problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Evaluate $\log_{\sqrt{6}} (216\sqrt{6})$. | 7 |
Figures $0$, $1$, $2$, and $3$ consist of $1$, $5$, $13$, and $25$ nonoverlapping unit squares, respectively. If the pattern were continued, how many nonoverlapping unit squares would there be in figure 100? | 20201 |
Two numbers are independently selected from the set of positive integers less than or equal to 6. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction. | \frac{2}{3} |
Given plane vectors $\overrightarrow{a}=(-1,2)$ and $\overrightarrow{b}=(1,-4)$.
$(1)$ If $4\overrightarrow{a}+\overrightarrow{b}$ is perpendicular to $k\overrightarrow{a}-\overrightarrow{b}$, find the value of the real number $k$.
$(2)$ If $\theta$ is the angle between $4\overrightarrow{a}+\overrightarrow{b}$ and ... | -\frac{3}{4} |
At an auction event, the price of a TV, whose cost was $500, increased by 2/5 times its initial price. The price of a phone, which was $400, also increased by 40% from its original price. If Bethany had taken both items to the auction, calculate the total amount she received for the items after the sale at the auction. | If the TV's original price was $500, at the auction, its price increased by 2/5*$500 = $<<2/5*500=200>>200
The total price at which the TV was sold at the auction is $500+$200 = $<<500+200=700>>700
The price of the phone also increased at the action, a 40% increase which is 40/100*$400 = $<<40/100*400=160>>160
The tota... |
Compute $\sin(-30^\circ)$ and verify by finding $\cos(-30^\circ)$, noticing the relationship, and confirming with the unit circle properties. | \frac{\sqrt{3}}{2} |
Regular decagon $P_1 P_2 \dotsb P_{10}$ is drawn in the coordinate plane with $P_1$ at $(1,0)$ and $P_6$ at $(3,0).$ If $P_n$ is the point $(x_n,y_n),$ compute the numerical value of the product
\[(x_1 + y_1 i)(x_2 + y_2 i)(x_3 + y_3 i) \dotsm (x_{10} + y_{10} i).\] | 1023 |
Given that $0 < x < \frac{\pi}{2}$ and $\sin(2x - \frac{\pi}{4}) = -\frac{\sqrt{2}}{10}$, find the value of $\sin x + \cos x$. | \frac{2\sqrt{10}}{5} |
Find the greatest real $k$ such that, for every tetrahedron $ABCD$ of volume $V$ , the product of areas of faces $ABC,ABD$ and $ACD$ is at least $kV^2$ . | 9/2 |
1. The focal distance of the parabola $4x^{2}=y$ is \_\_\_\_\_\_\_\_\_\_\_\_
2. The equation of the hyperbola that has the same asymptotes as the hyperbola $\frac{x^{2}}{2} -y^{2}=1$ and passes through $(2,0)$ is \_\_\_\_\_\_\_\_\_\_\_\_
3. In the plane, the distance formula between a point $(x_{0},y_{0})$ and a line $... | 2\sqrt{2} |
Find the number of ordered pairs of integers $(a,b)$ with $1 \leq a \leq 100$ and $b \geq 0$ such that the polynomial $x^2+ax+b$ can be factored into the product of two (not necessarily distinct) linear factors with integer coefficients. | 2600 |
Given that there are 6 teachers with IDs $A$, $B$, $C$, $D$, $E$, $F$ and 4 different schools, with the constraints that each school must have at least 1 teacher and $B$ and $D$ must be arranged in the same school, calculate the total number of different arrangements. | 240 |
The radius of a sphere is $p$ units and the radius of a hemisphere is $2p$ units. What is the ratio of the volume of the sphere to the volume of the hemisphere? | \frac{1}{4} |
Onum Lake has 25 more trout than Boast Pool. There are 75 fish in Boast Pool. Riddle Pond has half as many fish as Onum Lake. What is the average number of fish in all three bodies of water? | Onum lake has 25 trout + 75 = <<25+75=100>>100 fish.
