problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
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Quadrilateral $ABCD$ is a square. A circle with center $D$ has arc $AEC$. A circle with center $B$ has arc $AFC$. If $AB = 4$ cm, what is the total number of square centimeters in the football-shaped area of regions II and III combined? | 8\pi - 16 |
When plotted in the standard rectangular coordinate system, trapezoid $ABCD$ has vertices $A(1, -2)$, $B(1, 1)$, $C(5, 7)$ and $D(5, 1)$. What is the area of trapezoid $ABCD$? | 18 |
Pascal has a triangle. In the $n$th row, there are $n+1$ numbers $a_{n, 0}, a_{n, 1}, a_{n, 2}, \ldots, a_{n, n}$ where $a_{n, 0}=a_{n, n}=1$. For all $1 \leq k \leq n-1, a_{n, k}=a_{n-1, k}-a_{n-1, k-1}$. What is the sum of all numbers in the 2018th row? | 2 |
How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\times 5$ square array of dots? | 100 |
Let the polynomial be defined as $$Q(x) = \left(\frac{x^{20} - 1}{x-1}\right)^2 - x^{20}.$$ Calculate the sum of the first five distinct $\alpha_k$ values where each zero of $Q(x)$ can be expressed in the complex form $z_k = r_k [\cos(2\pi \alpha_k) + i\sin(2\pi \alpha_k)]$, with $\alpha_k \in (0, 1)$ and $r_k > 0$. | \frac{3}{4} |
Given a sequence $\{a\_n\}$ that satisfies $a\_1=1$ and $a\_n= \frac{2S\_n^2}{2S\_n-1}$ for $n\geqslant 2$, where $S\_n$ is the sum of the first $n$ terms of the sequence, find the value of $S\_{2016}$. | \frac{1}{4031} |
Given that $\log_{10} \sin x + \log_{10} \cos x = -1$ and that $\log_{10} (\sin x + \cos x) = \frac{1}{2} (\log_{10} n - 1),$ find $n.$
| 12 |
Shift the graph of the function $f(x)=2\sin(2x+\frac{\pi}{6})$ to the left by $\frac{\pi}{12}$ units, and then shift it upwards by 1 unit to obtain the graph of $g(x)$. If $g(x_1)g(x_2)=9$, and $x_1, x_2 \in [-2\pi, 2\pi]$, then find the maximum value of $2x_1-x_2$. | \frac {49\pi}{12} |
In a recent test, $15\%$ of the students scored $60$ points, $20\%$ got $75$ points, $30\%$ scored $85$ points, $10\%$ scored $90$ points, and the rest scored $100$ points. Find the difference between the mean and the median score on this test. | -1.5 |
There are 4 carriages in a train and each carriage has 25 seats. If each carriage could accommodate 10 more passengers, how many passengers would fill up 3 trains? | If each carriage had another 10 seats, each carriage would have 25 + 10 = 35 seats.
The train comprises of 4 carriages and each carriage now has 35 seats; therefore a train has 35 x 4 = <<4*35=140>>140 seats.
3 trains with 140 seats each in total can take up 3 x 140 = <<3*140=420>>420 passengers.
#### 420 |
Consider pairs $(f,g)$ of functions from the set of nonnegative integers to itself such that
[list]
[*]$f(0) \geq f(1) \geq f(2) \geq \dots \geq f(300) \geq 0$
[*]$f(0)+f(1)+f(2)+\dots+f(300) \leq 300$
[*]for any 20 nonnegative integers $n_1, n_2, \dots, n_{20}$, not necessarily distinct, we have $$g(n_1+n_2+\dots+n_{... | 115440 |
Given that positive real numbers a and b satisfy $a^{2}+2ab+4b^{2}=6$, calculate the maximum value of a+2b. | 2\sqrt{2} |
If $a$ and $b$ are positive integers and the equation \( ab - 8a + 7b = 395 \) holds true, what is the minimal possible value of \( |a - b| \)? | 15 |
The matrix
\[\mathbf{M} = \begin{pmatrix} 0 & 2y & z \\ x & y & -z \\ x & -y & z \end{pmatrix}\]satisfies $\mathbf{M}^T \mathbf{M} = \mathbf{I}.$ Find $x^2 + y^2 + z^2.$
Note: For a matrix $\mathbf{A},$ $\mathbf{A}^T$ is the transpose of $\mathbf{A},$ which is generated by reflecting the matrix $\mathbf{A}$ over the ... | 1 |
What is $\frac56$ of 30? | 25 |
Given the function $f(x)=a\ln x + x - \frac{1}{x}$, where $a$ is a real constant.
