problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Fifty numbers have an average of 76. Forty of these numbers have an average of 80. What is the average of the other ten numbers? | 60 |
Given that there is 1 path from point A to the first red arrow, 2 paths from the first red arrow to the second red arrow, 3 paths from the first red arrow to each of the first two blue arrows, 4 paths from the second red arrow to each of the first two blue arrows, 5 paths from each of the first two blue arrows to each ... | 4312 |
If it takes 10 people 10 days to shovel 10,000 pounds of coal, how many days will it take half of these ten people to shovel 40,000 pounds of coal? | The 10 people shovel 10,000 pounds per 10 days, or 10,000/10=<<10000/10=1000>>1000 pounds per day per 10 people.
And 1000 pounds per day per 10 people is 1000/10=<<1000/10=100>>100 pounds per day per person.
Half of ten people is 10/2=<<10/2=5>>5 people.
At 100 pounds per day per person, 5 people can shovel 5*100=<<5*1... |
Six men and some number of women stand in a line in random order. Let $p$ be the probability that a group of at least four men stand together in the line, given that every man stands next to at least one other man. Find the least number of women in the line such that $p$ does not exceed 1 percent. | 594 |
Mustafa buys a fridge, a TV and a computer with a total budget of $1600. The TV costs $600 and the fridge costs $500 more than the computer. How much does the computer cost? | The fridge and computer cost 1600-600=<<1600-600=1000>>1000 dollars.
The computer cost (1000-500)/2=250 dollars.
#### 250 |
Compute $\dbinom{133}{133}$. | 1 |
Given three points in space: A(0,1,5), B(1,5,0), and C(5,0,1), if the vector $\vec{a}=(x,y,z)$ is perpendicular to both $\overrightarrow{AB}$ and $\overrightarrow{AC}$, and the magnitude of vector $\vec{a}$ is $\sqrt{15}$, then find the value of $x^2y^2z^2$. | 125 |
Find the rightmost non-zero digit of the expansion of (20)(13!). | 6 |
Compute $0.18\div0.003.$ | 60 |
Two right triangles, $ABC$ and $ACD$, are joined as shown. Squares are drawn on four of the sides. The areas of three of the squares are 9, 16 and 36 square units. What is the number of square units in the area of the fourth square?
Note that the diagram is not drawn to scale.
[asy]
defaultpen(linewidth(0.7));
draw((... | 61 |
What is the value of the product
\[\left(1+\frac{1}{1}\right)\cdot\left(1+\frac{1}{2}\right)\cdot\left(1+\frac{1}{3}\right)\cdot\left(1+\frac{1}{4}\right)\cdot\left(1+\frac{1}{5}\right)\cdot\left(1+\frac{1}{6}\right)?\] | 7 |
A rectangle PQRS has a perimeter of 24 meters and side PQ is fixed at 7 meters. Find the minimum diagonal PR of the rectangle. | \sqrt{74} |
The graph relates the distance traveled [in miles] to the time elapsed [in hours] on a trip taken by an experimental airplane. During which hour was the average speed of this airplane the largest? | second (1-2) |
What is the sum of the greatest common divisor of $50$ and $5005$ and the least common multiple of $50$ and $5005$? | 50055 |
The product $(8)(888\dots8)$, where the second factor has $k$ digits, is an integer whose digits have a sum of $1000$. What is $k$? | 991 |
A random number selector can only select one of the nine integers 1, 2, ..., 9, and it makes these selections with equal probability. Determine the probability that after $n$ selections ( $n>1$ ), the product of the $n$ numbers selected will be divisible by 10. | \[ 1 - \left( \frac{8}{9} \right)^n - \left( \frac{5}{9} \right)^n + \left( \frac{4}{9} \right)^n \] |
In right triangle $A B C$, a point $D$ is on hypotenuse $A C$ such that $B D \perp A C$. Let $\omega$ be a circle with center $O$, passing through $C$ and $D$ and tangent to line $A B$ at a point other than $B$. Point $X$ is chosen on $B C$ such that $A X \perp B O$. If $A B=2$ and $B C=5$, then $B X$ can be expressed ... | 8041 |
