problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Given that the sum of the first $n$ terms of the sequence $\{a_{n}\}$ is $S_{n}$, and $a_{1}=4$, $a_{n}+a_{n+1}=4n+2$ for $n\in \mathbb{N}^{*}$, calculate the maximum value of $n$ that satisfies $S_{n} \lt 2023$. | 44 |
If $\frac{1}{9}$ of 60 is 5, what is $\frac{1}{20}$ of 80? | 6 |
Given that point $P$ is a moving point on the parabola $y^{2}=4x$, the minimum value of the sum of the distance from point $P$ to line $l$: $2x-y+3=0$ and the $y$-axis is ___. | \sqrt{5}-1 |
Marisela is putting on a juggling show! She starts with $1$ ball, tossing it once per second. Lawrence tosses her another ball every five seconds, and she always tosses each ball that she has once per second. Compute the total number of tosses Marisela has made one minute after she starts juggling. | 390 |
Find the maximum value of
\[\cos \theta_1 \sin \theta_2 + \cos \theta_2 \sin \theta_3 + \cos \theta_3 \sin \theta_4 + \cos \theta_4 \sin \theta_5 + \cos \theta_5 \sin \theta_1,\]over all real numbers $\theta_1,$ $\theta_2,$ $\theta_3,$ $\theta_4,$ and $\theta_5.$ | \frac{5}{2} |
Define an odd function f(x) on ℝ that satisfies f(x+1) is an even function, and when x ∈ [0,1], f(x) = x(3-2x). Evaluate f(31/2). | -1 |
A university has 120 foreign language teachers. Among them, 50 teach English, 45 teach Japanese, and 40 teach French. There are 15 teachers who teach both English and Japanese, 10 who teach both English and French, and 8 who teach both Japanese and French. Additionally, 4 teachers teach all three languages: English, Japanese, and French. How many foreign language teachers do not teach any of these three languages? | 14 |
The increasing [sequence](https://artofproblemsolving.com/wiki/index.php/Sequence) $3, 15, 24, 48, \ldots\,$ consists of those [positive](https://artofproblemsolving.com/wiki/index.php/Positive) multiples of 3 that are one less than a [perfect square](https://artofproblemsolving.com/wiki/index.php/Perfect_square). What is the [remainder](https://artofproblemsolving.com/wiki/index.php/Remainder) when the 1994th term of the sequence is divided by 1000? | 063 |
Select 3 people from 3 boys and 2 girls to participate in a speech competition.
(1) Calculate the probability that the 3 selected people are all boys;
(2) Calculate the probability that exactly 1 of the 3 selected people is a girl;
(3) Calculate the probability that at least 1 of the 3 selected people is a girl. | \dfrac{9}{10} |
The diagram shows a smaller rectangle made from three squares, each of area \(25 \ \mathrm{cm}^{2}\), inside a larger rectangle. Two of the vertices of the smaller rectangle lie on the mid-points of the shorter sides of the larger rectangle. The other two vertices of the smaller rectangle lie on the other two sides of the larger rectangle. What is the area, in \(\mathrm{cm}^{2}\), of the larger rectangle? | 150 |
What is the largest positive integer that is not the sum of a positive integral multiple of $42$ and a positive composite integer? | 215 |
A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vertex exactly once? | \frac{2}{243} |
In the triangle \(A B C\), angle \(C\) is a right angle, and \(AC: AB = 3: 5\). A circle with its center on the extension of leg \(AC\) beyond point \(C\) is tangent to the extension of hypotenuse \(AB\) beyond point \(B\) and intersects leg \(BC\) at point \(P\), with \(BP: PC = 1: 4\). Find the ratio of the radius of the circle to leg \(BC\). | 37/15 |
What is the value of $\frac13\cdot\frac92\cdot\frac1{27}\cdot\frac{54}{1}\cdot\frac{1}{81}\cdot\frac{162}{1}\cdot\frac{1}{243}\cdot\frac{486}{1}$? | 12 |
A natural number is abundant if it is less than the sum of its proper divisors. What is the smallest abundant number? | 12 |
In the diagram, the equilateral triangle has a base of $8$ m. What is the perimeter of the triangle? [asy]
size(100);
draw((0,0)--(8,0)--(4,4*sqrt(3))--cycle);
label("8 m",(4,0),S);
draw((4,-.2)--(4,.2));
draw((1.8,3.5)--(2.2,3.3));
draw((6.3,3.5)--(5.8,3.3));
[/asy] | 24 |
