problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Given that $x$ is a root of the equation $x^{2}+x-6=0$, simplify $\frac{x-1}{\frac{2}{{x-1}}-1}$ and find its value. | \frac{8}{3} |
Which of the following numbers is an odd integer, contains the digit 5, is divisible by 3, and lies between \(12^2\) and \(13^2\)? | 165 |
If the roots of the quadratic equation $\frac32x^2+11x+c=0$ are $x=\frac{-11\pm\sqrt{7}}{3}$, then what is the value of $c$? | 19 |
Calculate the lengths of the arcs of the curves given by the parametric equations.
$$
\begin{aligned}
& \left\{\begin{array}{l}
x=\left(t^{2}-2\right) \sin t+2 t \cos t \\
y=\left(2-t^{2}\right) \cos t+2 t \sin t
\end{array}\right. \\
& 0 \leq t \leq 2 \pi
\end{aligned}
$$ | \frac{8\pi^3}{3} |
Determine the sum of all prime numbers $p$ for which there exists no integer solution in $x$ to the congruence $3(6x+1)\equiv 4\pmod p$. | 5 |
The square root of $x$ is greater than 2 and less than 4. How many integer values of $x$ satisfy this condition? | 11 |
Six boys stood equally spaced on a circle of radius 40 feet. Each boy walked to all of the other non-adjacent persons on the circle, shook their hands and then returned to his original spot on the circle before the next boy started his trip to shake hands with all of the other non-adjacent boys on the circle. After all... | 480 + 480\sqrt{3}\text{ feet} |
Determine all quadruplets ($x, y, z, t$) of positive integers, such that $12^x + 13^y - 14^z = 2013^t$. | (1, 3, 2, 1) |
A boy has the following seven coins in his pocket: $2$ pennies, $2$ nickels, $2$ dimes, and $1$ quarter. He takes out two coins, records the sum of their values, and then puts them back with the other coins. He continues to take out two coins, record the sum of their values, and put them back. How many different sums c... | 9 |
Triangle $ABC$ is a right isosceles triangle. Points $D$, $E$ and $F$ are the midpoints of the sides of the triangle. Point $G$ is the midpoint of segment $DF$ and point $H$ is the midpoint of segment $FE$. What is the ratio of the shaded area to the non-shaded area in triangle $ABC$? Express your answer as a common fr... | \frac{5}{11} |
How many perfect squares divide $10^{10}$? | 36 |
For a positive integer $M$, if there exist integers $a$, $b$, $c$ and $d$ so that:
\[ M \leq a < b \leq c < d \leq M+49, \qquad ad=bc \]
then we call $M$ a GOOD number, if not then $M$ is BAD. Please find the greatest GOOD number and the smallest BAD number. | 576 |
Harry has 50 books in his library. His sister Flora has twice as many books and their cousin Gary has half the books Harry has. How many books do the three of them own together? | Flora has 50*2= <<50*2=100>>100 books
Gary has 50/2= <<50/2=25>>25 books
Together they own 100+25+50= <<100+25+50=175>>175 books
#### 175 |
Oleg is an event organizer. He is organizing an event with 80 guests where 40 of them are men, half the number of men are women, and the rest are children. If he added 10 children to the guest list, how many children will there be at the event? | There are 40/2= <<40/2=20>>20 women at the event.
Hence, there are a total of 40 + 20 = <<40+20=60>>60 men and women at the event.
So, the total number of children at the event is 80 - 60 = <<80-60=20>>20.
After Oleg added 10 children to the guest list, there are a total of 20 + 10 = <<20+10=30>>30 children.
#### 30 |
You can buy 4 apples or 1 watermelon for the same price. You bought 36 fruits evenly split between oranges, apples and watermelons, and the price of 1 orange is $0.50. How much does 1 apple cost if your total bill was $66? | If 36 fruits were evenly split between 3 types of fruits, then I bought 36/3 = <<36/3=12>>12 units of each fruit
If 1 orange costs $0.50 then 12 oranges will cost $0.50 * 12 = $<<0.5*12=6>>6
If my total bill was $66 and I spent $6 on oranges then I spent $66 - $6 = $<<66-6=60>>60 on the other 2 fruit types.
