problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Given that the terminal side of angle $\alpha$ passes through the fixed point $P$ on the function $y=\log _{a}(x-3)+2$, find the value of $\sin 2\alpha+\cos 2\alpha$. | \frac{7}{5} |
In the sequence ${a_{n}}$, $a_{1}=1$, $a_{n+2}+(-1)^{n}a_{n}=1$. Let $s_{n}$ be the sum of the first $n$ terms of the sequence ${a_{n}}$. Find $s_{100}$ = \_\_\_\_\_\_. | 1300 |
A cooperative receives apple and grape juice in identical containers and produces an apple-grape drink in identical cans. One container of apple juice is enough for exactly 6 cans of the drink, and one container of grape juice is enough for exactly 10 cans. When the recipe of the drink was changed, one container of app... | 15 |
Compute $-8\cdot 4-(-6\cdot -3)+(-10\cdot -5)$. | 0 |
Determine the product of all constants $t$ such that the quadratic $x^2 + tx - 24$ can be factored in the form $(x+a)(x+b)$, where $a$ and $b$ are integers. | 5290000 |
From 6 sprinters, 4 are to be selected to participate in a 4×100 m relay. If among them, Athlete A cannot run the first leg, and Athlete B cannot run the fourth leg, how many different ways are there to form the team? | 252 |
Petya wants to color some cells of a $6 \times 6$ square so that there are as many vertices as possible that belong to exactly three colored squares. What is the maximum number of such vertices he can achieve? | 25 |
Given the fraction $\frac{987654321}{2^{30}\cdot 5^6}$, determine the minimum number of digits to the right of the decimal point required to express this fraction as a decimal. | 30 |
There exists a constant $c,$ so that among all chords $\overline{AB}$ of the parabola $y = x^2$ passing through $C = (0,c),$
\[t = \frac{1}{AC^2} + \frac{1}{BC^2}\]is a fixed constant. Find the constant $t.$
[asy]
unitsize(1 cm);
real parab (real x) {
return(x^2);
}
pair A, B, C;
A = (1.7,parab(1.7));
B = (-1,pa... | 4 |
Count all the distinct anagrams of the word "YOANN". | 60 |
When a die is thrown twice in succession, the numbers obtained are recorded as $a$ and $b$, respectively. The probability that the line $ax+by=0$ and the circle $(x-3)^2+y^2=3$ have no points in common is ______. | \frac{2}{3} |
Find $XY$ in the triangle below.
[asy]
unitsize(1inch);
pair P,Q,R;
P = (0,0);
Q= (1,0);
R = (0,1);
draw (P--Q--R--P,linewidth(0.9));
draw(rightanglemark(Q,P,R,3));
label("$X$",P,S);
label("$Y$",Q,S);
label("$Z$",R,N);
label("$12\sqrt{2}$",R/2,W);
label("$45^\circ$",(0.7,0),N);
[/asy] | 12\sqrt{2} |
Let $\mathbf{M}$ be a matrix such that
\[\mathbf{M} \begin{pmatrix} 2 \\ -1 \end{pmatrix} = \begin{pmatrix} 3 \\ 0 \end{pmatrix} \quad \text{and} \quad \mathbf{M} \begin{pmatrix} -3 \\ 5 \end{pmatrix} = \begin{pmatrix} -1 \\ -1 \end{pmatrix}.\]Compute $\mathbf{M} \begin{pmatrix} 5 \\ 1 \end{pmatrix}.$ | \begin{pmatrix} 11 \\ -1 \end{pmatrix} |
Lana and Mike are taking their dog and renting a cabin in the mountains for 2 weeks. The daily rate is $125.00 There is a $100.00 pet fee. There is also a 20% service/cleaning fee for the rental. They need to pay 50% of the entire bill as a security deposit. How much is their security deposit? | There are 7 days in a week and they're going for 2 weeks so that's 7*2 = <<7*2=14>>14 days
The daily rate is $125.00 and they are staying for 14 days so that's 125*14 = $<<125*14=1750.00>>1,750.00
There is also a $100.00 pet fee on top of their rental fee of $1,750.00 so that's 100+1750 = $<<100+1750=1850.00>>1850.00
T... |
Abby, Bart, Cindy and Damon weigh themselves in pairs. Together Abby and Bart weigh 260 pounds, Bart and Cindy weigh 245 pounds, and Cindy and Damon weigh 270 pounds. How many pounds do Abby and Damon weigh together? | 285 |
Consider an $8 \times 8$ grid of squares. A rook is placed in the lower left corner, and every minute it moves to a square in the same row or column with equal probability (the rook must move; i.e. it cannot stay in the same square). What is the expected number of minutes until the rook reaches the upper right corner? | 70 |
Donald went to a computer store. He saw a 15% reduction in the price of the laptop he wants to buy. If the laptop originally costs $800, how much will he pay for the laptop? | There was $800 x 15/100 = $<<800*15/100=120>>120.
