problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
$-14-(-2)^{3}\times \dfrac{1}{4}-16\times \left(\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{3}{8}\right)$. | -22 |
In a positive term geometric sequence ${a_n}$, ${a_5 a_6 =81}$, calculate the value of ${\log_{3}{a_1} + \log_{3}{a_5} +...+\log_{3}{a_{10}}}$. | 20 |
Let $f(x)=x^4+14x^3+52x^2+56x+16$. Let $z_1,z_2,z_3,z_4$ be the four roots of $f$. Find the smallest possible value of $|z_{a}z_{b}+z_{c}z_{d}|$ where $\{a,b,c,d\}=\{1,2,3,4\}$. | 8 |
3 years ago James turned 27. In 5 years Matt will be twice James age. How old is Matt now? | James is now 27+3=<<27+3=30>>30 years old
In 5 years James will be 30+5=<<30+5=35>>35 years old
So in 5 years Matt will be 35*2=<<35*2=70>>70 years old
That means Matt is now 70-5=<<70-5=65>>65 years old
#### 65 |
Translate the graph of $y= \sqrt {2}\sin (2x+ \frac {\pi}{3})$ to the right by $\phi(0 < \phi < \pi)$ units to obtain the graph of the function $y=2\sin x(\sin x-\cos x)-1$. Then, $\phi=$ ______. | \frac {13\pi}{24} |
A student accidentally added five to both the numerator and denominator of a fraction, changing the fraction's value to $\frac12$. If the original numerator was a 2, what was the original denominator? | 9 |
The probability that the random variable $X$ follows a normal distribution $N\left( 3,{{\sigma }^{2}} \right)$ and $P\left( X\leqslant 4 \right)=0.84$ can be expressed in terms of the standard normal distribution $Z$ as $P(Z\leqslant z)=0.84$, where $z$ is the z-score corresponding to the upper tail probability $1-0.84... | 0.68 |
On a sheet of graph paper, two rectangles are outlined. The first rectangle has a vertical side shorter than the horizontal side, and for the second rectangle, the opposite is true. Find the maximum possible area of their intersection if the first rectangle contains 2015 cells and the second one contains 2016 cells. | 1302 |
Mario's salary increased by 40% to $4000 this year. Bob's salary from last year was equal to three times Mario's salary this year. If Bob's current salary is 20% more than his salary last year, what is his current salary? | Last year, Bob's salary was equal to three times Mario's salary this year, meaning Bob was earning 3*$4000 = $<<3*4000=12000>>12000
If Bob's salary increased by 20% this year, he is earning 20/100*$12000= $<<20/100*12000=2400>>2400 more
Bob's total salary this year is $12000+$2400= $<<12000+2400=14400>>14400
#### 14400 |
The increasing sequence $1, 3, 4, 9, 10, 12, 13, \dots$ consists of all those positive integers which are powers of 3 or sums of distinct powers of 3. Find the $150^{\text{th}}$ term of this sequence. | 2280 |
Find all the real solutions to
\[\frac{(x - 1)(x - 2)(x - 3)(x - 4)(x - 3)(x - 2)(x - 1)}{(x - 2)(x - 4)(x - 2)} = 1.\]Enter all the solutions, separated by commas. | 2 + \sqrt{2}, 2 - \sqrt{2} |
Expand the following expression: $(9x+4)\cdot 2x^2$ | 18x^3+8x^2 |
The equation
\[\frac{1}{x} + \frac{1}{x + 2} - \frac{1}{x + 4} - \frac{1}{x + 6} - \frac{1}{x + 8} - \frac{1}{x + 10} + \frac{1}{x + 12} + \frac{1}{x + 14} = 0\]has four roots of the form $-a \pm \sqrt{b \pm c \sqrt{d}},$ where $a,$ $b,$ $c,$ $d$ are positive integers, and $d$ is not divisible by the square of a prime.... | 37 |
Given the ellipse $C$: $mx^{2}+3my^{2}=1$ ($m > 0$) with a major axis length of $2\sqrt{6}$, and $O$ is the origin.
