problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
In a 60-item exam, Liza got 90% of the items correctly. Her best friend, Rose, got 2 more correct answers than her. How many incorrect answers did Rose have? | Liza got 60 x 90/100 = <<60*90/100=54>>54 of the items correctly.
So, Rose got 54 + 2 = <<54+2=56>>56 of the items correctly.
Thus, Rose had 60 - 56 = <<60-56=4>>4 incorrect answers.
#### 4 |
The numbers 1, 3, 6, 10, $\ldots$, are called triangular numbers, as shown geometrically here. What is the $20^{\text{th}}$ triangular number?
[asy]
dot((0,0));
label("1",(0,-1.5));
dot((3,0));
dot((4,0));
dot((3,1));
label("3",(3.5,-1.5));
dot((7,0));
dot((8,0));
dot((9,0));
dot((7,1));
dot((7,2));
dot((8,1));
la... | 210 |
A right triangle has legs measuring 20 inches and 21 inches. What is the length of the hypotenuse, in inches? | 29 |
In the final stage of a professional bowling competition, the top five players compete as follows:
- The fifth place player competes against the fourth place player.
- The loser of the match receives the 5th place award.
- The winner then competes against the third place player.
- The loser of this match receives the... | 16 |
$A B C$ is a triangle with points $E, F$ on sides $A C, A B$, respectively. Suppose that $B E, C F$ intersect at $X$. It is given that $A F / F B=(A E / E C)^{2}$ and that $X$ is the midpoint of $B E$. Find the ratio $C X / X F$. | \sqrt{5} |
Given $|x|=4$, $|y|=2$, and $x<y$, then the value of $x\div y$ is ______. | -2 |
Let \( f(x) = x^2 + px + q \). It is known that the inequality \( |f(x)| > \frac{1}{2} \) has no solutions on the interval \([1, 3]\). Find \( \underbrace{f(f(\ldots f}_{2017}\left(\frac{3+\sqrt{7}}{2}\right)) \ldots) \). If necessary, round your answer to two decimal places. | 0.18 |
Let $a_1,$ $a_2,$ $a_3,$ $\dots$ be a sequence of real numbers satisfying
\[a_n = a_{n - 1} a_{n + 1}\]for all $n \ge 2.$ If $a_1 = 1 + \sqrt{7}$ and $a_{1776} = 13 + \sqrt{7},$ then determine $a_{2009}.$ | -1 + 2 \sqrt{7} |
A quadrilateral is inscribed in a circle of radius $200\sqrt{2}$. Three of the sides of this quadrilateral have length $200$. What is the length of the fourth side? | 500 |
Lydia likes a five-digit number if none of its digits are divisible by 3. Find the total sum of the digits of all five-digit numbers that Lydia likes. | 174960 |
What is the smallest positive value of $m$ so that the equation $10x^2 - mx + 360 = 0$ has integral solutions with one root being a multiple of the other? | 120 |
The difference between two numbers is 9, and the sum of the squares of each number is 153. What is the value of the product of the two numbers? | 36 |
In Chemistry class, Samantha finds that she can make a certain solution by mixing $.04$ liters of chemical A with $.02$ liters of water (this gives her $.06$ liters of the solution). She wants to make a total of $.48$ liters of this new solution. To do so, how many liters of water will she use? | 0.16 |
If a rectangular prism has a length of $l$, a width of $w$, and a height of $h$, then the length of its diagonal is equal to $\sqrt{l^2 + w^2 + h^2}$. Suppose $l = 3$ and $h = 12$; if the length of the diagonal is $13$, what is the width? | 4 |
A cryptographer designed the following method to encode natural numbers: first, represent the natural number in base 5, then map the digits in the base 5 representation to the elements of the set $\{V, W, X, Y, Z\}$ in a one-to-one correspondence. Using this correspondence, he found that three consecutive increasing na... | 108 |