Thus, Donald will pay $800 - $120 = $<<800-120=680>>680.
#### 680 |
On grid paper, a step-like right triangle was drawn with legs equal to 6 cells. Then all the grid lines inside the triangle were traced. What is the maximum number of rectangles that can be found in this drawing? | 126 |
Solve the equation: $4x^2 - (x^2 - 2x + 1) = 0$. | -1 |
15. If \( a = 1.69 \), \( b = 1.73 \), and \( c = 0.48 \), find the value of
$$
\frac{1}{a^{2} - a c - a b + b c} + \frac{2}{b^{2} - a b - b c + a c} + \frac{1}{c^{2} - a c - b c + a b}.
$$ | 20 |
What is the value of $2-(-2)^{-2}$? | \frac{7}{4} |
A rectangular table of size \( x \) cm \( \times 80 \) cm is covered with identical sheets of paper of size 5 cm \( \times 8 \) cm. The first sheet is placed in the bottom-left corner, and each subsequent sheet is placed 1 cm higher and 1 cm to the right of the previous one. The last sheet is adjacent to the top-right ... | 77 |
What is the smallest positive integer $n$ such that $\frac{n}{n+50}$ is equal to a terminating decimal? | 14 |
Consider a $10\times10$ checkerboard with alternating black and white squares. How many distinct squares, with sides on the grid lines of the checkerboard (horizontal and vertical) and containing at least 6 black squares, can be drawn on the checkerboard? | 140 |
Evaluate the product $\frac{1}{2}\cdot\frac{4}{1}\cdot\frac{1}{8}\cdot\frac{16}{1} \dotsm \frac{1}{16384}\cdot\frac{32768}{1}$. | 256 |
Suppose that $a,b,c$ are real numbers such that $a < b < c$ and $a^3-3a+1=b^3-3b+1=c^3-3c+1=0$ . Then $\frac1{a^2+b}+\frac1{b^2+c}+\frac1{c^2+a}$ can be written as $\frac pq$ for relatively prime positive integers $p$ and $q$ . Find $100p+q$ .
