problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
What are the rightmost three digits of $7^{1984}$? | 401 |
Four balls of radius $1$ are mutually tangent, three resting on the floor and the fourth resting on the others.
A tetrahedron, each of whose edges have length $s$, is circumscribed around the balls. Then $s$ equals | 2+2\sqrt{6} |
Manny has a tree that grows at the rate of fifty centimeters every two weeks. If the tree is currently 2 meters tall, how tall, in centimeters, will the tree be in 4 months? | A month has four weeks, so there are 4*4= <<4*4=16>>16 weeks in four months.
Since the tree grows for five centimeters every two weeks, it will grow by fifty centimeters every week for 16/2 = 8 weeks.
The tree will increase in height by 8*50 = <<8*50=400>>400 centimeters after four months.
Since 1 meter equals 100 cent... |
In the plane rectangular coordinate system $xOy$, the parametric equations of the line $l$ are $\left\{{\begin{array}{l}{x=t}\\{y=-1+\sqrt{3}t}\end{array}}\right.$ (where $t$ is a parameter). Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive half-axis of the $x$-axis as the... | \frac{2\sqrt{3} + 1}{3} |
For distinct positive integers $a, b<2012$, define $f(a, b)$ to be the number of integers $k$ with $1\le k<2012$ such that the remainder when $ak$ divided by $2012$ is greater than that of $bk$ divided by $2012$. Let $S$ be the minimum value of $f(a, b)$, where $a$ and $b$ range over all pairs of distinct positive inte... | 502 |
Find a nonzero $p$ such that $px^2-12x+4=0$ has only one solution. | 9 |
John decides to go back to school to get his PhD. He first takes 1 year of courses to get acclimated back to school life before starting his PhD. After that, he spends 2 years learning the basics of his field. He then starts research, and he spends 75% more time on research than he did learning the basics. He then ... | It took him 2*.75=<<2*.75=1.5>>1.5 years longer to do his research than it took to learn the basics
So it took him a total of 2+1.5=<<2+1.5=3.5>>3.5 years to do research
His dissertation took 1/2 = <<1/2=.5>>.5 years to write
So everything together took 1+2+3.5+.5=<<1+2+3.5+.5=7>>7 years
#### 7 |
Allison, Brian and Noah each have a 6-sided cube. All of the faces on Allison's cube have a 5. The faces on Brian's cube are numbered 1, 2, 3, 4, 5 and 6. Three of the faces on Noah's cube have a 2 and three of the faces have a 6. All three cubes are rolled. What is the probability that Allison's roll is greater than e... | \frac{1}{3} |
Hallie is working as a waitress for $10/hour. On Monday, she works for 7 hours, and she receives $18 in tips. On Tuesday she works for 5 hours, and she receives $12 in tips. On Wednesday she works for 7 hours, and she receives $20 in tips. How much money does she earn in total from Monday to Wednesday? | Hallie works a total of 7 + 5 + 7 = <<7+5+7=19>>19 hours
For her hourly pay, she earns 19 * $10 = $<<19*10=190>>190
In tips, she receives a total of $18 + $12 + $20 = $<<18+12+20=50>>50
The total amount of money Hallie earns is $190 + $50 = $<<190+50=240>>240
#### 240 |
Given vectors $\overrightarrow {m}=(\sin x，-1)$ and $\overrightarrow {n}=( \sqrt {3}\cos x，- \frac {1}{2})$, and the function $f(x)= \overrightarrow {m}^{2}+ \overrightarrow {m}\cdot \overrightarrow {n}-2$.
(I) Find the maximum value of $f(x)$ and the set of values of $x$ at which the maximum is attained.
(II) Given th... | \frac{2\sqrt{3}}{3} |
An object in the plane moves from the origin and takes a ten-step path, where at each step the object may move one unit to the right, one unit to the left, one unit up, or one unit down. How many different points could be the final point? | 221 |
A set of 10 distinct integers $S$ is chosen. Let $M$ be the number of nonempty subsets of $S$ whose elements have an even sum. What is the minimum possible value of $M$ ?
<details><summary>Clarifications</summary>
- $S$ is the ``set of 10 distinct integers'' from the first sentence.