(I) If $x=\frac{1}{2}$ is a local maximum point of $f(x)$, find the local minimum value of $f(x)$.
(II) If the inequality $a\ln x - \frac{1}{x} \leqslant b - x$ holds for any $-\frac{5}{2} \leqslant a \leqslant 0$ and $\frac{1}{2} \leqsla... | \frac{3}{2} |
A store sells jellybeans at a fixed price per gram. The price for 250 g of jellybeans is $\$ 7.50$. What mass of jellybeans sells for $\$ 1.80$? | 60 \mathrm{~g} |
In the representation of three two-digit numbers, there are no zeros, and in each of them, both digits are different. Their sum is 41. What could their sum be if the digits in them are swapped? | 113 |
Todd bought a pair of jeans that cost $125 at full price. The jeans were on sale for 20% off. He then applied a coupon that took off $10. He paid with a store credit card that gave him another 10% off the remaining amount. How many dollars did he save on the original price of the jeans? | The jeans were on sale for 20% = 20 / 100 = 1 /5 off, so the sale took 125 / 5 = $<<125/5=25>>25 off.
Thus, after the sale, the jeans cost $125 - $25 = $<<125-25=100>>100.
Todd’s coupon lowered the price to $100 - $10 = $<<100-10=90>>90.
The store credit card gave him 10% = 10 / 100 = 1 / 10 off, so the card took $90 /... |
What is the area of the triangle bounded by the lines $y=x,$ $y=-x,$ and $y=6$? | 36 |
Calculate:<br/>$(1)-9+5-\left(-12\right)+\left(-3\right)$;<br/>Calculate:<br/>$(2)-(+1.5)-(-4\frac{1}{4})+3.75-(-8\frac{1}{2})$;<br/>$(3)$Read the following solution process and answer the question:<br/>Calculate:$\left(-15\right)\div (-\frac{1}{2}×\frac{25}{3}$）$÷\frac{1}{6}$<br/>Solution: Original expression $=\left(... | \frac{108}{5} |
Jun Jun is looking at an incorrect single-digit multiplication equation \( A \times B = \overline{CD} \), where the digits represented by \( A \), \( B \), \( C \), and \( D \) are all different from each other. Clever Jun Jun finds that if only one digit is changed, there are 3 ways to correct it, and if only the orde... | 17 |
Given $f(\alpha) = \frac{\sin(\pi - \alpha)\cos(\pi + \alpha)\sin(-\alpha + \frac{3\pi}{2})}{\cos(-\alpha)\cos(\alpha + \frac{\pi}{2})}$.
$(1)$ Simplify $f(\alpha)$;
$(2)$ If $\alpha$ is an angle in the third quadrant, and $\cos(\alpha - \frac{3\pi}{2}) = \frac{1}{5}$, find the value of $f(\alpha)$; | \frac{2\sqrt{6}}{5} |
Find a positive integer that is divisible by 18 and has a square root between 26 and 26.2. | 684 |
Phil started his day with $40. He bought a slice of pizza for $2.75, a soda for $1.50 and a pair of jeans for $11.50. If he has nothing but quarters left of his original money, how many quarters does he now have? | The total cost of pizza, soda and jeans is $2.75 + $1.50 + $11.50 = $<<2.75+1.5+11.5=15.75>>15.75
He now has $40 - $15.75 = $<<40-15.75=24.25>>24.25 left.
There are 4 quarters in $1.00 so $24 is equal to 24 x 4 = <<24*4=96>>96 quarters.
25 cents is equal to 1 quarter.
Therefore, Phil has 96 quarters + 1 quarter = <<96+... |
Find the remainder when\[\binom{\binom{3}{2}}{2} + \binom{\binom{4}{2}}{2} + \dots + \binom{\binom{40}{2}}{2}\]is divided by $1000$.
