problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Ten more than twice the number of birds on the fence is 50. How many birds are on the fence? | Let x be the number of birds on the fence. We know that 10 + 2*x = 50.
Subtracting 10 from both sides, we get 2*x = 50 - 10 = 40
Dividing both sides by 2, we get x = 20
#### 20 |
If the cotangents of the three interior angles \(A, B, C\) of triangle \(\triangle ABC\), denoted as \(\cot A, \cot B, \cot C\), form an arithmetic sequence, then the maximum value of angle \(B\) is \(\frac{\pi}{3}\). | \frac{\pi}{3} |
There are two ponds side by side in a park. Pond A has twice as many frogs as Pond B. If there are 32 frogs in Pond A, how many frogs are there in both ponds combined? | Pond B has 32/2 frogs = <<32/2=16>>16 frogs.
In total the ponds have 32 frogs + 16 frogs = <<32+16=48>>48 frogs.
#### 48 |
Numbers from 1 to 100 are written in a vertical row in ascending order. Fraction bars of different sizes are inserted between them. The calculation starts with the smallest fraction bar and ends with the largest one, for example, $\frac{1}{\frac{5}{3}}=\frac{15}{4}$. What is the greatest possible value that the resulti... | 100 |
When a water tank is $30\%$ full, it contains 27 gallons less than when it is $20\%$ empty. How many gallons of water does the tank hold when it is full? | 54\text{ gallons} |
How many different routes can Samantha take by biking on streets to the southwest corner of City Park, then taking a diagonal path through the park to the northeast corner, and then biking on streets to school? | 400 |
Fred wants to order a variety pack of sliced meats for the upcoming game. He can order a 4 pack of fancy, sliced meat for $40.00 and add rush delivery for an additional 30%. With rush shipping added, how much will each type of sliced meat cost? | The variety pack costs $40.00 and rush shipping is an additional 30% so that’s 40*.30 = $12.00
The variety pack costs $40.00 and rush shipping is $12.00 so they will cost 40+12 = $<<40+12=52.00>>52.00
The order will cost $52.00 and he will get 4 types of fancy, sliced meats so each type costs 52/4 = $<<52/4=13.00>>13.0... |
On the number line, points $M$ and $N$ divide $L P$ into three equal parts. What is the value at $M$? | \frac{1}{9} |
Three cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a $\clubsuit$, the second card is a $\heartsuit$, and the third card is a king? | \frac{13}{2550} |
A parking garage near Nora's house is 4 stories tall. There are 100 spots per level. There are 58 open parking spots on the first level. There are 2 more open parking spots on the second level than on the first level, and there are 5 more open parking spots on the third level than on the second level. There are 31 open... | There are 4 levels * 100 spots = <<4*100=400>>400 possible parking spots.
The second level has 58 open spots + 2 = <<58+2=60>>60 open spots.
The third level has 60 open spots + 5 = <<60+5=65>>65 open spots.
Total open spots are 58 + 60 + 65 + 31 = <<58+60+65+31=214>>214 open spots.
If there are 400 possible spots – 214... |
Given an arithmetic sequence $\{a_n\}$ with a positive common difference $d > 0$, and $a_2$, $a_5 - 1$, $a_{10}$ form a geometric sequence. If $a_1 = 5$, and $S_n$ is the sum of the first $n$ terms of the sequence $\{a_n\}$, determine the minimum value of $$\frac{2S_{n} + n + 32}{a_{n} + 1}$$ | \frac{20}{3} |
If \(x\) and \(y\) are positive real numbers such that \(6x^2 + 12xy + 6y^2 = x^3 + 3x^2 y + 3xy^2\), find the value of \(x\). | \frac{24}{7} |
There are 160 tissues inside a tissue box. If Tucker bought 3 boxes, and used 210 tissues while he was sick with the flu, how many tissues would he have left? | When Tucker buys the tissue boxes, he has a total of 160 x 3 = <<160*3=480>>480 tissues.
When he is sick with the flu and uses some of the tissues, he has 480 - 210 = <<480-210=270>>270 tissues left.
#### 270 |
The local school is holding a big fair to raise money for 5 classes that want to go on a trip. 150 people came to the party and paid a total of $368 for entrance tickets. There was a raffle which brought in $343. The sale of cakes and drinks brought $279. At the end of the fair, the principal of the school shared the m... | First, let’s calculate the total amount of money collected, which is 368 + 343 + 279 = $<<368+343+279=990>>990.
Let us now share this sum between the 5 classes: 990 / 5 = $<<990/5=198>>198.