Riddle Pond has 1/2 * 100 fish in Onum lake = <<1/2*100=50>>50 fish.
Combined the three bodies of water have 100 fish + 50 fish + 75 fish = <<100+50+75=225>>225 fish.
The average number of fish in the bodies of water is 225 fish / 3 = <<225/3=75>>75 fish.
#### 75 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $2B=A+C$ and $a+\sqrt{2}b=2c$, find the value of $\sin C$. | \frac{\sqrt{6}+\sqrt{2}}{4} |
A bag contains 3 red balls, 2 black balls, and 1 white ball. All 6 balls are identical in every aspect except for color and are well mixed. Balls are randomly drawn from the bag.
(1) With replacement, find the probability of drawing exactly 1 red ball in 2 consecutive draws;
(2) Without replacement, find the probabilit... | \frac{3}{5} |
Given a 3x3 matrix where each row and each column forms an arithmetic sequence, and the middle element $a_{22} = 5$, find the sum of all nine elements. | 45 |
Robyn has 4 tasks to do and Sasha has 14 tasks to do. How many of Sasha's tasks should Robyn do in order for them to have the same number of tasks? | 5 |
In triangle \(ABC\), the median \(BK\), the angle bisector \(BE\), and the altitude \(AD\) are given.
Find the side \(AC\), if it is known that the lines \(BK\) and \(BE\) divide the segment \(AD\) into three equal parts, and \(AB=4\). | \sqrt{13} |
From a regular octagon, a triangle is formed by connecting three randomly chosen vertices of the octagon. What is the probability that at least one of the sides of the triangle is also a side of the octagon? | \frac{5}{7} |
In a box, 10 smaller boxes are placed. Some of the boxes are empty, and some contain another 10 smaller boxes each. Out of all the boxes, exactly 6 contain smaller boxes. How many empty boxes are there? | 55 |
Using five nines (9), arithmetic operations, and exponentiation, create the numbers from 1 to 13. | 13 |
Given a line $l$ whose inclination angle $\alpha$ satisfies the condition $\sin \alpha +\cos \alpha = \frac{1}{5}$, determine the slope of $l$. | -\frac{4}{3} |
Let $x$ and $y$ be positive real numbers such that $4x + 9y = 60.$ Find the maximum value of $xy.$ | 25 |
Trapezoid $ABCD$ has base $AB = 20$ units and base $CD = 30$ units. Diagonals $AC$ and $BD$ intersect at $X$. If the area of trapezoid $ABCD$ is $300$ square units, what is the area of triangle $BXC$? | 72 |
Given acute angles \\(\alpha\\) and \\(\beta\\) satisfy \\((\tan \alpha-1)(\tan \beta-1)=2\\), then the value of \\(\alpha+\beta\\) is \_\_\_\_\_\_. | \dfrac {3\pi}{4} |
Compute $\arccos (\sin 2).$ All functions are in radians. | 2 - \frac{\pi}{2} |
The function $f$ defined by $f(x)= \frac{ax+b}{cx+d}$, where $a$,$b$,$c$ and $d$ are nonzero real numbers, has the properties $f(19)=19$, $f(97)=97$ and $f(f(x))=x$ for all values except $\frac{-d}{c}$. Find the unique number that is not in the range of $f$. | 58 |
Village Foods sells good food at a fair price. Their specialty is fresh vegetables. If they get 500 customers per month, and each customer purchases 2 heads of lettuce for $1 each and 4 tomatoes for $0.5 apiece, then how much money, in dollars, will the store receive in sales of lettuce and tomatoes per month? | 2 heads of lettuce at $1 apiece is 2*$1=$<<2*1=2>>2.
4 tomatoes at $0.50 apiece is 4*$0.5=$<<4*0.5=2>>2.
Thus, each customer purchases $2+$2=$<<2+2=4>>4 in lettuce and tomatoes per month.