~ pi_is_3.14 | 4 |
There were 148 peanuts in a jar. Brock ate one-fourth of the peanuts and Bonita ate 29 peanuts. How many peanuts remain in the jar? | 148 * (1/4) = <<148*(1/4)=37>>37
148 - 37 - 29 = <<148-37-29=82>>82
There are 82 peanuts left in the jar.
#### 82 |
A perfect power is an integer $n$ that can be represented as $a^{k}$ for some positive integers $a \geq 1$ and $k \geq 2$. Find the sum of all prime numbers $0<p<50$ such that $p$ is 1 less than a perfect power. | 41 |
On a circle, 103 natural numbers are written. It is known that among any 5 consecutive numbers, there will be at least two even numbers. What is the minimum number of even numbers that can be in the entire circle? | 42 |
Find $x$ such that $\lceil x \rceil \cdot x = 210$. Express $x$ as a decimal. | 14.0 |
At the end of the year, the Math Club decided to hold an election for which 5 equal officer positions were available. However, 16 candidates were nominated, of whom 7 were past officers. Of all possible elections of the officers, how many will have at least 1 of the past officers? | 4242 |
Given the island of Zenith has 32500 acres of usable land, each individual requires 2 acres for sustainable living, and the current population of 500 people increases by a factor of 4 every 30 years, calculate the number of years from 2022 when the population reaches its maximum capacity. | 90 |
Let $\mathbf{a} = \begin{pmatrix} 1 \\ -2 \\ -5 \end{pmatrix},$ $\mathbf{b} = \begin{pmatrix} \sqrt{7} \\ 4 \\ -1 \end{pmatrix},$ and $\mathbf{c} = \begin{pmatrix} 13 \\ -4 \\ 17 \end{pmatrix}.$ Find the angle between the vectors $\mathbf{a}$ and $(\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}... | 90^\circ |
If $3+\triangle=5$ and $\triangle+\square=7$, what is the value of $\triangle+\Delta+\Delta+\square+\square$? | 16 |
Trevor and two of his neighborhood friends go to the toy shop every year to buy toys. Trevor always spends $20 more than his friend Reed on toys, and Reed spends 2 times as much money as their friend Quinn on the toys. If Trevor spends $80 every year to buy his toys, calculate how much money in total the three spend in... | If Trevor spends $20 more than Reed every year to buy toys, Reed spends $80-$20 =$60.
Since Reed uses $60 every year to buy toys, his friend Quinn uses 60/2 = $<<60/2=30>>30 every year to buy toys.
The total amount of money the three spend in a year together to buy toys is $30+$60+$80 = $<<30+60+80=170>>170.
The total ... |
Let $a_1, a_2, \ldots$ be a sequence with the following properties.
(i) $a_1 = 1$, and
(ii) $a_{2n}=n\cdot a_n$ for any positive integer $n$.
What is the value of $a_{2^{100}}$? | 2^{4950} |
In 2010, the ages of a brother and sister were 16 and 10 years old, respectively. In what year was the brother's age twice that of the sister's? | 2006 |
The hypotenuse of a right triangle measures 10 inches and one angle is $45^{\circ}$. What is the number of square inches in the area of the triangle? | 25 |
Let the set \( \mathrm{S} = \{1, 2, 3, \ldots, 10\} \). The subset \( \mathrm{A} \) of \( \mathrm{S} \) satisfies \( \mathrm{A} \cap \{1, 2, 3\} \neq \emptyset \) and \( \mathrm{A} \cup \{4, 5, 6\} \neq \mathrm{S} \). Find the number of such subsets \( \mathrm{A} \). | 888 |
Let $Q$ be a point outside of circle $C.$ A line from $Q$ is tangent to circle $C$ at point $R.$ A secant from $Q$ intersects $C$ at $X$ and $Y,$ such that $QX < QY.$ If $QX = 5$ and $QR = XY - QX,$ what is $QY$? | 20 |
A path of length $n$ is a sequence of points $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)$ with integer coordinates such that for all $i$ between 1 and $n-1$ inclusive, either (1) $x_{i+1}=x_{i}+1$ and $y_{i+1}=y_{i}$ (in which case we say the $i$th step is rightward) or (2) $x... | 1024 |
In triangle $ABC,$ we have $\angle C = 3\angle A,$ $a = 27,$ and $c = 48.$ What is $b$?
