problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
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Alex has 12 friends and 63 coins. What is the minimum number of additional coins he needs so that he can give each friend at least one coin and no two friends receive the same number of coins? | 15 |
Given $f\left(x\right)=e^{x}-ax+\frac{1}{2}{x}^{2}$, where $a \gt -1$.<br/>$(Ⅰ)$ When $a=0$, find the equation of the tangent line to the curve $y=f\left(x\right)$ at the point $\left(0,f\left(0\right)\right)$;<br/>$(Ⅱ)$ When $a=1$, find the extreme values of the function $f\left(x\right)$;<br/>$(Ⅲ)$ If $f(x)≥\frac{1}{... | 1 + \frac{1}{e} |
There are 12 carpets in house 1, 20 carpets in house 2, and 10 carpets in house 3. If house 4 has twice as many carpets as house 3, how many carpets do all 4 houses have in total? | House 4 has 2 * 10 house 3 carpets = <<2*10=20>>20 carpets
The total number of carpets across all houses is 12 + 20 + 10 + 20 = <<12+20+10+20=62>>62
#### 62 |
Two circles with a radius of 15 cm overlap such that each circle passes through the center of the other. Determine the length of the common chord (dotted segment) in centimeters between these two circles. Express your answer in simplest radical form. | 15\sqrt{3} |
How many even three-digit integers have the property that their digits, read left to right, are in strictly increasing order (each digit is greater than the previous digit)? | 34 |
Dr. Math's four-digit house number $WXYZ$ contains no zeroes and can be split into two different two-digit primes ``$WX$'' and ``$YZ$'' where the digits $W$, $X$, $Y$, and $Z$ are not necessarily distinct. If each of the two-digit primes is less than 60, how many such house numbers are possible? | 156 |
Kennedy’s car can drive 19 miles per gallon of gas. She was able to drive 15 miles to school, 6 miles to the softball park, 2 miles to a burger restaurant, 4 miles to her friend’s house, and 11 miles home before she ran out of gas. How many gallons of gas did she start with? | Kennedy drove 15 + 6 + 2 + 4 + 11 = <<15+6+2+4+11=38>>38 miles.
She gets 19 miles per gallon of gas, so she started with 38 / 19 = <<38/19=2>>2 gallons of gas.
#### 2 |
There are 10 rows of 15 chairs set up for the award ceremony. The first row is reserved for the awardees while the second and third rows are for the administrators and teachers. The last two rows are then reserved for the parents and the rest of the rows are for the students. If only 4/5 of the seats reserved for the s... | There are 1 + 2 + 2 = <<1+2+2=5>>5 rows that are not reserved for the students.
Hence, 10 - 5 = <<10-5=5>>5 rows are reserved for the students.
That is equal to 5 x 15 = <<5*15=75>>75 seats.
But only 75 x 4/5 = <<75*4/5=60>>60 seats are occupied by the students.
Therefore, 75 - 60 = <<75-60=15>>15 seats can be given to... |
Find the matrix $\mathbf{M}$ that swaps the columns of a matrix. In other words,
\[\mathbf{M} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} b & a \\ d & c \end{pmatrix}.\]If no such matrix $\mathbf{M}$ exists, then enter the zero matrix. | \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} |
$ABCD$ is a rectangle. $E$ is a point on $AB$ between $A$ and $B$ , and $F$ is a point on $AD$ between $A$ and $D$ . The area of the triangle $EBC$ is $16$ , the area of the triangle $EAF$ is $12$ and the area of the triangle $FDC$ is 30. Find the area of the triangle $EFC$ . | 38 |
Aaron wants to purchase a guitar under a payment plan of $100.00 per month for 12 months. His father has agreed to lend him the entire amount for the guitar, plus a one-time 10% interest fee for the entire loan. With interest, how much money will Aaron owe his dad? | The guitar costs $100 every month for 12 months so it costs 100*12 = $<<100*12=1200>>1200
His dad is charging him 10% interest on the $1200 so interest comes to .10*1200 = $<<1200*.10=120.00>>120.00
All total, Aaron will owe his father the price of the guitar, $1200 and the interest, $120 so 1200+120 = $<<1200+120=1320... |
The numbers \(1, 2, 3, \ldots, 10\) are written in some order around a circle. Peter computed the sums of all 10 triples of neighboring numbers and wrote the smallest of these sums on the board. What is the maximum possible number that could have been written on the board? | 15 |
The teacher wrote a two-digit number on the board. Each of the three boys made two statements.