*Proposed by Michael Ren* | 301 |
Given the expression \(\frac{a}{b}+\frac{c}{d}+\frac{e}{f}\), where each letter is replaced by a different digit from \(1, 2, 3, 4, 5,\) and \(6\), determine the largest possible value of this expression. | 9\frac{5}{6} |
If $f(1)=5$, $f(2)=8$ and $f(x)=ax+bx+2$, what is the value of $f(3)$? | 11 |
Given a rhombus with diagonals of length $12$ and $30$, find the radius of the circle inscribed in this rhombus. | \frac{90\sqrt{261}}{261} |
Fill each cell in the given grid with a number from 1 to 4 so that no number repeats within any row or column. Each "L" shaped block spans two rows and two columns. The numbers inside the circles on the line indicate the sum of the numbers in the two adjacent cells (as shown in the provided example, where the third row... | 2143 |
Xibing is a local specialty in Haiyang, with a unique flavor, symbolizing joy and reunion. Person A and person B went to the market to purchase the same kind of gift box filled with Xibing at the same price. Person A bought $2400$ yuan worth of Xibing, which was $10$ boxes less than what person B bought for $3000$ yuan... | 50 |
A man walked a certain distance at a constant rate. If he had gone $\frac{1}{2}$ mile per hour faster, he would have walked the distance in four-fifths of the time; if he had gone $\frac{1}{2}$ mile per hour slower, he would have been $2\frac{1}{2}$ hours longer on the road. The distance in miles he walked was | 15 |
Let \( d = \overline{xyz} \) be a three-digit number that cannot be divisible by 10. If the sum of \( \overline{xyz} \) and \( \overline{zyx} \) is divisible by \( c \), find the largest possible value of this integer \( d \). | 979 |
The number of values of $x$ satisfying the equation
\[\frac {2x^2 - 10x}{x^2 - 5x} = x - 3\]is: | 0 |
We know about a convex pentagon that each side is parallel to one of its diagonals. What can be the ratio of the length of a side to the length of the diagonal parallel to it? | \frac{\sqrt{5} - 1}{2} |
In triangle $PQR,$ $PQ = 4,$ $PR = 9,$ $QR = 10,$ and a point $S$ lies on $\overline{QR}$ such that $\overline{PS}$ bisects $\angle QPR.$ Find $\cos \angle QPS.$ | \sqrt{\frac{23}{48}} |
John is lifting weights. He bench presses 15 pounds for 10 reps and does 3 sets. How much total weight does he move? | He presses 15 pounds x 10 reps = <<15*10=150>>150 pounds
He moves 3 sets x 150 pounds = <<3*150=450>>450 pounds in total.
#### 450 |
A pyramid is constructed using twenty cubical blocks: the first layer has 10 blocks arranged in a square, the second layer contains 6 blocks arranged in a larger square centered on the 10, the third layer has 3 blocks arranged in a triangle, and finally one block sits on top of the third layer. Each block in layers 2, ... | 54 |
In \( \triangle ABC \), \( AB = 4 \), \( BC = 7 \), \( CA = 5 \). Let \(\angle BAC = \alpha\). Find the value of \( \sin^6 \frac{\alpha}{2} + \cos^6 \frac{\alpha}{2} \). | 7/25 |
The school now introduces a new color, silver, for the flag design. Crestview's school colors are now purple, gold, and silver. The students are designing a flag using three solid-colored horizontal stripes. Using one, two, or all three of the school colors, how many different flags are possible if adjacent stripes may... | 27 |
Sandi had $600. She spent half of it at a farmer’s market. Gillian spent $150 more than three times Sandi's total. What is the total that Gillian spent at the farmer’s market? | Sandi spent 600/2 = <<600/2=300>>300 dollars at the farmer’s market.
Gillian spent 300*3+150 = <<300*3+150=1050>>1050 dollars at the farmer’s market.