</details>
*Ray Li* | 511 |
Alice starts to make a list, in increasing order, of the positive integers that have a first digit of 2. She writes $2, 20, 21, 22, \ldots$ but by the 1000th digit she (finally) realizes that the list would contain an infinite number of elements. Find the three-digit number formed by the last three digits she wrote (th... | 216 |
Note that there are exactly three ways to write the integer $4$ as a sum of positive odd integers where the order of the summands matters:
\begin{align*}
1+1+1+1&=4,
1+3&=4,
3+1&=4.
\end{align*}
Let $f(n)$ be the number of ways to write a natural number $n$ as a sum of positive odd integers where the order of the... | 71 |
What is the ratio of the volume of a cube with edge length six inches to the volume of a cube with edge length one foot? Express your answer as a common fraction. | \frac{1}{8} |
Missy has an obedient dog and a stubborn dog. She has to yell at the stubborn dog four times for every one time she yells at the obedient dog. If she yells at the obedient dog 12 times, how many times does she yell at both dogs combined? | First find the number of times Missy yells at the stubborn dog: 4 * 12 times = <<4*12=48>>48 times
Then add the number of times she yells at the obedient dog to find the total number of times she yells: 48 times + 12 times = <<48+12=60>>60 times
#### 60 |
A plane intersects a right circular cylinder of radius $2$ forming an ellipse. If the major axis of the ellipse is $60\%$ longer than the minor axis, find the length of the major axis. | 6.4 |
Let $a_0 = 2,$ $b_0 = 3,$ and
\[a_{n + 1} = \frac{a_n^2}{b_n} \quad \text{and} \quad b_{n + 1} = \frac{b_n^2}{a_n}\]for all $n \ge 0.$ Then $b_8 = \frac{3^m}{2^n}$ for some integers $m$ and $n.$ Enter the ordered pair $(m,n).$ | (3281,3280) |
Let $a, b, c$, and $d$ be positive real numbers such that
\[\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}a^2+b^2&=&c^2+d^2&=&2008,\\ ac&=&bd&=&1000.\end{array}\]
If $S=a+b+c+d$, compute the value of $\lfloor S\rfloor$.
| 126 |
Cristian has 50 more black cookies in his cookie jar than white cookies. He eats half of the black cookies and 3/4 of the white cookies. If he initially had 80 white cookies, how many cookies are remaining within the cookie jar altogether? | Cristian eats 3/4*80 = <<3/4*80=60>>60 white cookies from the jar, 3/4 of the total number of white cookies in the cookie jar.
The number of white cookies remaining in the cookie jar is 80-60 =<<80-60=20>>20
In his cookie jar, Cristian also has 80+50 = <<80+50=130>>130 black cookies.
He eats 1/2*130 = <<130*1/2=65>>65 ... |
Find the number of positive integers $n,$ $1 \le n \le 2600,$ for which the polynomial $x^2 + x - n$ can be factored as the product of two linear factors with integer coefficients. | 50 |
Randomly assign numbers 1 to 400 to 400 students. Then decide to use systematic sampling to draw a sample of size 20 from these 400 students. By order of their numbers, evenly divide them into 20 groups (1-20, 21-40, ..., 381-400). If the number drawn from the first group is 11 by lottery, the number drawn from the thi... | 51 |
There exist constants $a$ and $b$ so that
\[\cos^3 \theta = a \cos 3 \theta + b \cos \theta\]for all angles $\theta.$ Enter the ordered pair $(a,b).$ | \left( \frac{1}{4}, \frac{3}{4} \right) |
There were 10 apples and 5 oranges in the basket. If Emily adds 5 more oranges to the basket, what percentage of the fruit in the basket would be apples? | If Emily adds 5 more oranges, there will be a total of 10 + 5 + 5 = <<10+5+5=20>>20 fruits in the basket.
So, the percentage of the fruit that is apples is 10/20 x 100% = 50%.
