problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Given the line $x-y+2=0$ and the circle $C$: $(x-3)^{2}+(y-3)^{2}=4$ intersect at points $A$ and $B$. The diameter through the midpoint of chord $AB$ is $MN$. Calculate the area of quadrilateral $AMBN$. | 4\sqrt{2} |
For each prime $p$, a polynomial $P(x)$ with rational coefficients is called $p$-good if and only if there exist three integers $a, b$, and $c$ such that $0 \leq a<b<c<\frac{p}{3}$ and $p$ divides all the numerators of $P(a)$, $P(b)$, and $P(c)$, when written in simplest form. Compute the number of ordered pairs $(r, s... | 12 |
Evaluate $(7 + 5 + 3) \div 3 - 2 \div 3$. | 4 \frac{1}{3} |
Determine the smallest positive integer $m$ such that $11m-3$ and $8m + 5$ have a common factor greater than $1$. | 108 |
The vectors $\mathbf{a} = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix},$ $\mathbf{b} = \begin{pmatrix} 2 \\ -3 \\ 1 \end{pmatrix},$ and $\mathbf{c} = \begin{pmatrix} 4 \\ 1 \\ -5 \end{pmatrix}$ are mutually orthogonal. There exist scalars $p,$ $q,$ and $r$ such that
\[\begin{pmatrix} -4 \\ 7 \\ 3 \end{pmatrix} = p \mathb... | \left( 2, -\frac{13}{7}, -\frac{4}{7} \right) |
Completely factor the following expression: \[(15x^3+80x-5)-(-4x^3+4x-5).\] | 19x(x^2+4) |
What is the sum of the reciprocals of the natural-number factors of 6? | 2 |
An isosceles trapezoid is circumscribed around a circle. The longer base of the trapezoid is $16$, and one of the base angles is $\arcsin(.8)$. Find the area of the trapezoid.
$\textbf{(A)}\ 72\qquad \textbf{(B)}\ 75\qquad \textbf{(C)}\ 80\qquad \textbf{(D)}\ 90\qquad \textbf{(E)}\ \text{not uniquely determined}$
| 80 |
In triangle \(ABC\), side \(AB\) is 21, the bisector \(BD\) is \(8 \sqrt{7}\), and \(DC\) is 8. Find the perimeter of the triangle \(ABC\). | 60 |
Find the product of the divisors of $50$. | 125,\!000 |
Given $cosθ+cos(θ+\frac{π}{3})=\frac{\sqrt{3}}{3},θ∈(0,\frac{π}{2})$, find $\sin \theta$. | \frac{-1 + 2\sqrt{6}}{6} |
In a cafeteria line, the number of people ahead of Kaukab is equal to two times the number of people behind her. There are $n$ people in the line. What is a possible value of $n$? | 25 |
If the surface area of a cone is $3\pi$, and its lateral surface unfolds into a semicircle, then the diameter of the base of the cone is ___. | \sqrt{6} |
If $23=x^4+\frac{1}{x^4}$, then what is the value of $x^2+\frac{1}{x^2}$? | 5 |
Given that $m \angle A= 60^\circ$, $BC=12$ units, $\overline{BD} \perp \overline{AC}$, $\overline{CE} \perp \overline{AB}$ and $m \angle DBC = 3m \angle ECB$, the length of segment $EC$ can be expressed in the form $a(\sqrt{b}+\sqrt{c})$ units where $b$ and $c$ have no perfect-square factors. What is the value of $a+b... | 11 |
Given the planar vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy $\overrightarrow{a} \cdot (\overrightarrow{a} + \overrightarrow{b}) = 3$, and $|\overrightarrow{a}| = 2$, $|\overrightarrow{b}| = 1$, find the angle between the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{2\pi}{3} |
Let \( f(x) = x - \frac{x^3}{2} + \frac{x^5}{2 \cdot 4} - \frac{x^7}{2 \cdot 4 \cdot 6} + \cdots \), and \( g(x) = 1 + \frac{x^2}{2^2} + \frac{x^4}{2^2 \cdot 4^2} + \frac{x^6}{2^2 \cdot 4^2 \cdot 6^2} + \cdots \). Find \( \int_{0}^{\infty} f(x) g(x) \, dx \). | \sqrt{e} |
Every morning when Tim wakes up, he groggily searches around his sock drawer and picks two socks randomly. If he has 10 gray-bottomed socks and 8 white-bottomed socks in his drawer, what is the probability that he picks a matching pair? | \frac{73}{153} |
For any interval $\mathcal{A}$ in the real number line not containing zero, define its *reciprocal* to be the set of numbers of the form $\frac 1x$ where $x$ is an element in $\mathcal{A}$ . Compute the number of ordered pairs of positive integers $(m,n)$ with $m< n$ such that the length of the interval $[... | 60 |