Assuming th... |
A rectangle is divided into four smaller rectangles, labelled W, X, Y, and Z. The perimeters of rectangles W, X, and Y are 2, 3, and 5, respectively. What is the perimeter of rectangle Z? | 6 |
Given the function $f(x) = \sqrt{2}\cos(2x - \frac{\pi}{4})$, where $x \in \mathbb{R}$,
1. Find the smallest positive period of the function $f(x)$ and its intervals of monotonically increasing values.
2. Find the minimum and maximum values of the function $f(x)$ on the interval $\left[-\frac{\pi}{8}, \frac{\pi}{2}\rig... | -1 |
In the positive term geometric sequence $\\{a_n\\}$, $a\_$ and $a\_{48}$ are the two roots of the equation $2x^2 - 7x + 6 = 0$. Find the value of $a\_{1} \cdot a\_{2} \cdot a\_{25} \cdot a\_{48} \cdot a\_{49}$. | 9\sqrt{3} |
A square is inscribed in a circle. A smaller square has one side coinciding with a side of the larger square and has two vertices on the circle, as shown. What percent of the area of the larger square is the area of the smaller square?
[asy]
draw(Circle((0,0),1.4142));
draw((1,1)--(1,-1)--(-1,-1)--(-1,1)--cycle);
dr... | 4\% |
Given the new operation $n\heartsuit m=n^{3+m}m^{2+n}$, evaluate $\frac{2\heartsuit 4}{4\heartsuit 2}$. | \frac{1}{2} |
Barry wants to make a huge salad using only cucumbers and tomatoes. He will use a total of 280 pieces of vegetables. If there are thrice as many tomatoes as cucumbers, how many cucumbers will be used in the salad? | Let c be the number of cucumbers and t the number of tomatoes.
The total is the sum of the number of cucumbers and that of tomatoes, which is c + t = 280 piece of vegetables
Since there are thrice as many tomatoes as cucumbers, this means c =t/3
Therefore 3*c = t.
The sum will therefore be c + 3*c = 280 pieces of veget... |
Hooligan Vasya loves to run on the escalator in the metro, and he runs down twice as fast as he runs up. If the escalator is not working, it takes Vasya 6 minutes to run up and down. If the escalator is running downwards, it takes Vasya 13.5 minutes to run up and down. How many seconds will it take Vasya to run up and ... | 324 |
A bouncy ball is dropped from a height of 100 meters. After each bounce, it reaches a height that is half of the previous one. What is the total distance the ball has traveled when it hits the ground for the 10th time? (Round the answer to the nearest whole number) | 300 |
Two fair 8-sided dice, with sides numbered from 1 to 8, are rolled once. The sum of the numbers rolled determines the radius of a circle. Calculate the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference. | \frac{1}{64} |
Compute the exact value of the expression
\[|\pi - |\pi - 7||.\]Write your answer using only integers and $\pi,$ without any absolute value signs. | 7 - 2 \pi |
A cylinder has a radius of 3 cm and a height of 8 cm. What is the longest segment, in centimeters, that would fit inside the cylinder? | 10 |
A large rectangular garden contains two flower beds in the shape of congruent isosceles right triangles and a trapezoidal playground. The parallel sides of the trapezoid measure $30$ meters and $46$ meters. Determine the fraction of the garden occupied by the flower beds.
