problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
What is the largest divisor by which $29 \cdot 14$ leaves the same remainder when divided by $13511, 13903,$ and $14589$? | 98 |
Given numbers \(a, b, c\) satisfy \(a b c+a+c-b\). Then the maximum value of the algebraic expression \(\frac{1}{1+a^{2}}-\frac{1}{1+b^{2}}+\frac{1}{1+c^{2}}\) is | \frac{5}{4} |
A rectangle has a perimeter of 30 units and its dimensions are whole numbers. What is the maximum possible area of the rectangle in square units? | 56 |
The operation $\star$ is defined as $a \star b = a^2 \div b$. For how many integer values of $x$ will the value of $10 \star x$ be a positive integer? | 9 |
Solve the inequality
\[|x - 1| + |x + 2| < 5.\] | (-3,2) |
An apartment building consists of 20 rooms numbered $1,2, \ldots, 20$ arranged clockwise in a circle. To move from one room to another, one can either walk to the next room clockwise (i.e. from room $i$ to room $(i+1)(\bmod 20))$ or walk across the center to the opposite room (i.e. from room $i$ to room $(i+10)(\bmod 2... | 257 |
A cowboy is initially 6 miles south and 2 miles west of a stream that flows due northeast. His cabin is located 12 miles east and 9 miles south of his initial position. He wants to water his horse at the stream and then return to his cabin. What is the shortest distance he can travel to accomplish this?
A) $\sqrt{289} ... | 8 + \sqrt{545} |
Calculate the area, in square units, of the triangle formed by the $x$ and $y$ intercepts of the curve $y = (x-3)^2 (x+2) (x-1)$. | 45 |
Two distinct non-consecutive positive integers $x$ and $y$ are factors of 48. If $x\cdot y$ is not a factor of 48, what is the smallest possible value of $x\cdot y$? | 18 |
20 shareholders are seated around a round table. What is the minimum total number of their shares if it is known that:
a) any three of them together have more than 1000 shares,
b) any three consecutive shareholders together have more than 1000 shares? | 6674 |
For how many integers $n$ between $1$ and $50$, inclusive, is $\frac{(n^2-1)!}{(n!)^n}$ an integer? | 34 |
Line segment $\overline{AB}$ is extended past $B$ to $P$ such that $AP:PB = 10:3.$ Then
\[\overrightarrow{P} = t \overrightarrow{A} + u \overrightarrow{B}\]for some constants $t$ and $u.$ Enter the ordered pair $(t,u).$
[asy]
unitsize(1 cm);
pair A, B, P;
A = (0,0);
B = (5,1);
P = interp(A,B,10/7);
draw(A--P);
d... | \left( -\frac{3}{7}, \frac{10}{7} \right) |
Find \( g(2021) \) if for any real numbers \( x \) and \( y \) the following equality holds:
\[ g(x-y) = 2021(g(x) + g(y)) - 2022xy \] | 2043231 |
Two cars start from the same location at the same time, moving in the same direction at a constant speed. Each car can carry a maximum of 24 barrels of gasoline, and each barrel of gasoline allows a car to travel 60km. Both cars must return to the starting point, but they do not have to return at the same time. The car... | 360 |
We call a number a mountain number if its middle digit is larger than any other digit. For example, 284 is a mountain number. How many 3-digit mountain numbers are there? | 240 |
I take variable $b$, double it, and add four. I subtract $4b$ from this new expression, and divide the resulting difference by two. What is my final expression in simplest form? | 2 - b |
Find the smallest positive integer \( n \) that is not less than 9, such that for any \( n \) integers (which can be the same) \( a_{1}, a_{2}, \cdots, a_{n} \), there always exist 9 numbers \( a_{i_{1}}, a_{i_{2}}, \cdots, a_{i_{9}} \) (where \(1 \leq i_{1} < i_{2} < \cdots < i_{9} \leq n \)) and \( b_{i} \in \{4,7\} ... | 13 |
Given vectors \(\overrightarrow{O P}=\left(2 \cos \left(\frac{\pi}{2}+x\right),-1\right)\) and \(\overrightarrow{O Q}=\left(-\sin \left(\frac{\pi}{2}- x\right), \cos 2 x\right)\), and the function \(f(x)=\overrightarrow{O P} \cdot \overrightarrow{O Q}\). If \(a, b, c\) are the sides opposite angles \(A, B, C\) respecti... | 15/2 |
Find the largest constant $m,$ so that for any positive real numbers $a,$ $b,$ $c,$ and $d,$
\[\sqrt{\frac{a}{b + c + d}} + \sqrt{\frac{b}{a + c + d}} + \sqrt{\frac{c}{a + b + d}} + \sqrt{\frac{d}{a + b + c}} > m.\] | 2 |
In triangle $ABC$, angle $B$ equals $120^\circ$, and $AB = 2 BC$. The perpendicular bisector of side $AB$ intersects $AC$ at point $D$. Find the ratio $CD: DA$. | 3:2 |