#### 198 |
What is the probability that two people, A and B, randomly choosing their rooms among 6 different rooms in a family hotel, which has two rooms on each of the three floors, will stay in two rooms on the same floor? | \frac{1}{5} |
In triangle $\triangle ABC$, the sides opposite angles A, B, and C are denoted as $a$, $b$, and $c$ respectively. Given that $C = \frac{2\pi}{3}$ and $a = 6$:
(Ⅰ) If $c = 14$, find the value of $\sin A$;
(Ⅱ) If the area of $\triangle ABC$ is $3\sqrt{3}$, find the value of $c$. | 2\sqrt{13} |
Given a regular triangular pyramid \( S A B C \). Point \( S \) is the apex of the pyramid, \( AB = 1 \), \( AS = 2 \), \( BM \) is the median of triangle \( ABC \), and \( AD \) is the angle bisector of triangle \( SAB \). Find the length of segment \( DM \). | \frac{\sqrt{31}}{6} |
Omkar, \mathrm{Krit}_{1}, \mathrm{Krit}_{2}, and \mathrm{Krit}_{3} are sharing $x>0$ pints of soup for dinner. Omkar always takes 1 pint of soup (unless the amount left is less than one pint, in which case he simply takes all the remaining soup). Krit $_{1}$ always takes \frac{1}{6}$ of what is left, Krit ${ }_{2}$ alw... | \frac{49}{3} |
In triangle $ABC$, a midline $MN$ that connects the sides $AB$ and $BC$ is drawn. A circle passing through points $M$, $N$, and $C$ touches the side $AB$, and its radius is equal to $\sqrt{2}$. The length of side $AC$ is 2. Find the sine of angle $ACB$. | \frac{1}{2} |
A regular hexagon of side length $1$ is inscribed in a circle. Each minor arc of the circle determined by a side of the hexagon is reflected over that side. What is the area of the region bounded by these $6$ reflected arcs? | 3\sqrt{3}-\pi |
Given that $f(x+5)=4x^3 + 5x^2 + 9x + 6$ and $f(x)=ax^3 + bx^2 + cx + d$, find the value of $a+b+c+d$. | -206 |
Given that the location of the military camp is $A(1,1)$, and the general sets off from point $B(4,4)$ at the foot of the mountain, with the equation of the riverbank line $l$ being $x-y+1=0$, find the shortest total distance of the "General Drinking Horse" problem. | 2\sqrt{5} |
A stationery store sells a certain type of pen bag for $18$ yuan each. Xiao Hua went to buy this pen bag. When checking out, the clerk said, "If you buy one more, you can get a 10% discount, which is $36 cheaper than now." Xiao Hua said, "Then I'll buy one more, thank you." According to the conversation between the two... | 486 |
The endpoints of a diameter of circle $M$ are $(-1,-4)$ and $(-7,6)$. What are the coordinates of the center of circle $M$? Express your answer as an ordered pair. | (-4,1) |
When $\{a,0,-1\} = \{4,b,0\}$, find the values of $a$ and $b$. | -1 |
A gardener wants to plant 3 maple trees, 4 oak trees, and 5 birch trees in a row. He will randomly determine the order of these trees. What is the probability that no two birch trees are adjacent? | 7/99 |
Determine the number of 0-1 binary sequences of ten 0's and ten 1's which do not contain three 0's together. | 24068 |
The difference of the logarithms of the hundreds digit and the tens digit of a three-digit number is equal to the logarithm of the difference of the same digits, and the sum of the logarithms of the hundreds digit and the tens digit is equal to the logarithm of the sum of the same digits, increased by 4/3. If you subtr... | 421 |
In a 6 by 6 grid of points, what fraction of the larger rectangle's area is inside the shaded right triangle? The vertices of the shaded triangle correspond to grid points and are located at (1,1), (1,5), and (4,1). | \frac{1}{6} |
If an irrational number $a$ multiplied by $\sqrt{8}$ is a rational number, write down one possible value of $a$ as ____. | \sqrt{2} |
Given the parametric equation of line C1 as $$\begin{cases} x=2+t \\ y=t \end{cases}$$ (where t is the parameter), and the polar coordinate equation of the ellipse C2 as ρ²cos²θ + 9ρ²sin²θ = 9. Establish a rectangular coordinate system with the origin O as the pole and the positive semi-axis of the x-axis as the polar ... | \frac{6\sqrt{3}}{5} |
In the village of Matitika, five friends live along a straight road in the following order: Alya, Bella, Valya, Galya, and Dilya. Each of them calculated the sum of distances (in meters) from her house to the houses of the others. Bella reported the number 700, Valya reported 600, Galya reported 650. How many meters ar... | 150 |
Numbers from 1 to 9 are arranged in the cells of a \(3 \times 3\) table such that the sum of the numbers on one diagonal is 7, and the sum on the other diagonal is 21. What is the sum of the numbers in the five shaded cells?