Triangle $\triangle DEF$ has a right angle at $F$, $\angle D = 60^\circ$, and $DF=12$. Find the radius of the incircle of $\triangle DEF$. | 6(\sqrt{3}-1) |
In the rectangular coordinate system $xOy$, the equation of line $C_1$ is $y=-\sqrt{3}x$, and the parametric equations of curve $C_2$ are given by $\begin{cases}x=-\sqrt{3}+\cos\varphi\\y=-2+\sin\varphi\end{cases}$. Establish a polar coordinate system with the coordinate origin as the pole and the positive half of the ... | \sqrt{3} |
A basketball player scored 18, 22, 15, and 20 points respectively in her first four games of a season. Her points-per-game average was higher after eight games than it was after these four games. If her average after nine games was greater than 19, determine the least number of points she could have scored in the ninth... | 21 |
The diagonal lengths of a rhombus are 24 units and 10 units. What is the area of the rhombus, in square units? | 120 |
Let $f(x) = x^2-3x$. For what values of $x$ is $f(f(x)) = f(x)$? Enter all the solutions, separated by commas. | 0, 3, -1, 4 |
Find all real solutions to
\[\frac{1}{(x - 1)(x - 2)} + \frac{1}{(x - 2)(x - 3)} + \frac{1}{(x - 3)(x - 4)} = \frac{1}{6}.\]Enter all solutions, separated by commas. | 7,-2 |
When one ounce of water is added to a mixture of acid and water, the new mixture is $20\%$ acid. When one ounce of acid is added to the new mixture, the result is $33\frac13\%$ acid. The percentage of acid in the original mixture is | 25\% |
How can 13 rectangles of sizes $1 \times 1, 2 \times 1, 3 \times 1, \ldots, 13 \times 1$ be combined to form a rectangle, where all sides are greater than 1? | 13 \times 7 |
Define a $\it{good\ word}$ as a sequence of letters that consists only of the letters A, B, C, and D — some of these letters may not appear in the sequence — where A is never immediately followed by B or D, B is never immediately followed by C, C is never immediately followed by A, and D is never immediately followed b... | 512 |
The sequence $(x_n)$ is defined by $x_1 = 115$ and $x_k = x_{k - 1}^2 + x_{k - 1}$ for all $k \ge 2.$ Compute
\[\frac{1}{x_1 + 1} + \frac{1}{x_2 + 1} + \frac{1}{x_3 + 1} + \dotsb.\] | \frac{1}{115} |
Solve for $y$: $4+2.3y = 1.7y - 20$ | -40 |
The sequence 1,3,1,3,3,1,3,3,3,1,3,3,3,3,1,3,... follows a certain rule. What is the sum of the first 44 terms in this sequence? | 116 |
Let $A, M$, and $C$ be digits with $(100A+10M+C)(A+M+C) = 2005$. What is $A$? | 4 |
Given $$\alpha, \beta \in (0, \frac{\pi}{2})$$, and $$\alpha + \beta \neq \frac{\pi}{2}, \sin\beta = \sin\alpha\cos(\alpha + \beta)$$.
(1) Express $\tan\beta$ in terms of $\tan\alpha$;
(2) Find the maximum value of $\tan\beta$. | \frac{\sqrt{2}}{4} |
Ryan wants to take 5 peanut butter sandwiches to the beach. If each sandwich consists of 3 slices of bread how many slices does Ryan need to make 5 sandwiches? | To make 1 sandwich, you need 3 slices so Ryan needs 1*3= <<1*3=3>>3 slices to make one sandwich.
Since it takes 3 slices to make a sandwich, and Ryan wants 5 sandwiches to take to the beach he needs 3*5= <<3*5=15>>15 slices.
#### 15 |
Farmer Red has three milk cows: Bess, Brownie, and Daisy. Bess, the smallest cow, gives him two pails of milk every day. Brownie, the largest cow, produces three times that amount. Then Daisy makes one pail more than Bess. How many pails of milk does Farmer Red get from them each week? | Bess produces 2 pails every day.