A) $\frac{1}{10}$
B) $\frac{1}{11}$
C) $\frac{4... | \frac{4}{23} |
Parallelogram $ABCD$ has vertices $A(3,3)$, $B(-3,-3)$, $C(-9,-3)$, and $D(-3,3)$. If a point is selected at random from the region determined by the parallelogram, what is the probability that the point is not above the $x$-axis? Express your answer as a common fraction. | \frac{1}{2} |
The consignment shop received for sale cameras, clocks, pens, and receivers totaling 240 rubles. The sum of the prices of the receiver and one clock is 4 rubles more than the sum of the prices of the camera and the pen, and the sum of the prices of one clock and the pen is 24 rubles less than the sum of the prices of t... | 18 |
With $400$ members voting the House of Representatives defeated a bill. A re-vote, with the same members voting, resulted in the passage of the bill by twice the margin by which it was originally defeated. The number voting for the bill on the revote was $\frac{12}{11}$ of the number voting against it originally. How m... | 60 |
If \( 0 \leq p \leq 1 \) and \( 0 \leq q \leq 1 \), define \( H(p, q) \) by
\[
H(p, q) = -3pq + 4p(1-q) + 4(1-p)q - 5(1-p)(1-q).
\]
Define \( J(p) \) to be the maximum of \( H(p, q) \) over all \( q \) (in the interval \( 0 \leq q \leq 1 \)). What is the value of \( p \) (in the interval \( 0 \leq p \leq 1 \)) that m... | \frac{9}{16} |
In the eight-term sequence $A,B,C,D,E,F,G,H$, the value of $C$ is $5$ and the sum of any three consecutive terms is $30$. What is $A+H$? | 25 |
Sam the butcher made sausage by grinding up 10 pounds of spicy meat mix, loading it into a tube casing, and creating a string of 40 sausage links. Then, she hung up the string of sausage links in her cooler. Later that evening, Brandy, Sam’s Golden Retriever, broke into the cooler and ate 12 links of sausage. After B... | There are 16 ounces in one pound, and thus in 10 pounds of meat mix there are 10*16=<<10*16=160>>160 ounces of sausage meat.
Thus, the 40 sausage links contained 160 ounces of meat, or 160/40=<<160/40=4>>4 ounces per link.
If Brandy ate 12 links, then 40-12=<<40-12=28>>28 links remained.
Thus, 28 remaining links, at 4 ... |
At time $t=0,$ a ball is thrown downward at 24 feet per second from a height of 160 feet above the ground. The equation $h = -16t^2 - 24t +160$ describes the height (in feet) of the ball. In how many seconds will the ball hit the ground? Express your answer as a decimal. | 2.5 |
Evaluate the expression $2^{3}-2+3$. | 9 |
Mr. Reader has six different Spiderman comic books, five different Archie comic books and four different Garfield comic books. When stacked, all of the Spiderman comic books are grouped together, all of the Archie comic books are grouped together and all of the Garfield comic books are grouped together. In how many dif... | 12,\!441,\!600 |
1. Let $[x]$ denote the greatest integer less than or equal to the real number $x$. Given a sequence of positive integers $\{a_{n}\}$ such that $a_{1} = a$, and for any positive integer $n$, the sequence satisfies the recursion
$$
a_{n+1} = a_{n} + 2 \left[\sqrt{a_{n}}\right].
$$
(1) If $a = 8$, find the smallest posi... | 82 |
We are given $2n$ natural numbers
\[1, 1, 2, 2, 3, 3, \ldots, n - 1, n - 1, n, n.\]
Find all $n$ for which these numbers can be arranged in a row such that for each $k \leq n$, there are exactly $k$ numbers between the two numbers $k$. | $n=3,4,7,8$ |
The escalator of the department store, which at any given time can be seen at $75$ steps section, moves up one step in $2$ seconds. At time $0$ , Juku is standing on an escalator step equidistant from each end, facing the direction of travel. He goes by a certain rule: one step forward, two steps back, then again ... | 23 |
Peter, Emma, and Kyler played chess with each other. Peter won 4 games and lost 2 games. Emma won 3 games and lost 3 games. If Kyler lost 3 games, how many games did he win? | 1 |
On a checkerboard composed of 64 unit squares, what is the probability that a randomly chosen unit square does not touch the outer edge of the board? | \frac{9}{16} |
Let $a,$ $b,$ $c,$ $d$ be nonzero integers such that
\[\begin{pmatrix} a & b \\ c & d \end{pmatrix}^2 = \begin{pmatrix} 7 & 0 \\ 0 & 7 \end{pmatrix}.\]Find the smallest possible value of $|a| + |b| + |c| + |d|.$ | 7 |
Steve has a bank account that earns 10% interest every year. He puts $100 in it, and then 10 each year. How much money is in it after two years? | After one year it earns $10 in interest because 100 x .1 = <<100*.1=10>>10
It has $120 in it after a year because 100 + 10 + 10 = <<100+10+10=120>>120
In the second year it earns $12 in interest because 120 x .1 = <<120*.1=12>>12
In the second year it has $142 in it because 120 + 12 + 10 = <<120+12+10=142>>142
#### 142 |
Compute
\[\sum_{n = 1}^\infty \frac{1}{n(n + 2)}.\] | \frac{3}{4} |
A shipping boat's crew consisted of 17 sailors, with five inexperienced sailors. Each experienced sailor was paid 1/5 times more than the inexperienced sailors. If the inexperienced sailors were paid $10 per hour for a 60-hour workweek, calculate the total combined monthly earnings of the experienced sailors. | If the experienced sailors received 1/5 times more money than the inexperienced sailors, they were paid 1/5*10 = $<<1/5*10=2>>2 more.