$(1)$ Find the equation of the ellipse $C$.
$(2)$ Let point $A(3,0)$, point $B$ be on the $y$-axis, and point $P$ be on the ellipse $C$ and to the right of the $y$-axis. If $BA=BP$, find the minimum val... | 3\sqrt{3} |
10 people attend a meeting. Everyone at the meeting exchanges business cards with everyone else. How many exchanges of business cards occur? | 45 |
When Hannah was 6 years old, her age was double the age of her sister July. Now, 20 years later, if July's husband is 2 years older than her, how old is he? | July was 6 / 2 = <<6/2=3>>3 years old when Hannah was 6.
Now, July is 3 + 20 = <<3+20=23>>23 years old
July's husband is 23 + 2 = <<23+2=25>>25 years old
#### 25 |
Given that $3\sin \alpha - 2\cos \alpha = 0$, find the value of the following expressions:
$$(1)\ \frac{\cos \alpha - \sin \alpha}{\cos \alpha + \sin \alpha} + \frac{\cos \alpha + \sin \alpha}{\cos \alpha - \sin \alpha};$$
$$(2)\ \sin^2\alpha - 2\sin \alpha\cos \alpha + 4\cos^2\alpha.$$ | \frac{28}{13} |
We have an $n$-gon, and each of its vertices is labeled with a number from the set $\{1, \ldots, 10\}$. We know that for any pair of distinct numbers from this set there is at least one side of the polygon whose endpoints have these two numbers. Find the smallest possible value of $n$. | 50 |
*How many odd numbers between 200 and 999 have distinct digits, and no digit greater than 7?* | 120 |
Compute: \( 4.165 \times 4.8 + 4.165 \times 6.7 - 4.165 \div \frac{2}{3} = \) | 41.65 |
There is one odd integer \( N \) between 400 and 600 that is divisible by both 5 and 11. What is the sum of the digits of \( N \)? | 18 |
For a sale, a store owner reduces the price of a $\$10$ scarf by $30\%$. Later the price is lowered again, this time by $50\%$ of the reduced price. What is the current price, in dollars? | \$3.50 |
For a finite sequence \(P = \left(p_1, p_2, \cdots, p_n\right)\), the Cesaro sum (named after the mathematician Cesaro) is defined as \(\frac{1}{n}(S_1 + S_2 + \cdots + S_n)\), where \(S_k = p_1 + p_2 + \cdots + p_k\) for \(1 \leq k \leq n\). If a sequence \(\left(p_1, p_2, \cdots, p_{99}\right)\) of 99 terms has a Ces... | 991 |
The height of a rhombus, drawn from the vertex of its obtuse angle, divides the side of the rhombus in the ratio $1:3$, measured from the vertex of its acute angle. What fraction of the area of the rhombus is occupied by the area of a circle inscribed in it? | \frac{\pi \sqrt{15}}{16} |
Vera has a set of weights, each of which has a distinct mass and weighs an integer number of grams. It is known that the lightest weight in the set weighs 71 times less than the sum of the weights of all other weights in the set. It is also known that the two lightest weights together weigh 34 times less than the sum o... | 35 |
Find the minimum value of
\[
y^2 + 9y + \frac{81}{y^3}
\]
for \(y > 0\). | 39 |