Note: $a$ is the side length opposite $\angle A,$ etc. | 35 |
If Tony normally has a temperature of 95 degrees, and he comes down with a sickness that raises his temperature by 10 degrees, how many degrees above the threshold of fever is his temperature if a fever is anything over 100 degrees? | First, we find what Tony's temperature is by adding his normal temperature of 95 to the increase in temperature from the sickness of 10, performing 95+10=<<95+10=105>>105 degrees.
Since the threshold for having a fever is 100, we subtract 100 from 105 to find out the answer, performing 105-100=<<105-100=5>>5 degrees ab... |
The chances of making the junior high basketball team start at 10% if you're 66 inches and increase 10% for every additional inch of height. Devin starts out as 65 inches tall, then grows 3 inches. What are his chances of making the basketball team? | First find Devin's new height by adding his growth to the original height: 65 inches + 3 inches = <<65+3=68>>68 inches
Then subtract 66 inches from Devin's height to find how many extra inches of height he has: 68 inches - 66 inches = <<68-66=2>>2 inches
Then multiply the number of extra inches by the percentage increa... |
Given that $\cos(\pi+\theta) = -\frac{1}{2}$, find the value of $\tan(\theta - 9\pi)$. | \sqrt{3} |
Let \(a, b, c, d\) be positive integers such that \(a^5 =\) | 757 |
Emma and Briana invested some capital into a business. Emma's investment is supposed to yield 15% of the capital annually while Briana's should yield 10% annually. If Emma invested $300 and Briana invested $500, what will be the difference between their return-on-investment after 2 years? | 15% of $300 is (15/100)*$300 = $<<(15/100)*300=45>>45
Emma will earn an annual ROI of $45 for 2 years for a total of $45*2 = $<<45*2=90>>90
10% of $500 is (10/100)*$500 = $<<(10/100)*500=50>>50
Briana will earn an annual ROI of $50 for 2 years for a total of $50*2 = $<<50*2=100>>100
The difference between their ROI aft... |
1. Given $$\cos\left(\alpha+ \frac {\pi}{6}\right)-\sin\alpha= \frac {3 \sqrt {3}}{5}$$, find the value of $$\sin\left(\alpha+ \frac {5\pi}{6}\right)$$;
2. Given $$\sin\alpha+\sin\beta= \frac {1}{2}$$ and $$\cos\alpha+\cos\beta= \frac {\sqrt {2}}{2}$$, find the value of $$\cos(\alpha-\beta)$$. | -\frac {5}{8} |
Given a sequence ${a_n}$ whose first $n$ terms have a sum of $S_n$, and the point $(n, \frac{S_n}{n})$ lies on the line $y = \frac{1}{2}x + \frac{11}{2}$. Another sequence ${b_n}$ satisfies $b_{n+2} - 2b_{n+1} + b_n = 0$ ($n \in \mathbb{N}^*$), and $b_3 = 11$, with the sum of the first 9 terms being 153.
(I) Find the g... | 18 |
On his calculator, August had solved a math problem with an answer of 600. The following math problem had an answer twice as big as the answer of the first math problem, and the third math problem had an answer 400 less than the combined total answers of the first and the second math problems. What's the total of Augus... | August solved a math problem that had an answer twice as big as the answer of the first math problem, meaning the next answer was 600*2 = <<600*2=1200>>1200
The first math problem and the second math problem has answers totaling 1200+600 = <<1200+600=1800>>1800
The third math problem had an answer 400 less than the com... |
Find, as a function of $\, n, \,$ the sum of the digits of \[9 \times 99 \times 9999 \times \cdots \times \left( 10^{2^n} - 1 \right),\] where each factor has twice as many digits as the previous one. | \[ 9 \cdot 2^n \] |
Let $f_0(x)=x+|x-100|-|x+100|$, and for $n\geq 1$, let $f_n(x)=|f_{n-1}(x)|-1$. For how many values of $x$ is $f_{100}(x)=0$? | 301 |
Let $a,b,c$ be positive real numbers such that $a+b+c=10$ and $ab+bc+ca=25$. Let $m=\min\{ab,bc,ca\}$. Find the largest possible value of $m$. | \frac{25}{9} |
Given an ellipse C: $$\frac{x^2}{a^2}+ \frac{y^2}{b^2}=1 \quad (a>b>0)$$ which passes through the point $(1, \frac{2\sqrt{3}}{3})$, with its foci denoted as $F_1$ and $F_2$. The circle $x^2+y^2=2$ intersects the line $x+y+b=0$ forming a chord of length 2.