The total hourly earnings for the experienced sailors each is $10+$2= $<<10+2=12>>12
In a 60-hour workweek, an experienced sailor is paid 60*$12 = $<<60*12=720>>720
In a month, an experi... |
There are 6 dogs and 2 ducks in the garden. How many feet are there in the garden? | 6 dogs have 6 x 4 = <<6*4=24>>24 feet.
2 ducks have 2 x 2 = <<2*2=4>>4 feet
In total there are 24 + 4 = <<24+4=28>>28 feet in the garden
#### 28 |
Rationalize the denominator of $\frac{7}{3+\sqrt{8}}$. The answer can be expressed as $\frac{P\sqrt{Q}+R}{S}$, where $P$, $Q$, $R$, and $S$ are integers, $S$ is positive, and $Q$ is not divisible by the square of any prime. If the greatest common divisor of $P$, $R$, and $S$ is 1, find $P+Q+R+S$. | 23 |
In the Cartesian coordinate system $(xOy)$, a curve $C_{1}$ is defined by the parametric equations $x=\cos{\theta}$ and $y=\sin{\theta}$, and a line $l$ is defined by the polar equation $\rho(2\cos{\theta} - \sin{\theta}) = 6$.
1. Find the Cartesian equations for the curve $C_{1}$ and the line $l$.
2. Find a point $P$... | \frac{6\sqrt{5}}{5} - 1 |
Simplify $(2x^3)^3$. | 8x^9 |
A farmer contracted several acres of fruit trees. This year, he invested 13,800 yuan, and the total fruit yield was 18,000 kilograms. The fruit sells for a yuan per kilogram in the market and b yuan per kilogram when sold directly from the orchard (b < a). The farmer transports the fruit to the market for sale, selling... | 20\% |
There are 101 people participating in a Secret Santa gift exchange. As usual each person is randomly assigned another person for whom (s)he has to get a gift, such that each person gives and receives exactly one gift and no one gives a gift to themself. What is the probability that the first person neither gives gifts ... | 0.96039 |
In the figure, $ABCD$ is a square of side length $1$. The rectangles $JKHG$ and $EBCF$ are congruent. What is $BE$? | 2-\sqrt{3} |
A magician has $5432_{9}$ tricks for his magical performances. How many tricks are there in base 10? | 3998 |
A high school offers three separate elective classes for the senior two-grade mathematics course. After the selection process, four students request to change their math class. However, each class can accept at most two more students. Determine the number of different ways the students can be redistributed among the cl... | 54 |
In the Cartesian coordinate plane $(xOy)$, the curve $y=x^{2}-6x+1$ intersects the coordinate axes at points that lie on circle $C$.