| 25 |
For a positive integer $p$, define the positive integer $n$ to be $p$-safe if $n$ differs in absolute value by more than $2$ from all multiples of $p$. For example, the set of $10$-safe numbers is $\{ 3, 4, 5, 6, 7, 13, 14, 15, 16, 17, 23, \ldots\}$. Find the number of positive integers less than or equal to $10,000$ w... | 958 |
Triangle $ABC$ has a perimeter of 2007 units. The sides have lengths that are all integer values with $AB< BC \leq AC$. What is the smallest possible value of $BC - AB$? | 1 |
Ryosuke is picking up his friend from work. The odometer reads 74,568 when he picks his friend up, and it reads 74,592 when he drops his friend off at his house. Ryosuke's car gets 28 miles per gallon and the price of one gallon of gas is $\$4.05$. What was the cost of the gas that was used for Ryosuke to drive his fri... | \$3.47 |
Given that $i$ is the imaginary unit, $a\in\mathbb{R}$, if $\frac{1-i}{a+i}$ is a pure imaginary number, calculate the modulus of the complex number $z=(2a+1)+ \sqrt{2}i$. | \sqrt{11} |
Factor $t^2-121$. | (t-11)(t+11) |
Alli rolls a standard $6$-sided die twice. What is the probability of rolling integers that differ by $2$ on her first two rolls? Express your answer as a common fraction. | \dfrac{2}{9} |
A particle with charge $8.0 \, \mu\text{C}$ and mass $17 \, \text{g}$ enters a magnetic field of magnitude $\text{7.8 mT}$ perpendicular to its non-zero velocity. After 30 seconds, let the absolute value of the angle between its initial velocity and its current velocity, in radians, be $\theta$ . Find $100\thet... | 1.101 |
An electrician was called to repair a garland of four light bulbs connected in series, one of which has burned out. It takes 10 seconds to unscrew any bulb from the garland and 10 seconds to screw it back in. The time spent on other actions is negligible. What is the minimum time in which the electrician can definitely... | 60 |
Find the total area of the region outside of an equilateral triangle but inside three circles each with radius 1, centered at the vertices of the triangle. | \frac{2 \pi-\sqrt{3}}{2} |
Given two groups of numerical sequences, each containing 15 arithmetic progressions with 10 terms each. The first terms of the progressions in the first group are $1, 2, 3, \ldots, 15$, and their differences are respectively $2, 4, 6, \ldots, 30$. The second group of progressions has the same first terms $1, 2, 3, \ldo... | 160/151 |
Eight circles of diameter 1 are packed in the first quadrant of the coordinate plane as shown. Let region $\mathcal{R}$ be the union of the eight circular regions. Line $l,$ with slope 3, divides $\mathcal{R}$ into two regions of equal area. Line $l$'s equation can be expressed in the form $ax=by+c,$ where $a, b,$ and ... | 65 |
Let $z$ be a complex number with $|z| = \sqrt{2}.$ Find the maximum value of
\[|(z - 1)^2 (z + 1)|.\] | 4 \sqrt{2} |
Given a sequence of numbers arranged according to a certain rule: $-\frac{3}{2}$, $1$, $-\frac{7}{10}$, $\frac{9}{17}$, $\ldots$, based on the pattern of the first $4$ numbers, the $10$th number is ____. | \frac{21}{101} |
The positive integers $A, B$, and $C$ form an arithmetic sequence, while the integers $B, C$, and $D$ form a geometric sequence. If $\frac{C}{B} = \frac{7}{3},$ what is the smallest possible value of $A + B + C + D$? | 76 |
Find the real solutions of $(2 x+1)(3 x+1)(5 x+1)(30 x+1)=10$. | \frac{-4 \pm \sqrt{31}}{15} |
In the country of Francisca, there are 2010 cities, some of which are connected by roads. Between any two cities, there is a unique path which runs along the roads and which does not pass through any city twice. What is the maximum possible number of cities in Francisca which have at least 3 roads running out of them? | 1004 |
Circle $\Gamma$ is the incircle of $\triangle ABC$ and is also the circumcircle of $\triangle XYZ$. The point $X$ is on $\overline{BC}$, point $Y$ is on $\overline{AB}$, and the point $Z$ is on $\overline{AC}$. If $\angle A=40^\circ$, $\angle B=60^\circ$, and $\angle C=80^\circ$, what is the measure of $\angle AYX$? | 120^\circ |
Ocho has 8 friends and half are girls. His friends who are boys like to play theater with him. How many boys play theater with him? | Half his friends are boys because 1 - 1/2 = 1/2