(I) Determine the standard equation of ellipse C;
(II) Let Q be ... | \frac{2\sqrt{3}}{3} |
The inclination angle $\alpha$ of the line $l: \sqrt{3}x+3y+1=0$ is $\tan^{-1}\left( -\frac{\sqrt{3}}{3} \right)$. Calculate the value of the angle $\alpha$. | \frac{5\pi}{6} |
If two distinct members of the set $\{ 3, 7, 21, 27, 35, 42, 51 \}$ are randomly selected and multiplied, what is the probability that the product is a multiple of 63? Express your answer as a common fraction. | \frac{3}{7} |
Compute the argument of the sum:
\[ e^{2\pi i/40} + e^{6\pi i/40} + e^{10\pi i/40} + e^{14\pi i/40} + e^{18\pi i/40} + e^{22\pi i/40} + e^{26\pi i/40} + e^{30\pi i/40} + e^{34\pi i/40} + e^{38\pi i/40} \]
and express it in the form \( r e^{i \theta} \), where \( 0 \le \theta < 2\pi \). | \frac{\pi}{2} |
A standard deck of 54 playing cards (with four cards of each of thirteen ranks, as well as two Jokers) is shuffled randomly. Cards are drawn one at a time until the first queen is reached. What is the probability that the next card is also a queen? | \frac{2}{27} |
In the engineering department, 70% of the students are men and 180 are women. How many men are there? | The percentage for women is 100% - 70% = 30%.
Since 180 represents 30%, then 180/30 = 6 students represent 1%.
Hence, 6 x 70 = <<6*70=420>>420 students are men.
#### 420 |
In a convex pentagon \(ABCDE\), \(AB = BC\), \(CD = DE\), \(\angle ABC = 100^\circ\), \(\angle CDE = 80^\circ\), and \(BD^2 = \frac{100}{\sin 100^\circ}\). Find the area of the pentagon. | 50 |
The expression $x^2 + 17x + 70$ can be rewritten as $(x + a)(x + b)$, and the expression $x^2 - 18x + 80$ written as $(x - b)(x - c)$, where a, b, and c are integers. Calculate the value of $a + b + c$. | 28 |
A telephone pole is supported by a steel cable which extends from the top of the pole to a point on the ground 3 meters from its base. When Leah walks 2.5 meters from the base of the pole toward the point where the cable is attached to the ground, her head just touches the cable. Leah is 1.5 meters tall. How many meter... | 9 |
A hyperbola is centered at the origin and opens either horizontally or vertically. It passes through the points $(-3, 4),$ $(-2, 0),$ and $(t, 2).$ Find $t^2.$ | \frac{21}{4} |
The greatest common divisor of two integers is $(x+3)$ and their least common multiple is $x(x+3)$, where $x$ is a positive integer. If one of the integers is 36, what is the smallest possible value of the other one? | 108 |
Jackson collects 45 hermit crabs, 3 spiral shells per hermit crab, and 2 starfish per spiral shell. How many souvenirs does he collect total? | First find the number of spiral shells Jackson collects: 45 hermit crabs * 3 spiral shells/hermit crab = <<45*3=135>>135 spiral shells
Then find the number of starfish he collects: 135 spiral shells * 2 starfish/spiral shell = <<135*2=270>>270 starfish
Then add the number of each animal he collects to find the total nu... |
An ideal gas is used as the working substance of a heat engine operating cyclically. The cycle consists of three stages: isochoric pressure reduction from $3 P_{0}$ to $P_{0}$, isobaric density increase from $\rho_{0}$ to $3 \rho_{0}$, and a return to the initial state, represented as a quarter circle in the $P / P_{0}... | 1/9 |
Find all polynomials $P$ with integer coefficients such that $P (0)\ne 0$ and $$P^n(m)\cdot P^m(n)$$ is a square of an integer for all nonnegative integers $n, m$. | P(x) = x + 1 |
The taxi fare in Metropolis City is $3.00 for the first $\frac{3}{4}$ mile and additional mileage charged at the rate $0.30 for each additional 0.1 mile. You plan to give the driver a $3 tip. Calculate the number of miles you can ride for $15. | 3.75 |
Let \[f(x) =
\begin{cases}
2x + 9 &\text{if }x<-2, \\
5-2x&\text{if }x\ge -2.