Brownie produces 3 times as much * 2 = <<3*2=6>>6 pails every day.
Daisy produces 2 + 1 more pail than Bess = <<2+1=3>>3 pails every day.
Bess, Brownie, and Daisy together produce 2 + 6 + 3 = <<2+6+3=11>>11 pails every day.
A week is 7 days, so Farmer Red gets 11 * 7 = <<11*7=77>>77 pai... |
There exists a complex number of the form $z = x + yi,$ where $x$ and $y$ are positive integers, such that
\[z^3 = -74 + ci,\]for some integer $c.$ Find $z.$ | 1 + 5i |
The area of the square is $s^2$ and the area of the rectangle is $3s \times \frac{9s}{2}$. | 7.41\% |
Jefferson Middle School has the same number of boys and girls. $\frac{3}{4}$ of the girls and $\frac{2}{3}$ of the boys went on a field trip. What fraction of the students on the field trip were girls? | \frac{9}{17} |
One dress requires 5.5 yards of fabric. Amare needs to make 4 dresses for the wedding and she has 7 feet of fabric. How many feet of fabric does Amare still need for the dresses? | Dresses = 5.5 * 4 = <<5.5*4=22>>22 yards
22 yards = <<22*3=66>>66 feet
66 - 7 = <<66-7=59>>59
Amare still needs 59 feet of fabric.
#### 59 |
It can be shown that for any positive integer $n,$
\[\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^n = \begin{pmatrix} F_{n + 1} & F_n \\ F_n & F_{n - 1} \end{pmatrix},\]where $F_n$ denotes the $n$th Fibonacci number.
Compute $F_{784} F_{786} - F_{785}^2.$ | -1 |
Points $A$ , $B$ , $C$ , $D$ , and $E$ are on the same plane such that $A,E,C$ lie on a line in that order, $B,E,D$ lie on a line in that order, $AE = 1$ , $BE = 4$ , $CE = 3$ , $DE = 2$ , and $\angle AEB = 60^\circ$ . Let $AB$ and $CD$ intersect at $P$ . The square of the area of quadrilateral $PA... | 967 |
What is the value of $(625^{\log_5 2015})^{\frac{1}{4}}$? | 2015 |
There are two alloys of copper and zinc. In the first alloy, there is twice as much copper as zinc, and in the second alloy, there is five times less copper than zinc. In what ratio should these alloys be combined to obtain a new alloy in which zinc is twice as much as copper? | 1 : 2 |
Determine the form of $n$ such that $2^n + 2$ is divisible by $n$ where $n$ is less than 100. | n=6, 66, 946 |
What is the units digit of $13^{2003}$? | 7 |
The function $f(x),$ defined for $0 \le x \le 1,$ has the following properties:
(i) $f(0) = 0.$
(ii) If $0 \le x < y \le 1,$ then $f(x) \le f(y).$
(iii) $f(1 - x) = 1 - f(x)$ for all $0 \le x \le 1.$
(iv) $f \left( \frac{x}{3} \right) = \frac{f(x)}{2}$ for $0 \le x \le 1.$
Find $f \left( \frac{2}{7} \right).$ | \frac{3}{8} |
In triangle $ABC$, $AB = 5$, $BC = 4$, and $CA = 3$.
[asy]
defaultpen(1);
pair C=(0,0), A = (0,3), B = (4,0);
draw(A--B--C--cycle);
label("\(A\)",A,N);
label("\(B\)",B,E);
label("\(C\)",C,SW);
[/asy]
Point $P$ is randomly selected inside triangle $ABC$. What is the probability that $P$ is closer to $C$ than it is ... | \frac{1}{2} |
The route not passing through the Zoo is 11 times shorter. | 11 |
If $a = \log_8 225$ and $b = \log_2 15$, then $a$, in terms of $b$, is: | \frac{2b}{3} |
Belle eats 4 dog biscuits and 2 rawhide bones every evening. If each rawhide bone is $1, and each dog biscuit is $0.25, then how much does it cost, in dollars, to feed Belle these treats for a week? | A week's worth of rawhide bones cost 1*2*7=<<1*2*7=14>>14 dollars.
