problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
A regular hexagon $ABCDEF$ has sides of length 2. Find the area of $\bigtriangleup ADF$. Express your answer in simplest radical form. | 4\sqrt{3} |
In the fall of 1996, a total of 800 students participated in an annual school clean-up day. The organizers of the event expect that in each of the years 1997, 1998, and 1999, participation will increase by 50% over the previous year. The number of participants the organizers will expect in the fall of 1999 is | 2700 |
Calculate the volumes of the bodies bounded by the surfaces.
$$
z = 2x^2 + 18y^2, \quad z = 6
$$ | 6\pi |
Alex, Bonnie, and Chris each have $3$ blocks, colored red, blue, and green; and there are $3$ empty boxes. Each person independently places one of their blocks into each box. Each block placement by Bonnie and Chris is picked such that there is a 50% chance that the color matches the color previously placed by Alex or ... | \frac{37}{64} |
The equations of the asymptotes of the hyperbola $\frac{x^2}{2}-y^2=1$ are ________, and its eccentricity is ________. | \frac{\sqrt{6}}{2} |
Compute
\[\frac{(10^4+400)(26^4+400)(42^4+400)(58^4+400)}{(2^4+400)(18^4+400)(34^4+400)(50^4+400)}.\] | 962 |
Let $N$ be the number of positive integers that are less than or equal to $5000$ and whose base-$3$ representation has more $1$'s than any other digit. Find the remainder when $N$ is divided by $1000$. | 379 |
A person has 13 pieces of a gold chain containing 80 links. Separating one link costs 1 cent, and attaching a new one - 2 cents.
What is the minimum amount needed to form a closed chain from these pieces?
Remember, larger and smaller links must alternate. | 30 |
Each face of a regular tetrahedron is labeled with one of the numbers $1, 2, 3, 4$. Four identical regular tetrahedrons are simultaneously rolled onto a table. What is the probability that the product of the four numbers on the faces touching the table is divisible by 4? | $\frac{13}{16}$ |
On the island of Misfortune with a population of 96 people, the government decided to implement five reforms. Each reform is disliked by exactly half of the citizens. A citizen protests if they are dissatisfied with more than half of all the reforms. What is the maximum number of people the government can expect at th... | 80 |
Find the last three digits of $9^{105}.$ | 049 |
The circumcircle of acute $\triangle ABC$ has center $O$ . The line passing through point $O$ perpendicular to $\overline{OB}$ intersects lines $AB$ and $BC$ at $P$ and $Q$ , respectively. Also $AB=5$ , $BC=4$ , $BQ=4.5$ , and $BP=\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ . | 23 |
A store owner purchases merchandise at a discount of 30% off the original list price. To ensure a profit, the owner wants to mark up the goods such that after offering a 15% discount on the new marked price, the final selling price still yields a 30% profit compared to the cost price. What percentage of the original li... | 107\% |
Suppose that the graph of a certain function, $y=f(x)$, has the property that if it is shifted $20$ units to the right, then the resulting graph is identical to the original graph of $y=f(x)$.
What is the smallest positive $a$ such that if the graph of $y=f\left(\frac x5\right)$ is shifted $a$ units to the right, then... | 100 |
How many paths are there from point $A$ to point $B$ in a $7 \times 6$ grid, if every step must be up or to the right, and you must not pass through the cell at position $(3,3)$?
[asy]size(4cm,4cm);int w=7;int h=6;int i;pen p=fontsize(9);for (i=0; i<h; ++i){draw((0,i) -- (w-1,i));}for (i=0; i<w; ++i){draw((i, 0)--(i,h-... | 262 |
During the first eleven days, 700 people responded to a survey question. Each respondent chose exactly one of the three offered options. The ratio of the frequencies of each response was \(4: 7: 14\). On the twelfth day, more people participated in the survey, which changed the ratio of the response frequencies to \(6:... | 75 |
The four points $A(-4,0), B(0,-4), X(0,8),$ and $Y(14,k)$ are grouped on the Cartesian plane. If segment $AB$ is parallel to segment $XY$ what is the value of $k$? | -6 |
There are 88 dogs in a park. 12 of the dogs are running. Half of them are playing with toys. A fourth of them are barking. How many dogs are not doing anything? | 88/2 = <<88/2=44>>44 dogs are playing with toys.