(1) Find the equation of circle $C$;
(2) Given point $A(3,0)$, and point $B$ is a moving point on circle $C$, find the maximum value of $\overrightarrow{OA} \cdot \overrightarrow{OB}$, an... | \frac{36\sqrt{37}}{37} |
Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs? | \frac{2}{5} |
In triangle \( \triangle ABC \), the angles are \( \angle B = 30^\circ \) and \( \angle A = 90^\circ \). Point \( K \) is marked on side \( AC \), and points \( L \) and \( M \) are marked on side \( BC \) such that \( KL = KM \) (point \( L \) lies on segment \( BM \)).
Find the length of segment \( LM \), given that... | 14 |
Determine the value of $\frac{3b^{-1} - \frac{b^{-1}}{3}}{b^2}$ when $b = \tfrac{1}{3}$. | 72 |
Select two distinct numbers simultaneously and at random from the set $\{1, 2, 3, 4, 5, 6\}$. What is the probability that the smaller one divides the larger one and both numbers are either both even or both odd? | \frac{4}{15} |
Find the largest integer $n$ satisfying the following conditions:
(i) $n^2$ can be expressed as the difference of two consecutive cubes;
(ii) $2n + 79$ is a perfect square.
| 181 |
Five people of different heights are standing in line from shortest to tallest. As it happens, the tops of their heads are all collinear; also, for any two successive people, the horizontal distance between them equals the height of the shorter person. If the shortest person is 3 feet tall and the tallest person is 7 f... | \sqrt{21} |
In any isosceles triangle $ABC$ with $AB=AC$, the altitude $AD$ bisects the base $BC$ so that $BD=DC$.
Determine the area of $\triangle ABC$.
[asy]
draw((0,0)--(14,0)--(7,24)--cycle,black+linewidth(1));
draw((7,24)--(7,0),black+linewidth(1)+dashed);
draw((7,0)--(7,1)--(6,1)--(6,0)--cycle,black+linewidth(1));
draw((5.... | 168 |
In parallelogram $ABCD$, angle $B$ measures $110^\circ$. What is the number of degrees in the measure of angle $C$? | 70^\circ |
The perimeter of triangle $APM$ is $152$, and the angle $PAM$ is a right angle. A circle of radius $19$ with center $O$ on $\overline{AP}$ is drawn so that it is tangent to $\overline{AM}$ and $\overline{PM}$. Given that $OP=m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+n$.
| 98 |
Five identical rectangles are arranged to form a larger rectangle $PQRS$, as shown. The area of $PQRS$ is $4000$. What is the length, $x$, rounded off to the nearest integer? [asy]
real x = 1; real w = 2/3;
// Draw outer square and labels
pair s = (0, 0); pair r = (2 * x, 0); pair q = (3 * w, x + w); pair p = (0, x ... | 35 |
Compute $\sin(-60^\circ)$. | -\frac{\sqrt{3}}{2} |
When a car's brakes are applied, it travels 7 feet less in each second than the previous second until it comes to a complete stop. A car goes 28 feet in the first second after the brakes are applied. How many feet does the car travel from the time the brakes are applied to the time the car stops? | 70 |
Find the increments of the argument and the function for \( y = 2x^2 + 1 \) when the argument \( x \) changes from 1 to 1.02. | 0.0808 |
Let $g(x) = x^5 + x^4 + x^3 + x^2 + x + 1.$ What is the remainder when the polynomial $g(x^{12})$ is divided by the polynomial $g(x)$? | 6 |
20 different villages are located along the coast of a circular island. Each of these villages has 20 fighters, with all 400 fighters having different strengths.
Two neighboring villages $A$ and $B$ now have a competition in which each of the 20 fighters from village $A$ competes with each of the 20 fighters from vill... | 290 |
Abigail has a report due tomorrow which needs to be 1000 words in length. Abigail can type 300 words in half an hour. If she has already written 200 words, how many more minutes will it take her to finish the report? | She needs to type an additional 1000 – 200 = <<1000-200=800>>800 words.
In one minute she can type 300 words / 30 = <<300/30=10>>10 words.
It will take her 800 / 10 = <<800/10=80>>80 more minutes to finish typing the report.