A dragon has 40 piles of gold coins, with the number of coins in any two piles differing. After the dragon plundered a neighboring city and brought back more gold, the number of coins in each pile increased by either 2, 3, or 4 times. What is the minimum number of different piles of coins that could result? | 14 |
What is the smallest positive integer with exactly 16 positive divisors? | 384 |
A student research group at a school found that the attention index of students during class changes with the listening time. At the beginning of the lecture, students' interest surges; then, their interest remains in a relatively ideal state for a while, after which students' attention begins to disperse. Let $f(x)$ r... | \dfrac {85}{3} |
Jeff bought 6 pairs of shoes and 4 jerseys for $560. Jerseys cost 1/4 price of one pair of shoes. Find the shoe's price total price. | Let X be the shoe price. The jersey's price is 1/4*X.
Jeff bought 6 pairs of shoes and 4 jerseys for 6*X + 4*(1/4X) = $560.
Multiplying through the parentheses produces 6X + X = $560
Combining like terms produces 7X = $560
Dividing both sides by 7 produces X = $80, so the shoe price is $80
Since Jeff bought 6 pairs of ... |
Express the sum as a common fraction: $.1 + .02 + .003 + .0004 + .00005.$ | \dfrac{2469}{20,\!000} |
The Absent-Minded Scientist had a sore knee. The doctor prescribed him 10 pills for his knee: take one pill daily. The pills are effective in $90\%$ of cases, and in $2\%$ of cases, there is a side effect—absent-mindedness disappears, if present.
Another doctor prescribed the Scientist pills for absent-mindedness—also... | 0.69 |
Calculate $180 \div \left( 12 + 9 \times 3 - 4 \right)$. | \frac{36}{7} |
Let \( a_1, a_2, \dots \) be a sequence of positive real numbers such that
\[ a_n = 7a_{n-1} - 2n \] for all \( n > 1 \). Find the smallest possible value of \( a_1 \). | \frac{13}{18} |
On a circle, points \(B\) and \(D\) are located on opposite sides of the diameter \(AC\). It is known that \(AB = \sqrt{6}\), \(CD = 1\), and the area of triangle \(ABC\) is three times the area of triangle \(BCD\). Find the radius of the circle. | 1.5 |
Travis is hopping around on the vertices of a cube. Each minute he hops from the vertex he's currently on to the other vertex of an edge that he is next to. After four minutes, what is the probability that he is back where he started? | 7/27 |
The average of the numbers $1, 2, 3,\dots, 98, 99,$ and $x$ is $100x$. What is $x$? | \frac{50}{101} |
Simplify $\sqrt{\frac{1}{{49}}}=$____; $|{2-\sqrt{5}}|=$____. | \sqrt{5}-2 |
Given the function $f(x)= \begin{cases} 2x-10, & x\leqslant 7 \\ \frac {1}{f(x-2)}, & x > 7 \end{cases}$, and the sequence ${a_{n}}={f(n)}$ where $n\in\mathbb{N}^{*}$, find the sum of the first 50 terms of the sequence ${a_{n}}$. | \frac {225}{4} |
Segment \( BD \) is the median of an isosceles triangle \( ABC \) (\( AB = BC \)). A circle with a radius of 4 passes through points \( B \), \( A \), and \( D \), and intersects side \( BC \) at point \( E \) such that \( BE : BC = 7 : 8 \). Find the perimeter of triangle \( ABC \). | 20 |
A point $P$ is outside a circle and is $13$ inches from the center. A secant from $P$ cuts the circle at $Q$ and $R$ so that the external segment of the secant $PQ$ is $9$ inches and $QR$ is $7$ inches. The radius of the circle is: | 5 |
For any positive real numbers \(a\) and \(b\), define \(a \circ b=a+b+2 \sqrt{a b}\). Find all positive real numbers \(x\) such that \(x^{2} \circ 9x=121\). | \frac{31-3\sqrt{53}}{2} |
Each of the $20$ balls is tossed independently and at random into one of the $5$ bins. Let $p$ be the probability that some bin ends up with $3$ balls, another with $5$ balls, and the other three with $4$ balls each. Let $q$ be the probability that every bin ends up with $4$ balls. What is $\frac{p}{q}$? | 4 |
Find the smallest positive integer solution to $\tan{19x^{\circ}}=\dfrac{\cos{96^{\circ}}+\sin{96^{\circ}}}{\cos{96^{\circ}}-\sin{96^{\circ}}}$. | 159 |
Given the sequence ${a_n}$, $a_1=1$ and $a_n a_{n+1} + \sqrt{3}(a_n - a_{n+1}) + 1 = 0$. Determine the value of $a_{2016}$. | 2 - \sqrt{3} |
Points $K$, $L$, $M$, and $N$ lie in the plane of the square $ABCD$ so that $AKB$, $BLC$, $CMD$, and $DNA$ are equilateral triangles. If $ABCD$ has an area of 16, find the area of $KLMN$. Express your answer in simplest radical form.