#### 1050 |
What is the sum of all numbers $q$ which can be written in the form $q=\frac{a}{b}$ where $a$ and $b$ are positive integers with $b \leq 10$ and for which there are exactly 19 integers $n$ that satisfy $\sqrt{q}<n<q$? | 777.5 |
Given \( 1991 = 2^{\alpha_{1}} + 2^{\alpha_{2}} + \cdots + 2^{\alpha_{n}} \), where \( \alpha_{1}, \alpha_{2}, \cdots, \alpha_{n} \) are distinct non-negative integers, find the sum \( \alpha_{1} + \alpha_{2} + \cdots + \alpha_{n} \). | 43 |
Let $n$ be largest number such that \[ \frac{2014^{100!}-2011^{100!}}{3^n} \] is still an integer. Compute the remainder when $3^n$ is divided by $1000$ . | 83 |
At Jefferson High School, there are 500 students enrolled. One hundred twenty students are in the orchestra, 190 are in band, and 220 are in chorus. If only 400 students are in orchestra, band, and/or chorus, how many students are in exactly two of these groups? | 130 |
Each principal of Lincoln High School serves exactly one $3$-year term. What is the maximum number of principals this school could have during an $8$-year period? | 4 |
In right triangle $ABC$ with right angle $C$, $CA = 30$ and $CB = 16$. Its legs $CA$ and $CB$ are extended beyond $A$ and $B$. Points $O_1$ and $O_2$ lie in the exterior of the triangle and are the centers of two circles with equal radii. The circle with center $O_1$ is tangent to the hypotenuse and to the extension of... | 737 |
Four pens and three pencils cost $\$2.24$. Two pens and five pencils cost $\$1.54$. No prices include tax. In cents, what is the cost of a pencil? | 12 |
In the quadrilateral \(ABCD\), it is known that \(AB = BD\), \(\angle ABD = \angle DBC\), and \(\angle BCD = 90^\circ\). On the segment \(BC\), there is a point \(E\) such that \(AD = DE\). What is the length of segment \(BD\) if it is known that \(BE = 7\) and \(EC = 5\)? | 17 |
Let $N$ be the sum of the divisors of $200$. What is the largest prime factor of $N$? | 31 |
A baseball cap factory made 320 caps the first week, 400 the second week, and 300 the third week. If the company makes their average number of caps from the first 3 weeks during the fourth week, how many total caps will they make? | During the first 3 weeks the factory made 320 caps + 400 caps + 300 caps = <<320+400+300=1020>>1020 caps total.
The average for the first 3 weeks is 1020 caps / 3 weeks = <<1020/3=340>>340.
If the factor makes the average during the fourth week they will make 1020 caps + 340 = <<1020+340=1360>>1360 caps.
#### 1360 |
Find the point on the line
\[y = -3x + 5\]that is closest to the point $(-4,-2).$ | \left( \frac{17}{10}, -\frac{1}{10} \right) |
A sequence of integers is defined as follows: $a_i = i$ for $1 \le i \le 5,$ and
\[a_i = a_1 a_2 \dotsm a_{i - 1} - 1\]for $i > 5.$ Evaluate $a_1 a_2 \dotsm a_{2011} - \sum_{i = 1}^{2011} a_i^2.$ | -1941 |
Given an increasing geometric sequence $\{a_{n}\}$ with a common ratio greater than $1$ such that $a_{2}+a_{4}=20$, $a_{3}=8$.<br/>$(1)$ Find the general formula for $\{a_{n}\}$;<br/>$(2)$ Let $b_{m}$ be the number of terms of $\{a_{n}\}$ in the interval $\left(0,m\right]\left(m\in N*\right)$. Find the sum of the first... | 480 |
Let $a$, $b$, and $c$ be solutions of the equation $x^3 - 6x^2 + 11x = 12$.
Compute $\frac{ab}{c} + \frac{bc}{a} + \frac{ca}{b}$. | -\frac{23}{12} |
Terry drives at a speed of 40 miles per hour. He drives daily forth and back from his home to his workplace which is 60 miles away from his home. How many hours does Terry spend driving from home to the workplace and then back? | The miles that Terry drives daily from home and back are 60 * 2 = <<60*2=120>>120 miles.
The hours he takes driving from home and back are 120 / 40 = <<120/40=3>>3 hours.