Therefore, if 500 customers per month spend $4 on lettuce and tomatoes, then the store will receive $4*500=$<<4*500=2000>>2000 per m... |
Let \(a\) and \(b\) be two natural numbers. If the remainder of the product \(a \cdot b\) divided by 15 is 1, then \(b\) is called the multiplicative inverse of \(a\) modulo 15. Based on this definition, find the sum of all multiplicative inverses of 7 modulo 15 that lie between 100 and 200. | 1036 |
Find the value of the arithmetic series $1-3+5-7+9-11+\cdots +2021-2023+2025$. | 1013 |
In triangle $XYZ$ where $XY=60$ and $XZ=15$, the area of the triangle is given as $225$. Let $W$ be the midpoint of $\overline{XY}$, and $V$ be the midpoint of $\overline{XZ}$. The angle bisector of $\angle YXZ$ intersects $\overline{WV}$ and $\overline{YZ}$ at points $P$ and $Q$, respectively. Determine the area of qu... | 123.75 |
What is the largest integer less than or equal to \(\sqrt[3]{(2010)^{3}+3 \times(2010)^{2}+4 \times 2010+1}\)? | 2011 |
The solutions to the equation $x^2 - 3|x| - 2 = 0$ are. | \frac{-3 - \sqrt{17}}{2} |
Let $Q$ be the product of the first $50$ positive even integers. Find the largest integer $l$ such that $Q$ is divisible by $2^l$. | 97 |
Observe the following equations:
1. $\cos 2\alpha = 2\cos^2\alpha - 1$;
2. $\cos 4\alpha = 8\cos^4\alpha - 8\cos^2\alpha + 1$;
3. $\cos 6\alpha = 32\cos^6\alpha - 48\cos^4\alpha + 18\cos^2\alpha - 1$;
4. $\cos 8\alpha = 128\cos^8\alpha - 256\cos^6\alpha + 160\cos^4\alpha - 32\cos^2\alpha + 1$;
5. $\cos 10\alpha = m\co... | 962 |
Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$ and $Q$ are distinct points in the plane with nonnegative integer coordinates $(x,y)$ such that $41x + y = 2009$. Find the number of such distinct triangles whose area is a positive integer.
| 600 |
By starting with a million and alternatively dividing by 2 and multiplying by 5, Anisha created a sequence of integers that starts 1000000, 500000, 2500000, 1250000, and so on. What is the last integer in her sequence? Express your answer in the form $a^b$, where $a$ and $b$ are positive integers and $a$ is as small as... | 5^{12} |
For the quadrilateral $ABCD$, it is known that $\angle BAC = \angle CAD = 60^\circ$ and $AB + AD = AC$. It is also given that $\angle ACD = 23^\circ$. What is the measure of angle $ABC$ in degrees? | 83 |
Given $f(x)= \frac{1}{4^{x}+2}$, use the method of deriving the sum formula for an arithmetic sequence to find the value of $f( \frac{1}{10})+f( \frac{2}{10})+…+f( \frac{9}{10})$. | \frac{9}{4} |
The chess club has 20 members: 12 boys and 8 girls. A 4-person team is chosen at random. What is the probability that the team has at least 2 boys and at least 1 girl? | \frac{4103}{4845} |
Maria is 54 inches tall, and Samuel is 72 inches tall. Using the conversion 1 inch = 2.54 cm, how tall is each person in centimeters? Additionally, what is the difference in their heights in centimeters? | 45.72 |
How many right-angled triangles can Delia make by joining three vertices of a regular polygon with 18 sides? | 144 |
John plays at the arcade for 3 hours. He uses $.50 for every 6 minutes. How much money did he spend, in dollars? | He was at the arcade for 3*60=<<3*60=180>>180 minutes
So he put coins in 180/6=<<180/6=30>>30 times
That means he spent 30*.5=$<<30*.5=15>>15
#### 15 |
The acronym AMC is shown in the rectangular grid below with grid lines spaced $1$ unit apart. In units, what is the sum of the lengths of the line segments that form the acronym AMC$? | 13 + 4\sqrt{2} |
Gulliver arrives in the land of the Lilliputians with 7,000,000 rubles. He uses all the money to buy kefir at a price of 7 rubles per bottle (an empty bottle costs 1 ruble at that time). After drinking all the kefir, he returns the bottles and uses the refunded money to buy more kefir. During this process, he notices t... | 1166666 |
Find the value of $x$ such that $\sqrt{3x + 7} = 10$. | 31 |
Fred had 212 sheets of paper. He received another 307 sheets of paper from Jane and gave Charles 156 sheets of paper. How many sheets of paper does Fred have left? | He had 212 sheets and received 307 more for a total of 212+307 = <<212+307=519>>519 sheets
He gave out 156 so he has 519-156 = <<519-156=363>>363 sheets
#### 363 |
An adult panda can eat 138 pounds of bamboo each day. A baby panda can eat 50 pounds of bamboo a day. How many pounds of bamboo will the pandas eat in a week? | An adult panda will eat 138 pounds of bamboo * 7 days = <<138*7=966>>966 pounds.