[asy]
pair K,L,M,I,A,B,C,D;
D=(0,0);
C=(10,0);
B=(10,10);
A=(0,10);
I=(-8.7,5);
L=(18... | 32 + 16\sqrt{3} |
Natural numbers of the form $F_n=2^{2^n} + 1 $ are called Fermat numbers. In 1640, Fermat conjectured that all numbers $F_n$, where $n\neq 0$, are prime. (The conjecture was later shown to be false.) What is the units digit of $F_{1000}$? | 7 |
Each of the integers 334 and 419 has digits whose product is 36. How many 3-digit positive integers have digits whose product is 36? | 21 |
Ten positive integers are arranged around a circle. Each number is one more than the greatest common divisor of its two neighbors. What is the sum of the ten numbers? | 28 |
Max and Minnie each add up sets of three-digit positive integers. Each of them adds three different three-digit integers whose nine digits are all different. Max creates the largest possible sum. Minnie creates the smallest possible sum. What is the difference between Max's sum and Minnie's sum? | 1845 |
The regular octagon $ABCDEFGH$ has its center at $J$. Each of the vertices and the center are to be associated with one of the digits $1$ through $9$, with each digit used once, in such a way that the sums of the numbers on the lines $AJE$, $BJF$, $CJG$, and $DJH$ are equal. In how many ways can this be done? [asy]
siz... | 1152 |
The numbers 60, 221, and 229 are the legs and hypotenuse of a right triangle. Find the multiplicative inverse to 450 modulo 3599. (Express your answer as an integer $n$ with $0\leq n<3599$.) | 8 |
Let \( F_{1} \) and \( F_{2} \) be the two foci of an ellipse. A circle with center \( F_{2} \) is drawn, which passes through the center of the ellipse and intersects the ellipse at point \( M \). If the line \( ME_{1} \) is tangent to circle \( F_{2} \) at point \( M \), find the eccentricity \( e \) of the ellipse. | \sqrt{3}-1 |
A line segment is divided into four parts by three randomly selected points. What is the probability that these four parts can form the four sides of a quadrilateral? | 1/2 |
Vitya and Masha were born in the same year in June. Find the probability that Vitya is at least one day older than Masha. | 29/60 |
The number 519 is formed using the digits 5, 1, and 9. The three digits of this number are rearranged to form the largest possible and then the smallest possible three-digit numbers. What is the difference between these largest and smallest numbers? | 792 |
Given that a blue ball and an orange ball are randomly and independently tossed into bins numbered with the positive integers, where for each ball the probability that it is tossed into bin k is 3^(-k) for k = 1, 2, 3, ..., determine the probability that the blue ball is tossed into a higher-numbered bin than the orang... | \frac{7}{16} |
John has five children. What is the probability that at least half of them are girls? (We can assume a boy is equally likely to be born as is a girl, and vice-versa.) | \frac{1}{2} |
Katrina saw an ad that said if she signed up for her local recycling program, she could earn $5.00. When she signed up, they told her for every friend that she referred, the friend would receive $5.00 and she would receive another $5.00 per friend. That day, she had 5 friends sign up and another 7 friends by the end ... | She signed up 5 friends, then an additional 7 for a total of 5+7 =<<5+7=12>>12 friends
Each friend received $5 so 12*5 = $<<12*5=60.00>>60.00
And she received $5 for each friend so 12*5 = $<<12*5=60.00>>60.00
Her initial reward was $5 and her friends made $60 for signing up and she received $60 when they signed up for ... |
Given vectors $\overrightarrow{a}=(\cos \alpha,\sin \alpha)$, $\overrightarrow{b}=(\cos x,\sin x)$, $\overrightarrow{c}=(\sin x+2\sin \alpha,\cos x+2\cos \alpha)$, where $(0 < \alpha < x < \pi)$.