#### 3 |
A solid cube of side length $1$ is removed from each corner of a solid cube of side length $3$. How many edges does the remaining solid have? | 84 |
A huge number $y$ is given by $2^33^24^65^57^88^39^{10}11^{11}$. What is the smallest positive integer that, when multiplied with $y$, results in a product that is a perfect square? | 110 |
What is the largest four-digit negative integer congruent to $1 \pmod{23}?$ | -1011 |
Determine the value of $l$ for which
\[\frac{9}{x + y + 1} = \frac{l}{x + z - 1} = \frac{13}{z - y + 2}.\] | 22 |
1 chocolate bar costs $1.50 and can be broken into 3 sections to make 3 s'mores. Ron is hosting a boy scout camp out in his backyard for 15 scouts. He wants to make sure that there are enough chocolate bars for everyone to have 2 s'mores each. How much will he spend on chocolate bars? | There are 15 scouts and Ron wants to make sure they each get 2 s'mores a piece, for a total of 15*2 = <<15*2=30>>30 s'mores
1 chocolate bar can make 3 s'mores and he wants to have enough for 30 s'mores, so he needs 30/3 = <<10=10>>10 chocolate bars
Each chocolate bar costs $1.50 and he needs 10, for a total of 1.5*10 =... |
We wrote the numbers from 1 to 2009 on a piece of paper. In the second step, we also wrote down twice each of these numbers on the paper, and then we erased the numbers that appeared twice. We repeat this process such that, in the $i$-th step, we write $i$ times each of the numbers from 1 to 2009 on the paper and then... | 2009 |
Let \(a\), \(b\), \(c\), and \(d\) be distinct positive integers such that \(a+b\), \(a+c\), and \(a+d\) are all odd and are all squares. Let \(L\) be the least possible value of \(a + b + c + d\). What is the value of \(10L\)? | 670 |
In triangle $ABC, \angle A=2 \angle C$. Suppose that $AC=6, BC=8$, and $AB=\sqrt{a}-b$, where $a$ and $b$ are positive integers. Compute $100 a+b$. | 7303 |
If the direction vector of line $l$ is $\overrightarrow{d}=(1,\sqrt{3})$, then the inclination angle of line $l$ is ______. | \frac{\pi}{3} |
In right triangle $ABC$ with $\angle BAC = 90^\circ$, we have $AB = 15$ and $BC = 17$. Find $\tan A$ and $\sin A$. | \frac{8}{17} |
If the function $f(x)=x^{2}-m\cos x+m^{2}+3m-8$ has a unique zero, then the set of real numbers $m$ that satisfy this condition is \_\_\_\_\_\_. | \{2\} |
Victory and Sam have saved money for their holiday spending. Victory saves $100 less than Sam. If Sam saves $1000, how much is the total money saved for holiday spending? | Victory saves $1000 - $100 = $<<1000-100=900>>900 for holiday spending.
The total money saved is $1000 + $900 = $<<1000+900=1900>>1900.
#### 1900 |
What is the value of $\left(\sqrt{4!\cdot 3!}\right)^2$? | 144 |
$M$ is an $8 \times 8$ matrix. For $1 \leq i \leq 8$, all entries in row $i$ are at least $i$, and all entries on column $i$ are at least $i$. What is the minimum possible sum of the entries of $M$ ? | 372 |
Leila spent $40 which is 1/4 of her money on a sweater. She was then left with $20 after spending the rest of her money on buying pieces of jewelry. How much more did Leila spend on jewelry than on the sweater? | Since 1/4 of her money is $40, then Leila originally had $40 x 4 = $<<40*4=160>>160.
$160 - $40 = $<<160-40=120>>120 was left after buying a sweater.
Since she is left with $20, Leila spent $120 - $20 = $<<120-20=100>>100 on jewelry.
Therefore, she spent $100 - $40 = $<<100-40=60>>60 more for jewelry than on a sweater.... |
Write any natural number on a piece of paper, and rotate the paper 180 degrees. If the value remains the same, such as $0$, $11$, $96$, $888$, etc., we call such numbers "神马数" (magical numbers). Among all five-digit numbers, how many different "magical numbers" are there? | 60 |
What is the smallest four-digit positive integer that is divisible by 47? | 1034 |
Alexa and Emily open up a lemonade stand in the front yard. They spent $10 for lemons, $5 for sugar and $3 for cups. The lemonade is $4 a cup. They sell a total of 21 cups. How much profit did Alexa and Emily make after paying off expenses? | Alexa and Emily spend a total of $10 + $5 + $3 = $<<10+5+3=18>>18.