Auston is 60 inches tall. Using the conversion 1 inch = 2.54 cm, how tall is Auston in centimeters? Express your answer as a decimal to the nearest tenth. | 152.4 |
In Morse code, each symbol is represented by a sequence of dashes and dots. How many distinct symbols can be represented using sequences of 1, 2, 3, or 4 total dots and/or dashes? | 30 |
A box contains one hundred multicolored balls: 28 red, 20 green, 13 yellow, 19 blue, 11 white, and 9 black. What is the minimum number of balls that must be drawn from the box, without looking, to ensure that at least 15 balls of one color are among them? | 76 |
How many different ways can the five vertices S, A, B, C, and D of a square pyramid S-ABCD be colored using four distinct colors so that each vertex is assigned one color and no two vertices sharing an edge have the same color? | 72 |
Calculate the sum of the sequence $2 - 6 + 10 - 14 + 18 - \cdots - 98 + 102$. | -52 |
Three vertices of parallelogram $ABCD$ are $A(-1, 3), B(2, -1), D(7, 6)$ with $A$ and $D$ diagonally opposite. Calculate the product of the coordinates of vertex $C$. | 40 |
Oleg drew an empty 50×50 table and wrote a number above each column and next to each row.
It turned out that all 100 written numbers are different, with 50 of them being rational and the remaining 50 being irrational. Then, in each cell of the table, he wrote the sum of the numbers written next to its row and its col... | 1250 |
If 10 people need 45 minutes and 20 people need 20 minutes to repair a dam, how many minutes would 14 people need to repair the dam? | 30 |
What is GCF(LCM(16, 21), LCM(14, 18))? | 14 |
To meet the shopping needs of customers during the "May Day" period, a fruit supermarket purchased cherries and cantaloupes from the fruit production base for $9160$ yuan, totaling $560$ kilograms. The purchase price of cherries is $35$ yuan per kilogram, and the purchase price of cantaloupes is $6 yuan per kilogram. ... | 35 |
A rigid board with a mass \( m \) and a length \( l = 20 \) meters partially lies on the edge of a horizontal surface, overhanging it by three quarters of its length. To prevent the board from falling, a stone with a mass of \( 2m \) was placed at its very edge. How far from the stone can a person with a mass of \( m /... | 15 |
In the diagram, $\triangle ABC$ is right-angled at $A$ with $AB = 3$ and $AC = 3\sqrt{3}$. Altitude $AD$ intersects median $BE$ at point $G$. Find the length of $AG$. Assume the diagram has this configuration:
[Contextual ASY Diagram]
```
draw((0,0)--(9,0)--(0,10*sqrt(3))--cycle);
draw((0,0)--(7.5,4.33)); draw((0,10*... | -\frac{1.5\sqrt{3}}{3} |
Integers $0 \leq a, b, c, d \leq 9$ satisfy $$\begin{gathered} 6 a+9 b+3 c+d=88 \\ a-b+c-d=-6 \\ a-9 b+3 c-d=-46 \end{gathered}$$ Find $1000 a+100 b+10 c+d$ | 6507 |
Dani wrote the integers from 1 to \( N \). She used the digit 1 fifteen times. She used the digit 2 fourteen times.
What is \( N \) ? | 41 |
What is $(-1)^1+(-1)^2+\cdots+(-1)^{2006}$ ? | 0 |
Andy gets a cavity for every 4 candy canes he eats. He gets 2 candy canes from his parents and 3 candy canes each from 4 teachers. Then he uses his allowance to buy 1/7 as many candy canes as he was given. How many cavities does he get from eating all his candy canes? | First find how many candy canes Andy gets from his teachers: 3 canes/teacher * 4 teachers = <<3*4=12>>12 canes
Then add the number of candy canes he gets from his parents: 12 canes + 2 canes = <<12+2=14>>14 canes
Then divide that number by 7 to find the number of canes he buys: 14 canes / 7 = <<14/7=2>>2 canes
Then add... |
A triangle with angles \( A, B, C \) satisfies the following conditions:
\[
\frac{\sin A + \sin B + \sin C}{\cos A + \cos B + \cos C} = \frac{12}{7},
\]
and
\[
\sin A \sin B \sin C = \frac{12}{25}.