A thousand points form the vertices of a convex polygon with 1000 sides. Inside this polygon, there are another 500 points placed such that no three of these 500 points are collinear. The polygon is triangulated in such a way that all of these 1500 points are vertices of the triangles, and none of the triangles have a... | 1998 |
Simplify \[\frac{1}{\dfrac{1}{\sqrt{2}+1} + \dfrac{2}{\sqrt{3}-1}}.\] | \sqrt3-\sqrt2 |
If the fractional equation in terms of $x$, $\frac{x-2}{x-3}=\frac{n+1}{3-x}$ has a positive root, then $n=\_\_\_\_\_\_.$ | -2 |
Alice is sitting in a teacup ride with infinitely many layers of spinning disks. The largest disk has radius 5. Each succeeding disk has its center attached to a point on the circumference of the previous disk and has a radius equal to $2 / 3$ of the previous disk. Each disk spins around its center (relative to the dis... | 18 \pi |
\(x, y\) are real numbers, \(z_{1}=x+\sqrt{11}+yi\), \(z_{6}=x-\sqrt{11}+yi\) (where \(i\) is the imaginary unit). Find \(|z_{1}| + |z_{6}|\). | 30(\sqrt{2} + 1) |
What is the nonnegative difference between the roots for the equation $x^2+30x+180=-36$? | 6 |
Given that $F_1$ and $F_2$ are the two foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a>0$, $b>0$), an isosceles right triangle $MF_1F_2$ is constructed with $F_1$ as the right-angle vertex. If the midpoint of the side $MF_1$ lies on the hyperbola, calculate the eccentricity of the hyperbola. | \frac{\sqrt{5} + 1}{2} |
Circles of radius 4 and 5 are externally tangent and are circumscribed by a third circle. Calculate the area of the region outside the smaller circles but inside the larger circle. | 40\pi |
How many positive 3-digit numbers are multiples of 25, but not of 60? | 33 |
What is the value of the sum $\frac{2}{3}+\frac{2^2}{3^2}+\frac{2^3}{3^3}+ \ldots +\frac{2^{10}}{3^{10}}$? Express your answer as a common fraction. | \frac{116050}{59049} |
In the plane rectangular coordinate system $xOy$, the parametric equations of the line $l_{1}$ are $\left\{\begin{array}{l}{x=t}\\{y=kt}\end{array}\right.$ (where $t$ is the parameter), and the parametric equations of the line $l_{2}$ are $\left\{\begin{array}{l}{x=-km+2}\\{y=m}\end{array}\right.$ (where $m$ is the par... | 1+\frac{5\sqrt{2}}{2} |
Let $p(x) = 2x - 7$ and $q(x) = 3x - b$. If $p(q(4)) = 7$, what is $b$? | 5 |
Amy and Ben need to eat 1000 total carrots and 1000 total muffins. The muffins can not be eaten until all the carrots are eaten. Furthermore, Amy can not eat a muffin within 5 minutes of eating a carrot and neither can Ben. If Amy eats 40 carrots per minute and 70 muffins per minute and Ben eats 60 carrots per minute a... | 23.5 |
A unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure 1. The cube is then cut in the same manner along the dashed lines shown in Figure 2. This creates nine pieces. What is the volume of the piece that contains vertex $W$?
[asy]
path a=(0,0)--(10,0)--(10,10)--(0,10)--... | \frac{1}{12} |
$\triangle PQR$ is similar to $\triangle STU$. The length of $\overline{PQ}$ is 10 cm, $\overline{QR}$ is 12 cm, and the length of $\overline{ST}$ is 5 cm. Determine the length of $\overline{TU}$ and the perimeter of $\triangle STU$. Express your answer as a decimal. | 17 |
A point $(x, y)$ is randomly selected from inside the rectangle with vertices $(0, 0)$, $(4, 0)$, $(4, 3)$, and $(0, 3)$. What is the probability that both $x < y$ and $x + y < 5$? | \frac{3}{8} |
In triangle \( \triangle ABC \), \(E\) and \(F\) are the midpoints of \(AC\) and \(AB\) respectively, and \( AB = \frac{2}{3} AC \). If \( \frac{BE}{CF} < t \) always holds, then the minimum value of \( t \) is ______. | \frac{7}{8} |
Each of the first $150$ positive integers is painted on a different marble, and the $150$ marbles are placed in a bag. If $n$ marbles are chosen (without replacement) from the bag, what is the smallest value of $n$ such that we are guaranteed to choose three marbles with consecutive numbers? | 101 |