- Andrey: "This number ends in the digit 6" and "This number is divisible by 7."
- Borya: "This number is greater than 26" and "This number ends in the digit 8."
- Sasha: "This number is divisible by 13" and "This number is ... | 91 |
It costs 2.5 cents to copy a page. How many pages can you copy for $\$20$? | 800 |
Austin bought his seven friends each a robot. Each robot costs $8.75. He was charged $7.22 total for tax. He left with $11.53 in change. How much did Austin start with? | First we need to find the total amount for the robots. 7 robots * $8.75 per robot = $<<7*8.75=61.25>>61.25 in total.
Now we need to add in the tax $61.25 for the robots + $7.22 for tax = $<<61.25+7.22=68.47>>68.47 total spent in store.
Now to find what was the beginning amount we take the $68.47 total spent in-store + ... |
Let $ ABC$ be a triangle with $ AB \equal{} AC$ . The angle bisectors of $ \angle C AB$ and $ \angle AB C$ meet the sides $ B C$ and $ C A$ at $ D$ and $ E$ , respectively. Let $ K$ be the incentre of triangle $ ADC$. Suppose that $ \angle B E K \equal{} 45^\circ$ . Find all possible values of $ \angle C AB$ .
[i]Jan ... | 60^\circ \text{ and } 90^\circ |
Find $t$ such that $(t,5)$ lies on the line through $(0,3)$ and $(-8,0)$. | \frac{16}{3} |
Cindy was asked by her teacher to subtract 3 from a certain number and then divide the result by 9. Instead, she subtracted 9 and then divided the result by 3, giving an answer of 43. What would her answer have been had she worked the problem correctly? | 15 |
Simplify: $(\sqrt{5})^4$. | 25 |
A cell phone plan costs $20$ dollars each month, plus $5$ cents per text message sent, plus $10$ cents for each minute used over $30$ hours. In January Michelle sent $100$ text messages and talked for $30.5$ hours. How much did she have to pay? | 28.00 |
Given a sequence $\left\{a_{n}\right\}$, where $a_{1}=a_{2}=1$, $a_{3}=-1$, and $a_{n}=a_{n-1} a_{n-3}$, find $a_{1964}$. | -1 |
Given that the vertex of the parabola C is O(0,0), and the focus is F(0,1).
(1) Find the equation of the parabola C;
(2) A line passing through point F intersects parabola C at points A and B. If lines AO and BO intersect line l: y = x - 2 at points M and N respectively, find the minimum value of |MN|. | \frac {8 \sqrt {2}}{5} |
$P(x)=ax^2+bx+c$ has exactly $1$ different real root where $a,b,c$ are real numbers. If $P(P(P(x)))$ has exactly $3$ different real roots, what is the minimum possible value of $abc$ ? | -2 |
Roger initially has 20 socks in a drawer, each of which is either white or black. He chooses a sock uniformly at random from the drawer and throws it away. He repeats this action until there are equal numbers of white and black socks remaining. Suppose that the probability he stops before all socks are gone is $p$. If ... | 20738 |
The numeral $65$ in base $c$ represents the same number as $56$ in base $d$. Assuming that both $c$ and $d$ are positive integers, find the least possible value of $c+d$. | 13 |
For the line $l_1: ax - y - a + 2 = 0$ (where $a \in \mathbb{R}$), the line $l_2$ passing through the origin $O$ is perpendicular to $l_1$, and the foot of the perpendicular from $O$ is $M$. Then, the maximum value of $|OM|$ is ______. | \sqrt{5} |
James gets paid $0.50/mile to drive a truck carrying hazardous waste. He has to pay $4.00/gallon for gas and his truck gets 20 miles per gallon. How much profit does he make from a 600 mile trip? | First find the total payment James receives: $0.50/mile * 600 miles = $<<0.50*600=300>>300
Then find how many gallons of gas he needs to buy: 600 miles / 20 miles/gallon = <<600/20=30>>30 gallons
Then multiply that number by the price of gas per gallon to find the total cost of the gas: $4.00/gallon * 30 gallons = $<<4... |
When a certain biased coin is flipped five times, the probability of getting heads exactly once is not equal to $0$ and is the same as that of getting heads exactly twice. Let $\frac ij$, in lowest terms, be the probability that the coin comes up heads in exactly $3$ out of $5$ flips. Find $i+j$.