88/4 = <<88/4=22>>22 dogs are barking.
88-12-44-22 = <<88-12-44-22=10>>10 dogs are not doing anything.
#### 10 |
Given that Jackie has $40$ thin rods, one of each integer length from $1 \text{ cm}$ through $40 \text{ cm}$, with rods of lengths $5 \text{ cm}$, $12 \text{ cm}$, and $20 \text{ cm}$ already placed on a table, find the number of the remaining rods that she can choose as the fourth rod to form a quadrilateral with posi... | 30 |
There are six identical red balls and three identical green balls in a pail. Four of these balls are selected at random and then these four balls are arranged in a line in some order. Find the number of different-looking arrangements of the selected balls. | 15 |
Given $\{a_{n}\}\left(n\in N*\right)$ is an arithmetic sequence with a common difference of $-2$, and $a_{6}$ is the geometric mean of $a_{2}$ and $a_{8}$. Let $S_{n}$ be the sum of the first $n$ terms of $\{a_{n}\}$. Find the value of $S_{10}$. | 90 |
A store prices an item so that when 5% sales tax is added to the price in cents, the total cost rounds naturally to the nearest multiple of 5 dollars. What is the smallest possible integer dollar amount $n$ to which the total cost could round?
A) $50$
B) $55$
C) $60$
D) $65$
E) $70$ | 55 |
The points $P,$ $Q,$ and $R$ are represented by the complex numbers $z,$ $(1 + i) z,$ and $2 \overline{z},$ respectively, where $|z| = 1.$ When $P,$ $Q$, and $R$ are not collinear, let $S$ be the fourth vertex of the parallelogram $PQSR.$ What is the maximum distance between $S$ and the origin of the complex plane? | 3 |
What is the greatest possible number of digits in the product of a 4-digit whole number and a 3-digit whole number? | 7 |
Let $a$ be the proportion of teams that correctly answered problem 1 on the Guts round. Estimate $A=\lfloor 10000a\rfloor$. An estimate of $E$ earns $\max (0,\lfloor 20-|A-E| / 20\rfloor)$ points. If you have forgotten, question 1 was the following: Two hexagons are attached to form a new polygon $P$. What is the minim... | 2539 |
How many perfect squares are divisors of the product $1! \cdot 2! \cdot 3! \cdot \hdots \cdot 9!$? | 672 |
A certain fruit store deals with two types of fruits, A and B. The situation of purchasing fruits twice is shown in the table below:
| Purchase Batch | Quantity of Type A Fruit ($\text{kg}$) | Quantity of Type B Fruit ($\text{kg}$) | Total Cost ($\text{元}$) |
|----------------|---------------------------------------|-... | 22 |
While waiting for their food at a restaurant in Harvard Square, Ana and Banana draw 3 squares $\square_{1}, \square_{2}, \square_{3}$ on one of their napkins. Starting with Ana, they take turns filling in the squares with integers from the set $\{1,2,3,4,5\}$ such that no integer is used more than once. Ana's goal is t... | \[ 541 \] |
In the Cartesian coordinate system $xOy$, the parametric equation of curve $C$ is $$\begin{cases} x=3\cos \theta \\ y=2\sin \theta \end{cases} (\theta \text{ is the parameter}),$$ and the parametric equation of the line $l$ is $$\begin{cases} x=t-1 \\ y=2t-a-1 \end{cases} (t \text{ is the parameter}).$$
(Ⅰ) If $a=1$,... | 2\sqrt{5}-2\sqrt{2} |
Let the complex number $z=-3\cos \theta + i\sin \theta$ (where $i$ is the imaginary unit).