A week's worth of dog biscuits cost 0.25*4*7=<<0.25*4*7=7>>7 dollars,
Thus, in total, it cost 14+7=<<14+7=21>>21 dollars
#### 21 |
In triangle $ABC$, altitudes $AD$, $BE$, and $CF$ intersect at the orthocenter $H$. If $\angle ABC = 49^\circ$ and $\angle ACB = 12^\circ$, then find the measure of $\angle BHC$, in degrees. | 61^\circ |
If $4x\equiv 8\pmod{20}$ and $3x\equiv 16\pmod{20}$, then what is the remainder when $x^2$ is divided by $20$? | 4 |
The function $f(x)$ takes positive real numbers to real numbers, such that
\[xf(y) - yf(x) = f \left( \frac{x}{y} \right)\]for all positive real numbers $x$ and $y.$ Find all possible values of $f(100).$ Enter all possible values, separated by commas. | 0 |
Emily ordered her playing cards by suit in the order $$A,2,3,4,5,6,7,8,9,10,J,Q,K,A,2,3,\cdots.$$What is the $42$nd card? | 3 |
Isabella's house has $3$ bedrooms. Each bedroom is $12$ feet long, $10$ feet wide, and $8$ feet high. Isabella must paint the walls of all the bedrooms. Doorways and windows, which will not be painted, occupy $60$ square feet in each bedroom. How many square feet of walls must be painted? | 876 |
Sasha wrote the numbers $7, 8, 9, \ldots, 17$ on the board and then erased one or more of them. It turned out that the remaining numbers on the board cannot be divided into several groups such that the sums of the numbers in the groups are equal. What is the maximum value that the sum of the remaining numbers on the bo... | 121 |
Bryan has some stamps of 3 cents, 4 cents, and 6 cents. What is the least number of stamps he can combine so the value of the stamps is 50 cents? | 10 |
Let $x$ and $y$ be two-digit positive integers with mean $60$. What is the maximum value of the ratio $\frac{x}{y}$? | \frac{33}{7} |
There are 20 hands in Peter’s class, not including his. Assume every student in the class has 2 arms and 2 hands. How many students are in Peter’s class including him? | There are 20 + 2 of Peter’s hands = <<20+2=22>>22 hands in Peter’s class including his
There are 22 hands / 2 hands per student = <<22/2=11>>11 students
#### 11 |
When Harriett vacuumed the sofa and chair she found 10 quarters, 3 dimes, 3 nickels, and 5 pennies. How much money did Harriett find? | She found 10 quarters that are $0.25 each so she found 10*.25 = $<<10*.25=2.50>>2.50
She found 3 dimes that are $0.10 each so she found 3*.10 = $<<3*.10=0.30>>0.30
She found 3 nickels that are $0.05 each so she found 3*.05 = $<<3*.05=0.15>>0.15
She found 5 pennies that are $0.01 each so she found 5*.01 = $<<5*.01=0.05>... |
Let $ABCD$ be a quadrilateral with an inscribed circle, centre $O$ . Let
\[AO = 5, BO =6, CO = 7, DO = 8.\]
If $M$ and $N$ are the midpoints of the diagonals $AC$ and $BD$ , determine $\frac{OM}{ON}$ . | 35/48 |
How many natural-number factors does $\textit{N}$ have if $\textit{N} = 2^3 \cdot 3^2 \cdot 5^1$? | 24 |
Detached calculation.
327 + 46 - 135
1000 - 582 - 128
(124 - 62) × 6
500 - 400 ÷ 5 | 420 |
The sequence $\left\{a_{n}\right\}_{n \geq 1}$ is defined by $a_{n+2}=7 a_{n+1}-a_{n}$ for positive integers $n$ with initial values $a_{1}=1$ and $a_{2}=8$. Another sequence, $\left\{b_{n}\right\}$, is defined by the rule $b_{n+2}=3 b_{n+1}-b_{n}$ for positive integers $n$ together with the values $b_{1}=1$ and $b_{2}... | 89 |
Let $m, n \in \mathbb{N}$, and $f(x) = (1+x)^m + (1+x)^n$.