\]
Given that \( \sin C \) takes on three possible values \( s_1, s_2 \), and \( s_3 \), find the value of \( 100 s_1 s_2... | 48 |
Let $M$ be the number of $8$-digit positive integers such that the digits are in both increasing order and even. Determine the remainder obtained when $M$ is divided by $1000$. (Repeated digits are allowed.) | 165 |
Jackson is making pancakes with three ingredients: flour, milk and eggs. 20% of the bottles of milk are spoiled and the rest are fresh. 60% of the eggs are rotten. 1/4 of the cannisters of flour have weevils in them. If Jackson picks a bottle of milk, an egg and a canister of flour at random, what are the odds all thre... | First find the percentage of milk bottles that are fresh: 100% - 20% = 80%
Then find the percentage of the eggs that aren't rotten: 100% - 60% = 40%
Then find the fraction of the flour canisters that don't have weevils: 1 - 1/4 = 3/4
Divide the numerator of this fraction by the denominator and multiply by 100% to conve... |
Define a sequence of integers by $T_1 = 2$ and for $n\ge2$ , $T_n = 2^{T_{n-1}}$ . Find the remainder when $T_1 + T_2 + \cdots + T_{256}$ is divided by 255.
*Ray Li.* | 20 |
Point $C(0,p)$ lies on the $y$-axis between $Q(0,12)$ and $O(0,0)$ as shown. Determine an expression for the area of $\triangle COB$ in terms of $p$. Your answer should be simplified as much as possible. [asy]
size(5cm);defaultpen(fontsize(9));
pair o = (0, 0); pair q = (0, 12); pair b = (12, 0);
pair a = (2, 12); pair... | 6p |
Find the least positive integer $n$ such that
$\frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}.$ | 1 |
A mother is making her own bubble mix out of dish soap and water for her two-year-old son. The recipe she is following calls for 3 tablespoons of soap for every 1 cup of water. If the container she hopes to store the bubble mix in can hold 40 ounces of liquid, and there are 8 ounces in a cup of water, how many tables... | The 40-ounce container can hold 40/8 = <<40/8=5>>5 cups of water.
This means that 5*3 = <<5*3=15>>15 tablespoons of soap should be used, since each cup of water requires 3 tablespoons of soap.
#### 15 |
$100_{10}$ in base $b$ has exactly $5$ digits. What is the value of $b$? | 3 |
A triangle has vertices $A(0,0)$, $B(12,0)$, and $C(8,10)$. The probability that a randomly chosen point inside the triangle is closer to vertex $B$ than to either vertex $A$ or vertex $C$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. | 409 |
Calculate the number of minutes in a week. | 10000 |
A ball is dropped from 1000 feet high and always bounces back up half the distance it just fell. After how many bounces will the ball first reach a maximum height less than 1 foot? | 10 |
For positive integers $N$ and $k$, define $N$ to be $k$-nice if there exists a positive integer $a$ such that $a^{k}$ has exactly $N$ positive divisors. Find the number of positive integers less than $1000$ that are neither $7$-nice nor $8$-nice.