In triangle $XYZ$, $XY=25$ and $XZ=14$. The angle bisector of $\angle X$ intersects $YZ$ at point $E$, and point $N$ is the midpoint of $XE$. Let $Q$ be the point of the intersection of $XZ$ and $YN$. The ratio of $ZQ$ to $QX$ can be expressed in the form $\dfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. | 39 |
If $\frac{137}{a}=0.1 \dot{2} 3 \dot{4}$, find the value of $a$. | 1110 |
A natural number, when raised to the sixth power, has digits which, when arranged in ascending order, are:
$$
0,2,3,4,4,7,8,8,9
$$
What is this number? | 27 |
What is the smallest possible real value of $x^2 + 8x$? | -16 |
Suppose that $f(x+3)=3x^2 + 7x + 4$ and $f(x)=ax^2 + bx + c$. What is $a+b+c$? | 2 |
On the board, the number 0 is written. Two players take turns appending to the expression on the board: the first player appends a + or - sign, and the second player appends one of the natural numbers from 1 to 1993. The players make 1993 moves each, and the second player uses each of the numbers from 1 to 1993 exactly once. At the end of the game, the second player receives a reward equal to the absolute value of the algebraic sum written on the board. What is the maximum reward the second player can guarantee for themselves? | 1993 |
A king summoned two wise men. He gave the first one 100 blank cards and instructed him to write a positive number on each (the numbers do not have to be different), without showing them to the second wise man. Then, the first wise man can communicate several distinct numbers to the second wise man, each of which is either written on one of the cards or is a sum of the numbers on some cards (without specifying exactly how each number is derived). The second wise man must determine which 100 numbers are written on the cards. If he cannot do this, both will be executed; otherwise, a number of hairs will be plucked from each of their beards equal to the amount of numbers the first wise man communicated. How can the wise men, without colluding, stay alive and lose the minimum number of hairs? | 101 |
Simplify $\sin (x - y) \cos y + \cos (x - y) \sin y.$ | \sin x |
If $x$ is an odd number, then find the largest integer that always divides the expression\[(10x+2)(10x+6)(5x+5)\]
| 960 |
Twelve chess players played a round-robin tournament. Each player then wrote 12 lists. In the first list, only the player himself was included, and in the $(k+1)$-th list, the players included those who were in the $k$-th list as well as those whom they defeated. It turned out that each player's 12th list differed from the 11th list. How many draws were there? | 54 |
Joel is 5 years old and his dad is 32 years old. How old will Joel be when his dad is twice as old as him? | The difference in age between Joel and his dad is 32 - 5 = <<32-5=27>>27 years.
When you double this gap in years, you have 27 x 2 = <<27*2=54>>54.
When his dad is 54, Joel will be 54 / 2 = <<54/2=27>>27 years old.
#### 27 |
There is only one value of $k$ for which the line $x=k$ intersects the graphs of $y=x^2+6x+5$ and $y=mx+b$ at two points which are exactly $5$ units apart. If the line $y=mx+b$ passes through the point $(1,6)$, and $b\neq 0$, find the equation of the line. Enter your answer in the form "$y = mx + b$". | y=10x-4 |
Let $a,$ $b,$ and $c$ be constants, and suppose that the inequality \[\frac{(x-a)(x-b)}{x-c} \le 0\]is true if and only if either $x < -4$ or $|x-25| \le 1.$ Given that $a < b,$ find the value of $a + 2b + 3c.$ | 64 |
In the arithmetic sequence $\{a_n\}$, $S_{10} = 10$, $S_{20} = 30$, then $S_{30} = \ ?$ | 60 |
If Alex does not sing on Saturday, then she has a $70 \%$ chance of singing on Sunday; however, to rest her voice, she never sings on both days. If Alex has a $50 \%$ chance of singing on Sunday, find the probability that she sings on Saturday. | \frac{2}{7} |
For each positive integer $n$ , consider the highest common factor $h_n$ of the two numbers $n!+1$ and $(n+1)!$ . For $n<100$ , find the largest value of $h_n$ . | 97 |
Create three-digit numbers without repeating digits using the numbers 0, 1, 2, 3, 4, 5:
(1) How many of them have a ones digit smaller than the tens digit?