1. When $m=n=7$, $f(x) = a_7x^7 + a_6x^6 + \ldots + a_1x + a_0$, find $a_0 + a_2 + a_4 + a_6$.
2. If the coefficient of the expanded form of $f(x)$ is 19 when $m, n$ vary, find the minimum value of the coefficient of $x^2$. | 81 |
Consider a coordinate plane where at each lattice point, there is a circle with radius $\frac{1}{8}$ and a square with sides of length $\frac{1}{4}$, whose sides are parallel to the coordinate axes. A line segment runs from $(0,0)$ to $(729, 243)$. Determine how many of these squares and how many of these circles are i... | 972 |
A patient is receiving treatment through a saline drip which makes 20 drops per minute. If the treatment lasts 2 hours, and every 100 drops equal 5 ml of liquid, how many milliliters of treatment will the patient receive after the 2 hours have passed? | The total of minutes the treatment will take is 2 hours x 60 minutes/hour = <<2*60=120>>120 minutes
After 120 minutes has passed the total amount of drops that the patient will receive is 20 drops/minute x 120 minutes = <<20*120=2400>>2400 drops
Knowing the number of drops, we know that the patient has taken 2400 drops... |
Let \[f(x) =
\begin{cases}
3x + 5 &\text{if }x<-3, \\
7-4x&\text{if }x\ge -3.
\end{cases}
\]Find $f(5)$. | -13 |
Given that Mr. A initially owns a home worth $\$15,000$, he sells it to Mr. B at a $20\%$ profit, then Mr. B sells it back to Mr. A at a $15\%$ loss, then Mr. A sells it again to Mr. B at a $10\%$ profit, and finally Mr. B sells it back to Mr. A at a $5\%$ loss, calculate the net effect of these transactions on Mr. A. | 3541.50 |
Form a four-digit number without repeating digits using the numbers 0, 1, 2, 3, 4, 5, 6, where the sum of the digits in the units, tens, and hundreds places is even. How many such four-digit numbers are there? (Answer with a number) | 324 |
To investigate the growth inhibitory effect of a certain drug on mice, $40$ mice were divided into two groups, a control group (without the drug) and an experimental group (with the drug).<br/>$(1)$ Suppose the number of mice in the control group among two mice is $X$, find the probability distribution and mathematical... | 95\% |
If $\frac{2+3+4}{3}=\frac{1990+1991+1992}{N}$, then $N=$ | 1991 |
Let $A = \{x \mid x^2 - ax + a^2 - 19 = 0\}$, $B = \{x \mid x^2 - 5x + 6 = 0\}$, and $C = \{x \mid x^2 + 2x - 8 = 0\}$.