#### 80 |
Let $A,B,C$ be angles of a triangle, where angle $B$ is obtuse, and \begin{align*}
\cos^2 A + \cos^2 B + 2 \sin A \sin B \cos C &= \frac{15}{8} \text{ and} \\
\cos^2 B + \cos^2 C + 2 \sin B \sin C \cos A &= \frac{14}{9}.
\end{align*}There are positive integers $p$, $q$, $r$, and $s$ for which \[ \cos^2 C + \cos^2 A + 2... | 222 |
The figure shows rectangle $ABCD$ with segment $PQ$ dividing the rectangle into two congruent squares. How many right triangles can be drawn using three of the points $\{A,P,B,C,Q,D\}$ as vertices? [asy]
draw((0,0)--(8,0)--(8,4)--(0,4)--cycle);
draw((4,0)--(4,4));
label("D",(0,0),S);
label("Q",(4,0),S);
label("C",(8,0)... | 14 \text{ right triangles} |
What is the distance between the two (non-intersecting) face diagonals on adjacent faces of a unit cube? | \frac{\sqrt{3}}{3} |
The hypotenuse of a right triangle measures $6\sqrt{2}$ inches and one angle is $45^{\circ}$. What is the number of square inches in the area of the triangle? | 18 |
Given vectors \( \boldsymbol{a} \), \( \boldsymbol{b} \), and \( \boldsymbol{c} \) satisfying
$$
|\boldsymbol{a}|:|\boldsymbol{b}|:|\boldsymbol{c}|=1: k: 3 \quad \left(k \in \mathbf{Z}_{+}\right),
$$
and \( \boldsymbol{b} - \boldsymbol{a} = 2(\boldsymbol{c} - \boldsymbol{b}) \),
if \( \alpha \) is the angle between \( ... | -\frac{1}{12} |
There were 180 apples in each crate. 12 such crates of apples were delivered to a factory. 160 apples were rotten and had to be thrown away. The remaining apples were packed into boxes of 20 apples each. How many boxes of apples were there? | The total number of apples delivered was: 180 apples x 12 crates = <<180*12=2160>>2160 apples
So, the number of remaining apples after throwing away the rotten ones was: 2160 apples – 160 apples = <<2160-160=2000>>2000 apples
And now, we can find the total number of boxes of apples by dividing the remaining number of a... |
In the diagram, what is the measure of $\angle ACB$ in degrees? [asy]
size(250);
draw((-60,0)--(0,0));
draw((0,0)--(64.3,76.6)--(166,0)--cycle);
label("$A$",(64.3,76.6),N);
label("$93^\circ$",(64.3,73),S);
label("$130^\circ$",(0,0),NW);
label("$B$",(0,0),S);
label("$D$",(-60,0),S);
label("$C$",(166,0),S);
[/asy] | 37^\circ |
A list of five positive integers has a median of 3 and a mean of 11. What is the maximum possible value of the list's largest element? | 47 |
Solve for $m$: $(m-4)^3 = \left(\frac 18\right)^{-1}$. | 6 |
Carla adds a can of chilis, two cans of beans, and 50% more tomatoes than beans to a normal batch of chili. If she makes a quadruple batch, how many cans of food does she need? | First find the number of cans of tomatoes in a normal batch of chili: 2 cans * 150% = <<2*150*.01=3>>3 cans
Then add the number of cans of each type of food to find the total number of cans per batch: 3 cans + 2 cans + 1 can = <<3+2+1=6>>6 cans
Then multiply the number of cans per batch by the number of batches to find... |
In convex quadrilateral $ABCD, \angle A \cong \angle C, AB = CD = 180,$ and $AD \neq BC.$ The perimeter of $ABCD$ is $640$. Find $\lfloor 1000 \cos A \rfloor.$ (The notation $\lfloor x \rfloor$ means the greatest integer that is less than or equal to $x.$)
| 777 |
Ali and Leila reserve their places for a trip to Egypt. The price is $147 per person, but they were each given a discount of $14 since there are two of them. How much does their trip cost? | After the discount, the price will be $147 - 14 = $<<147-14=133>>133.
For both Ali and Leila together, it will cost $133* 2 = $<<133*2=266>>266.