#### 50 |
A decorative garden is designed with a rectangular lawn with semicircles of grass at either end. The ratio of the length of the rectangle to its width is 5:4, and the total length including the semicircles is 50 feet. Calculate the ratio of the area of the rectangle to the combined area of the semicircles. | \frac{5}{\pi} |
How many positive integers less than 2023 are congruent to 7 modulo 13? | 156 |
Find the largest positive integer $n>10$ such that the residue of $n$ when divided by each perfect square between $2$ and $\dfrac n2$ is an odd number. | 505 |
Our water polo team has 15 members. I want to choose a starting team consisting of 7 players, one of whom will be the goalie (the other six positions are interchangeable, so the order in which they are chosen doesn't matter). In how many ways can I choose my starting team? | 45,\!045 |
In a large bag of decorative ribbons, $\frac{1}{4}$ are yellow, $\frac{1}{3}$ are purple, $\frac{1}{8}$ are orange, and the remaining 45 ribbons are silver. How many of the ribbons are orange? | 19 |
Given the function $f(x)=\sin ( \frac {7π}{6}-2x)-2\sin ^{2}x+1(x∈R)$,
(1) Find the period and the monotonically increasing interval of the function $f(x)$;
(2) In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. The graph of function $f(x)$ passes through points $(A, \frac {1... | 3 \sqrt {2} |
Given a small cube block, each face is painted with a different color. If you want to carve 1, 2, 3, 4, 5, 6 small dots on the faces of the block, and the dots 1 and 6, 2 and 5, 3 and 4 are carved on opposite faces respectively, determine the number of different carving methods. | 48 |
Gilbert, the bearded dragon, eats 4 crickets per week when the temperature averages 90 degrees F per day, but he eats twice as many crickets per week when the temperature averages 100 degrees F. How many crickets will he eat over 15 weeks if the temperature averages 90 degrees F for 80% of the time, and 100 degrees F ... | It will be 90 degrees for 80% of the 15 weeks, for a total of 0.8*15=12 weeks.
It will be 100 degrees F for the remainder of the 15 weeks, or 15-12=<<15-12=3>>3 weeks.
During the 12 weeks when it averages 90 degrees F, he will eat 4 crickets per week for a total of 12*4=<<12*4=48>>48 crickets.
When it averages 100 degr... |
Tony read 23 books, Dean read 12 books and Breanna read 17 books. Tony and Dean read 3 of the same books and all three had read the same book as well. In total how many different books have Tony, Dean, and Breanna read? | Tony and Dean together read 23 + 12 = <<23+12=35>>35 books
3 of these books were duplicates so Tony and Dean together read 35 - 3 = <<35-3=32>>32 unique books
All three read 32 + 17 = <<32+17=49>>49 books
Because the books are being counted three times, the book is counted 3 - 1 = <<3-1=2>>2 extra times.
All together t... |
Find the focus of the parabola $y = 4x^2 - 3.$ | \left( 0, -\frac{47}{16} \right) |
How many positive integers less than 1000 are congruent to 6 (mod 11)? | 91 |
Suppose that $x$ is an integer that satisfies the following congruences: \begin{align*}
3+x &\equiv 2^2 \pmod{3^3} \\
5+x &\equiv 3^2 \pmod{5^3} \\
7+x &\equiv 5^2 \pmod{7^3}
\end{align*}What is the remainder when $x$ is divided by $105$? | 4 |
Consider a sequence of consecutive integer sets where each set starts one more than the last element of the preceding set and each set has one more element than the one before it. For a specific n where n > 0, denote T_n as the sum of the elements in the nth set. Find T_{30}. | 13515 |
If $x$ is a positive number such that \[\sqrt{8x}\cdot\sqrt{10x}\cdot\sqrt{3x}\cdot\sqrt{15x}=15,\]find all possible values for $x$. | \frac{1}{2} |
An entry in a grid is called a saddle point if it is the largest number in its row and the smallest number in its column. Suppose that each cell in a $3 \times 3$ grid is filled with a real number, each chosen independently and uniformly at random from the interval $[0,1]$. Compute the probability that this grid has at... | \frac{3}{10} |
In order to train for his fights Rocky would run 4 miles on day one of training. Rocky would double the miles for day 2, and triple the miles from day 2 for day 3. How many miles did Rocky run in the first three days of training? | Day 1:4