| 283 |
In a certain competition, the rules are as follows: among the 5 questions preset by the organizer, if a contestant can answer two consecutive questions correctly, they will stop answering and advance to the next round. Assuming the probability of a contestant correctly answering each question is 0.8, and the outcomes o... | 0.128 |
Given \( m > n \geqslant 1 \), find the smallest value of \( m + n \) such that
\[ 1000 \mid 1978^{m} - 1978^{n} . \ | 106 |
Given circle $O$, points $E$ and $F$ are on the same side of diameter $\overline{AB}$, $\angle AOE = 60^\circ$, and $\angle FOB = 90^\circ$. Calculate the ratio of the area of the smaller sector $EOF$ to the area of the circle. | \frac{1}{12} |
If α is in the interval (0, π) and $\frac{1}{2}\cos2α = \sin\left(\frac{π}{4} + α\right)$, then find the value of $\sin2α$. | -1 |
A pipe with inside diameter 10'' is to carry water from a reservoir to a small town in an arid land. Neglecting the friction and turbulence of the water against the inside of the pipes, what is the minimum number of 2''-inside-diameter pipes of the same length needed to carry the same volume of water to the arid town? | 25 |
Let $\zeta=\cos \frac{2 \pi}{13}+i \sin \frac{2 \pi}{13}$. Suppose $a>b>c>d$ are positive integers satisfying $$\left|\zeta^{a}+\zeta^{b}+\zeta^{c}+\zeta^{d}\right|=\sqrt{3}$$ Compute the smallest possible value of $1000 a+100 b+10 c+d$. | 7521 |
Points $A,B,C$ and $D$ lie on a line, in that order, with $AB = CD$ and $BC = 12$. Point $E$ is not on the line, and $BE = CE = 10$. The perimeter of $\triangle AED$ is twice the perimeter of $\triangle BEC$. Find $AB$. | 9 |
Let $z$ be a complex number such that $|z| = 3.$ Find the largest possible distance between $(1 + 2i)z^3$ and $z^4$ when plotted in the complex plane. | 216 |
How many non-congruent squares can be drawn, such that their vertices are lattice points on the 5 by 5 grid of lattice points shown? [asy]
dot((0,0));dot((1,0));dot((2,0));dot((3,0));dot((4,0));
dot((0,1));dot((1,1));dot((2,1));dot((3,1));dot((4,1));
dot((0,2));dot((1,2));dot((2,2));dot((3,2));dot((4,2));
dot((0,3));do... | 8 |
Find the product of all positive integral values of $x$ such that $x^2 - 40x + 399 = q$ for some prime number $q$. Note that there must be at least one such $x$. | 396 |
Egor, Nikita, and Innokentiy took turns playing chess with each other (two play, one watches). After each game, the loser gave up their place to the spectator (there were no draws). As a result, Egor participated in 13 games, and Nikita participated in 27 games. How many games did Innokentiy play? | 14 |
Let $S$ be the set of all positive integer divisors of $100,000.$ How many numbers are the product of two distinct elements of $S?$ | 117 |
Consider the set of 30 parabolas defined as follows: all parabolas have as focus the point $(0,0)$ and the directrix lines have the form $y=ax+b$ with $a$ and $b$ integers such that $a\in \{-2,-1,0,1,2\}$ and $b\in \{-3,-2,-1,1,2,3\}$. No three of these parabolas have a common point. How many points in the plane are on... | 810 |
Find the maximum value of the following expression:
$$
|\cdots|\left|x_{1}-x_{2}\right|-x_{3}\left|-\cdots-x_{1990}\right|,
$$
where \( x_{1}, x_{2}, \cdots, x_{1990} \) are distinct natural numbers from 1 to 1990. | 1989 |
Unconventional dice are to be designed such that the six faces are marked with numbers from $1$ to $6$ with $1$ and $2$ appearing on opposite faces. Further, each face is colored either red or yellow with opposite faces always of the same color. Two dice are considered to have the same design if one of them can... | 48 |
Suppose $f(z)$ and $g(z)$ are polynomials in $z$, and the degree of $g(z)$ is less than the degree of $f(z)$. If the degree of $f(z)$ is two, what is the degree of $f(z)+g(z)$? | 2 |
A triangle with sides of 5, 12, and 13 has both an inscribed and a circumscribed circle. What is the distance between the centers of those circles? | \frac{\sqrt{65}}{2} |
Madeline wants to drink 100 ounces of water in a day. Her water bottle can hold 12 ounces of water. She refills her water bottle 7 times. How much more water does she need to drink? | She has already had 12*7 =<<12*7=84>>84 ounces of water.