(2) How many of them are divisible by 5? | 36 |
A soccer team has three goalies and ten defenders. The team also has twice as many midfielders as defenders, and the rest of the players are strikers. If the team has 40 players, how many strikers are in the team? | The team's total number of goalies and defenders is 3+10 = <<3+10=13>>13.
The number of midfielders in the team is twice the number of defenders, which means there are 2*10=<<2*10=20>>20 defenders in the team.
The total number of players in the team playing the positions of goalies, defenders and midfielders is 20+13 = <<20+13=33>>33
If there are 40 players in the team, then the number of strikes in the team is 40-33 = <<40-33=7>>7
#### 7 |
Evaluate the absolute value of the expression $|7 - \sqrt{53}|$.
A) $7 - \sqrt{53}$
B) $\sqrt{53} - 7$
C) $0.28$
D) $\sqrt{53} + 7$
E) $-\sqrt{53} + 7$ | \sqrt{53} - 7 |
Jeff wants to calculate the product $0.52 \times 7.35$ using a calculator. However, he mistakenly inputs the numbers as $52 \times 735$ without the decimal points. The calculator then shows a product of $38220$. What would be the correct product if Jeff had correctly entered the decimal points?
A) $0.3822$
B) $38.22$
C) $3.822$
D) $0.03822$
E) $382.2$ | 3.822 |
Find the absolute value of the difference of single-digit integers $A$ and $B$ such that $$ \begin{array}{c@{}c@{\;}c@{}c@{}c@{}c} & & & B& B & A_6\\ & & & \mathbf{4} & \mathbf{1} & B_6\\& & + & A & \mathbf{1} & \mathbf{5_6}\\ \cline{2-6} & & A & \mathbf{1} & \mathbf{5} & \mathbf{2_6} \\ \end{array} $$Express your answer in base $6$. | 1_6 |
Given the functions $f(x)=x+e^{x-a}$ and $g(x)=\ln (x+2)-4e^{a-x}$, where $e$ is the base of the natural logarithm. If there exists a real number $x_{0}$ such that $f(x_{0})-g(x_{0})=3$, find the value of the real number $a$. | -\ln 2-1 |
Find the maximum and minimum values of the function $f(x)=2\cos^2x+3\sin x$ on the interval $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$. $\boxed{\text{Fill in the blank}}$ | -3 |
Napoleon has 17 jelly beans and Sedrich has 4 more jelly beans than Napoleon. If twice the sum of Napoleon and Sedrich's jelly beans is 4 times the number of jelly beans that Mikey has, how many jelly beans does Mikey have? | Sedrich has 17 + 4 = <<17+4=21>>21 jelly beans.
The sum of Napoleon and Sedrich's jelly beans is 17 + 21 = <<17+21=38>>38 jelly beans.
Twice the sum is 38 x 2 = <<38*2=76>>76 jelly beans.
Mikey has 76/4 = <<76/4=19>>19 jelly beans.
#### 19 |
If eight liters is 20% the capacity of a container filled with water, calculate the total capacity of 40 such containers filled with water. | If eight liters is 20% the capacity of a container filled with water, when the container is full, which is 100%, the container will contain 100%/20% * 8 liters= 40 liters.
The total capacity of 40 such containers filled with water is 40*40 = <<40*40=1600>>1600 liters
#### 1600 |
Corveus sleeps 4 hours a day and his doctor recommended for him to sleep 6 hours a day. How many hours of sleep does Corveus lack in a week? | Corveus lacks 6 - 4 = <<6-4=2>>2 hours of sleep every day.
Therefore, Corveus' total hours of lack of sleep every week is 2 x 7 = <<2*7=14>>14.