They make a total of $4 x 21 = $<<4*21=84>>84.
They make a profit of $84 - $18 = $<<84-18=66>>66.
#### 66 |
An ellipse satisfies the property that a light ray emitted from one focus of the ellipse, after reflecting off the ellipse, will pass through the other focus. Consider a horizontally placed elliptical billiards table that satisfies the equation $\frac{x^2}{16} + \frac{y^2}{9} = 1$. Let points A and B correspond to its ... | 16 |
Compute $\displaystyle \sum_{n=2}^\infty \sum_{k=1}^{n-1} \frac{k}{2^{n+k}}$. | \frac{4}{9} |
Jason bought 4 dozen cupcakes. He plans to give 3 cupcakes each to his cousins. How many cousins does Jason have? | Since a dozen is equal to 12, then Jason bought 12 x 4 = <<12*4=48>>48 cupcakes.
So, Jason has 48/3 = <<48/3=16>>16 cousins.
#### 16 |
Samanta is planning a party for her friend Marta. She decided to raise some money among the guests she invited, to buy Marta a gift. Every participant gave Samanta $5 for this purpose and she herself put in $10. The gift cost was lower than expected, so there was $15 leftover. What was the price of the gift, if there w... | From all the guests Samanta got $5/guest * 12 guests = $<<5*12=60>>60.
Including money Samanta put herself, she had $60 + $10 = $<<60+10=70>>70.
The cost of the gift was $70 - $15 = $<<70-15=55>>55.
#### 55 |
Triangle $ABC$, $ADE$, and $EFG$ are all equilateral. Points $D$ and $G$ are midpoints of $\overline{AC}$ and $\overline{AE}$, respectively. If $AB=4$, what is the perimeter of figure $ABCDEFG$? [asy]
/* AMC8 2000 #15 Problem */
draw((0,0)--(4,0)--(5,2)--(5.5,1)--(4.5,1));
draw((0,0)--(2,4)--(4,0));
draw((3,2)--(5,2));... | 15 |
In a book, the pages are numbered from 1 through $n$. When summing the page numbers, one page number was mistakenly added three times instead of once, resulting in an incorrect total sum of $2046$. Identify the page number that was added three times. | 15 |
What is the product of all real numbers that are doubled when added to their reciprocals? | -1 |
Triangle $ABC$ has vertices $A(0, 8)$, $B(2, 0)$, $C(8, 0)$. A horizontal line with equation $y=t$ intersects line segment $ \overline{AB} $ at $T$ and line segment $ \overline{AC} $ at $U$, forming $\triangle ATU$ with area 13.5. Compute $t$. | 2 |
To reach the Solovyov family's dacha from the station, one must first travel 3 km on the highway and then 2 km on a path. Upon arriving at the station, the mother called her son Vasya at the dacha and asked him to meet her on his bicycle. They started moving towards each other at the same time. The mother walks at a co... | 800 |
The expression $(5 \times 5)+(5 \times 5)+(5 \times 5)+(5 \times 5)+(5 \times 5)$ is equal to what? | 125 |
How many distinct sequences of four letters can be made from the letters in EQUALS if each sequence must begin with L, end with Q, and no letter can appear in a sequence more than once? | 12 |
Factor $x^2+4x+4-81x^4$ into two quadratic polynomials with integer coefficients. Submit your answer in the form $(ax^2+bx+c)(dx^2+ex+f)$, with $a<d$. | (-9x^2+x+2)(9x^2+x+2) |
Elon Musk's Starlink project belongs to his company SpaceX. He plans to use tens of thousands of satellites to provide internet services to every corner of the Earth. A domestic company also plans to increase its investment in the development of space satellite networks to develop space internet. It is known that the r... | 50 |