(1) When $\theta= \frac {4}{3}\pi$, find the value of $|z|$;
(2) When $\theta\in\left[ \frac {\pi}{2},\pi\right]$, the complex number $z_{1}=\cos \theta - i\sin \theta$, and $z_{1}z$ is a pure imaginary number, find the value... | \frac {2\pi}{3} |
The inscribed circle of triangle $ABC$ is tangent to $\overline{AB}$ at $P,$ and its radius is $21$. Given that $AP=23$ and $PB=27,$ find the perimeter of the triangle.
| 345 |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 6.$ Find the minimum value of
\[\frac{x^2 + y^2}{x + y} + \frac{x^2 + z^2}{x + z} + \frac{y^2 + z^2}{y + z}.\] | 6 |
Given the value \(\left(\frac{11}{12}\right)^{2}\), determine the interval in which this value lies. | \frac{1}{2} |
In the middle of the school year, $40\%$ of Poolesville magnet students decided to transfer to the Blair magnet, and $5\%$ of the original Blair magnet students transferred to the Poolesville magnet. If the Blair magnet grew from $400$ students to $480$ students, how many students does the Poolesville magnet ha... | 170 |
Simplify the following expression:
$$5x + 6 - x + 12$$ | 4x + 18 |
If $\left(2x-a\right)^{7}=a_{0}+a_{1}(x+1)+a_{2}(x+1)^{2}+a_{3}(x+1)^{3}+\ldots +a_{7}(x+1)^{7}$, and $a_{4}=-560$.<br/>$(1)$ Find the value of the real number $a$;<br/>$(2)$ Find the value of $|a_{1}|+|a_{2}|+|a_{3}|+\ldots +|a_{6}|+|a_{7}|$. | 2186 |
Suppose that $a$ is inversely proportional to $b$. Let $a_1,a_2$ be two nonzero values of $a$ such that $\frac{a_1}{a_2}=\frac{2}{3}$. Let the corresponding $b$ values be $b_1,b_2$. If $b_1,b_2$ are nonzero, find the value of $\frac{b_1}{b_2}$. | \frac{3}{2} |
Suppose
\[\frac{1}{x^3 - 3x^2 - 13x + 15} = \frac{A}{x+3} + \frac{B}{x-1} + \frac{C}{(x-1)^2}\]
where $A$, $B$, and $C$ are real constants. What is $A$? | \frac{1}{16} |
Concentric circles $\Omega_1$ and $\Omega_2$ with radii $1$ and $100$ , respectively, are drawn with center $O$ . Points $A$ and $B$ are chosen independently at random on the circumferences of $\Omega_1$ and $\Omega_2$ , respectively. Denote by $\ell$ the tangent line to $\Omega_1$ passing through $... | 10004 |
Person A and person B start simultaneously from points A and B, respectively, and move towards each other. When person A reaches the midpoint C of A and B, person B is still 240 meters away from point C. When person B reaches point C, person A has already moved 360 meters past point C. What is the distance between poin... | 144 |
Find the product of the roots of the equation \[(2x^3 + x^2 - 8x + 20)(5x^3 - 25x^2 + 19) = 0.\] | 38 |
Chelsea made 4 batches of cupcakes for the bake sale. The cupcakes took 20 minutes to bake and 30 minutes to ice per batch. How long did it take Chelsea to make the cupcakes? | Chelsea baked 4 batches of cupcakes x 20 minutes each = <<4*20=80>>80 minutes of baking.
She iced 4 batches of cupcakes x 30 minutes each = <<4*30=120>>120 minutes.
It took Chelsea 80 + 120 = <<80+120=200>>200 minutes to make the cupcakes for the bake sale.