[asy] pair A = (0,0), B = (7,4.2), C = (10, 0), D = (3, -5), E = (3, 0), F = (7,0); draw(A--B--C--D--cycle,dot); draw(A--E--F--C,dot); draw(D--E--F--B,dot); markscalefactor = 0.1; draw(rightanglemark(B, A, D)); draw(rightanglemark(D, E, C)); draw(rightanglemark(B, F, A)); draw(rightanglemark(D, C, B)); MP("A",(0,0),W);... | 4.2 |
In parallelogram $ABCD$, let $O$ be the intersection of diagonals $\overline{AC}$ and $\overline{BD}$. Angles $CAB$ and $DBC$ are each twice as large as angle $DBA$, and angle $ACB$ is $r$ times as large as angle $AOB$. Find $r.$ | \frac{7}{9} |
On Blacks, Martha goes to the mall to buy clothes on sale. For every 2 jackets she buys, she gets 1 free jacket. For every 3 t-shirts she buys, she gets 1 free t-shirt. Martha decides to buy 4 jackets and 9 t-shirts. How many clothes will she take home? | For buying 4 jackets Martha gets 4 jackets / 2 = <<4/2=2>>2 jackets for free
For buying 9 t-shirts, Martha gets 9 t-shirts/ 3 = <<9/3=3>>3 t-shirts for free
Adding the free jackets, Martha gets 4 jackets + 2 jackets = <<4+2=6>>6 jackets
Adding the free t-shirts, Martha gets 9 t-shirts + 3 t-shirts = <<9+3=12>>12 t-shir... |
Given $| \mathbf{e} |=1$, and it satisfies $|\mathbf{a} + \mathbf{e}|=|\mathbf{a} - 2\mathbf{e}|$, then the projection of vector $\mathbf{a}$ in the direction of $\mathbf{e}$ is | \frac{1}{2} |
Given a triangular prism \( S-ABC \) with a base that is an isosceles right triangle with \( AB \) as the hypotenuse, and \( SA = SB = SC = AB = 2 \). If the points \( S, A, B, C \) all lie on the surface of a sphere centered at \( O \), what is the surface area of this sphere? | \frac{16 \pi}{3} |
In $\triangle ABC$ lines $CE$ and $AD$ are drawn so that $\dfrac{CD}{DB}=\dfrac{3}{1}$ and $\dfrac{AE}{EB}=\dfrac{3}{2}$. Let $r=\dfrac{CP}{PE}$ where $P$ is the intersection point of $CE$ and $AD$. Then $r$ equals:
[asy] size(8cm); pair A = (0, 0), B = (9, 0), C = (3, 6); pair D = (7.5, 1.5), E = (6.5, 0); pair P = in... | 5 |
Al and Bert must arrive at a town 22.5 km away. They have one bicycle between them and must arrive at the same time. Bert sets out riding at 8 km/h, leaves the bicycle, and then walks at 5 km/h. Al walks at 4 km/h, reaches the bicycle, and rides at 10 km/h. For how many minutes was the bicycle not in motion? | 75 |
Emily paid for a $\$2$ sandwich using 50 coins consisting of pennies, nickels, and dimes, and received no change. How many dimes did Emily use? | 10 |
2002 is a palindromic year, meaning it reads the same backward and forward. The previous palindromic year was 11 years ago (1991). What is the maximum number of non-palindromic years that can occur consecutively (between the years 1000 and 9999)? | 109 |
Given that Bill's age in two years will be three times his current age, and the digits of both Jack's and Bill's ages are reversed, find the current age difference between Jack and Bill. | 18 |
What is the base ten equivalent of $12345_{6}$? | 1865 |
In a paper, a $4 \times 6$ grid was drawn, and then the diagonal from $A$ to $B$ was traced.
Observe that the diagonal $AB$ intersects the grid at 9 points.