A baby panda will eat 50 pounds * 7 days = <<50*7=350>>350 pounds.
Total the pandas will eat 966 pounds + 350 = <<966+350=1316>>1316 pounds of bamboo.
#### 1316 |
There are 50 more pens than notebooks on Wesley's school library shelf. If there are 30 notebooks on the shelf, how many notebooks and pens, does he have altogether? | If there are 30 notebooks on the shelf, there are 50+30 = <<30+50=80>>80 pens on the shelf.
Altogether, Wesley has 80+30 = <<80+30=110>>110 notebooks and pens on the shelf.
#### 110 |
What is the largest $n$ for which the numbers $1,2, \ldots, 14$ can be colored in red and blue so that for any number $k=1,2, \ldots, n$, there are a pair of blue numbers and a pair of red numbers, each pair having a difference equal to $k$? | 11 |
Summer and Jolly both went to the same middle school. However, when they finished college, Summer had five more degrees than Jolly. If Summer has a total of 150 degrees, what's the combined number of degrees they both have? | Jolly has 150 - 5 = <<150-5=145>>145 degrees
Since Summer has 150 degrees, the combined total for both is 150+145 = <<150+145=295>>295
#### 295 |
When the binary number $100101110010_2$ is divided by 4, what is the remainder (give your answer in base 10)? | 2 |
Suppose $x$ satisfies $x^{3}+x^{2}+x+1=0$. What are all possible values of $x^{4}+2 x^{3}+2 x^{2}+2 x+1 ?$ | 0 |
An HMMT party has $m$ MIT students and $h$ Harvard students for some positive integers $m$ and $h$, For every pair of people at the party, they are either friends or enemies. If every MIT student has 16 MIT friends and 8 Harvard friends, and every Harvard student has 7 MIT enemies and 10 Harvard enemies, compute how ma... | 342 |
How many four-digit positive integers are multiples of 7? | 1286 |
Triangle \(ABC\) has \(AB = 10\) and \(BC:AC = 35:36\). What is the largest area that this triangle can have? | 1260 |
Given \( n \in \mathbf{N}, n > 4 \), and the set \( A = \{1, 2, \cdots, n\} \). Suppose there exists a positive integer \( m \) and sets \( A_1, A_2, \cdots, A_m \) with the following properties:
1. \( \bigcup_{i=1}^{m} A_i = A \);
2. \( |A_i| = 4 \) for \( i=1, 2, \cdots, m \);
3. Let \( X_1, X_2, \cdots, X_{\mathrm{C... | 13 |
On a normal day, Julia can run a mile in 10 minutes. However, today she decided to wear her new shoes to run. They were uncomfortable and slowed her mile down to 13 minutes. How much longer would it take Julia to run 5 miles in her new shoes than if she wore her old ones? | With her regular shoes, it should take 5*10=<<5*10=50>>50 minutes to run five miles
If she wore her new shoes it would take 13*5=<<13*5=65>>65 minutes to run five miles
65-50=<<65-50=15>>15 more minutes
#### 15 |
Cities $A$, $B$, $C$, $D$, and $E$ are connected by roads $\widetilde{AB}$, $\widetilde{AD}$, $\widetilde{AE}$, $\widetilde{BC}$, $\widetilde{BD}$, $\widetilde{CD}$, and $\widetilde{DE}$. How many different routes are there from $A$ to $B$ that use each road exactly once? (Such a route will necessarily visit some citie... | 16 |