#### 200 |
If the sum of the squares of nonnegative real numbers $a,b,$ and $c$ is $39$, and $ab + bc + ca = 21$, then what is the sum of $a,b,$ and $c$? | 9 |
How many integer solutions does the inequality
$$
|x| + |y| < 1000
$$
have, where \( x \) and \( y \) are integers? | 1998001 |
The ratio of the magnitudes of two angles of a triangle is 2, and the difference in lengths of the sides opposite these angles is 2 cm; the length of the third side of the triangle is 5 cm. Calculate the area of the triangle. | \frac{15 \sqrt{7}}{4} |
Given the set $X=\left\{1,2,3,4\right\}$, consider a function $f:X\to X$ where $f^1=f$ and $f^{k+1}=\left(f\circ f^k\right)$ for $k\geq1$. Determine the number of functions $f$ that satisfy $f^{2014}\left(x\right)=x$ for all $x$ in $X$. | 13 |
Ryan got $80\%$ of the problems correct on a $25$-problem test, $90\%$ on a $40$-problem test, and $70\%$ on a $10$-problem test. What percent of all the problems did Ryan answer correctly? | 84 |
Let $f(x) = 4x - 9$ and $g(f(x)) = x^2 + 6x - 7$. Find $g(-8)$. | \frac{-87}{16} |
Kelvin the Frog is hopping on a number line (extending to infinity in both directions). Kelvin starts at 0. Every minute, he has a $\frac{1}{3}$ chance of moving 1 unit left, a $\frac{1}{3}$ chance of moving 1 unit right and $\frac{1}{3}$ chance of getting eaten. Find the expected number of times Kelvin returns to 0 (n... | \frac{3\sqrt{5}-5}{5} |
Two regular tetrahedrons $A$ and $B$ are made with the 8 vertices of a unit cube. (this way is unique)
What's the volume of $A\cup B$ ? | 1/2 |
Suraya picked 12 apples more than Caleb, and Caleb picked 5 apples less than Kayla. If Kayla picked 20 apples, how many more apples did Suraya pick than Kayla? | Caleb picked 20 - 5 = <<20-5=15>>15 apples.
Suraya picked 12 + 15 = <<12+15=27>>27 apples.
Suraya picked 27 - 20 = <<27-20=7>>7 apples more than Kayla
#### 7 |
Five cards have the numbers 101, 102, 103, 104, and 105 on their fronts. On the reverse, each card has one of five different positive integers: \(a, b, c, d,\) and \(e\) respectively. We know that \(a + 2 = b - 2 = 2c = \frac{d}{2} = e^2\).
Gina picks up the card which has the largest integer on its reverse. What numb... | 105 |
Find the smallest natural number $A$ that satisfies the following conditions:
a) Its notation ends with the digit 6;
b) By moving the digit 6 from the end of the number to its beginning, the number increases fourfold. | 153846 |
Find the maximum value of the real constant $C$ such that $x^{2}+y^{2}+1\geq C(x+y)$ , and $ x^{2}+y^{2}+xy+1\geq C(x+y)$ for all reals $x,y$ . | \sqrt{2} |
For the consumer, a single discount of $n\%$ is more advantageous than any of the following discounts:
(1) two successive $15\%$ discounts
(2) three successive $10\%$ discounts
(3) a $25\%$ discount followed by a $5\%$ discount
What is the smallest possible positive integer value of $n$? | 29 |
Carmen burns a candle for 1 hour every night. A candle will last her 8 nights. If she burns the candle for 2 hours a night, how many candles will she use over 24 nights? | Burning a candle for 1 hour lasts her 8 nights so if she burns for 2 hours it will last her 8/2 = <<1*8/2=4>>4 nights
1 candle now lasts 4 nights and she needs enough for 24 nights so she needs 24/4 = <<24/4=6>>6 candles
#### 6 |
A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the side... | 2(w+h)^2 |
What is the value of $\left(\frac{2}{3}\right)\left(\frac{3}{4}\right)\left(\frac{4}{5}\right)\left(\frac{5}{6}\right)$? Express your answer as a common fraction. | \frac{1}{3} |
Let $PQRS$ be an isosceles trapezoid with bases $PQ=120$ and $RS=25$. Suppose $PR=QS=y$ and a circle with center on $\overline{PQ}$ is tangent to segments $\overline{PR}$ and $\overline{QS}$. If $n$ is the smallest possible value of $y$, then $n^2$ equals what? | 2850 |
A kindergarten received cards for learning to read: some are labeled "МА", and the rest are labeled "НЯ".