$(1)$ If $\alpha= \frac {\pi}{4}$, find the minimum value of the function $f(x)= \overrightarrow{b} \cdot \overrightarrow{c}... | - \frac { \sqrt {3}}{5} |
Triangle \( ABC \) has a right angle at \( B \). Point \( D \) lies on side \( BC \) such that \( 3 \angle BAD = \angle BAC \). Given \( AC = 2 \) and \( CD = 1 \), compute \( BD \). | \frac{3}{8} |
At a crossroads, if vehicles are not allowed to turn back, calculate the total number of possible driving routes. | 12 |
Let $A_1,B_1,C_1,D_1$ be the midpoints of the sides of a convex quadrilateral $ABCD$ and let $A_2, B_2, C_2, D_2$ be the midpoints of the sides of the quadrilateral $A_1B_1C_1D_1$ . If $A_2B_2C_2D_2$ is a rectangle with sides $4$ and $6$ , then what is the product of the lengths of the diagonals of $ABCD$ ... | 96 |
If one vertex and the two foci of an ellipse form an equilateral triangle, determine the eccentricity of this ellipse. | \dfrac{1}{2} |
Stacy bought two packs of printer paper for the office. Each pack has 240 sheets of paper. Her office prints 80 one-page documents per day. How many days will the printer paper last her office? | Stacy bought 2 * 240 = <<2*240=480>>480 sheets of paper.
At 80 pages per day, the printer paper will last her office 480 / 80 = <<480/80=6>>6 days.
#### 6 |
The total number of whales in the sea this year is double what it was last year. If the World Animals Organization predicts that there will be 800 more whales in the sea next year, and the number of whales in the sea last year was 4000, calculate the total number of whales in the sea next year if the predictions are ac... | If there were 4000 whales in the sea three years ago, and their number doubled this year, then there are 2*4000 = <<4000*2=8000>>8000 whales in the sea this year.
If the predictions come true, there will be 8000+800 = <<8000+800=8800>>8800 whales in the water next year.
#### 8800 |
Anna flips an unfair coin 10 times. The coin has a $\frac{1}{3}$ probability of coming up heads and a $\frac{2}{3}$ probability of coming up tails. What is the probability that she flips exactly 7 tails? | \frac{5120}{19683} |
The perpendicular to the side $AB$ of the trapezoid $ABCD$, passing through its midpoint $K$, intersects the side $CD$ at point $L$. It is known that the area of quadrilateral $AKLD$ is five times greater than the area of quadrilateral $BKLC$. Given $CL=3$, $DL=15$, and $KC=4$, find the length of segment $KD$. | 20 |
Alice, Bob, and Carol play a game in which each of them chooses a real number between 0 and 1. The winner of the game is the one whose number is between the numbers chosen by the other two players. Alice announces that she will choose her number uniformly at random from all the numbers between 0 and 1, and Bob announce... | \frac{13}{24} |
A chess piece called "the four-liner" attacks two vertical and two horizontal squares adjacent to the square on which it stands. What is the maximum number of non-attacking four-liners that can be placed on a $10 \times 10$ board? | 25 |
An equilateral triangle and a square are inscribed in a circle as shown. $ABC$ is isosceles. The triangle and square share a common vertex. What is the number of degrees in the measure of the angle indicated by the question mark?
[asy]
import markers; defaultpen(linewidth(0.8));
pair A,B,C,D,E,F,G;
draw(unitcircle)... | 75^\circ |
Find an integer $n$ such that the decimal representation of the number $5^{n}$ contains at least 1968 consecutive zeros. | 1968 |
Given that point $M(x\_0, y\_0)$ moves on the circle $x^{2}+y^{2}=4$, $N(4,0)$, and point $P(x,y)$ is the midpoint of segment $MN$.