How many of the numbers from the set $\{1,\ 2,\ 3,\ldots,\ 50\}$ have a perfect square factor other than one? | 19 |
Two positive integers differ by 8 and their product is 168. What is the larger integer? | 14 |
A class of 10 students took a math test. Each problem was solved by exactly 7 of the students. If the first nine students each solved 4 problems, how many problems did the tenth student solve? | 6 |
At his cafe, Milton sells apple pie and peach pie slices. He cuts the apple pie into 8 slices. He cuts the peach pie into 6 slices. On the weekend, 56 customers ordered apple pie slices and 48 customers ordered peach pie slices. How many pies did Milton sell during the weekend? | Milton sold 56 / 8 = <<56/8=7>>7 apple pies.
He sold 48 / 6 = <<48/6=8>>8 peach pies.
He sold a total of 7 + 8 = <<7+8=15>>15 pies.
#### 15 |
What is the perimeter of pentagon $ABCDE$ in this diagram? [asy]
pair cis(real r,real t) { return (r*cos(t),r*sin(t)); }
pair a=(0,0);
pair b=cis(1,-pi/2);
pair c=cis(sqrt(2),-pi/4);
pair d=cis(sqrt(3),-pi/4+atan(1/sqrt(2)));
pair e=cis(2,-pi/4+atan(1/sqrt(2))+atan(1/sqrt(3)));
dot(a); dot(b); dot(c); dot(d); dot(e);
d... | 6 |
Find the number of permutations of \( n \) distinct elements \( a_1, a_2, \cdots, a_n \) (where \( n \geqslant 2 \)) such that \( a_1 \) is not in the first position and \( a_2 \) is not in the second position. | 32,527,596 |
Given that $y$ is a multiple of $45678$, what is the greatest common divisor of $g(y)=(3y+4)(8y+3)(14y+9)(y+14)$ and $y$? | 1512 |
$(HUN 6)$ Find the positions of three points $A,B,C$ on the boundary of a unit cube such that $min\{AB,AC,BC\}$ is the greatest possible. | \sqrt{2} |
Given that an odd function \( f(x) \) satisfies the condition \( f(x+3) = f(x) \). When \( x \in [0,1] \), \( f(x) = 3^x - 1 \). Find the value of \( f\left(\log_1 36\right) \). | -1/3 |
How many whole numbers between $200$ and $500$ contain the digit $3$? | 138 |
The lines containing the altitudes of the scalene triangle \( ABC \) intersect at point \( H \). Let \( I \) be the incenter of triangle \( ABC \), and \( O \) be the circumcenter of triangle \( BHC \). It is known that point \( I \) lies on the segment \( OA \). Find the angle \( BAC \). | 60 |
For how many pairs $(m, n)$ with $m$ and $n$ integers satisfying $1 \leq m \leq 100$ and $101 \leq n \leq 205$ is $3^{m}+7^{n}$ divisible by 10? | 2625 |
The sum of the numerical coefficients in the complete expansion of $(x^2 - 2xy + y^2)^7$ is: | 0 |
Two cylindrical poles, with diameters of $10$ inches and $30$ inches respectively, are placed side by side and bound together with a wire. Calculate the length of the shortest wire that will go around both poles.