| 749 |
In unit square $ABCD,$ the inscribed circle $\omega$ intersects $\overline{CD}$ at $M,$ and $\overline{AM}$ intersects $\omega$ at a point $P$ different from $M.$ What is $AP?$ | \frac{\sqrt{5}}{10} |
Each of the first eight prime numbers is placed in a bowl. Two primes are drawn without replacement. What is the probability, expressed as a common fraction, that the sum of the two numbers drawn is a prime number? | \frac17 |
Square $PQRS$ has sides of length 1. Points $T$ and $U$ are on $\overline{QR}$ and $\overline{RS}$, respectively, so that $\triangle PTU$ is equilateral. A square with vertex $Q$ has sides that are parallel to those of $PQRS$ and a vertex on $\overline{PT}.$ The length of a side of this smaller square is $\frac{a-\sqrt... | 12 |
Evaluate $3x^y + 4y^x$ when $x=2$ and $y=3$. | 60 |
Given a regular tetrahedron with an edge length of \(2 \sqrt{6}\), a sphere is centered at the centroid \(O\) of the tetrahedron. The total length of the curves where the sphere intersects with the four faces of the tetrahedron is \(4 \pi\). Find the radius of the sphere centered at \(O\). | \frac{\sqrt{5}}{2} |
After sharing 100 stickers with her friends, Xia had five sheets of stickers left. If each sheet had ten stickers on it, how many stickers did Xia have at the beginning? | Five sheet have 5 x 10 = <<5*10=50>>50 stickers.
Therefore, Xia had 100 + 50 = <<100+50=150>>150 stickers at the beginning.
#### 150 |
A teacher was leading a class of four perfectly logical students. The teacher chose a set $S$ of four integers and gave a different number in $S$ to each student. Then the teacher announced to the class that the numbers in $S$ were four consecutive two-digit positive integers, that some number in $S$ was divisible by $... | 258 |
Let $P(x) = (x-1)(x-4)(x-5)$. Determine how many polynomials $Q(x)$ there exist such that there exists a polynomial $R(x)$ of degree 3 with $P(Q(x)) = P(x) \cdot R(x)$, and the coefficient of $x$ in $Q(x)$ is 6. | 22 |
A lot of snow has fallen, and the kids decided to make snowmen. They rolled 99 snowballs with masses of 1 kg, 2 kg, 3 kg, ..., up to 99 kg. A snowman consists of three snowballs stacked on top of each other, and one snowball can be placed on another if and only if the mass of the first is at least half the mass of the ... | 24 |
Given a number \\(x\\) randomly selected from the interval \\(\left[-\frac{\pi}{4}, \frac{2\pi}{3}\right]\\), find the probability that the function \\(f(x)=3\sin\left(2x- \frac{\pi}{6}\right)\\) is not less than \\(0\\). | \frac{6}{11} |
What is the smallest positive integer $n$ such that $\frac{n}{n+101}$ is equal to a terminating decimal? | 24 |
Jenny wants to sell some girl scout cookies and has the choice of two neighborhoods to visit. Neighborhood A has 10 homes which each will buy 2 boxes of cookies. Neighborhood B has 5 homes, each of which will buy 5 boxes of cookies. Assuming each box of cookies costs $2, how much will Jenny make at the better choice... | First, we need to determine which neighborhood is the better choice by comparing the number of boxes sold. A has 10 houses which will buy 2 boxes each, meaning A will give Jenny 10*2=<<10*2=20>>20 box sales
Neighborhood B has 5 homes which will each buy 5 boxes, meaning she will sell 5*5=<<5*5=25>>25 boxes by going to ... |
What prime is 4 greater than a perfect square and 7 less than the next perfect square? | 29 |
Given two points A (-2, 0), B (0, 2), and point C is any point on the circle $x^2+y^2-2x=0$, determine the minimum area of $\triangle ABC$. | 3- \sqrt{2} |
Evaluate the expression $\sqrt{25\sqrt{15\sqrt{9}}}$. | 5\sqrt{15} |
Every evening, Laura gathers three socks randomly to pack for her gym session next morning. She has 12 black socks, 10 white socks, and 6 striped socks in her drawer. What is the probability that all three socks she picks are of the same type? | \frac{60}{546} |
Let $\Delta ABC$ be an acute triangle with circumcenter $O$ and centroid $G$. Let $X$ be the intersection of the line tangent to the circumcircle of $\Delta ABC$ at $A$ and the line perpendicular to $GO$ at $G$. Let $Y$ be the intersection of lines $XG$ and $BC$. Given that the measures of $\angle ABC, \angle BCA,$ and... | 592 |
In a truck, there are 26 pink hard hats, 15 green hard hats, and 24 yellow hard hats. If Carl takes away 4 pink hard hats, and John takes away 6 pink hard hats and twice as many green hard hats as the number of pink hard hats that he removed, then calculate the total number of hard hats that remained in the truck. | If there were 26 pink hard hats and Carl took away 4 pink hard hats, the number of pink hard hats that remained is 26-4 = <<26-4=22>>22
John also took away 6 pink hard hats, leaving 22-6 = <<22-6=16>>16 pink hard hats in the truck.