She needs to drink 100-84 = <<100-84=16>>16 more ounces to make it to 100 ounces.
#### 16 |
Find $x$ such that $\lfloor x \rfloor + x = \dfrac{13}{3}$. Express $x$ as a common fraction. | \dfrac{7}{3} |
Find $r$ such that $\log_{81} (2r-1) = -1/2$. | \frac{5}{9} |
The pattern of Pascal's triangle is illustrated in the diagram shown. What is the fourth element in Row 15 of Pascal's triangle? $$
\begin{array}{ccccccccccccc}\vspace{0.1in}
\textrm{Row 0}: & \qquad & & & & & 1 & & & & & & \\ \vspace{0.1in}
\textrm{Row 1}: & \qquad & & & & 1 & & 1 & & & & &\\ \vspace{0.1in}
\textrm{R... | 455 |
Place four balls numbered 1, 2, 3, and 4 into three boxes labeled A, B, and C.
(1) If none of the boxes are empty and ball number 3 must be in box B, how many different arrangements are there?
(2) If ball number 1 cannot be in box A and ball number 2 cannot be in box B, how many different arrangements are there? | 36 |
In $\triangle ABC$, $A=30^{\circ}$, $2 \overrightarrow{AB}\cdot \overrightarrow{AC}=3 \overrightarrow{BC}^{2}$, find the cosine value of the largest angle in $\triangle ABC$. | -\frac{1}{2} |
A trapezium is given with parallel bases having lengths $1$ and $4$ . Split it into two trapeziums by a cut, parallel to the bases, of length $3$ . We now want to divide the two new trapeziums, always by means of cuts parallel to the bases, in $m$ and $n$ trapeziums, respectively, so that all the $m + n$ trap... | 15 |
In triangle \(ABC\), the sides \(AC = 14\) and \(AB = 6\) are given. A circle with center \(O\), constructed on side \(AC\) as the diameter, intersects side \(BC\) at point \(K\). It is given that \(\angle BAK = \angle ACB\). Find the area of triangle \(BOC\). | 21 |
In a round-robin tournament with 6 teams, each team plays one game against each other team, and each game results in one team winning and one team losing. At the end of the tournament, the teams are ranked by the number of games won. What is the maximum number of teams that could be tied for the most wins at the end of... | 5 |
How many ways can a student schedule $3$ mathematics courses -- algebra, geometry, and number theory -- in a $6$-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other $3$ periods is of no concern here.) | 24 |
In quadrilateral \(ABCD\), \(\angle ABD = 70^\circ\), \(\angle CAD = 20^\circ\), \(\angle BAC = 48^\circ\), \(\angle CBD = 40^\circ\). Find \(\angle ACD\). | 22 |
For how many of the given drawings can the six dots be labelled to represent the links between suspects? | 2 |
Tas and his friends put up a t-shirt for sale. They ended up selling 200 t-shirts in 25 minutes. Half of the shirts were black and cost $30, while the other half were white and cost $25. How much money did they make per minute during the sale? | The total revenue from black shirts was 100 * $30 = $<<100*30=3000>>3000.