**A)** $20\sqrt{3} + 24\pi$
**B)** $20\sqrt{3} + \frac{70\pi}{3}$
**C)** $30\sqrt{3} + 22\pi$
**D)** $16\sq... | 20\sqrt{3} + \frac{70\pi}{3} |
The numbers from 1 to 9 are placed at the vertices of a cube such that the sum of the four numbers on each face is the same. Find the common sum. | 22.5 |
Among the following functions, identify which pairs represent the same function.
1. $f(x) = |x|, g(x) = \sqrt{x^2}$;
2. $f(x) = \sqrt{x^2}, g(x) = (\sqrt{x})^2$;
3. $f(x) = \frac{x^2 - 1}{x - 1}, g(x) = x + 1$;
4. $f(x) = \sqrt{x + 1} \cdot \sqrt{x - 1}, g(x) = \sqrt{x^2 - 1}$. | (1) |
\[\log_{10} x + \log_{\sqrt{10}} x + \log_{\sqrt[3]{10}} x + \ldots + \log_{\sqrt[1]{10}} x = 5.5\] | \sqrt[10]{10} |
[asy]
draw((0,1)--(4,1)--(4,2)--(0,2)--cycle);
draw((2,0)--(3,0)--(3,3)--(2,3)--cycle);
draw((1,1)--(1,2));
label("1",(0.5,1.5));
label("2",(1.5,1.5));
label("32",(2.5,1.5));
label("16",(3.5,1.5));
label("8",(2.5,0.5));
label("6",(2.5,2.5));
[/asy]
The image above is a net of a unit cube. Let $n$ be a positive intege... | 16 |
A truck driver’s heavy semi truck can go 3 miles per gallon of gas. The truck driver needs to put gas in his truck at one gas station, but wants to put the minimum amount he needs to get him to a much cheaper gas station 90 miles away. He already has 12 gallons of gas in his tank. How many more gallons does the truck d... | The truck driver can go 3 * 12 = <<3*12=36>>36 miles on the gas in his tank.
He needs to go 90 - 36 = <<90-36=54>>54 more miles.
Thus, the truck driver needs to put 54 / 3 = <<54/3=18>>18 more gallons of gas in his semi.
#### 18 |
Calculate the infinite sum:
\[
\sum_{n=1}^\infty \frac{n^3 - n}{(n+3)!}
\] | \frac{1}{6} |
Let $A B C D E$ be a convex pentagon such that $\angle A B C=\angle A C D=\angle A D E=90^{\circ}$ and $A B=B C=C D=D E=1$. Compute $A E$. | 2 |
Any five points are taken inside or on a square with side length $1$. Let a be the smallest possible number with the property that it is always possible to select one pair of points from these five such that the distance between them is equal to or less than $a$. Then $a$ is:
$\textbf{(A)}\ \sqrt{3}/3\qquad \textbf{(B)... | \frac{\sqrt{2}}{2} |
Mary has 300 sheep and Bob has double the number of sheep as Mary plus another 35. How many sheep must Mary buy to have 69 fewer sheep than Bob? | Bob has (300*2) + 35 = <<300*2+35=635>>635 sheep
For Mary to have 69 sheep fewer than Bob she must own 635 – 69 = 566 sheep
As Mary has already 300 sheep she must buy 566 – 300 = <<566-300=266>>266 sheep
#### 266 |
Jake splits 8 shots of vodka with his friend. Each shot of vodka is 1.5 ounces. If the vodka is 50% pure alcohol, how much pure alcohol did Jake drink? | Jake drank 8/2=<<8/2=4>>4 shots
So he drank 4*1.5=<<4*1.5=6>>6 ounces of vodka
That means he drank 6*.5=<<6*.5=3>>3 ounces of pure alcohol
#### 3 |
Each pack of hot dogs contains 10 hot dogs (no buns), but each pack of hot dog buns contains 8 buns. Phil buys a number of these packs for a barbecue. After the barbecue, Phil finds that he has 4 hot dogs left over. What is the SECOND smallest number of packs of hot dogs he could have bought? | 6 |
If $\left(a + \frac{1}{a}\right)^2 = 3$, then $a^3 + \frac{1}{a^3}$ equals: | 0 |