(1) Find the trajectory equation of point $P(x,y)$;
(2) Find the maximum and minimum distances from point $P(x,y)$ to the line $3x+4y-86=0$. | 15 |
A line is parameterized by a parameter $t,$ so that the vector on the line at $t = -1$ is $\begin{pmatrix} 1 \\ 3 \\ 8 \end{pmatrix},$ and the vector on the line at $t = 2$ is $\begin{pmatrix} 0 \\ -2 \\ -4 \end{pmatrix}.$ Find the vector on the line at $t = 3.$ | \begin{pmatrix} -1/3 \\ -11/3 \\ -8 \end{pmatrix} |
The measures of angles $X$ and $Y$ are both positive, integer numbers of degrees. The measure of angle $X$ is a multiple of the measure of angle $Y$, and angles $X$ and $Y$ are supplementary angles. How many measures are possible for angle $X$? | 17 |
Given that three balls are randomly and independently tossed into bins numbered with the positive integers such that for each ball, the probability that it is tossed into bin i is $3^{-i}$ for i = 1,2,3,..., find the probability that all balls end up in consecutive bins. | 1/702 |
The sum of the first $n$ terms of the sequence $\{a_n\}$ is denoted as $S_n$. Given $a_1= \frac{1}{2}$ and $2a_{n+1}+S_n=0$ for $n=1, 2, 3, \ldots$. If $S_n \leqslant k$ always holds, then the minimum value of $k$ is \_\_\_\_\_\_. | \frac{1}{2} |
A certain point has rectangular coordinates $(10,3)$ and polar coordinates $(r, \theta).$ What are the rectangular coordinates of the point with polar coordinates $(r^2, 2 \theta)$? | (91,60) |
Both roots of the quadratic equation $x^2 - 63x + k = 0$ are prime numbers. Find the number of possible values of $k.$ | 1 |
Let $n \geq 3$ be an odd number and suppose that each square in a $n \times n$ chessboard is colored either black or white. Two squares are considered adjacent if they are of the same color and share a common vertex and two squares $a,b$ are considered connected if there exists a sequence of squares $c_1,\ldots,c_k$ wi... | \left(\frac{n+1}{2}\right)^2 + 1 |
Given an ellipse $E: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a>b>0)$ with an eccentricity of $\frac{\sqrt{2}}{2}$ and upper vertex at B. Point P is on E, point D is at (0, -2b), and the maximum area of △PBD is $\frac{3\sqrt{2}}{2}$.
(I) Find the equation of E;
(II) If line DP intersects E at another point Q, and lin... | \frac{2}{3} |
Niall's four children have different integer ages under 18. The product of their ages is 882. What is the sum of their ages? | 31 |
Two sectors of a circle of radius $12$ overlap as shown, with $P$ and $R$ as the centers of the respective circles. Determine the area of the shaded region.
[asy]
draw((0,0)--(10.3923,-6)--(20.7846,0)--(10.3923,6)--cycle,black+linewidth(1));
filldraw((10.3923,6)..(12,0)..(10.3923,-6)--cycle,gray,black+linewidth(1));
f... | 48\pi-72\sqrt{3} |
A mountain range has 200 active volcanoes. In a particular year, 20% of the volcanoes exploded in the first two months, 40% of the remaining exploded by the half of the year, and at the end of the year, another 50% of the volcanoes that hadn't already erupted also exploded. How many mountains are still intact at the en... | By the first two months, 20/100*200 = <<20/100*200=40>>40 mountains had erupted.
The total number of mountains remaining after the first round of explosions is 200-40= <<200-40=160>>160
When 40% of the remaining mountains exploded, the number of mountains that were still intact decreased by 40/100*160 = <<40/100*160=64... |
If $A$, $B$, and $C$ are the three interior angles of $\triangle ABC$, then the minimum value of $$\frac {4}{A}+ \frac {1}{B+C}$$ is \_\_\_\_\_\_. | \frac {9}{\pi} |
Abigail thinks she has some lost some money out of her purse. She had $11 in her purse at the start of the day, and she spent $2 in a store. If she now has $3 left, how much money has she lost? | After going shopping, Abigail had 11 – 2 = <<11-2=9>>9 dollars left.
This means she lost 9 – 3 = <<9-3=6>>6 dollars.