(1) If $A = B$, find the value of $a$;
(2) If $B \cap A \neq \emptyset$ and $C \cap A = \emptyset$, find the value of $a$. | -2 |
In the rectangular coordinate system on the plane, a polar coordinate system is established with the coordinate origin as the pole and the positive semi-axis of the x-axis as the polar axis. The polar coordinate equation of the curve C₁ is ρ²-6ρcosθ+5=0, and the parametric equation of the curve C₂ is $$\begin{cases} x=... | \sqrt {7} |
Calculate the value of $({-\frac{4}{5}})^{2022} \times ({\frac{5}{4}})^{2021}$. | \frac{4}{5} |
Given that \(a_{1}, a_{2}, \cdots, a_{n}\) are \(n\) people corresponding to \(A_{1}, A_{2}, \cdots, A_{n}\) cards (\(n \geq 2\), \(a_{i}\) corresponds to \(A_{i}\)). Now \(a_{1}\) picks a card from the deck randomly, and then each person in sequence picks a card. If their corresponding card is still in the deck, they ... | \frac{1}{2} |
The diagonals of a regular hexagon have two possible lengths. What is the ratio of the shorter length to the longer length? Express your answer as a common fraction in simplest radical form. | \frac{\sqrt{3}}{2} |
Let $x$, $y$, and $z$ be positive real numbers such that $(x \cdot y) + z = (x + z) \cdot (y + z)$. What is the maximum possible value of $xyz$? | \frac{1}{27} |
Let $G, A_{1}, A_{2}, A_{3}, A_{4}, B_{1}, B_{2}, B_{3}, B_{4}, B_{5}$ be ten points on a circle such that $G A_{1} A_{2} A_{3} A_{4}$ is a regular pentagon and $G B_{1} B_{2} B_{3} B_{4} B_{5}$ is a regular hexagon, and $B_{1}$ lies on minor arc $G A_{1}$. Let $B_{5} B_{3}$ intersect $B_{1} A_{2}$ at $G_{1}$, and let ... | 12^{\circ} |
Two congruent right circular cones each with base radius $3$ and height $8$ have the axes of symmetry that intersect at right angles at a point in the interior of the cones a distance $3$ from the base of each cone. A sphere with radius $r$ lies withing both cones. The maximum possible value of $r^2$ is $\frac{m}{n}$, ... | 298 |
You are trying to cross a 400 foot wide river. You can jump at most 4 feet, but you have many stones you can throw into the river. You will stop throwing stones and cross the river once you have placed enough stones to be able to do so. You can throw straight, but you can't judge distance very well, so each stone ends ... | 712.811 |
Given \(0 \leqslant x \leqslant 2\), the function \(y=4^{x-\frac{1}{2}}-3 \cdot 2^{x}+5\) reaches its minimum value at? | \frac{1}{2} |
A and B start from points A and B simultaneously, moving towards each other and meet at point C. If A starts 2 minutes earlier, then their meeting point is 42 meters away from point C. Given that A's speed is \( a \) meters per minute, B's speed is \( b \) meters per minute, where \( a \) and \( b \) are integers, \( a... | 21 |
A rancher owns a mixture of 8 sheep and 5 cattle that graze on his land. In a typical year, the rancher will allow his animals to feed off his pastures for as long as possible before they run out of grass. After the pastures run out of grass, he must buy feed corn for $10 per bag. Each cow eats 2 acres of grass per ... | First, the 144 acres of grass will be eaten according to the equation 5*2*T + 8*1*T = 144.
Solving this equation for T, we find 18*T = 144, or T = 8 months of food are covered by the animals grazing on the rancher's pasture.
This means the farmer must buy feed corn for the remaining 12 - 8 = <<12-8=4>>4 months.
Next, t... |
Given that the random variable $X$ follows a normal distribution $N(0,\sigma^{2})$, if $P(X > 2) = 0.023$, determine the probability $P(-2 \leqslant X \leqslant 2)$. | 0.954 |
Suppose that $a$ and $b$ are nonzero integers such that two of the roots of
\[x^3 + ax^2 + bx + 9a\]coincide, and all three roots are integers. Find $|ab|.$ | 1344 |
Jackson is buying chairs for his restaurant. He has 6 tables with 4 seats and 12 tables with 6 seats. How many chairs total does Jackson need to buy? | First find how many chairs Jackson needs for the four-seat tables by multiplying the number of four-seat tables by 4: 4 chairs/table * 6 tables = <<4*6=24>>24 chairs
Next find how many chairs Jackson needs for the six-seat tables by multiplying the number of six-seat tables by 6: 6 chairs/table * 12 tables = <<6*12=72>... |