The total revenue from white shirts was 100 * $25 = $<<100*25=2500>>2500.
The total revenue from all shirts sold was $3000 + $2500 = $<<3000+2500=5500>>5500.
Tas and his friends ended up making $5500 / 25 minutes = $220 per minute.
#### 220 |
Circles centered at $A$ and $B$ each have radius 2, as shown. Point $O$ is the midpoint of $\overline{AB}$, and $OA=2\sqrt{2}$. Segments $OC$ and $OD$ are tangent to the circles centered at $A$ and $B$, respectively, and $\overline{EF}$ is a common tangent. What is the area of the shaded region $ECODF$?
[asy]unitsiz... | 8\sqrt{2}-4-\pi |
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} |
How many multiples of 10 are between 11 and 103? | 9 |
What is the smallest four-digit number that is divisible by $33$? | 1023 |
Lexi wants to run a total of three and one-fourth miles. One lap on a particular outdoor track measures a quarter of a mile around. How many complete laps must she run? | There are 3/ 1/4 = 12 one-fourth miles in 3 miles.
So, Lexi will have to run 12 (from 3 miles) + 1 (from 1/4 mile) = <<12+1=13>>13 complete laps.
#### 13 |
Expand $-(3-c)(c+2(3-c))$. What is the sum of the coefficients of the expanded form? | -10 |
(Ⅰ) Find the equation of the line that passes through the intersection point of the two lines $2x-3y-3=0$ and $x+y+2=0$, and is perpendicular to the line $3x+y-1=0$.
(Ⅱ) Given the equation of line $l$ in terms of $x$ and $y$ as $mx+y-2(m+1)=0$, find the maximum distance from the origin $O$ to the line $l$. | 2 \sqrt {2} |
For all values of $x$ for which it is defined, $f(x) = \cot \frac{x}{4} - \cot x$ can be written as
\[f(x) = \frac{\sin kx}{\sin \frac{x}{4} \sin x}.\]Find the value of $k.$ | \frac{3}{4} |
Two players, A and B, play a game called "draw the joker card". In the beginning, Player A has $n$ different cards. Player B has $n+1$ cards, $n$ of which are the same with the $n$ cards in Player A's hand, and the rest one is a Joker (different from all other $n$ cards). The rules are i) Player A first draws a card fr... | n=32 |
For how many integer values of $n$ between 1 and 349 inclusive does the decimal representation of $\frac{n}{350}$ terminate? | 49 |
Let $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c}$ be vectors such that $\|\mathbf{a}\| = \|\mathbf{b}\| = 1$ and $\|\mathbf{c}\| = 2.$ Find the maximum value of
\[\|\mathbf{a} - 2 \mathbf{b}\|^2 + \|\mathbf{b} - 2 \mathbf{c}\|^2 + \|\mathbf{c} - 2 \mathbf{a}\|^2.\] | 42 |
Let $a,$ $b,$ and $c$ be complex numbers such that $|a| = |b| = |c| = 1$ and
\[\frac{a^2}{bc} + \frac{b^2}{ac} + \frac{c^2}{ab} = -1.\]Find all possible values of $|a + b + c|.$
Enter all the possible values, separated by commas. | 1,2 |
Given two lines $l_{1}$: $mx+2y-2=0$ and $l_{2}$: $5x+(m+3)y-5=0$, if $l_{1}$ is parallel to $l_{2}$, find the value of $m$. | -5 |
On a particular day in Salt Lake, UT, the temperature was given by $-t^2 +12t+50$ where $t$ is the time in hours past noon. What is the largest $t$ value at which the temperature was exactly 77 degrees? | 9 |
The internal angles of quadrilateral $ABCD$ form an arithmetic progression. Triangles $ABD$ and $DCB$ are similar with $\angle DBA = \angle DCB$ and $\angle ADB = \angle CBD$. Moreover, the angles in each of these two triangles also form an arithmetic progression. In degrees, what is the largest possible sum of the two... | 240 |
How many diagonals does a regular seven-sided polygon contain? | 14 |
Convex hexagon $ABCDEF$ has exactly two distinct side lengths. Side $AB$ measures 5 units, and side $BC$ measures 6 units. The perimeter of hexagon $ABCDEF$ is 34 units. How many sides of hexagon $ABCDEF$ have measure 6 units? | 4 |