#### 14 |
Three numbers, $a_1, a_2, a_3$, are drawn randomly and without replacement from the set $\{1, 2, 3,\ldots, 1000\}$. Three other numbers, $b_1, b_2, b_3$, are then drawn randomly and without replacement from the remaining set of $997$ numbers. Let $p$ be the probability that, after suitable rotation, a brick of dimensions $a_1 \times a_2 \times a_3$ can be enclosed in a box of dimension $b_1 \times b_2 \times b_3$, with the sides of the brick parallel to the sides of the box. If $p$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator? | 5 |
A semicircle and a circle each have a radius of 5 units. A square is inscribed in each. Calculate the ratio of the perimeter of the square inscribed in the semicircle to the perimeter of the square inscribed in the circle. | \frac{\sqrt{10}}{5} |
Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[f(x(x + f(y))) = (x + y)f(x),\]
for all $x, y \in\mathbb{R}$. | f(x) = 0 \text{ and } f(x) = x. |
Sarah baked 4 dozen pies for a community fair. Out of these pies:
- One-third contained chocolate,
- One-half contained marshmallows,
- Three-fourths contained cayenne pepper,
- One-eighth contained walnuts.
What is the largest possible number of pies that had none of these ingredients? | 12 |
For what value of $x$ will $\frac{3+x}{5+x}$ and $\frac{1+x}{2+x}$ be equal? | 1 |
Ellie and Sam run on the same circular track but in opposite directions, with Ellie running counterclockwise and Sam running clockwise. Ellie completes a lap every 120 seconds, while Sam completes a lap every 75 seconds. They both start from the same starting line simultaneously. Ten to eleven minutes after the start, a photographer located inside the track takes a photo of one-third of the track, centered on the starting line. Determine the probability that both Ellie and Sam are in this photo.
A) $\frac{1}{4}$
B) $\frac{5}{12}$
C) $\frac{1}{3}$
D) $\frac{7}{18}$
E) $\frac{1}{5}$ | \frac{5}{12} |
Let $n$ be the smallest positive integer such that the remainder of $3n+45$ , when divided by $1060$ , is $16$ . Find the remainder of $18n+17$ upon division by $1920$ . | 1043 |
Five people are at a party. Each pair of them are friends, enemies, or frenemies (which is equivalent to being both friends and enemies). It is known that given any three people $A, B, C$ : - If $A$ and $B$ are friends and $B$ and $C$ are friends, then $A$ and $C$ are friends; - If $A$ and $B$ are enemies and $B$ and $C$ are enemies, then $A$ and $C$ are friends; - If $A$ and $B$ are friends and $B$ and $C$ are enemies, then $A$ and $C$ are enemies. How many possible relationship configurations are there among the five people? | 17 |
The instructor of a summer math camp brought several shirts, several pairs of trousers, several pairs of shoes, and two jackets for the entire summer. In each lesson, he wore trousers, a shirt, and shoes, and he wore a jacket only on some lessons. On any two lessons, at least one piece of his clothing or shoes was different. It is known that if he had brought one more shirt, he could have conducted 18 more lessons; if he had brought one more pair of trousers, he could have conducted 63 more lessons; if he had brought one more pair of shoes, he could have conducted 42 more lessons. What is the maximum number of lessons he could conduct under these conditions? | 126 |
A professional company is hiring for a new position. They have two qualified applicants. The first applicant will accept a salary of $42000 and make the company $93000 in the first year, but needs 3 months of additional training that costs $1200 a month. The second applicant does not need training and will make the company $92000 in the first year, but is requesting a salary of $45000 and a hiring bonus of 1% of his salary. Less the amount it will cost to pay for each candidate, how many more dollars will one candidate make the company than the other in the first year? | The first candidate’s training will cost 1200 * 3 = $<<1200*3=3600>>3600.
That candidate will make the company 93000 - 42000 - 3600 = $<<93000-42000-3600=47400>>47400 in the first year.
The second candidate’s bonus will cost 45000 * 1 / 100 = $<<45000*1/100=450>>450.
That candidate will make the company 92000 - 45000 - 450 = $<<92000-45000-450=46550>>46550 in the first year.
Thus, the first candidate will make the company 47400 - 46550 = $<<47400-46550=850>>850 more in the first year.