Day 2:2(4)=8
Day 3:8(3)=24
Total:4+8+24=<<4+8+24=36>>36 miles
#### 36 |
Determine the monotonicity of the function $f(x) = \frac{x}{x^2 + 1}$ on the interval $(1, +\infty)$, and find the maximum and minimum values of the function when $x \in [2, 3]$. | \frac{3}{10} |
Four people are sitting at four sides of a table, and they are dividing a 32-card Hungarian deck equally among themselves. If one selected player does not receive any aces, what is the probability that the player sitting opposite them also has no aces among their 8 cards? | 130/759 |
How many integers between $2020$ and $2400$ have four distinct digits arranged in increasing order? (For example, $2347$ is one integer.) | 15 |
Let $C$ be the graph of $xy = 1$, and denote by $C^*$ the reflection of $C$ in the line $y = 2x$. Let the equation of $C^*$ be written in the form
\[12x^2 + bxy + cy^2 + d = 0.\]
Find the product $bc$. | 84 |
The school's boys basketball team has 16 players, including a set of twins, Bob and Bill, and a set of triplets, Chris, Craig, and Carl. In how many ways can we choose 7 starters if all three triplets must be in the starting lineup and both twins must either both be in the lineup or both not be in the lineup? | 385 |
How many four-digit numbers contain one even digit and three odd digits, with no repeated digits? | 1140 |
In the diagram, $PQRS$ is a trapezoid with an area of $12.$ $RS$ is twice the length of $PQ.$ What is the area of $\triangle PQS?$
[asy]
draw((0,0)--(1,4)--(7,4)--(12,0)--cycle);
draw((7,4)--(0,0));
label("$S$",(0,0),W);
label("$P$",(1,4),NW);
label("$Q$",(7,4),NE);
label("$R$",(12,0),E);
[/asy] | 4 |
Natalie's father has saved up $10,000 to split up between his kids. Natalie will get half, as she is the oldest. Rick will get 60 percent of the remaining money, and Lucy will get whatever is left after Natilie and Rick get paid. How much money does Lucy get? | Natalie: 10000/2=<<10000/2=5000>>5000$
10000-5000=<<10000-5000=5000>>5000
Rick:5000(.60)=3000$
5000-3000=<<5000-3000=2000>>2000$ left for Lucy.
#### 2000 |
In Mr. Abraham's class, $10$ of the $15$ students received an $A$ on the latest exam. If the same ratio of students received an $A$ on Mrs. Berkeley's latest exam, and if Mrs. Berkeley has $24$ students total, how many students in Mrs. Berkeley's class received an $A$? | 16 |
The stem-and-leaf plot shows the number of minutes and seconds of one ride on each of the $17$ top-rated roller coasters in the world. In the stem-and-leaf plot, $2 \ 20$ represents $2$ minutes, $20$ seconds, which is the same as $140$ seconds. What is the median of this data set? Express your answer in seconds.
\begi... | 163 |
Given that the function $f(x)= \frac{1}{2}(m-2)x^{2}+(n-8)x+1$ is monotonically decreasing in the interval $\left[ \frac{1}{2},2\right]$ where $m\geqslant 0$ and $n\geqslant 0$, determine the maximum value of $mn$. | 18 |
An integer is called snakelike if its decimal representation $a_1a_2a_3\cdots a_k$ satisfies $a_i<a_{i+1}$ if $i$ is odd and $a_i>a_{i+1}$ if $i$ is even. How many snakelike integers between 1000 and 9999 have four distinct digits? | 882 |
If point P is one of the intersections of the hyperbola with foci A(-√10,0), B(√10,0) and a real axis length of 2√2, and the circle x^2 + y^2 = 10, calculate the value of |PA| + |PB|. | 6\sqrt{2} |
Given that $(2x)_((-1)^{5}=a_0+a_1x+a_2x^2+...+a_5x^5$, find:
(1) $a_0+a_1+...+a_5$;
(2) $|a_0|+|a_1|+...+|a_5|$;
(3) $a_1+a_3+a_5$;
(4) $(a_0+a_2+a_4)^2-(a_1+a_3+a_5)^2$. | -243 |
Lennon is a sales rep and is paid $0.36 in mileage reimbursement when he travels to meet with clients. On Monday he drove 18 miles. Tuesday he drove 26 miles. Wednesday and Thursday he drove 20 miles each day and on Friday he drove 16 miles. How much money will he be reimbursed? | In one week he drove 18+26+20+20+16 = <<18+26+20+20+16=100>>100 miles
He is reimbursed $0.36 per mile of travel, and he travels 100 miles so he will receive .36*100 = $<<.36*100=36.00>>36.00
#### 36 |
Let $S = \{5^k | k \in \mathbb{Z}, 0 \le k \le 2004 \}$. Given that $5^{2004} = 5443 \cdots 0625$ has $1401$ digits, how many elements of $S$ begin with the digit $1$?