Each child took three cards and started to form words from them. It turned out that 20 children could form the word "МАМА" from their cards, 30 children could form the word "НЯНЯ", and 40 children could form the w... | 10 |
Given that 3 people are to be selected from 5 girls and 2 boys, if girl A is selected, determine the probability that at least one boy is selected. | \frac{3}{5} |
In right triangle $ABC$ with $\angle B = 90^\circ$, sides $AB=1$ and $BC=3$. The bisector of $\angle BAC$ meets $\overline{BC}$ at $D$. Calculate the length of segment $BD$.
A) $\frac{1}{2}$
B) $\frac{3}{4}$
C) $1$
D) $\frac{5}{4}$
E) $2$ | \frac{3}{4} |
During the weekends, Eli delivers milk in the complex plane. On Saturday, he begins at $z$ and delivers milk to houses located at $z^{3}, z^{5}, z^{7}, \ldots, z^{2013}$, in that order; on Sunday, he begins at 1 and delivers milk to houses located at $z^{2}, z^{4}, z^{6}, \ldots, z^{2012}$, in that order. Eli always wa... | \frac{1005}{1006} |
Jake can wash his car with 1 bottle of car wash soap 4 times. If each bottle costs $4.00, and he washes his car once a week for 20 weeks, how much does he spend on car soap? | 1 bottle of soap will last for 4 washes and he needs enough bottles for 20 weeks so 20/4 = <<20/4=5>>5 bottles
Each bottle cost $4.00 and he needs 5 bottles so he will spend $4*5 = $<<4*5=20.00>>20.00 on car soap
#### 20 |
The planet Xavier follows an elliptical orbit with its sun at one focus. At its nearest point (perigee), it is 2 astronomical units (AU) from the sun, while at its furthest point (apogee) it is 12 AU away. When Xavier is midway along its orbit, as shown, how far is it from the sun, in AU?
[asy]
unitsize(1 cm);
path... | 7 |
Let \(\mathbf{v}\) be a vector such that
\[
\left\| \mathbf{v} + \begin{pmatrix} 4 \\ 2 \end{pmatrix} \right\| = 10.
\]
Find the smallest possible value of \(\|\mathbf{v}\|\). | 10 - 2\sqrt{5} |
To buy a book, you pay $20 for each of the first 5 books at the supermarket, and for each additional book you buy over $20, you receive a discount of $2. If Beatrice bought 20 books, how much did she pay at the supermarket? | For the first five books, the total cost is $20*5 = $<<20*5=100>>100
For every additional book over $5, you pay $2 less, totaling to $20-$2 = $18
Beatrice bought 20 books, so the number of books at which she received the $2 discount is 20-5 = <<20-5=15>>15 books.
For the 15 books, Beatrice paid 15*$18 = $<<15*18=270>>2... |
Circles $A,B$, and $C$ are externally tangent to each other and internally tangent to circle $D$. Circles $B$ and $C$ are congruent. Circle $A$ has radius 1 and passes through the center of $D$. What is the radius of circle $B$?
[asy]unitsize(1cm);
pair A,B,C,D;
A=(-1,0);
B=(0.66,0.88);
C=(0.66,-0.88);
D=(0,0);
draw(C... | \frac{8}{9} |
In $\triangle ABC$, $P$ is a point on the side $BC$ such that $\overrightarrow{BP} = \frac{1}{2}\overrightarrow{PC}$. Points $M$ and $N$ lie on the line passing through $P$ such that $\overrightarrow{AM} = \lambda \overrightarrow{AB}$ and $\overrightarrow{AN} = \mu \overrightarrow{AC}$ where $\lambda, \mu > 0$. Find th... | \frac{8}{3} |
Let $\theta$ be the angle between the planes $2x + y - 2z + 3 = 0$ and $6x + 3y + 2z - 5 = 0.$ Find $\cos \theta.$ | \frac{11}{21} |
Points $P$ and $Q$ are on a circle with radius $7$ and $PQ = 8$. Point $R$ is the midpoint of the minor arc $PQ$. Calculate the length of line segment $PR$. | \sqrt{32} |
If $f(x)=\frac{x(x-1)}{2}$, then $f(x+2)$ equals: | \frac{(x+2)f(x+1)}{x} |
Side $\overline{AB}$ of $\triangle ABC$ has length $10$. The bisector of angle $A$ meets $\overline{BC}$ at $D$, and $CD = 3$. The set of all possible values of $AC$ is an open interval $(m,n)$. What is $m+n$? | 18 |
Padma is trading cards with Robert. Padma started with 75 cards and traded 2 or her valuable ones for 10 of Robert's cards. Robert started with 88 of his own cards and traded another 8 of his cards for 15 of Padma's cards. How many cards were traded between both Padma and Robert? | Padma traded 2 + 15 of her cards = <<2+15=17>>17 cards.