Calculate the value of $n$ such that
\[(1 + \tan 1^\circ)(1 + \tan 2^\circ)(1 + \tan 3^\circ) \dotsm (1 + \tan 30^\circ) = 2^n.\] | 15 |
Let $a$, $b$, $c$, $d$, and $e$ be positive integers with $a+b+c+d+e=2010$ and let $M$ be the largest of the sum $a+b$, $b+c$, $c+d$ and $d+e$. What is the smallest possible value of $M$? | 671 |
Let $Z$ denote the set of points in $\mathbb{R}^n$ whose coordinates are 0 or 1. (Thus $Z$ has $2^n$ elements, which are the vertices of a unit hypercube in $\mathbb{R}^n$.) Given a vector subspace $V$ of $\mathbb{R}^n$, let $Z(V)$ denote the number of members of $Z$ that lie in $V$. Let $k$ be given, $0 \leq k \leq n$... | 2^k |
Calculate the definite integral:
$$
\int_{0}^{\frac{\pi}{2}} \frac{\sin ^{2} x \, dx}{(1+\cos x+\sin x)^{2}}
$$ | \frac{1}{2} - \frac{1}{2} \ln 2 |
A permutation $(a_1,a_2,a_3,a_4,a_5)$ of $(1,2,3,4,5)$ is heavy-tailed if $a_1 + a_2 < a_4 + a_5$. What is the number of heavy-tailed permutations? | 48 |
The solution to the inequality
$$
(x-1)^{[\sqrt{1}]}(x-2)^{[\sqrt{2}]} \ldots(x-k)^{[\sqrt{k}]} \ldots(x-150)^{[\sqrt{150}]}<0
$$
is a union of several non-overlapping intervals. Find the sum of their lengths. If necessary, round the answer to the nearest 0.01.
Recall that $[x]$ denotes the greatest integer less tha... | 78.00 |
Given the parabola $y^2=2px$ ($p>0$) with focus $F(1,0)$, and the line $l: y=x+m$ intersects the parabola at two distinct points $A$ and $B$. If $0\leq m<1$, determine the maximum area of $\triangle FAB$. | \frac{8\sqrt{6}}{9} |
Alice and Bob each arrive at a meeting at a random time between 8:00 and 9:00 AM. If Alice arrives after Bob, what is the probability that Bob arrived before 8:45 AM? | \frac{9}{16} |
What is the largest prime factor of $3328$? | 13 |
Ben will receive a bonus of $1496. He chooses to allocate this amount as follows: 1/22 for the kitchen, 1/4 for holidays and 1/8 for Christmas gifts for his 3 children. How much money will he still have left after these expenses? | Ben's spending for the kitchen is $1496 x 1/22 = $<<1496*1/22=68>>68.
Ben's spending for holidays is $1496 x 1/4 = $<<1496*1/4=374>>374.
Ben's spending for children's gifts is $1496 x 1/8 = $<<1496*1/8=187>>187.
The total amount spent is $68 + $374 + $187 = $<<68+374+187=629>>629.
So, Ben still has $1496 - $629 = $<<14... |
Determine the value of $x$ that satisfies $\sqrt[5]{x\sqrt{x^3}}=3$. | 9 |
Given that $\binom{23}{5}=33649$, $\binom{23}{6}=42504$, and $\binom{23}{7}=33649$, find $\binom{25}{7}$. | 152306 |
Given that $\binom{24}{5}=42504$, and $\binom{24}{6}=134596$, find $\binom{26}{6}$. | 230230 |
I planned to work 25 hours a week for 15 weeks to earn $3750$ for a vacation. However, due to a family emergency, I couldn't work for the first three weeks. How many hours per week must I work for the remaining weeks to still afford the vacation? | 31.25 |
Find the minimum value of $m$ such that any $m$ -element subset of the set of integers $\{1,2,...,2016\}$ contains at least two distinct numbers $a$ and $b$ which satisfy $|a - b|\le 3$ . | 505 |
What is the smallest positive value of $m$ such that the equation $10x^2 - mx + 660 = 0$ has integral solutions? | 170 |
James has a total of 66 dollars in his piggy bank. He only has one dollar bills and two dollar bills in his piggy bank. If there are a total of 49 bills in James's piggy bank, how many one dollar bills does he have? | 32 |
If the integers \( a, b, \) and \( c \) satisfy:
\[
a + b + c = 3, \quad a^3 + b^3 + c^3 = 3,
\]
then what is the maximum value of \( a^2 + b^2 + c^2 \)? | 57 |
If $f(1) = 3$, $f(2)= 12$, and $f(x) = ax^2 + bx + c$, what is the value of $f(3)$? | 21 |
There are integers $a, b,$ and $c,$ each greater than $1,$ such that
\[\sqrt[a]{N\sqrt[b]{N\sqrt[c]{N}}} = \sqrt[36]{N^{25}}\]
for all $N \neq 1$. What is $b$? | 3 |