| 604 |
Given that the sequence $\left\{\frac{1}{b_{n}}\right\}$ is a "dream sequence" defined by $\frac{1}{a_{n+1}}- \frac{2}{a_{n}}=0$, and that $b_1+b_2+b_3=2$, find the value of $b_6+b_7+b_8$. | 64 |
An ancient Greek was born on January 7, 40 B.C., and died on January 7, 40 A.D. Calculate the number of years he lived. | 79 |
If $f(x) = 4-3x$ and $g(x) = x^2 +1$, find $f(g(\sqrt{2}))$. | -5 |
Find the domain of the expression $\frac{\sqrt{x-2}}{\sqrt{5-x}}$. | [2,5) |
Given that $F_{1}$ and $F_{2}$ are the left and right foci of the hyperbola $C: \frac {x^{2}}{a^{2}} - \frac {y^{2}}{b^{2}} = 1$ $(a > 0, b > 0)$, point $P$ is on the hyperbola $C$, $PF_{2}$ is perpendicular to the x-axis, and $\sin \angle PF_{1}F_{2} = \frac {1}{3}$, determine the eccentricity of the hyperbola $C$. | \sqrt{2} |
Given the numbers 2 and 8, find the product of three numbers that form a geometric sequence with these two numbers. | 64 |
In triangle \(ABC\), the side \(BC\) is 19 cm. The perpendicular \(DF\), drawn to side \(AB\) through its midpoint \(D\), intersects side \(BC\) at point \(F\). Find the perimeter of triangle \(AFC\) if side \(AC\) is 10 cm. | 29 |
Romeo buys five bars of chocolate at $5 each and sells these chocolates for a total of $90. If he uses packaging material that costs him $2 for each bar of chocolate, how much profit does he make for all the chocolates he sells? | The selling price for each bar of chocolate is 90/5=$<<90/5=18>>18.
The profit for each bar of chocolate is 18-5-2=$<<18-5-2=11>>11.
The total profit is 11*5=$<<11*5=55>>55.
#### 55 |
Among the following statements, the correct one is:
(1) The probability of event A or B happening is definitely greater than the probability of exactly one of A or B happening;
(2) The probability of events A and B happening simultaneously is definitely less than the probability of exactly one of A or B happening; ... | (4) |
Given a triangle $\triangle ABC$ with its three interior angles $A$, $B$, and $C$ satisfying: $$A+C=2B, \frac {1}{\cos A}+ \frac {1}{\cos C}=- \frac { \sqrt {2}}{\cos B}$$, find the value of $$\cos \frac {A-C}{2}$$. | \frac { \sqrt {2}}{2} |
Svitlana writes the number 147 on a blackboard. Then, at any point, if the number on the blackboard is $n$, she can perform one of the following three operations: - if $n$ is even, she can replace $n$ with $\frac{n}{2}$; - if $n$ is odd, she can replace $n$ with $\frac{n+255}{2}$; and - if $n \geq 64$, she can replace ... | 163 |
Evaluate $101 \times 101$ using a similar mental math technique. | 10201 |
There are 110 calories in a serving of cheese. Rick buys the large blocks that have 16 servings per block. If Rick has already eaten 5 servings of cheese, how many calories are remaining in the block? | The block has 16 servings and he has eaten 5 servings leaving 16-5 = <<16-5=11>>11 servings of cheese
There are 11 servings of cheese left and each serving is 110 calories for a total of 11*110 = <<11*110=1210>>1,210 calories
#### 1210 |
Let
\[z = \frac{(-11 + 13i)^3 \cdot (24 - 7i)^4}{3 + 4i},\]and let $w = \frac{\overline{z}}{z}.$ Compute $|w|.$ | 1 |
Two fair octahedral dice, each with the numbers 1 through 8 on their faces, are rolled. Let $N$ be the remainder when the product of the numbers showing on the two dice is divided by 8. Find the expected value of $N$. | \frac{11}{4} |
There are 7 students standing in a row. How many different arrangements are there in the following situations?