\end{cases}
\]Find $f(-7).$ | -5 |
The average yield per unit area of a rice variety for five consecutive years was 9.4, 9.7, 9.8, 10.3, and 10.8 (unit: t/hm²). Calculate the variance of this sample data. | 0.244 |
Given $y=f(x)$ is a quadratic function, and $f(0)=-5$, $f(-1)=-4$, $f(2)=-5$,
(1) Find the analytical expression of this quadratic function.
(2) Find the maximum and minimum values of the function $f(x)$ when $x \in [0,5]$. | - \frac {16}{3} |
If \( x = \frac{2}{3} \) and \( y = \frac{3}{2} \), find the value of \( \frac{1}{3}x^8y^9 \). | \frac{1}{2} |
For each vertex of the triangle \(ABC\), the angle between the altitude and the angle bisector drawn from that vertex was determined. It turned out that these angles at vertices \(A\) and \(B\) are equal to each other and are less than the angle at vertex \(C\). What is the measure of angle \(C\) in the triangle? | 60 |
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 |
Fran writes the numbers \(1,2,3, \ldots, 20\) on a chalkboard. Then she erases all the numbers by making a series of moves; in each move, she chooses a number \(n\) uniformly at random from the set of all numbers still on the chalkboard, and then erases all of the divisors of \(n\) that are still on the chalkboard (inc... | \frac{131}{10} |
Nine hundred tickets were sold to a concert. Three-fourths of those who bought the ticket came before the start of the concert. Five-ninths of the remaining came few minutes after the first song. Eighty people arrived during the middle part of the concert while the rest did not go. How many of those who bought the tick... | 900 x 3/4 = <<900*3/4=675>>675 people came before the start of the concert.
900 - 675 = <<900-675=225>>225 people did not come before the start of the concert.
225 x 5/9 = <<225*5/9=125>>125 people came few minutes after the first song.
So, 125 + 80 = <<125+80=205>>205 people came to the concert that was not able to be... |
A certain high school has 1000 students in the first year. Their choices of elective subjects are shown in the table below:
| Subject | Physics | Chemistry | Biology | Politics | History | Geography |
|---------|---------|-----------|---------|----------|---------|-----------|
| Number of Students | 300 | 200 | 100 | ... | \frac{3}{10} |
Points \( A, B, C \) are situated sequentially, with the distance \( AB \) being 3 km and the distance \( BC \) being 4 km. A cyclist departed from point \( A \) heading towards point \( C \). Simultaneously, a pedestrian departed from point \( B \) heading towards point \( A \). It is known that the pedestrian and the... | 2.1 |
A number is randomly selected from the interval $[-π, π]$. Calculate the probability that the value of the function $y = \cos x$ falls within the range $[-\frac{\sqrt{3}}{2}, \frac{\sqrt{3}}{2}]$. | \frac{2}{3} |
How many four-digit positive integers are there with thousands digit $2?$ | 1000 |
(Experimental Class Question) Given that $\cos \alpha = \frac{1}{7}$ and $\cos (\alpha - \beta) = \frac{13}{14}$, with $0 < \beta < \alpha < \pi$.
