2022. How many ways are there to do this?

511566

Determine the value of $$\lim_{\Delta x \to 0} \frac{f(1 - 2\Delta x) - f(1)}{\Delta x}$$ for the function $f(x) = 2\ln(3x) + 8x$.

-20

Complex numbers $u,$ $v,$ and $w$ are zeros of a polynomial $Q(z) = z^3 + 2z^2 + sz + t,$ and $|u|^2 + |v|^2 + |w|^2 = 350.$ The points corresponding to $u,$ $v,$ and $w$ in the complex plane are the vertices of an isosceles right triangle with hypotenuse $k.$ Find $k^2.$

525

Find the numerical value of $k$ such that \[\frac{9}{x + y} = \frac{k}{x + z} = \frac{15}{z - y}.\]

24

Given two vectors $\overrightarrow{a} = (1 - \sin\theta, 1)$ and $\overrightarrow{b} = \left(\frac{1}{2}, 1 + \sin\theta\right)$ (where $\theta$ is an acute angle), and $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$, find the value of $\tan\theta$.

1

Two trains are moving in opposite directions with speeds of 210 km/hr and 90 km/hr respectively. One train has a length of 1.10 km and they cross each other in 24 seconds. What is the length of the other train?

899.92

Daniel is building a rectangular playground from 480 feet of fencing. The fencing must cover three sides of the playground (the fourth side is bordered by Daniel's house). What is the maximum area of this playground?

28800

How many distinct four-digit positive integers have only even digits?

500

Johnny makes $8.25 per hour at his work. If he earns $16.5, how many hours did he work?

2

A quadrilateral is divided into 1000 triangles. What is the maximum number of distinct points that can be the vertices of these triangles?

1002

The numbers $x$ and $y$ are inversely proportional. When the sum of $x$ and $y$ is 50, $x$ is three times $y$. What is the value of $y$ when $x=-12$?

-39.0625

Patsy is gearing up for this weekend’s graduation. She needs to have 6 appetizers per each of her 30 guests. She’s making 3 dozen deviled eggs, 2 dozen pigs in a blanket and 2 dozen kebabs. How many more dozen appetizers does she need to make?

8

Solve for $x$: $\dfrac{1}{3} + \dfrac{1}{x} = \dfrac{2}{3}$.

3

If $f(x)$ is a function whose domain is $[-12, 12]$, and $p(x) = f\left(\frac{x}{3}\right)$, then the domain of $p(x)$ is an interval of what width?

72

George now has a palette of 10 colors. He wants to choose 3 colors to paint his room, but he insists that blue must be one of the colors. How many ways can he do this?

36

A number is *interesting*if it is a $6$ -digit integer that contains no zeros, its first $3$ digits are strictly increasing, and its last $3$ digits are non-increasing. What is the average of all interesting numbers?

308253

The zoo keeps 35 female (a) animals. Males outnumber females by 7. How many (a) animals are there in all?

77

Find the sum of all integers $k$ such that $\binom{29}{4} + \binom{29}{5} = \binom{30}{k}$.

30

In the Bank of Shower, a bored customer lays $n$ coins in a row. Then, each second, the customer performs ``The Process." In The Process, all coins with exactly one neighboring coin heads-up before The Process are placed heads-up (in its initial location), and all other coins are placed tails-up. The customer stops once all coins are tails-up. Define the function $f$ as follows: If there exists some initial arrangement of the coins so that the customer never stops, then $f(n) = 0$ . Otherwise, $f(n)$ is the average number of seconds until the customer stops over all initial configurations. It is given that whenever $n = 2^k-1$ for some positive integer $k$ , $f(n) > 0$ . Let $N$ be the smallest positive integer so that \[ M = 2^N \cdot \left(f(2^2-1) + f(2^3-1) + f(2^4-1) + \cdots + f(2^{10}-1)\right) \]is a positive integer. If $M = \overline{b_kb_{k-1}\cdots b_0}$ in base two, compute $N + b_0 + b_1 + \cdots + b_k$ . *Proposed by Edward Wan and Brandon Wang*

8

Lisa, Robert and Claire have taken photos on their school trip. Lisa has taken some times as many photos as Claire and Robert has taken 20 more photos than Claire. Claire has taken 10 photos. How many photos has Lisa taken?

3

The operation $\star$ is defined as $a \star b = a^2 \div b$. For how many integer values of $x$ will the value of $12 \star x$ be a positive integer?