Suppose that a parabola has vertex $\left(\frac{1}{4},-\frac{9}{8}\right)$ and equation $y = ax^2 + bx + c$, where $a > 0$ and $a + b + c$ is an integer. Find the smallest possible value of $a.$ | \frac{2}{9} |
Find the maximum of
\[\sqrt{x + 27} + \sqrt{13 - x} + \sqrt{x}\]for $0 \le x \le 13.$ | 11 |
Let $D$ be the circle with equation $x^2 - 4y - 4 = -y^2 + 6x + 16$. Find the center $(c,d)$ and the radius $s$ of $D$, and compute $c+d+s$. | 5 + \sqrt{33} |
Given an arithmetic sequence $\{a_n\}$ with a common difference $d \neq 0$ and the first term $a_1 = d$, the sum of the first $n$ terms of the sequence $\{a_n^2\}$ is $S_n$. A geometric sequence $\{b_n\}$ has a common ratio $q$ less than 1 and consists of rational sine values, with the first term $b_1 = d^2$, and the s... | \frac{1}{2} |
Let $A,$ $R,$ $M,$ and $L$ be positive real numbers such that
\begin{align*}
\log_{10} (AL) + \log_{10} (AM) &= 2, \\
\log_{10} (ML) + \log_{10} (MR) &= 3, \\
\log_{10} (RA) + \log_{10} (RL) &= 4.
\end{align*}Compute the value of the product $ARML.$ | 1000 |
Given triangle PQR with PQ = 60 and PR = 20, the area is 240. Let M be the midpoint of PQ and N be the midpoint of PR. An altitude from P to side QR intersects MN and QR at X and Y, respectively. Find the area of quadrilateral XYMR. | 80 |
How many of the first $1000$ positive integers can be expressed in the form
\[\lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 6x \rfloor + \lfloor 8x \rfloor\]where $x$ is a real number, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$? | 600 |
V is the pyramidal region defined by the inequalities \( x, y, z \geq 0 \) and \( x + y + z \leq 1 \). Evaluate the integral:
\[ \int_V x y^9 z^8 (1 - x - y - z)^4 \, dx \, dy \, dz. \ | \frac{9! 8! 4!}{25!} |
If infinitely many values of $y$ satisfy the equation $2(4+cy) = 12y+8$, then what is the value of $c$? | 6 |
A metal bar with a temperature of $20{ }^{\circ} \mathrm{C}$ is placed into water that is initially at $80{ }^{\circ} \mathrm{C}$. After thermal equilibrium is reached, the temperature is $60{ }^{\circ} \mathrm{C}$. Without removing the first bar from the water, another metal bar with a temperature of $20{ }^{\circ} \m... | 50 |
There are seven cards in a hat, and on the card $k$ there is a number $2^{k-1}$ , $k=1,2,...,7$ . Solarin picks the cards up at random from the hat, one card at a time, until the sum of the numbers on cards in his hand exceeds $124$ . What is the most probable sum he can get? | 127 |
James has five huskies, two pitbulls and four golden retrievers, all female. They became pregnant and each golden retriever had two more pups than each husky. If the huskies and pitbulls had 3 pups each, How many more pups than adult dogs are there in total? | There are 5+2 = <<5+2=7>>7 huskies and pitbulls
7 huskies and pitbulls had 3 pups each for a total of 3*7 = <<7*3=21>>21 pups
Each golden retriever had 2 more pups each than the huskies who had 3 each so each golden retriever had 3+2 = <<3+2=5>>5 pups
4 golden retrievers had 5 pups each for a total of 20 pups
There are... |
A triline is a line with the property that three times its slope is equal to the sum of its \(x\)-intercept and its \(y\)-intercept. For how many integers \(q\) with \(1 \leq q \leq 10000\) is there at least one positive integer \(p\) so that there is exactly one triline through \((p, q)\)? | 57 |
Compute
\[\left( 1 + \cos \frac {\pi}{8} \right) \left( 1 + \cos \frac {3 \pi}{8} \right) \left( 1 + \cos \frac {5 \pi}{8} \right) \left( 1 + \cos \frac {7 \pi}{8} \right).\] | \frac{1}{8} |
On the first day of the journey, the Skipper sailed his ship halfway to the destination by traveling due east for 20 hours at a speed of 30 kilometers per hour, and then turned the ship's engines off to let them cool down. But while the engines were off, a wind storm blew his vessel backward in a westward direction. A... | Traveling 20 hours due east at 30 kilometers per hour, the ship sailed 20*30=<<20*30=600>>600 kilometers the first day.