Find the largest positive integer solution of the equation $\left\lfloor\frac{N}{3}\right\rfloor=\left\lfloor\frac{N}{5}\right\rfloor+\left\lfloor\frac{N}{7}\right\rfloor-\left\lfloor\frac{N}{35}\right\rfloor$. | 65 |
Two circles of radius $r$ are externally tangent to each other and internally tangent to the ellipse $x^2 + 5y^2 = 6,$ as shown below. Find $r.$
[asy]
size(7cm);
draw(scale(sqrt(6), sqrt(6)/sqrt(5))* unitcircle);
draw((0,-1.5)--(0,1.7),EndArrow);
draw((-3,0)--(3,0),EndArrow);
draw(Circle( (sqrt(0.96),0), sqrt(0.96) ));... | \frac{2\sqrt6}{5} |
For transportation between points located hundreds of kilometers apart on the Earth's surface, people of the future will likely dig straight tunnels through which capsules will travel frictionlessly under the influence of Earth's gravity. Let points \( A, B, \) and \( C \) lie on the same meridian, with the surface dis... | 42 |
Compute
\[\prod_{k = 1}^{12} \prod_{j = 1}^{10} (e^{2 \pi ji/11} - e^{2 \pi ki/13}).\] | 1 |
John attends a protest for 4 days. He then attends a second protest for 25% longer than the first. How many days did he spend protesting? | The second protest was 4*.25=<<4*.25=1>>1 day longer than the first.
So the second protest was 4+1=<<4+1=5>>5 days.
Thus he spent 4+5=<<4+5=9>>9 days protesting.
#### 9 |
Find all solutions to $aabb=n^4-6n^3$ , where $a$ and $b$ are non-zero digits, and $n$ is an integer. ( $a$ and $b$ are not necessarily distinct.) | 6655 |
The base 5 number $34x1_5$ is divisible by 31. What is the digit $x$? | 4 |
Petya approaches the entrance door with a combination lock, which has buttons numbered from 0 to 9. To open the door, three correct buttons need to be pressed simultaneously. Petya does not remember the code and tries combinations one by one. Each attempt takes Petya 2 seconds.
a) How much time will Petya need to defi... | \frac{29}{120} |
The coefficient of $x^2$ in the expansion of $(x-1) - (x-1)^2 + (x-1)^3 - (x-1)^4 + (x-1)^5$ is ____. | -20 |
From a large bottle containing 1 liter of alcohol, 1/3 liter of alcohol is poured out, an equal amount of water is added and mixed thoroughly. Then, 1/3 liter of the mixture is poured out, an equal amount of water is added and mixed thoroughly again. Finally, 1/3 liter of the mixture is poured out once more, and an equ... | 8/27 |
For some positive integer $k$, the repeating base-$k$ representation of the (base-ten) fraction $\frac{7}{51}$ is $0.\overline{23}_k = 0.232323..._k$. What is $k$? | 16 |
The second and fourth terms of a geometric sequence are 2 and 6. Which of the following is a possible first term? | $-\frac{2\sqrt{3}}{3}$ |
Whitney’s mom gave her two $20 bills to spend at the school book fair. Whitney has decided to buy 2 posters, 3 notebooks, and 2 bookmarks. Each poster costs $5, each notebook costs $4, and each bookmark costs $2. How much money, in dollars, will Whitney have left over after the purchase? | Whitney is buying 2 posters for $5 each, so the posters will cost 2*$5= $<<2*5=10>>10 total cost for posters.
Whitney is buying 3 notebooks for $4 each, so the notebooks will cost 3*$4= $<<3*4=12>>12 total cost for notebooks.
Whitney is buying 2 notebooks for $2, so the bookmarks 2*$2= $<<2*2=4>>4 total cost for bookma... |
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