If John also took twice as many green hard hats as pink hard hats, he took 2*6 = <<6*2=1... |
For a given positive integer \( k \), let \( f_{1}(k) \) represent the square of the sum of the digits of \( k \), and define \( f_{n+1}(k) = f_{1}\left(f_{n}(k)\right) \) for \( n \geq 1 \). Find the value of \( f_{2005}\left(2^{2006}\right) \). | 169 |
In how many ways can 5 people be seated around a round table? (Two seatings are considered the same if one is a rotation of the other.) | 24 |
We shuffle a deck of French playing cards and then draw the cards one by one. In which position is it most likely to draw the second ace? | 18 |
The sum of two numbers is 22. Their difference is 4. What is the greater of the two numbers? | 13 |
I am playing a walking game with myself. On move 1, I do nothing, but on move $n$ where $2 \le n \le 25$, I take one step forward if $n$ is prime and two steps backwards if the number is composite. After all 25 moves, I stop and walk back to my original starting point. How many steps long is my walk back? | 21 |
Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is 1/19 of the original integer. | 950 |
What is the largest value among $\operatorname{lcm}[12,2],$ $\operatorname{lcm}[12,4],$ $\operatorname{lcm}[12,6],$ $\operatorname{lcm}[12,8],$ $\operatorname{lcm}[12,10],$ and $\operatorname{lcm}[12,12]?$ Express your answer as an integer. | 60 |
Find all the pairs of prime numbers $ (p,q)$ such that $ pq|5^p\plus{}5^q.$ | (2, 3), (2, 5), (3, 2), (5, 2), (5, 5), (5, 313), (313, 5) |
Two subsets of the set $T = \{w, x, y, z, v\}$ need to be chosen so that their union is $T$ and their intersection contains exactly three elements. How many ways can this be accomplished, assuming the subsets are chosen without considering the order? | 20 |
Find the smallest integer $n$ such that $(x^2+y^2+z^2)^2 \le n(x^4+y^4+z^4)$ for all real numbers $x,y$, and $z$. | 3 |
If 4 daps are equivalent to 3 dops, and 2 dops are equivalent to 7 dips, how many daps are equivalent to 42 dips? | 16\text{ daps} |
A teacher gave a test to a class in which $10\%$ of the students are juniors and $90\%$ are seniors. The average score on the test was $84.$ The juniors all received the same score, and the average score of the seniors was $83.$ What score did each of the juniors receive on the test? | 93 |
Given the cyclist encounters red lights at each of 4 intersections with probability $\frac{1}{3}$ and the events of encountering red lights are independent, calculate the probability that the cyclist does not encounter red lights at the first two intersections and encounters the first red light at the third intersectio... | \frac{4}{27} |
On a straight segment of a one-way, single-lane highway, cars travel at the same speed and follow a safety rule where the distance from the back of one car to the front of the next is equal to the car’s speed divided by 10 kilometers per hour, rounded up to the nearest whole number (e.g., a car traveling at 52 kilomete... | 92 |
The equation $x^3 - 9x^2 + 8x +2 = 0$ has three real roots $p$, $q$, $r$. Find $\frac{1}{p^2} + \frac{1}{q^2} + \frac{1}{r^2}$. | 25 |
Joe found a new series to watch with a final season that will premiere in 10 days. The series has 4 full seasons already aired; each season has 15 episodes. To catch up with the season premiere he has to watch all episodes. How many episodes per day does Joe have to watch? | The total number of episodes is 4 seasons x 15 episodes/season = <<4*15=60>>60 episodes
Joe has to watch per day 60 episodes ÷ 10 days= <<60/10=6>>6 episodes/day
#### 6 |
If the height of an external tangent cone of a sphere is three times the radius of the sphere, determine the ratio of the lateral surface area of the cone to the surface area of the sphere. | \frac{3}{2} |
For how many integer values of $b$ does the equation $$x^2 + bx + 12b = 0$$ have integer solutions for $x$? | 16 |