If half the distance to the destination is 600 kilometers, then 2*600=<<2*600=1200>>1200 kilometers is the total distance from start to destination.
One-third of the distance from start to destination... |
The quantity
\[\frac{\tan \frac{\pi}{5} + i}{\tan \frac{\pi}{5} - i}\]is a tenth root of unity. In other words, it is equal to $\cos \frac{2n \pi}{10} + i \sin \frac{2n \pi}{10}$ for some integer $n$ between 0 and 9 inclusive. Which value of $n$? | 3 |
Suppose $105 \cdot 77 \cdot 132 \equiv m \pmod{25}$, where $0 \le m < 25$. | 20 |
In recent years, the awareness of traffic safety among citizens has gradually increased, leading to a greater demand for helmets. A certain store purchased two types of helmets, type A and type B. It is known that they bought 20 type A helmets and 30 type B helmets, spending a total of 2920 yuan. The unit price of type... | 1976 |
The domain of the function $q(x) = x^4 + 4x^2 + 4$ is $[0,\infty)$. What is the range? | [4,\infty) |
A boy has 12 oranges. He gives one-third of this number to his brother, one-fourth of the remainder to his friend and keeps the rest for himself. How many does his friend get? | He gives a third of 12 oranges to his brother which is 12*(1/3) = <<12*(1/3)=4>>4 oranges
He has 12-4 = <<12-4=8>>8 oranges left
He gives a fourth of 8 oranges to his friend which is 8*(1/4) = <<8*(1/4)=2>>2 oranges
#### 2 |
Find $2^{\frac{1}{2}} \cdot 4^{\frac{1}{4}} \cdot 8^{\frac{1}{8}} \cdot 16^{\frac{1}{16}} \dotsm.$ | 4 |
There are seven students taking a graduation photo in a row. Student A must stand in the middle, and students B and C must stand together. How many different arrangements are possible? | 192 |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $b\sin(C+\frac{π}{3})-c\sin B=0$.
$(1)$ Find the value of angle $C$.
$(2)$ If the area of $\triangle ABC$ is $10\sqrt{3}$ and $D$ is the midpoint of $AC$, find the minimum value of $BD$. | 2\sqrt{5} |
Solve for $x$ in the equation
\[2^{(16^x)} = 16^{(2^x)}.\] | \frac{2}{3} |
An ice cream cone has radius 1 inch and height 4 inches, What is the number of inches in the radius of a sphere of ice cream which has the same volume as the cone? | 1 |
A decagon is inscribed in a rectangle such that the vertices of the decagon divide each side of the rectangle into five equal segments. The perimeter of the rectangle is 160 centimeters, and the ratio of the length to the width of the rectangle is 3:2. What is the number of square centimeters in the area of the decagon... | 1413.12 |
Jenny is older than Charlie by five years, while Charlie is older than Bobby by three years. How old will Charlie be when Jenny becomes twice as old as Bobby? | Jenny is older than Charlie by 5 years who is older than Bobby by 3 years, so Jenny is older than Bobby by 5+3 = <<8=8>>8 years
Jenny will be twice as old as Bobby when he is 8*2 = 16 years old
Jenny is 5 years older than Charlie so when Jenny is 16, Charlie will be 16-5 = 11 years old
#### 11 |
John buys 3 t-shirts that cost $20 each. He also buys $50 in pants. How much does he spend? | He spends 20*3=$<<20*3=60>>60 on t-shirts
So he spends 60+50=$<<60+50=110>>110 in total
#### 110 |
Joan has 180 socks. Two thirds of the socks are white, and the rest of the socks are blue. How many blue socks does Joan have? | Joan has (180/3)*2 = <<(180/3)*2=120>>120 white socks.
The number of blue socks Joan has is 180-120 = <<180-120=60>>60.