If the grid were of size $12 \times 17$, at how many points would the diagonal $AB$ intersect the grid? | 29 |
Let \( A \) be a 4-digit integer. When both the first digit (left-most) and the third digit are increased by \( n \), and the second digit and the fourth digit are decreased by \( n \), the new number is \( n \) times \( A \). Find the value of \( A \). | 1818 |
For positive integer $n$, let $s(n)$ denote the sum of the digits of $n$. Find the smallest positive integer satisfying $s(n) = s(n+864) = 20$. | 695 |
What is the maximum value that the expression \(\frac{1}{a+\frac{2010}{b+\frac{1}{c}}}\) can take, where \(a, b, c\) are distinct non-zero digits? | 1/203 |
A sphere intersects the $xy$-plane in a circle centered at $(2, 3, 0)$ with radius 2. The sphere also intersects the $yz$-plane in a circle centered at $(0, 3, -8),$ with radius $r.$ Find $r.$ | 2\sqrt{15} |
For each positive integer $p$, let $b(p)$ denote the unique positive integer $k$ such that $|k-\sqrt{p}|<\frac{1}{2}$. For example, $b(6)=2$ and $b(23)=5$. Find $S=\sum_{p=1}^{2007} b(p)$. | 59955 |
Expand $-(4-c)(c+2(4-c) + c^2)$. What is the sum of the coefficients of the expanded form? | -24 |
John has a raw squat of 600 pounds without sleeves or wraps. Sleeves add 30 pounds to his lift. Wraps add 25% to his squat. How much more pounds does he get out of wraps versus sleeves. | Wraps add 600*.25=<<600*.25=150>>150 pounds to his lift
So they add 150-30=<<150-30=120>>120 pounds more than sleeves
#### 120 |
Given the function $f(x) = A\sin(x + \varphi)$ ($A > 0$, $0 < \varphi < \pi$) has a maximum value of 1, and its graph passes through point M ($\frac{\pi}{3}$, $\frac{1}{2}$), then $f(\frac{3\pi}{4})$ = \_\_\_\_\_\_. | -\frac{\sqrt{2}}{2} |
The only prime factors of an integer $n$ are 2 and 3. If the sum of the divisors of $n$ (including itself) is $1815$ , find $n$ . | 648 |
Let $N = 34 \cdot 34 \cdot 63 \cdot 270$. What is the ratio of the sum of the odd divisors of $N$ to the sum of the even divisors of $N$? | 1 : 14 |
Given that $\cos \alpha = -\frac{4}{5}$, and $\alpha$ is an angle in the third quadrant, find the values of $\sin \alpha$ and $\tan \alpha$. | \frac{3}{4} |
Find the least common multiple of 36 and 132. | 396 |
Given the ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$, and the line $l: 4x - 5y + 40 = 0$. Is there a point on the ellipse for which the distance to line $l$ is minimal? If so, what is the minimal distance? | \frac{15}{\sqrt{41}} |
A group of 11 people, including Ivanov and Petrov, are seated in a random order around a circular table. Find the probability that there will be exactly 3 people sitting between Ivanov and Petrov. | 1/10 |
Let $(x_1,y_1),$ $(x_2,y_2),$ $\dots,$ $(x_n,y_n)$ be the solutions to
\begin{align*}
|x - 3| &= |y - 9|, \\
|x - 9| &= 2|y - 3|.
\end{align*}Find $x_1 + y_1 + x_2 + y_2 + \dots + x_n + y_n.$ | -4 |
Find the integer that is closest to $1000\sum_{n=3}^{10000}\frac1{n^2-4}$. | 521 |
Let $ a$, $ b$, $ c$ be nonzero real numbers such that $ a+b+c=0$ and $ a^3+b^3+c^3=a^5+b^5+c^5$. Find the value of
$ a^2+b^2+c^2$. | \frac{6}{5} |
Maya wants to learn how to lift and right now she can only lift a fourth of what America can. America can lift 240 pounds. As Maya grows stronger, she can add 10 more pounds to what she previously could lift. America follows this and now she has hit her peak lift at 300 pounds. If Maya reaches her absolute peak and can... | Maya can lift 240*0.25=<<240*0.25=60>>60 pounds.
Gaining progress, she can lift up to 60+10=<<60+10=70>>70 pounds.
At her peak, she can lift 300*0.5=<<300*0.5=150>>150 pounds.
Maya's peak is 150 pounds - 60 pounds when she started = <<150-60=90>>90 more pounds than when she started.