#### 850 |
Find the coefficient of $x$ when $5(2x - 3) + 7(5 - 3x^2 + 4x) - 6(3x - 2)$ is simplified. | 56 |
A room is 19 feet long and 11 feet wide. Find the ratio of the length of the room to its perimeter. Express your answer in the form $a:b$. | 19:60 |
George purchases a sack of apples, a bunch of bananas, a cantaloupe, and a carton of dates for $ \$ 20$. If a carton of dates costs twice as much as a sack of apples and the price of a cantaloupe is equal to the price of a sack of apples minus a bunch of bananas, how much would it cost George to purchase a bunch of bananas and a cantaloupe? | \$ 5 |
While watching a circus show, I counted out the number of acrobats and elephants. I counted 40 legs and 15 heads. How many acrobats did I see in the show? | 10 |
Three boys and three girls are lined up for a photo. Boy A is next to boy B, and exactly two girls are next to each other. Calculate the total number of different ways they can be arranged. | 144 |
A polynomial $p(x)$ leaves a remainder of $-1$ when divided by $x - 1,$ a remainder of 3 when divided by $x - 2,$ and a remainder of 4 when divided by $x + 3.$ Let $r(x)$ be the remainder when $p(x)$ is divided by $(x - 1)(x - 2)(x + 3).$ Find $r(6).$ | 40 |
If the equation with respect to \( x \), \(\frac{x \lg^2 a - 1}{x + \lg a} = x\), has a solution set that contains only one element, then \( a \) equals \(\quad\) . | 10 |
Let $a,$ $b,$ $c$ be nonzero real numbers such that $a + b + c = 0,$ and $ab + ac + bc \neq 0.$ Find all possible values of
\[
\frac{a^7 + b^7 + c^7}{abc (ab + ac + bc)}.
\] | -7 |
Find the sum of $327_8$ and $73_8$ in base $8$. | 422_8 |
The railway between Station A and Station B is 840 kilometers long. Two trains start simultaneously from the two stations towards each other, with Train A traveling at 68.5 kilometers per hour and Train B traveling at 71.5 kilometers per hour. After how many hours will the two trains be 210 kilometers apart? | 7.5 |
ABC is a triangle. D is the midpoint of AB, E is a point on the side BC such that BE = 2 EC and ∠ADC = ∠BAE. Find ∠BAC. | 30 |
For some constants $x$ and $a$, the third, fourth, and fifth terms in the expansion of $(x + a)^n$ are 84, 280, and 560, respectively. Find $n.$ | 7 |
The number \( x \) is such that \( \log _{2}\left(\log _{4} x\right) + \log _{4}\left(\log _{8} x\right) + \log _{8}\left(\log _{2} x\right) = 1 \). Find the value of the expression \( \log _{4}\left(\log _{2} x\right) + \log _{8}\left(\log _{4} x\right) + \log _{2}\left(\log _{8} x\right) \). If necessary, round your answer to the nearest 0.01. | 0.87 |
Maria has 19 cookies. She decided to give her friend 5 of them, and half of the rest to her family. From the rest, Maria decided to eat 2 cookies. How many cookies will she have left? | Maria wants to give away 5 cookies, so she will be left with 19 - 5 = <<19-5=14>>14 cookies.
Half of these will go to her family, which means 14 / 2 = <<14/2=7>>7 cookies.
Maria decided to eat 2/7 * 7 = <<2/7*7=2>>2 cookies.
So she will be left with 7 - 2 = <<7-2=5>>5 cookies.