15

if n is an integer , f ( n ) = f ( n - 1 ) - n and f ( 4 ) = 12 . what is the value of f ( 6 ) ?

1

At Billy's Restaurant, a group with 2 adults and some children came in to eat. If each meal cost 3 dollars, the bill was $21. How many children were in the group?

5

Tim has 22 books. Mike has 20 books. How many books do they have together ?

42

A 220 m long train running at the speed of 120 km/hr crosses another train running in opposite direction at the speed of 80 km/hr. The length of the other train is 280.04 m. How long does it take for the trains to cross each other?

9.00

What is the remainder when $5^{303}$ is divided by $11$?

4

Given that $ab + bc + cd + da = 42$ and $b + d = 6$, find the value of $a + c$.

7

Two trains of different lengths are running towards each other on parallel lines at 42 kmph and 30 kmph respectively. The trains will be clear of each other in 12.998960083193344 seconds from the moment they meet. If the second train is 160 m long, how long is the first train?

99.98

During the first week of performances of a certain play, some tickets were sold, all at reduced price. During the remaining weeks of performances, 5 times as many tickets were sold at full price as were sold at reduced price. The total number of tickets sold was 25200, and 16500 of them were sold at full price. How many tickets were sold at reduced price during the first week?

8700

Prove that there exist infinitely many positive integers $ n $ such that the largest prime divisor of $ n^4 + n^2 + 1 $ is equal to the largest prime divisor of $ (n+1)^4 + (n+1)^2 + 1 $.

Given that the graph of the function $y=f(x)$ is symmetric to the graph of the function $y=2^{x+2}$ with respect to the line $y=-x$, find the value of $f(-2)$.

1

John has 20% more boxes than Jules. Jules has 5 more boxes than Joseph. Joseph has 80% fewer boxes than Stan. If Stan has 100 boxes, how many boxes does John have?

30

Suppose the sum of seven consecutive even numbers is 400. What is the smallest of these seven numbers?

406

The closest approximation of (a number × 0.004) / 0.03 is 9.237333333333334. What is the number?

69.28

Lisa, Robert, and Claire have taken photos on their school trip. Lisa has taken some multiple of the number of photos Claire has taken, and Robert has taken 16 more photos than Claire. Claire has taken 8 photos. How many photos has Lisa taken?

24

Given the hyperbola $C$: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ with an eccentricity of $\sqrt{3}$, its real axis is $AB$, and a line parallel to $AB$ intersects hyperbola $C$ at points $M$ and $N$. Calculate the product of the slopes of lines $AM$ and $AN$.

-2

A middle school's 7th-grade literary and arts representative team took a bus to a county middle school 21 kilometers away from their school to participate in a competition. When they reached place A, the driver received a phone call from the school: "Hello, is this Mr. Zhang?" "Yes, it's me." "Please return immediately to pick up the 8th-grade sports representative team and take them to the county middle school, and let the 7th-grade team walk there." After resting for 10 minutes, the 7th-grade team started walking, and the bus did not stop on its return journey. When the bus arrived at the county middle school with the 8th-grade team, the 7th-grade team also arrived at the same time. It is known that the walking speed is 4 kilometers per hour, and the bus speed is 60 kilometers per hour. Find the distance from the school to place A and the distance walked by the 7th-grade representative team.

2

A grain storage warehouse has a total of 30 bins. Some hold 20 tons of grain each, and the rest hold a certain amount of grain each. The warehouse has a capacity of 510 tons, and there are 12 20-ton bins. How many tons of grain do the remaining bins hold each?

15

Kylie makes 10 beaded necklaces on Monday and 2 beaded necklaces on Tuesday. Then Kylie makes 5 beaded bracelets and 7 beaded earrings on Wednesday. A certain number of beads are needed to make one beaded necklace. 10 beads are needed to make one beaded bracelet. 5 beads are needed to make one beaded earring. Kylie uses 325 beads in total to make her jewelry. How many beads are needed to make one beaded necklace?

20

When you play a car game, you have to choose one character from three characters, and then one car from three cars. How many cases are there?

9

Sara made a complete list of the prime numbers between 10 and 50. What is the sum of the smallest prime number and the largest prime number on her list?

58

Paige was helping her mom plant flowers, and they decided to plant in three different sections in their garden. In the first section, they planted 470 seeds; in the second section, they planted 320 seeds; and in the third section, they planted 210 seeds. If they put 10 seeds in each flower bed for the first and second sections and 8 seeds in each flower bed for the third section, how many flower beds did they have in total?

105