#### 90 |
Tim is stuck in traffic for twice as long as he was driving. He drove 5 hours. How long was the trip? | He was stuck in traffic for 5*2=<<5*2=10>>10 hours
So his trip took 10+5=<<10+5=15>>15 hours
#### 15 |
For how many bases between two and nine inclusive does the representation of $576_{10}$ have a final digit of 1? | 1 |
Rhonda, Sally, and Diane are members of their school's track team. The three of them run the 600-meter relay race together. Rhonda runs the first 200 meters of the race, Sally runs the second 200 meters of the race, and Diane runs the final 200 meters of the race. Rhonda can run 200 meters in 24 seconds. Sally takes ... | Sally takes 24+2=<<24+2=26>>26 seconds.
Diane takes 24-3=<<24-3=21>>21 seconds.
Thus, the three take 24+26+21=<<24+26+21=71>>71 seconds to run the 600-meter relay race.
#### 71 |
What is the ratio of the area of the shaded square to the area of the large square? (The figure is drawn to scale.) [asy]
/* AMC8 1998 #13P */
size(1inch,1inch);
pair r1c1=(0,0), r1c2=(10,0), r1c3=(20,0), r1c4=(30, 0), r1c5=(40, 0);
pair r2c1=(0,10), r2c2=(10,10), r2c3=(20,10), r2c4=(30, 10), r2c5=(40, 10);
pair r3c1=(... | \frac{1}{8} |
A point $Q$ is chosen in the interior of $\triangle DEF$ such that when lines are drawn through $Q$ parallel to the sides of $\triangle DEF$, the resulting smaller triangles $u_{1}$, $u_{2}$, and $u_{3}$ have areas $16$, $25$, and $36$, respectively. Furthermore, a circle centered at $Q$ inside $\triangle DEF$ cuts off... | 225 |
Doc's Pizza contains 6 pieces of pizza. Ten fourth-graders each bought 20 pizzas from Doc's Pizza and put them in their box. How many pieces of pizza are the children carrying in total? | If Each fourth-grader bought 20 pizzas, with each pizza having 6 pieces, each fourth-grader had a total of 20*6 = <<20*6=120>>120 pieces of pizza in their box.
Ten fourth-graders each bought 20 pizzas, and since the 20 pizzas had 120 pieces, the total number of pizzas in the students' boxes is 120*10 = <<120*10=1200>>1... |
In $\triangle ABC$, let $a$, $b$, and $c$ be the sides opposite to angles $A$, $B$, and $C$ respectively. Given that $\cos B = \frac{4}{5}$ and $b = 2$.
1. Find the value of $a$ when $A = \frac{\pi}{6}$.
2. Find the value of $a + c$ when the area of $\triangle ABC$ is $3$. | 2\sqrt{10} |
In triangle ABC, let the lengths of the sides opposite to angles A, B, and C be a, b, and c respectively, and b = 3, c = 1, A = 2B. Find the value of a. | \sqrt{19} |
Given that Gill leaves Lille at 09:00, the train travels the first 27 km at 96 km/h and then stops at Lens for 3 minutes before traveling the final 29 km to Lillers at 96 km/h, calculate the arrival time at Lillers. | 09:38 |
The numbers $1,2,\ldots,9$ are arranged so that the $1$ st term is not $1$ and the $9$ th term is not $9$ . Calculate the probability that the third term is $3$. | \frac{43}{399} |
There are 54 students in a class, and there are 4 tickets for the Shanghai World Expo. Now, according to the students' ID numbers, the tickets are distributed to 4 students through systematic sampling. If it is known that students with ID numbers 3, 29, and 42 have been selected, then the ID number of another student w... | 16 |
There are 1987 sets, each with 45 elements. The union of any two sets has 89 elements. How many elements are there in the union of all 1987 sets? | 87429 |
Let \( T \) be the set of positive real numbers. Let \( g : T \to \mathbb{R} \) be a function such that
\[ g(x) g(y) = g(xy) + 2006 \left( \frac{1}{x} + \frac{1}{y} + 2005 \right) \] for all \( x, y > 0 \).