When \( x^{2} \) is added to the quadratic polynomial \( f(x) \), its maximum value increases by \( \frac{27}{2} \), and when \( 4x^{2} \) is subtracted from it, its maximum value decreases by 9. How will the maximum value of \( f(x) \) change if \( 2x^{2} \) is subtracted from it? | \frac{27}{4} |
What is the remainder when $3x^7-x^6-7x^5+2x^3+4x^2-11$ is divided by $2x-4$? | 117 |
Given a function f(n) defined on the set of positive integers, where f(1) = 2: For even n, f(n) = f(n-1) + 2; For odd n > 1, f(n) = f(n-2) + 2. Calculate the value of f(2017). | 2018 |
In the complex plane, the points \( 0, z, \frac{1}{z}, z+\frac{1}{z} \) form a parallelogram with an area of \( \frac{35}{37} \). If the real part of \( z \) is greater than 0, find the minimum value of \( \left| z + \frac{1}{z} \right| \). | \frac{5 \sqrt{74}}{37} |
A wizard is crafting a magical elixir. For this, he requires one of four magical herbs and one of six enchanted gems. However, one of the gems cannot be used with three of the herbs. Additionally, another gem can only be used if it is paired with one specific herb. How many valid combinations can the wizard use to prep... | 18 |
Grandma Olga has 3 daughters and 3 sons. If all her daughters each have 6 sons, and each of her sons has 5 daughters, how many grandchildren does she have in total? | From her daughters, Olga has 3 x 6 = <<3*6=18>>18 grandsons.
From her sons, she has 3 x 5 = <<3*5=15>>15 granddaughters.
Altogether, Olga has 18 + 15 = <<18+15=33>>33 grandchildren.
#### 33 |
Tom has a quarter as much money as Nataly. Nataly has three times as much money as Raquel. How much money do Tom, Raquel, and Nataly have combined if Raquel has $40? | If Nataly has 3 times as much money as Raquel, she has 3 * $40 = $<<3*40=120>>120
Tom has 1/4 as much money as Nataly, so he has 1/4 * $120 = $<<1/4*120=30>>30
Combined, the three have $30 + $120 + $40 = $<<30+120+40=190>>190
#### 190 |
There exist $r$ unique nonnegative integers $n_1 > n_2 > \cdots > n_r$ and $r$ unique integers $a_k$ ($1\le k\le r$) with each $a_k$ either $1$ or $- 1$ such that \[a_13^{n_1} + a_23^{n_2} + \cdots + a_r3^{n_r} = 2008.\] Find $n_1 + n_2 + \cdots + n_r$. | 21 |
The diagram shows a large square divided into squares of three different sizes. What percentage of the large square is shaded?
A) 61%
B) 59%
C) 57%
D) 55%
E) 53% | 59\% |
If $x - 2x + 3x = 100$, what is the value of $x$? | 50 |
What is the largest possible length of an arithmetic progression formed of positive primes less than $1,000,000$? | 12 |
A performing magician has a disappearing act where he makes a random member of his audience disappear and reappear. Unfortunately, one-tenth of the time, the audience member never reappears. However, one-fifth of the time, two people reappear instead of only one. If the magician has put on 100 performances of the act t... | One-tenth of the time, no one reappears, so there have been 100 / 10 = <<100/10=10>>10 times no one has reappeared.
One-fifth of the time, two people reappear, so there have been 100 / 5 = <<100/5=20>>20 times 2 people have reappeared.
In those 20 times, 2 * 20 = <<2*20=40>>40 people have reappeared in all.
Thus, there... |
Find the relationship between \(\arcsin \cos \arcsin x\) and \(\arccos \sin \arccos x\). | \frac{\pi}{2} |
In the cube $ABCDEFGH$, find $\sin \angle BAC$ ensuring that the angle is uniquely determined and forms a right angle. | \frac{\sqrt{2}}{2} |
Hugo's mountain has an elevation of 10,000 feet above sea level. Boris' mountain has an elevation that is 2,500 feet shorter than Hugo's mountain. If Hugo climbed his mountain 3 times, how many times would Boris need to climb his mountain to have climbed the same number of feet as Hugo? | Boris' mountain has an elevation 2,500 feet shorter than Hugo's mountain, or 10,000-2,500=<<10000-2500=7500>>7500 feet above sea level.