Robert traded 10 + 8 of his cards = <<10+8=18>>18 cards.
In total, between both of them Padma and Robert traded 17 + 18 = <<17+18=35>>35 cards.
#### 35 |
Camden went swimming 16 times in March and Susannah went 24 times. If the number of times they went throughout the month was divided equally among 4 weeks, how many more times a week did Susannah swim than Camden? | Camden went swimming 16/4 = <<16/4=4>>4 times a week
Susannah went swimming 24/4 = <<24/4=6>>6 times a week
Susannah went 6 - 4 = <<6-4=2>>2 more times a week than Camden
#### 2 |
Let $n$ represent the smallest integer that satisfies the following conditions:
$\frac n2$ is a perfect square.
$\frac n3$ is a perfect cube.
$\frac n5$ is a perfect fifth.
How many divisors does $n$ have that are not multiples of 10?
| 242 |
Given that when $(a+b+c+d+e+1)^N$ is expanded and like terms are combined, the resulting expression contains exactly 2002 terms that include all five variables $a, b, c, d, e$, each to some positive power, find the value of $N$. | 16 |
The median to a 10 cm side of a triangle has length 9 cm and is perpendicular to a second median of the triangle. Find the exact value in centimeters of the length of the third median. | 3\sqrt{13} |
$361+2(19)(6)+36=x$. Solve for $x$. | 625 |
Given $3\sin \left(-3\pi +\theta \right)+\cos \left(\pi -\theta \right)=0$, then the value of $\frac{sinθcosθ}{cos2θ}$ is ____. | -\frac{3}{8} |
Point $A$ lies at $(0,4)$ and point $B$ lies at $(3,8)$. Find the $x$-coordinate of the point $X$ on the $x$-axis maximizing $\angle AXB$. | 5 \sqrt{2}-3 |
Given the set $M = \{1,2,3,4\}$, let $A$ be a subset of $M$. The product of all elements in set $A$ is called the "cumulative value" of set $A$. It is stipulated that if set $A$ has only one element, its cumulative value is the value of that element, and the cumulative value of the empty set is 0. Find the number of su... | 13 |
Given that $| \overrightarrow{a}|=3 \sqrt {2}$, $| \overrightarrow{b}|=4$, $\overrightarrow{m}= \overrightarrow{a}+ \overrightarrow{b}$, $\overrightarrow{n}= \overrightarrow{a}+λ \overrightarrow{b}$, $ < \overrightarrow{a}, \overrightarrow{b} > =135^{\circ}$, if $\overrightarrow{m} \perp \overrightarrow{n}$, find the v... | -\frac{3}{2} |
Forty slips of paper numbered $1$ to $40$ are placed in a hat. Alice and Bob each draw one number from the hat without replacement, keeping their numbers hidden from each other. Alice says, "I can't tell who has the larger number." Then Bob says, "I know who has the larger number." Alice says, "You do? Is your number p... | 27 |
A thief on a bus gets off at a bus stop and walks in the direction opposite to the bus’s travel direction. The bus continues its journey, and a passenger realizes they have been robbed. The passenger gets off at the next stop and starts chasing the thief. If the passenger's speed is twice that of the thief, the bus's s... | 440 |
A right cylinder with a height of 8 inches is enclosed inside another cylindrical shell of the same height but with a radius 1 inch greater than the inner cylinder. The radius of the inner cylinder is 3 inches. What is the total surface area of the space between the two cylinders, in square inches? Express your answer ... | 16\pi |
Let $n \geq 5$ be an integer. Find the largest integer $k$ (as a function of $n$ ) such that there exists a convex $n$ -gon $A_{1}A_{2}\dots A_{n}$ for which exactly $k$ of the quadrilaterals $A_{i}A_{i+1}A_{i+2}A_{i+3}$ have an inscribed circle. (Here $A_{n+j} = A_{j}$ .) | \[
k = \left\lfloor \frac{n}{2} \right\rfloor
\] |
In $\triangle ABC$, it is given that $BD:DC = 3:2$ and $AE:EC = 3:4$. Point $M$ is the intersection of $AD$ and $BE$. If the area of $\triangle ABC$ is 1, what is the area of $\triangle BMD$? | $\frac{4}{15}$ |
In the expansion of the binomial ${(\sqrt{x}-\frac{1}{{2x}}})^n$, only the coefficient of the 4th term is the largest. The constant term in the expansion is ______. | \frac{15}{4} |
Edmonton is 220 kilometers north of Red Deer. Calgary is 110 kilometers south of Red Deer. If you travel at 110 kilometers per hour, how long will it take to get from Edmonton to Calgary? | Total distance travelled is 220km + 110km = <<220+110=330>>330 km.