#### 266 |
Find the positive real number $x$ such that $\lfloor x \rfloor \cdot x = 70$. Express $x$ as a decimal. | 8.75 |
What is the sum of the integers that are both greater than 3 and less than 12? | 60 |
A mathematician is working on a geospatial software and comes across a representation of a plot's boundary described by the equation $x^2 + y^2 + 8x - 14y + 15 = 0$. To correctly render it on the map, he needs to determine the diameter of this plot. | 10\sqrt{2} |
A circle has a radius of $\log_{10}{(a^2)}$ and a circumference of $\log_{10}{(b^4)}$. What is $\log_{a}{b}$? | \pi |
For every positive real number $x$, let
\[g(x) = \lim_{r \to 0} ((x+1)^{r+1} - x^{r+1})^{\frac{1}{r}}.\]
Find $\lim_{x \to \infty} \frac{g(x)}{x}$. | e |
The harmonic mean of two positive integers is the reciprocal of the arithmetic mean of their reciprocals. For how many ordered pairs of positive integers $(x,y)$ with $x \neq y$ is the harmonic mean of $x$ and $y$ equal to $4^{15}$? | 29 |
The different ways to obtain the number of combinations of dice, as discussed in Example 4-15 of Section 4.6, can also be understood using the generating function form of Pólya’s enumeration theorem as follows:
$$
\begin{aligned}
P= & \frac{1}{24} \times\left[\left(x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}\right)^{6}\right.... | 30 |
There is a cube of size \(10 \times 10 \times 10\) made up of small unit cubes. A grasshopper is sitting at the center \(O\) of one of the corner cubes. It can jump to the center of a cube that shares a face with the one in which the grasshopper is currently located, provided that the distance to point \(O\) increases.... | \frac{27!}{(9!)^3} |
Given two lines $l_{1}:(3+m)x+4y=5-3m$ and $l_{2}:2x+(5+m)y=8$. If line $l_{1}$ is parallel to line $l_{2}$, then the real number $m=$ . | -7 |
In parallelogram \(ABCD\), the angle at vertex \(A\) is \(60^{\circ}\), \(AB = 73\) and \(BC = 88\). The angle bisector of \(\angle ABC\) intersects segment \(AD\) at point \(E\) and ray \(CD\) at point \(F\). Find the length of segment \(EF\). | 15 |
Given triangle ABC, the sides opposite to angles A, B, and C are denoted as a, b, and c, respectively, and a = 6. Find the maximum value of the area of triangle ABC given that $\sqrt{7}bcosA = 3asinB$. | 9\sqrt{7} |
Given the function \( f(x) = A \sin (\omega x + \varphi) \) where \( A \neq 0 \), \( \omega > 0 \), \( 0 < \varphi < \frac{\pi}{2} \), if \( f\left(\frac{5\pi}{6}\right) + f(0) = 0 \), find the minimum value of \( \omega \). | \frac{6}{5} |
Given that in the geometric sequence $\{a_n\}$ where all terms are positive, $a_1a_3=16$ and $a_3+a_4=24$, find the value of $a_5$. | 32 |
A Mersenne prime is defined to be a prime number of the form $2^n - 1$, where $n$ must itself be a prime. For example, since $2^3 - 1 = 7$, and 3 is a prime number, 7 is a Mersenne prime. What is the largest Mersenne prime less than 200? | 127 |
Given two arbitrary positive integers \( n \) and \( k \), let \( f(n, k) \) denote the number of unit squares that one of the diagonals of an \( n \times k \) grid rectangle passes through. How many such pairs \( (n, k) \) are there where \( n \geq k \) and \( f(n, k) = 2018 \)? | 874 |
Let $f(x)$ be the function such that $f(x)>0$ at $x\geq 0$ and $\{f(x)\}^{2006}=\int_{0}^{x}f(t) dt+1.$
Find the value of $\{f(2006)\}^{2005}.$ | 2006 |
If $A:B:C = 2:1:4$, what is the value of $(3A + 2B) \div (4C - A)$? Express your answer as a common fraction. | \frac{4}{7} |
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