(1) A and B must stand together;
(2) A is not at the head of the line, and B is not at the end of the line;
(3) There must be exactly one person between A and B. | 1200 |
A stone is dropped into a well and the report of the stone striking the bottom is heard $7.7$ seconds after it is dropped. Assume that the stone falls $16t^2$ feet in t seconds and that the velocity of sound is $1120$ feet per second. The depth of the well is: | 784 |
A game is played with tokens according to the following rule. In each round, the player with the most tokens gives one token to each of the other players and also places one token in the discard pile. The game ends when some player runs out of tokens. Players $A$, $B$, and $C$ start with $15$, $14$, and $13$ tokens,... | 37 |
Points \( A, B, C \), and \( D \) are located on a line such that \( AB = BC = CD \). Segments \( AB \), \( BC \), and \( CD \) serve as diameters of circles. From point \( A \), a tangent line \( l \) is drawn to the circle with diameter \( CD \). Find the ratio of the chords cut on line \( l \) by the circles with di... | \sqrt{6}: 2 |
Let $g: \mathbb{R} \to \mathbb{R}$ be a function such that
\[g((x + y)^2) = g(x)^2 - 2xg(y) + 2y^2\]
for all real numbers $x$ and $y.$ Find the number of possible values of $g(1)$ and the sum of all possible values of $g(1)$. | \sqrt{2} |
Given $\frac{x}{y}=\frac{10}{4}.$ If $y = 18$, what is the value of $x$? | 45 |
Merry had 50 boxes of apples on Saturday and 25 boxes on Sunday. There were 10 apples in each box. If she sold a total of 720 apples on Saturday and Sunday, how many boxes of apples are left? | Merry had a total of 50 + 25 = <<50+25=75>>75 boxes of apples.
These 75 boxes is equal to 75 x 10 = <<75*10=750>>750 apples.
There were 750 - 720 = <<750-720=30>>30 apples left.
Thus, 30/10 = <<30/10=3>>3 boxes of apples are left.
#### 3 |
How many positive even multiples of $3$ less than $2020$ are perfect squares? | 7 |
A polynomial \( P(x) \) of degree 10 with a leading coefficient of 1 is given. The graph of \( y = P(x) \) lies entirely above the x-axis. The polynomial \( -P(x) \) was factored into irreducible factors (i.e., polynomials that cannot be represented as the product of two non-constant polynomials). It is known that at ... | 243 |
An earthquake caused four buildings to collapse. Experts predicted that each following earthquake would have double the number of collapsing buildings as the previous one, since each one would make the foundations less stable. After three more earthquakes, how many buildings had collapsed including those from the first... | The second earthquake caused 2 * 4 = <<2*4=8>>8 buildings to collapse.
The third earthquake caused 2 * 8 = <<2*8=16>>16 buildings to collapse.
The fourth earthquake caused 16 * 2 = <<16*2=32>>32 buildings to collapse.
Including the first earthquake, the earthquakes caused 4 + 8 + 16 + 32 = <<4+8+16+32=60>>60 buildings ... |
An integer between $1000$ and $9999$, inclusive, is chosen at random. What is the probability that it is an odd integer whose digits are all distinct? | \frac{56}{225} |
Simplify $\sqrt{9800}$. | 70\sqrt{2} |
In the diagram, the area of triangle $ABC$ is 36 square units. What is the area of triangle $BCD$ if the length of segment $CD$ is 39 units?