#### 60 |
The diagram shows twenty congruent circles arranged in three rows and enclosed in a rectangle. The circles are tangent to one another and to the sides of the rectangle as shown in the diagram. The ratio of the longer dimension of the rectangle to the shorter dimension can be written as $\dfrac{1}{2}(\sqrt{p}-q)$ where ... | 154 |
A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is planning to offer a discount on the selling price, but the profit margin must not be less than $10\%$. Find the maximum discount rate that can be offered. | 12\% |
A train has five carriages, each containing at least one passenger. Two passengers are said to be 'neighbours' if either they are in the same carriage or they are in adjacent carriages. Each passenger has exactly five or exactly ten neighbours. How many passengers are there on the train? | 17 |
Ian used a grocery delivery app to have his groceries delivered. His original order was $25 before delivery and tip. He noticed that 3 items changed on his order. A $0.99 can of tomatoes was replaced by a $2.20 can of tomatoes, his $1.00 lettuce was replaced with $1.75 head of lettuce and his $1.96 celery was replac... | He ordered a can of tomatoes for $0.99 but they subbed with one that cost $2.20 for a difference of 2.20-.99 = $<<2.20-.99=1.21>>1.21
He ordered lettuce for $1.00 but they subbed with one that cost $1.75 for a difference of 1.75-1.00 = $<<1.75-1.00=0.75>>0.75
His celery for $1.96 but they subbed with one that cost $2.0... |
Let $T$ be the set of ordered triples $(x,y,z)$ of real numbers where
\[\log_{10}(2x+2y) = z \text{ and } \log_{10}(x^{2}+2y^{2}) = z+2.\]
Find constants $c$ and $d$ such that for all $(x,y,z) \in T$, the expression $x^{3} + y^{3}$ equals $c \cdot 10^{3z} + d \cdot 10^{z}.$ What is the value of $c+d$?
A) $\frac{1}{16}$... | \frac{5}{16} |
Two externally tangent circles $\omega_1$ and $\omega_2$ have centers $O_1$ and $O_2$, respectively. A third circle $\Omega$ passing through $O_1$ and $O_2$ intersects $\omega_1$ at $B$ and $C$ and $\omega_2$ at $A$ and $D$, as shown. Suppose that $AB = 2$, $O_1O_2 = 15$, $CD = 16$, and $ABO_1CDO_2$ is a convex hexagon... | 140 |
Find the area of triangle $JKL$ below.
[asy]
unitsize(1inch);
pair P,Q,R;
P = (0,0);
Q= (sqrt(3),0);
R = (0,1);
draw (P--Q--R--P,linewidth(0.9));
draw(rightanglemark(Q,P,R,3));
label("$J$",P,S);
label("$K$",Q,S);
label("$L$",R,N);
label("$20$",(Q+R)/2,NE);
label("$60^\circ$",(0,0.75),E);
[/asy] | 50\sqrt{3} |
As shown in the diagram, a cube with a side length of 12 cm is cut once. The cut is made along \( IJ \) and exits through \( LK \), such that \( AI = DL = 4 \) cm, \( JF = KG = 3 \) cm, and the section \( IJKL \) is a rectangle. The total surface area of the two resulting parts of the cube after the cut is \( \quad \) ... | 1176 |
Find the largest prime divisor of \( 16^2 + 81^2 \). | 53 |
Solve the congruence $11n \equiv 7 \pmod{43}$, as a residue modulo 43. (Give an answer between 0 and 42.) | 28 |
Given the mean of the data $x_1, x_2, \ldots, x_n$ is 2, and the variance is 3, calculate the mean and variance of the data $3x_1+5, 3x_2+5, \ldots, 3x_n+5$. | 27 |
What is the greatest integer less than 100 for which the greatest common factor of that integer and 18 is 3? | 93 |
Suppose $\sin N = \frac{2}{3}$ in the diagram below. What is $LN$?
[asy]
pair L,M,N;
M = (0,0);
N = (17.89,0);
L = (0,16);
draw(L--M--N--L);
draw(rightanglemark(L,M,N,18));
label("$M$",M,SW);
label("$N$",N,SE);
label("$L$",L,NE);
label("$16$",L/2,W);
[/asy] | 24 |
A circular cylindrical post has a circumference of 6 feet and a height of 18 feet. A string is wrapped around the post which spirals evenly from the bottom to the top, looping around the post exactly six times. What is the length of the string, in feet? | 18\sqrt{5} |
A rectangular photograph is placed in a frame that forms a border two inches wide on all sides of the photograph. The photograph measures $8$ inches high and $10$ inches wide. What is the area of the border, in square inches? | 88 |
In the Cartesian coordinate system $xOy$, the curve $C$ is given by $\frac{x^2}{4} + \frac{y^2}{3} = 1$. Taking the origin $O$ of the Cartesian coordinate system $xOy$ as the pole and the positive half-axis of $x$ as the polar axis, and using the same unit length, a polar coordinate system is established. It is known t... | 2\sqrt{5} |
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