Let \( m \) be the number of possible values of \( g(3) \), and let \( t \) be the sum of all possible values of... | \frac{6019}{3} |
Given a sequence $\{a_{n}\}$ where $a_{1}=1$ and $a_{n+1}=\left\{\begin{array}{l}{{a}_{n}+1, n \text{ is odd}}\\{{a}_{n}+2, n \text{ is even}}\end{array}\right.$
$(1)$ Let $b_{n}=a_{2n}$, write down $b_{1}$ and $b_{2}$, and find the general formula for the sequence $\{b_{n}\}$.
$(2)$ Find the sum of the first $20$ te... | 300 |
Let $T$ be a subset of $\{1,2,3,...,40\}$ such that no pair of distinct elements in $T$ has a sum divisible by $5$. What is the maximum number of elements in $T$? | 24 |
A subset \( H \) of the set of numbers \(\{1, 2, \ldots, 100\}\) has the property that if an element is in \( H \), then ten times that element is not in \( H \). What is the maximum number of elements that \( H \) can have? | 91 |
Find the number of solutions to
\[\sin x = \left( \frac{1}{3} \right)^x\] on the interval $(0,150 \pi).$ | 75 |
What is the product of the least common multiple and the greatest common factor of $20$ and $90$? | 1800 |
Find the number of integers \( n \) that satisfy
\[ 20 < n^2 < 200. \] | 20 |
Add \(53_8 + 27_8\). Express your answer in base \(8\). | 102_8 |
Determine the integer \( m \), \( -180 \leq m \leq 180 \), such that \(\sin m^\circ = \sin 945^\circ.\) | -135 |
Consider an arithmetic sequence $\{a_n\}$ with the sum of its first $n$ terms denoted as $S_n$. Given that $a_1=9$, $a_2$ is an integer, and $S_n \leq S_5$, find the sum of the first 9 terms of the sequence $\{\frac{1}{a_n a_{n+1}}\}$. | -\frac{1}{9} |
Draw a square of side length 1. Connect its sides' midpoints to form a second square. Connect the midpoints of the sides of the second square to form a third square. Connect the midpoints of the sides of the third square to form a fourth square. And so forth. What is the sum of the areas of all the squares in this infi... | 2 |
Evaluate: $64^2-36^2$ | 2800 |
Players A and B have a Go game match, agreeing that the first to win 3 games wins the match. After the match ends, assuming in a single game, the probability of A winning is 0.6, and the probability of B winning is 0.4, with the results of each game being independent. It is known that in the first 2 games, A and B each... | 2.48 |
In the final phase of a professional bowling competition, the top five players compete as follows: first, the fifth and fourth place players compete, and the loser gets the 5th place prize; the winner then competes with the third place player, and the loser gets the 4th place prize; the winner then competes with the se... | 16 |
Lucy is listening to her favorite album while jumping rope. She can jump the rope 1 time per second. If the album's songs are all 3.5 minutes long and there are 10 songs, how many times will she jump rope? | She can jump her rope 60 times a minute because 60 x 1 = <<60*1=60>>60
The album is 35 minutes long because 10 x 3.5 = <<10*3.5=35>>35
She will jump her rope 2100 times during the album because 35 x 60 = <<35*60=2100>>2100
#### 2100 |
In the trapezoid \(ABCD\), the lengths of the bases \(AD = 24\) cm and \(BC = 8\) cm, and the diagonals \(AC = 13\) cm, \(BD = 5\sqrt{17}\) cm are known. Calculate the area of the trapezoid. | 80 |
Given that $| \overrightarrow{a}|=1$, $| \overrightarrow{b}|= \sqrt {2}$, and $\overrightarrow{a} \perp ( \overrightarrow{a}- \overrightarrow{b})$, find the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac {\pi}{4} |
Compute without using a calculator: $8!-7!$ | 35,\!280 |
In the country of Anchuria, a day can either be sunny, with sunshine all day, or rainy, with rain all day. If today's weather is different from yesterday's, the Anchurians say that the weather has changed. Scientists have established that January 1st is always sunny, and each subsequent day in January will be sunny onl... | 2047 |
A function $f$ is defined for all real numbers and satisfies $f(2+x)=f(2-x)$ and $f(7+x)=f(7-x)$ for all $x.$ If $f(0) = 0,$ what is the least number of roots $f(x)=0$ must have in the interval $-1000\leq x \leq 1000$? | 401 |
If the legs of a right triangle are in the ratio $3:4$, find the ratio of the areas of the two triangles created by dropping an altitude from the right-angle vertex to the hypotenuse. | \frac{9}{16} |
Let $S$ denote the sum of all of the three digit positive integers with three distinct digits. Compute the remainder when $S$ is divided by $1000$.