[asy]
draw((0,0)--(39,0)--(10,18)--(0,0)); // Adjusted for new problem length
dot((0,0));
label("$A$",(0,0),SW);
label("9",(4.5,0),S); // New base length of ABC
dot((9,0));
label... | 156 |
Consider a wardrobe that consists of $6$ red shirts, $7$ green shirts, $8$ blue shirts, $9$ pairs of pants, $10$ green hats, $10$ red hats, and $10$ blue hats. Additionally, you have $5$ ties in each color: green, red, and blue. Every item is distinct. How many outfits can you make consisting of one shirt, one pair of ... | 18900 |
Given the set $A = \{x | x < 1 \text{ or } x > 5\}$, and $B = \{x | a \leq x \leq b\}$, and $A \cup B = \mathbb{R}$, $A \cap B = \{x | 5 < x \leq 6\}$, find the value of $2a - b$. | -4 |
The four consecutive digits $a$, $b$, $c$ and $d$ are used to form the four-digit numbers $abcd$ and $dcba$. What is the greatest common divisor of all numbers of the form $abcd+dcba$? | 1111 |
In the rectangular coordinate system xOy, the parametric equation of line l is $$\begin{cases} x=4+ \frac { \sqrt {2}}{2}t \\ y=3+ \frac { \sqrt {2}}{2}t\end{cases}$$ (t is the parameter), and the polar coordinate system is established with the coordinate origin as the pole and the positive semi-axis of the x-axis as t... | \frac{86}{7} |
A piece of string fits exactly once around the perimeter of a square whose area is 144. Rounded to the nearest whole number, what is the area of the largest circle that can be formed from the piece of string? | 183 |
In 2006, the revenues of an insurance company increased by 25% and the expenses increased by 15% compared to the previous year. The company's profit (revenue - expenses) increased by 40%. What percentage of the revenues were the expenses in 2006? | 55.2 |
A three-quarter sector of a circle of radius $4$ inches together with its interior can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches? | $3 \pi \sqrt{7}$ |
Find the remainder when $2x^6-x^4+4x^2-7$ is divided by $x^2+4x+3$. | -704x-706 |
Consider an octagonal lattice where each vertex is evenly spaced and one unit from its nearest neighbor. How many equilateral triangles have all three vertices in this lattice? Every side of the octagon is extended one unit outward with a single point placed at each extension, keeping the uniform distance of one unit b... | 24 |
Kristine has 7 more CDs than Dawn. If Dawn has 10 CDs, how many CDs do they have together? | Kristine has 10 + 7 = <<10+7=17>>17 CDs.
Thus, they both have 10 + 17 = <<10+17=27>>27 CDs together.
#### 27 |
Mice built an underground house consisting of chambers and tunnels:
- Each tunnel leads from one chamber to another (i.e., none are dead ends).
- From each chamber, exactly three tunnels lead to three different chambers.
- From each chamber, it is possible to reach any other chamber through tunnels.
- There is exactly... | 10 |
Let $[x]$ represent the greatest integer less than or equal to the real number $x$. Define the sets
$$
\begin{array}{l}
A=\{y \mid y=[x]+[2x]+[4x], x \in \mathbf{R}\}, \\
B=\{1,2, \cdots, 2019\}.
\end{array}
$$
Find the number of elements in the intersection $A \cap B$. | 1154 |
Given that $\sin\alpha + \cos\alpha = \frac{\sqrt{2}}{3}$ and $0 < \alpha < \pi$, find the value of $\tan(\alpha - \frac{\pi}{4})$. | 2\sqrt{2} |
Suppose that $n, n+1, n+2, n+3, n+4$ are five consecutive integers.
Determine a simplified expression for the sum of these five consecutive integers. | 5n+10 |
In the rectangular coordinate system xOy, it is known that 0 < α < 2π. Point P, with coordinates $(1 - \tan{\frac{\pi}{12}}, 1 + \tan{\frac{\pi}{12}})$, lies on the terminal side of angle α. Determine the value of α. | \frac{\pi}{3} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it is known that $c=a\cos B+b\sin A$.
(1) Find $A$;
(2) If $a=2$ and $b=c$, find the area of $\triangle ABC$. | \sqrt{2}+1 |
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