Granny Smith has $63. Elberta has $2 more than Anjou and Anjou has one-third as much as Granny Smith. How many dollars does Elberta have? | 23 |
Maya owns 16 pairs of shoes, consisting of 8 identical black pairs, 4 identical brown pairs, 3 identical grey pairs, and 1 pair of white shoes. If Maya randomly picks two shoes, what is the probability that they are the same color and that one is a left shoe and the other is a right shoe? | \frac{45}{248} |
If $(x,y)$ is a solution to the system
\begin{align*}
xy &= 6, \\
x^2 y + xy^2 + x + y &= 63,
\end{align*}find $x^2 + y^2.$ | 69 |
A factory produces a certain type of component, and the inspector randomly selects 16 of these components from the production line each day to measure their dimensions (in cm). The dimensions of the 16 components selected in one day are as follows:
10.12, 9.97, 10.01, 9.95, 10.02, 9.98, 9.21, 10.03, 10.04, 9.99, 9.98,... | \frac{1}{5} |
Find the sum of the infinite series $1+2\left(\dfrac{1}{1998}\right)+3\left(\dfrac{1}{1998}\right)^2+4\left(\dfrac{1}{1998}\right)^3+\cdots$. | \frac{3992004}{3988009} |
In the disaster relief donation, $\frac{1}{10}$ of the people in a company each donated 200 yuan, $\frac{3}{4}$ of the people each donated 100 yuan, and the remaining people each donated 50 yuan. Find the average donation per person in the company. | 102.5 |
A seven-digit natural number \( N \) is called interesting if:
- It consists of non-zero digits;
- It is divisible by 4;
- Any number obtained from \( N \) by permuting its digits is also divisible by 4.
How many interesting numbers exist? | 128 |
Find the largest six-digit number in which all digits are distinct, and each digit, except the first and last ones, is either the sum or the difference of its neighboring digits. | 972538 |
An integer $n$ is chosen uniformly at random from the set $\{1,2,3, \ldots, 2023!\}$. Compute the probability that $$\operatorname{gcd}\left(n^{n}+50, n+1\right)=1$$ | \frac{265}{357} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $\angle A=45^{\circ}$, $a=6$.
(1) If $\angle C=105^{\circ}$, find $b$;
(2) Find the maximum area of $\triangle ABC$. | 9(1+\sqrt{2}) |
A $4$-foot by $8$-foot rectangular piece of plywood will be cut into $4$ congruent rectangles with no wood left over and no wood lost due to the cuts. What is the positive difference, in feet, between the greatest possible perimeter of a single piece and the least possible perimeter of a single piece? | 6 |
Each day for four days, Linda traveled for one hour at a speed that resulted in her traveling one mile in an integer number of minutes. Each day after the first, her speed decreased so that the number of minutes to travel one mile increased by $5$ minutes over the preceding day. Each of the four days, her distance trav... | 25 |
Square the numbers \(a=1001\) and \(b=1001001\). Extract the square root of the number \(c=1002003004005004003002001\). | 1001001001001 |
Esmeralda has created a special knight to play on quadrilateral boards that are identical to chessboards. If a knight is in a square then it can move to another square by moving 1 square in one direction and 3 squares in a perpendicular direction (which is a diagonal of a $2\times4$ rectangle instead of $2\times3$ like... | 12 |
In a race, there are eight runners. The first five runners finish the race in 8 hours, while the rest of the runners finish the race 2 hours later. Calculate the total time the eight runners took to finish the race. | The first five runners took a combined total of 5*8 = <<5*8=40>>40 hours to finish the race.
The number of runners who finished the race 2 hours later after the first five is 8-5 = <<8-5=3>>3
The 3 runners who finished the race 2 hours later took a total of 8+2 =<<8+2=10>>10 hours to run the race.
Together, the three r... |
Let \( n \) be a positive integer not exceeding 1996. If there exists a \( \theta \) such that \( (\sin \theta + i \cos \theta)^{n} = \sin \theta + i \cos n \theta \), find the number of possible values for \( n \). | 499 |
Three balls are randomly placed into three boxes. Let the random variable $\xi$ denote the maximum number of balls in any one box. Determine the mathematical expectation $E(\xi)$ of $\xi$. | \frac{17}{9} |
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