There exist constants $a_1, a_2, a_3, a_4, a_5, a_6, a_7$ such that
\[
\cos^7 \theta = a_1 \cos \theta + a_2 \cos 2 \theta + a_3 \cos 3 \theta + a_4 \cos 4 \theta + a_5 \cos 5 \theta + a_6 \cos 6 \theta + a_7 \cos 7 \theta
\]
for all angles $\theta.$ Find $a_1^2 + a_2^2 + a_3^2 + a_4^2 + a_5^2 + a_6^2 + a_7^2.$ | \frac{1716}{4096} |
Given that the sum of the first $n$ terms of the sequence $\{a_n\}$ is $S_n=\ln (1+ \frac {1}{n})$, find the value of $e^{a_7+a_8+a_9}$. | \frac {20}{21} |
In the quadratic equation $3x^{2}-6x-7=0$, the coefficient of the quadratic term is ____ and the constant term is ____. | -7 |
Hayden has a tank with a small hole in the bottom. The tank starts with 40 gallons of water. It loses 2 gallons of water per hour. Hayden does not add any water for the first two hours. He adds 1 gallon of water to the tank in hour three. He adds three gallons of water to the tank in the fourth hour. How much water is ... | Over the four hours, the tank loses 2 * 4 = <<2*4=8>>8 gallons of water.
Hayden adds a total of 1 + 3 = <<1+3=4>>4 gallons of water.
At the end of the four hours, there is 40 - 8 + 4 = <<40-8+4=36>>36 gallons of water left in the tank.
#### 36 |
Find $n$ such that $2^6 \cdot 3^3 \cdot n = 10!$. | 350 |
Compute
\[\left( 1 - \frac{1}{\cos 23^\circ} \right) \left( 1 + \frac{1}{\sin 67^\circ} \right) \left( 1 - \frac{1}{\sin 23^\circ} \right) \left( 1 + \frac{1}{\cos 67^\circ} \right).\] | 1 |
A round-robin tennis tournament consists of each player playing every other player exactly once. How many matches will be held during an 8-person round-robin tennis tournament? | 28 |
Calculate the limit of the function:
\[ \lim _{x \rightarrow \frac{1}{2}} \frac{\sqrt[3]{\frac{x}{4}}-\frac{1}{2}}{\sqrt{\frac{1}{2}+x}-\sqrt{2x}} \] | -\frac{2}{3} |
A person has a probability of $\frac{1}{2}$ to hit the target in each shot. What is the probability of hitting the target 3 times out of 6 shots, with exactly 2 consecutive hits? (Answer with a numerical value) | \frac{3}{16} |
Emma is 7 years old. If her sister is 9 years older than her, how old will Emma be when her sister is 56? | Emma’s sister is 7 + 9 = <<7+9=16>>16 years old.
In this many years, Emma’s sister will be 56 years old: 56 - 16 = <<56-16=40>>40 years.
When her sister is 56 years old, Emma will be 7 + 40 = <<7+40=47>>47 years.
#### 47 |
Samuel took 30 minutes to finish his homework while Sarah took 1.3 hours to finish it. How many minutes faster did Samuel finish his homework than Sarah? | Since there are 60 minutes in 1 hour, then 1.3 hours is equal to 1.3 x 60 = <<1.3*60=78>>78 minutes.
Thus, Samuel is 78 – 30 = <<78-30=48>>48 minutes faster than Sarah.
#### 48 |
The left and right foci of a hyperbola are $F_{1}$ and $F_{2}$, respectively. A line passing through $F_{2}$ intersects the right branch of the hyperbola at points $A$ and $B$. If $\triangle F_{1} A B$ is an equilateral triangle, what is the eccentricity of the hyperbola? | \sqrt{3} |
Find the quotient when $x^5-23x^3+11x^2-14x+10$ is divided by $x+5$. | x^4-5x^3+2x^2+x-19 |
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 |
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