#### 5 |
Find the biggest positive integer $n$ such that $n$ is $167$ times the amount of it's positive divisors. | 2004 |
Given $\sin(\alpha - \beta) = \frac{1}{3}$ and $\cos \alpha \sin \beta = \frac{1}{6}$, calculate the value of $\cos(2\alpha + 2\beta)$. | \frac{1}{9} |
If $f(x)=\dfrac{x+1}{3x-4}$, what is the value of $f(5)$? | \dfrac{6}{11} |
For which value of $x$ does the function $f(x) = \frac{2x^2 - 5x - 7}{x^2 - 4x + 1}$ cross its horizontal asymptote? | 3 |
Given $\triangle PQR$ with $\overline{RS}$ bisecting $\angle R$, $PQ$ extended to $D$ and $\angle n$ a right angle, then: | \frac{1}{2}(\angle p + \angle q) |
Write $\frac{5}{8}$ as a decimal. | 0.625 |
A tetrahedron \( P-ABC \) has edge lengths \( PA = BC = \sqrt{6} \), \( PB = AC = \sqrt{8} \), and \( PC = AB = \sqrt{10} \). Find the radius of the circumsphere of this tetrahedron. | \sqrt{3} |
What is the sum of all odd integers between $400$ and $600$? | 50000 |
Suppose we roll a standard fair 6-sided die. What is the probability that a perfect square is rolled? | \dfrac13 |
A square has a diagonal of length $10\sqrt{2}$ centimeters. What is the number of square centimeters in the area of the square? | 100 |
Find distinct digits to replace the letters \(A, B, C, D\) such that the following division in the decimal system holds:
$$
\frac{ABC}{BBBB} = 0,\overline{BCDB \, BCDB \, \ldots}
$$
(in other words, the quotient should be a repeating decimal). | 219 |
Each point in the hexagonal lattice shown is one unit from its nearest neighbor. How many equilateral triangles have all three vertices in the lattice? [asy]size(75);
dot(origin);
dot(dir(0));
dot(dir(60));
dot(dir(120));
dot(dir(180));
dot(dir(240));
dot(dir(300));
[/asy] | 8 |
Given that $\alpha$ is an angle in the third quadrant, the function $f(\alpha)$ is defined as:
$$f(\alpha) = \frac {\sin(\alpha - \frac {\pi}{2}) \cdot \cos( \frac {3\pi}{2} + \alpha) \cdot \tan(\pi - \alpha)}{\tan(-\alpha - \pi) \cdot \sin(-\alpha - \pi)}.$$
1. Simplify $f(\alpha)$.
2. If $\cos(\alpha - \frac {3\pi}{2}) = \frac {1}{5}$, find $f(\alpha + \frac {\pi}{6})$. | \frac{6\sqrt{2} - 1}{10} |
In the given configuration, triangle $ABC$ has a right angle at $C$, with $AC=4$ and $BC=3$. Triangle $ABE$ has a right angle at $A$ where $AE=5$. The line through $E$ parallel to $\overline{AC}$ meets $\overline{BC}$ extended at $D$. Calculate the ratio $\frac{ED}{EB}$. | \frac{4}{5} |
Given the complex numbers \( z_{1} \) and \( z_{2} \) such that \( \left| z_{2} \right| = 4 \) and \( 4z_{1}^{2} - 2z_{1}z_{2} + z_{2}^{2} = 0 \), find the maximum value of \( \left| \left( z_{1} + 1 \right)^{2} \left( z_{1} - 2 \right) \right| \). | 6\sqrt{6} |
There are 11 of the number 1, 22 of the number 2, 33 of the number 3, and 44 of the number 4 on the blackboard. The following operation is performed: each time, three different numbers are erased, and the fourth number, which is not erased, is written 2 extra times. For example, if 1 of 1, 1 of 2, and 1 of 3 are erased, then 2 more of 4 are written. After several operations, there are only 3 numbers left on the blackboard, and no further operations can be performed. What is the product of the last three remaining numbers? | 12 |
In her last basketball game, Jackie scored 36 points. These points raised the average number of points that she scored per game from 20 to 21. To raise this average to 22 points, how many points must Jackie score in her next game? | 38 |
The noon temperatures for ten consecutive days were $78^{\circ}$, $80^{\circ}$, $82^{\circ}$, $85^{\circ}$, $88^{\circ}$, $90^{\circ}$, $92^{\circ}$, $95^{\circ}$, $97^{\circ}$, and $95^{\circ}$ Fahrenheit. The increase in temperature over the weekend days (days 6 to 10) is attributed to a local summer festival. What is the mean noon temperature, in degrees Fahrenheit, for these ten days? | 88.2 |
Megan’s grandma gave her $125 to start a savings account. She was able to increase the account by 25% from funds she earned babysitting. Then it decreased by 20% when she bought a new pair of shoes. Her final balance is what percentage of her starting balance? | Megan started with $125 and increased it by 25%, so $125 + (0.25)125 or $125 + $31.25 = $<<125+31.25=156.25>>156.25.
Then the balance decreased by 20%, so $156.25 – (0.20)156.25 or $156.25 - $31.25 = $<<156.25-31.25=125>>125.
$125 is the balance Megan started with, so her final balance is 100% of the starting balance.