What is the smallest positive integer $n$ for which $11n - 3$ and $8n + 2$ share a common factor greater than $1$? | 19 |
In the year 2023, the International Mathematical Olympiad will be hosted by a country. Let $A$, $B$, and $C$ be distinct positive integers such that the product $A \cdot B \cdot C = 2023$. What is the largest possible value of the sum $A + B + C$? | 297 |
There are 320 ducks in a pond. On the first night 1/4 of them get eaten by a fox. On the second night 1/6 of the remaining ducks fly away, and on the third night 30 percent are stolen. How many ducks remain after the three nights? | First night:320(1/4)=80
320-80=<<320-80=240>>240
Second night:240(1/6)=40
240-40=<<240-40=200>>200
Third night:200(.30)=60
200-60=<<200-60=140>>140 ducks remain
#### 140 |
Given the function $f(x) = a\sin(\pi x + \alpha) + b\cos(\pi x + \beta)$, and it is known that $f(2001) = 3$, find the value of $f(2012)$. | -3 |
A solid cube of side length \(4 \mathrm{~cm}\) is cut into two pieces by a plane that passed through the midpoints of six edges. To the nearest square centimetre, the surface area of each half cube created is: | 69 |
Calculate: $(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})^{2}=\_\_\_\_\_\_$. | \sqrt{3}-\sqrt{2} |
In right triangle $DEF$, where $DE=15$, $DF=9$, and $EF=12$ units, what is the distance from $D$ to the midpoint of segment $EF$? | 7.5 |
Tony wants to build the longest rope he possibly can, so he collects all the rope in his home. He finds an 8-foot rope, a 20-foot rope, three 2 foot ropes, and a 7-foot rope. each knot between the ropes makes him lose 1.2 foot per knot. How long is his rope when he's done tying them all together? | The combined length of all the rope is 41 feet because 8 + 20 + (2 x 3) + 7 = <<8+20+(2*3)+7=41>>41
He loses 6 feet from the knots because 5 x 1.2 = <<5*1.2=6>>6
The final length is 35 feet because 41 - 6 = <<41-6=35>>35
#### 35 |
Add $46.913$ to $58.27$ and round your answer to the nearest hundredth. | 105.18 |
Calculate the value of $(-3 \frac{3}{8})^{- \frac{2}{3}}$. | \frac{4}{9} |
Two integers are relatively prime if they have no common factors other than 1 or -1. What is the probability that a positive integer less than or equal to 30 is relatively prime to 30? Express your answer as a common fraction. | \frac{4}{15} |
How can you weigh 1 kg of grain on a balance scale using two weights, one weighing 300 g and the other 650 g? | 1000 |
The third and twentieth terms of an arithmetic sequence are 10 and 65, respectively. What is the thirty-second term? | 103.8235294118 |
It is known that the only solution to the equation
$$
\pi / 4 = \operatorname{arcctg} 2 + \operatorname{arcctg} 5 + \operatorname{arcctg} 13 + \operatorname{arcctg} 34 + \operatorname{arcctg} 89 + \operatorname{arcctg}(x / 14)
$$
is a natural number. Find it. | 2016 |
Circle $C_1$ has its center $O$ lying on circle $C_2$. The two circles meet at $X$ and $Y$. Point $Z$ in the exterior of $C_1$ lies on circle $C_2$ and $XZ=13$, $OZ=11$, and $YZ=7$. What is the radius of circle $C_1$? | \sqrt{30} |
Victor works at Clucks Delux, a restaurant specializing in chicken. An order of Chicken Pasta uses 2 pieces of chicken, an order of Barbecue Chicken uses 3 pieces of chicken, and a family-size Fried Chicken Dinner uses 8 pieces of chicken. Tonight, Victor has 2 Fried Chicken Dinner orders, 6 Chicken Pasta orders, and 3... | Victor needs 2 * 8 = <<2*8=16>>16 pieces of chicken for Fried Chicken Dinners.
He needs 6 * 2 = <<6*2=12>>12 pieces of chicken for Chicken Pasta.
He needs 3 * 3 = <<3*3=9>>9 pieces of chicken for Barbecue Chicken orders.
Thus, he needs 16 + 12 + 9 = <<16+12+9=37>>37 pieces of chicken.
#### 37 |
Let $\mathcal{T}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{ab}$ where $a$ and $b$ are distinct digits. Find the sum of the elements of $\mathcal{T}$. | \frac{90}{11} |
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