| 680 |
Mary just held tryouts for the high school band. 80% of the 20 flutes got in, half the 30 clarinets got in, 1/3 of the 60 trumpets got in, and 1/10th of the 20 pianists got in. How many people are in the band total? | First find the total number of flutes who were accepted: 20 flutes * .8 = <<20*.8=16>>16 flutes
Then find the total number of clarinets who were accepted: 30 clarinets * .5 = <<30*.5=15>>15 clarinets
Then find the total number of trumpets who were accepted: 60 trumpets * 1/3 = <<60*1/3=20>>20 trumpets
Then find the tot... |
In triangle \( \triangle ABC \), \( \angle BAC = 90^\circ \), \( AC = AB = 4 \), and point \( D \) is inside \( \triangle ABC \) such that \( AD = \sqrt{2} \). Find the minimum value of \( BD + CD \). | 2\sqrt{10} |
Tony goes on 5 rollercoasters while he is at the park. The first went 50 miles per hour. The second went 62 miles per hour. The third when 73 miles per hour. The fourth when 70 miles per hour. His average speed during the day was 59 miles per hour. How fast was the fifth coaster? | The total speed of all five coasters is 295 because 59 x 5 = <<295=295>>295
The final coaster went 40 miles per hour because 295 - 50 - 62 - 73 - 70 = <<295-50-62-73-70=40>>40
#### 40 |
In $\triangle ABC$, $a=1$, $B=45^{\circ}$, $S_{\triangle ABC}=2$, calculate the diameter of the circumcircle of $\triangle ABC$. | 5\sqrt{2} |
If $\frac{1}{9}+\frac{1}{18}=\frac{1}{\square}$, what is the number that replaces the $\square$ to make the equation true? | 6 |
Alberto, Bernardo, and Carlos are collectively listening to three different songs. Each is simultaneously listening to exactly two songs, and each song is being listened to by exactly two people. In how many ways can this occur? | 6 |
The product of three even consecutive positive integers is twenty times their sum. What is the sum of the three integers? | 24 |
Let $m$ be a fixed integer greater than $1$. The sequence $x_0$, $x_1$, $x_2$, $\ldots$ is defined as follows:
\[x_i = \begin{cases}2^i&\text{if }0\leq i \leq m - 1;\\\sum_{j=1}^mx_{i-j}&\text{if }i\geq m.\end{cases}\]
Find the greatest $k$ for which the sequence contains $k$ consecutive terms divisible by $m$ .
[i] | k=m-1 |
Real numbers \(a\), \(b\), and \(c\) and positive number \(\lambda\) make \(f(x) = x^3 + ax^2 + b x + c\) have three real roots \(x_1\), \(x_2\), \(x_3\), such that:
(1) \(x_2 - x_1 = \lambda\);
(2) \(x_3 > \frac{1}{2}(x_1 + x_2)\).
Find the maximum value of \(\frac{2 a^3 + 27 c - 9 a b}{\lambda^3}\). | \frac{3\sqrt{3}}{2} |
Bus stop \(B\) is located on a straight highway between stops \(A\) and \(C\). After some time driving from \(A\), the bus finds itself at a point on the highway where the distance to one of the three stops is equal to the sum of the distances to the other two stops. After the same amount of time, the bus again finds i... | 180 |
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