1. Find the value of $\sin (2\alpha - \frac{\pi}{6})$;
2. Find the value of $\beta$. | \frac{\pi}{3} |
What is the result of adding 12.8 to a number that is three times more than 608? | 2444.8 |
Let $x$ and $y$ be real numbers such that $\frac{\sin x}{\sin y} = 3$ and $\frac{\cos x}{\cos y} = \frac12$. Find the value of
\[\frac{\sin 2x}{\sin 2y} + \frac{\cos 2x}{\cos 2y}.\] | \frac{49}{58} |
The common ratio of the geometric sequence $a+\log_{2}3$, $a+\log_{4}3$, $a+\log_{8}3$ is __________. | \frac{1}{3} |
Let $a_{1}, a_{2}, \ldots$ be an arithmetic sequence and $b_{1}, b_{2}, \ldots$ be a geometric sequence. Suppose that $a_{1} b_{1}=20$, $a_{2} b_{2}=19$, and $a_{3} b_{3}=14$. Find the greatest possible value of $a_{4} b_{4}$. | \frac{37}{4} |
Jenna is hemming her prom dress. The dress's hem is 3 feet long. Each stitch Jenna makes is 1/4 inch long. If Jenna makes 24 stitches per minute, how many minutes does it take Jenna to hem her dress? | First find how many inches the hem is by multiplying the length in feet by the number of inches per foot: 3 feet * 12 inches/foot = <<3*12=36>>36 inches
Then divide the length in inches by the length of each stitch to find how many stitches Jenna makes: 36 inches / .25 inches = <<36/.25=144>>144 stitches
Then divide th... |
Points with integer coordinates (including zero) are called lattice points (or grid points). Find the total number of lattice points (including those on the boundary) in the region bounded by the x-axis, the line \(x=4\), and the parabola \(y=x^2\). | 35 |
The moon is made of 50% iron, 20% carbon, and the remainder is other elements. Mars weighs twice as much as the moon, but has the exact same composition. If Mars is 150 tons of other elements, how many tons does the moon weigh? | 30% of Mars is made up of other elements because 100 - 50 - 20 = <<100-50-20=30>>30
Mars weighs 500 tons because 150 / .3 = <<150/.3=500>>500
The moon weighs 250 tons because 500 / 2 = <<500/2=250>>250
#### 250 |
For all $x \in (0, +\infty)$, the inequality $(2x - 2a + \ln \frac{x}{a})(-2x^{2} + ax + 5) \leq 0$ always holds. Determine the range of values for the real number $a$. | \left\{ \sqrt{5} \right\} |
What is $(a^3+b^3)\div(a^2-ab+b^2)$ when $a=5$ and $b=4$? | 9 |
Given that $17^{-1} \equiv 26 \pmod{53}$, find $36^{-1} \pmod{53}$, as a residue modulo 53. (Give a number between 0 and 52, inclusive.) | 27 |
Given the function $f\left(x\right)=x-{e}^{-x}$, if the line $y=mx+n$ is a tangent line to the curve $y=f\left(x\right)$, find the minimum value of $m+n$. | 1-\dfrac{1}{e} |
What is the sum of the prime factors of 91? | 20 |
A warehouse store sells cartons of gum. Each carton contains 5 packs of gum, and there are 3 sticks of gum in each pack. Each brown box contains 4 cartons of gum. How many sticks of gum are there in 8 brown boxes? | 5 packs of gum contain 5*3=<<5*3=15>>15 sticks of gum.
If each brown box contains 4 cartons, 8 brown boxes contain 8 * 4= <<8*4=32>>32 cartons.
So, 8 brown boxes contain 32 * 15=<<32*15=480>>480 sticks of gum.
#### 480 |
Find all real numbers $p$ such that the cubic equation
$$
5 x^{3}-5(p+1) x^{2}+(71 p-1) x+1=66 p
$$
has two roots that are both natural numbers. | 76 |
For a constant $c,$ in spherical coordinates $(\rho,\theta,\phi),$ find the shape described by the equation
\[\theta = c.\](A) Line
(B) Circle
(C) Plane
(D) Sphere
(E) Cylinder
(F) Cone
Enter the letter of the correct option. | \text{(C)} |
Determine the value of \(\sum_{n=1}^{\infty} \ln \left(1+\frac{1}{n}\right) \cdot \ln \left(1+\frac{1}{2 n}\right) \cdot \ln \left(1+\frac{1}{2 n+1}\right)\). | \frac{1}{3} \ln ^{3}(2) |
Find the integer $n$, $0 \le n \le 7$, such that \[n \equiv -3737 \pmod{8}.\] | 7 |
A digital clock displays time in a 24-hour format (from 00:00 to 23:59). Find the largest possible sum of the digits in this time display. | 19 |
Given $A=\{x|x^{3}+3x^{2}+2x > 0\}$, $B=\{x|x^{2}+ax+b\leqslant 0\}$ and $A\cap B=\{x|0 < x\leqslant 2\}$, $A\cup B=\{x|x > -2\}$, then $a+b=$ ______. | -3 |
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