#### 100 |
Estimate the time it takes to send $60$ blocks of data over a communications channel if each block consists of $512$ "chunks" and the channel can transmit $120$ chunks per second. | 240 |
Given that \( f \) is a mapping from the set \( M = \{a, b, c\} \) to the set \( N = \{-3, -2, \cdots, 3\} \). Determine the number of mappings \( f \) that satisfy
$$
f(a) + f(b) + f(c) = 0
$$ | 37 |
Consider the region \(B\) in the complex plane consisting of all points \(z\) such that both \(\frac{z}{50}\) and \(\frac{50}{\overline{z}}\) have real and imaginary parts between 0 and 1, inclusive. Find the area of \(B\). | 2500 - 312.5 \pi |
Lottery. (For 7th grade, 3 points) It so happened that Absent-Minded Scientist has only 20 rubles left, but he needs to buy a bus ticket to get home. The bus ticket costs 45 rubles. Nearby the bus stop, instant lottery tickets are sold for exactly 10 rubles each. With a probability of $p = 0.1$, a ticket contains a win of 30 rubles, and there are no other prizes. The Scientist decided to take a risk since he has nothing to lose. Find the probability that the Absent-Minded Scientist will be able to win enough money to buy a bus ticket. | 0.19 |
Determine the value of the sum \[ \sum_{n=0}^{332} (-1)^{n} {1008 \choose 3n} \] and find the remainder when the sum is divided by $500$. | 54 |
The expression $x^2 - 16x + 60$ can be written in the form $(x - a)(x - b)$, where $a$ and $b$ are both nonnegative integers and $a > b$. What is the value of $3b - a$? | 8 |
A concert ticket costs $40. Mr. Benson bought 12 tickets and received a 5% discount for every ticket bought that exceeds 10. How much did Mr. Benson pay in all? | Mr. Benson had a 5% discount for each of the 12 - 10 = <<12-10=2>>2 tickets.
So, those two tickets had a $40 x 5/100 = $<<40*5/100=2>>2 discount each.
Hence, each ticket cost $40 - $2 = $<<40-2=38>>38 each.
Thus, two discounted tickets amount to $38 x 2 = $<<38*2=76>>76.
And the other ten tickets amount to $40 x 10 = $<<40*10=400>>400.
Hence, Mr. Benson paid a total of $400 + $76 = $<<400+76=476>>476.
#### 476 |
Liam builds two snowmen using snowballs of radii 4 inches, 6 inches, and 8 inches for the first snowman. For the second snowman, he uses snowballs that are 75% of the size of each corresponding ball in the first snowman. Assuming all snowballs are perfectly spherical, what is the total volume of snow used in cubic inches? Express your answer in terms of $\pi$. | \frac{4504.5}{3}\pi |
Let $ABC$ be a triangle with $\angle BAC=60^\circ$ . Consider a point $P$ inside the triangle having $PA=1$ , $PB=2$ and $PC=3$ . Find the maximum possible area of the triangle $ABC$ . | \frac{3\sqrt{3}}{2} |
A box contains six cards. Three of the cards are black on both sides, one card is black on one side and red on the other, and two of the cards are red on both sides. You pick a card uniformly at random from the box and look at a random side. Given that the side you see is red, what is the probability that the other side is red? Express your answer as a common fraction. | \frac{4}{5} |
Let $n$ be an integer of the form $a^{2}+b^{2}$, where $a$ and $b$ are relatively prime integers and such that if $p$ is a prime, $p \leq \sqrt{n}$, then $p$ divides $a b$. Determine all such $n$. | n = 2, 5, 13 |
Given the real sequence $-1$, $a$, $b$, $c$, $-2$ forms a geometric sequence, find the value of $abc$. | -2\sqrt{2} |
Let $S$ be the set of natural numbers that cannot be written as the sum of three squares. Legendre's three-square theorem states that $S$ consists of precisely the integers of the form $4^a(8b+7)$ where $a$ and $b$ are nonnegative integers. Find the smallest $n\in\mathbb N$ such that $n$ and $n+1$ are both in $S$ . | 111 |
What is the units digit of the sum of the nine terms of the sequence $1! + 1, \, 2! + 2, \, 3! + 3, \, ..., \, 8! + 8, \, 9! + 9$? | 8 |
A train moves along a straight track, and from the moment it starts braking to the moment it stops, the distance $S$ in meters that the train travels in $t$ seconds after braking is given by $S=27t-0.45t^2$. Find the time in seconds after braking when the train stops, and the distance in meters the train has traveled during this period. | 405 |
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