If Hugo climbed his mountain 3 times, his total climbing distance would be 3*10,000=<<3*10000=30000>>30,000 feet.
To climb a total of 30,000 feet, Boris would need to need to climb his... |
Given Lucy starts with an initial term of 8 in her sequence, where each subsequent term is generated by either doubling the previous term and subtracting 2 if a coin lands on heads, or halving the previous term and subtracting 2 if a coin lands on tails, determine the probability that the fourth term in Lucy's sequence... | \frac{3}{4} |
Let $\triangle A B C$ be a triangle with $A B=7, B C=1$, and $C A=4 \sqrt{3}$. The angle trisectors of $C$ intersect $\overline{A B}$ at $D$ and $E$, and lines $\overline{A C}$ and $\overline{B C}$ intersect the circumcircle of $\triangle C D E$ again at $X$ and $Y$, respectively. Find the length of $X Y$. | \frac{112}{65} |
Given that the graph of $$f(x)=-\cos^{2} \frac {ω}{2}x+ \frac { \sqrt {3}}{2}\sinωx$$ has a distance of $$\frac {π}{2}(ω＞0)$$ between two adjacent axes of symmetry.
(Ⅰ) Find the intervals where $f(x)$ is strictly decreasing;
(Ⅱ) In triangle ABC, a, b, and c are the sides opposite to angles A, B, and C, respectively... | \sqrt {13} |
Triangles $ABC$ and $ADC$ are isosceles with $AB=BC$ and $AD=DC$. Point $D$ is inside $\triangle ABC$, $\angle ABC = 40^\circ$, and $\angle ADC = 140^\circ$. What is the degree measure of $\angle
BAD$? | 50^{\circ} |
The value of $a$ is chosen so that the number of roots of the first equation $4^{x}-4^{-x}=2 \cos(a x)$ is 2007. How many roots does the second equation $4^{x}+4^{-x}=2 \cos(a x)+4$ have for the same value of $a$? | 4014 |
Find the value of $k$ for the ellipse $\frac{x^2}{k+8} + \frac{y^2}{9} = 1$ with an eccentricity of $\frac{1}{2}$. | -\frac{5}{4} |
Given an ellipse E: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ ($$a > b > 0$$) passing through point Q ($$\frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}$$), the product of the slopes of the lines connecting the moving point P on the ellipse to the two endpoints of the minor axis is $$-\frac{1}{2}$$.
1. Find the equation of the... | \frac{4}{3} |
Droid owns a coffee shop. He uses 3 bags of coffee beans every morning, he uses triple that number in the afternoon than in the morning, and he uses twice the morning number in the evening. How many bags of coffee beans does he use every week? | Droid uses 3 x 3 = <<3*3=9>>9 bags of coffee beans in the afternoon.
He uses 3 x 2 = <<3*2=6>>6 bags of coffee beans in the evening.
So, he uses a total of 3 + 9 + 6 = <<3+9+6=18>>18 bags of coffee beans every day.
Therefore, Droid uses a total of 18 x 7 = <<18*7=126>>126 bags of coffee beans every week.
#### 126 |
The vertices of a quadrilateral lie on the graph of $y=\ln{x}$, and the $x$-coordinates of these vertices are consecutive positive integers. The area of the quadrilateral is $\ln{\frac{91}{90}}$. What is the $x$-coordinate of the leftmost vertex? | 12 |
The expressions $A$ = $1 \times 2 + 3 \times 4 + 5 \times 6 + \cdots + 37 \times 38 + 39$ and $B$ = $1 + 2 \times 3 + 4 \times 5 + \cdots + 36 \times 37 + 38 \times 39$ are obtained by writing multiplication and addition operators in an alternating pattern between successive integers. Find the positive difference betwe... | 722 |
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