#### 6 |
Find $\csc 225^\circ.$ | -\sqrt{2} |
Alice has $24$ apples. In how many ways can she share them with Becky and Chris so that each of the three people has at least two apples? | 190 |
Given the function $f(x)=\frac{x}{ax+b}(a≠0)$, and its graph passes through the point $(-4,4)$, and is symmetric about the line $y=-x$, find the value of $a+b$. | \frac{3}{2} |
For each integer \( n \geq 2 \), let \( A(n) \) be the area of the region in the coordinate plane defined by the inequalities \( 1 \leq x \leq n \) and \( 0 \leq y \leq x \left\lfloor \log_2{x} \right\rfloor \), where \( \left\lfloor \log_2{x} \right\rfloor \) is the greatest integer not exceeding \( \log_2{x} \). Find... | 99 |
Given that Anne, Cindy, and Ben repeatedly take turns tossing a die in the order Anne, Cindy, Ben, find the probability that Cindy will be the first one to toss a five. | \frac{30}{91} |
Given the inequality $x^{2}-4ax+3a^{2} < 0 (a > 0)$ with respect to $x$, find the minimum value of $(x_{1}+x_{2}+\frac{a}{x_{1}x_{2}})$. | \frac{2\sqrt{3}}{3} |
Given that $\tan \alpha = 2$, where $\alpha$ is an angle in the first quadrant, find the value of $\sin 2\alpha + \cos \alpha$. | \dfrac{4 + \sqrt{5}}{5} |
Add $10_7 + 163_7.$ Express your answer in base 7. | 203_7 |
Jacqueline has 200 liters of a chemical solution. Liliane has 30% more of this chemical solution than Jacqueline, and Alice has 15% more than Jacqueline. Determine the percentage difference in the amount of chemical solution between Liliane and Alice. | 13.04\% |
In triangle $ABC$, the measure of $\angle A$ is $86$ degrees. The measure of $\angle B$ is $22$ degrees more than three times the measure of $\angle C$. What is the measure, in degrees, of $\angle C$? | 18 \text{ degrees} |
The ratio of cats to dogs at the pet store is 2:3. There are 14 cats. How many dogs are there at the pet store? | 21\text{ dogs} |
In a regular tetrahedron \( P-ABCD \) with lateral and base edge lengths both equal to 4, find the total length of all curve segments formed by a moving point on the surface at a distance of 3 from vertex \( P \). | 6\pi |
Each of the three cutlets needs to be fried on a pan on both sides for five minutes each side. The pan can hold only two cutlets at a time. Is it possible to fry all three cutlets in less than 20 minutes (ignoring the time for flipping and transferring the cutlets)? | 15 |
The parabolas $y = (x + 1)^2$ and $x + 4 = (y - 3)^2$ intersect at four points. All four points lie on a circle of radius $r.$ Find $r^2.$ | \frac{13}{2} |
The region consisting of all points in three-dimensional space within $4$ units of line segment $\overline{CD}$ has volume $544\pi$. Calculate the length of $CD$. | \frac{86}{3} |
A drawer contains a mixture of red socks and blue socks, at most $1991$ in all. It so happens that, when two socks are selected randomly without replacement, there is a probability of exactly $\frac{1}{2}$ that both are red or both are blue. What is the largest possible number of red socks in the drawer that is consist... | 990 |
Given that $E$ is the midpoint of the diagonal $BD$ of the square $ABCD$, point $F$ is taken on $AD$ such that $DF = \frac{1}{3} DA$. Connecting $E$ and $F$, the ratio of the area of $\triangle DEF$ to the area of quadrilateral $ABEF$ is: | 1: 5 |
For a permutation $\pi$ of the integers from 1 to 10, define
\[ S(\pi) = \sum_{i=1}^{9} (\pi(i) - \pi(i+1))\cdot (4 + \pi(i) + \pi(i+1)), \]
where $\pi (i)$ denotes the $i$ th element of the permutation. Suppose that $M$ is the maximum possible value of $S(\pi)$ over all permutations $\pi$ of the integers fr... | 40320 |
Find the product of the solutions of: $|y|=2(|y|-1)$. | -4 |
I bought a TV for $1700 excluding tax. What is the price of the television when calculating 15% of the value-added tax? | The value-added tax is 1700*15% = <<1700*15*.01=255>>255.
The price of the television including tax 1700+255 = $<<1700+255=1955>>1955.
#### 1955 |
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