Time to travel that far is 330km / 110km/hour = <<330/110=3>>3 hours.
#### 3 |
What is the length of side $y$ in the following diagram?
[asy]
import olympiad;
draw((0,0)--(2,0)--(0,2*sqrt(3))--cycle); // modified triangle lengths
draw((0,0)--(-2,0)--(0,2*sqrt(3))--cycle);
label("10",(-1,2*sqrt(3)/2),NW); // changed label
label("$y$",(2/2,2*sqrt(3)/2),NE);
draw("$30^{\circ}$",(2.5,0),NW); // modi... | 10\sqrt{3} |
Given $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=2$, and $\overrightarrow{a}⊥(\overrightarrow{a}+\overrightarrow{b})$, find the projection of $\overrightarrow{a}$ onto $\overrightarrow{b}$. | -\frac{1}{2} |
Calculate the limit of the function:
$$\lim_{x \rightarrow \frac{1}{3}} \frac{\sqrt[3]{\frac{x}{9}}-\frac{1}{3}}{\sqrt{\frac{1}{3}+x}-\sqrt{2x}}$$ | -\frac{2 \sqrt{2}}{3 \sqrt{3}} |
What is the sum of all the positive divisors of 91? | 112 |
Let $a,$ $b,$ $c$ be nonzero real numbers. Find the number of real roots of the equation
\[\begin{vmatrix} x & c & -b \\ -c & x & a \\ b & -a & x \end{vmatrix} = 0.\] | 1 |
Ivan Petrovich wants to save money for his retirement in 12 years. He decided to deposit 750,000 rubles in a bank account with an 8 percent annual interest rate. What will be the total amount in the account by the time Ivan Petrovich retires, assuming the interest is compounded annually using the simple interest formul... | 1470000 |
For the set \( \{1, 2, 3, \ldots, 8\} \) and each of its non-empty subsets, define a unique alternating sum as follows: arrange the numbers in the subset in decreasing order and alternately add and subtract successive numbers. For instance, the alternating sum for \( \{1, 3, 4, 7, 8\} \) would be \( 8-7+4-3+1=3 \) and ... | 1024 |
Given an ellipse $C:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with left and right foci $F_{1}$, $F_{2}$. There exists a point $M$ in the first quadrant of ellipse $C$ such that $|MF_{1}|=|F_{1}F_{2}|$. Line $F_{1}M$ intersects the $y$-axis at point $A$, and $F_{2}A$ bisects $\angle MF_{2}F_{1}$. Find the eccentricity o... | \frac{\sqrt{5} - 1}{2} |
A cowboy is 5 miles north of a stream which flows due west. He is also 10 miles east and 6 miles south of his cabin. He wishes to water his horse at the stream and then return home. Determine the shortest distance he can travel to accomplish this.
A) $5 + \sqrt{256}$ miles
B) $5 + \sqrt{356}$ miles
C) $11 + \sqrt{356}$... | 5 + \sqrt{356} |
What is the sum of the positive factors of 48? | 124 |
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