He plays with four boys because 8 x 1/2= <<8*1/2=4>>4
#### 4 |
Out of 500 participants in a remote math olympiad, exactly 30 did not like the problem conditions, exactly 40 did not like the organization of the event, and exactly 50 did not like the method used to determine the winners. A participant is called "significantly dissatisfied" if they were dissatisfied with at least two... | 60 |
Come up with at least one three-digit number PAU (all digits are different), such that \((P + A + U) \times P \times A \times U = 300\). (Providing one example is sufficient) | 235 |
Let \( a_n \) be the coefficient of the \( x \) term in the expansion of \( (3 - \sqrt{x})^n \) for \( n = 2, 3, 4, \ldots \). Find \(\lim _{n \rightarrow \infty}\left(\frac{3^{2}}{a_{2}}+\frac{3^{3}}{a_{3}}+\cdots+\frac{3^{n}}{a_{n}}\right)\). | 18 |
Piravena must make a trip from $A$ to $B$, then from $B$ to $C$, then from $C$ to $A$. Each of these three parts of the trip is made entirely by bus or entirely by airplane. The cities form a right-angled triangle as shown, with $C$ a distance of 3000 km from $A$ and with $B$ a distance of 3250 km from $A$. To take a... | \$1012.50 |
At the end of a professional bowling tournament, the top 5 bowlers have a playoff. First #5 bowls #4. The loser receives $5$th prize and the winner bowls #3 in another game. The loser of this game receives $4$th prize and the winner bowls #2. The loser of this game receives $3$rd prize and the winner bowls #1. The winn... | 16 |
There are 17 people at a party, and each has a reputation that is either $1,2,3,4$, or 5. Some of them split into pairs under the condition that within each pair, the two people's reputations differ by at most 1. Compute the largest value of $k$ such that no matter what the reputations of these people are, they are abl... | 7 |
Rationalize the denominator: $$\frac{1}{\sqrt[3]{2}+\sqrt[3]{16}}$$ | \frac{\sqrt[3]{4}}{6} |
A mathematician is working on two projects. He has one week to write 518 maths questions for one project and 476 questions for another project. If he completes the same number of questions every day, how many should he aim to complete each day? | The writer is going to write 518 questions + 476 questions = <<518+476=994>>994 questions in 7 days.
He should aim to do 994 questions ÷ 7 days = <<994/7=142>>142 questions/day.
#### 142 |
Given an isosceles trapezoid \(ABCD\), where \(AD \parallel BC\), \(BC = 2AD = 4\), \(\angle ABC = 60^\circ\), and \(\overrightarrow{CE} = \frac{1}{3} \overrightarrow{CD}\), calculate the value of \(\overrightarrow{CA} \cdot \overrightarrow{BE}\). | -10 |
Let $u_n$ be the $n^\text{th}$ term of the sequence
\[1,\,\,\,\,\,\,2,\,\,\,\,\,\,5,\,\,\,\,\,\,6,\,\,\,\,\,\,9,\,\,\,\,\,\,12,\,\,\,\,\,\,13,\,\,\,\,\,\,16,\,\,\,\,\,\,19,\,\,\,\,\,\,22,\,\,\,\,\,\,23,\ldots,\]
where the first term is the smallest positive integer that is $1$ more than a multiple of $3$, the next two ... | 5898 |
Given the function $f(x) = |\ln x|$, the solution set of the inequality $f(x) - f(x_0) \geq c(x - x_0)$ is $(0, +\infty)$, where $x_0 \in (0, +\infty)$, and $c$ is a constant. When $x_0 = 1$, the range of values for $c$ is \_\_\_\_\_\_; when $x_0 = \frac{1}{2}$, the value of $c$ is \_\_\_\_\_\_. | -2 |
A high school bowling team's 3 members scored a total of 810 points in their first match. The first bowler scored 1/3 as many points as the second, and the second bowler scored 3 times as high as the third bowler. How many points did the third bowler score? | Let x represent the score of the third bowler
Bowler 2:3x
Bowler 1:3x(1/3)=x
Total:x+3x+x=810
5x=810
x=<<162=162>>162 points
#### 162 |
Find the ordered pair $(a,b)$ of real numbers such that the cubic polynomials $x^3 + ax^2 + 11x + 6 = 0$ and $x^3 + bx^2 + 14x + 8 = 0$ have two distinct roots in common. | (6,7) |
Find the integer $x$ that satisfies the equation $10x + 3 \equiv 7 \pmod{18}$. | 13 |
Let the integer part and decimal part of $2+\sqrt{6}$ be $x$ and $y$ respectively. Find the values of $x$, $y$, and the square root of $x-1$. | \sqrt{3} |
A solid right prism $ABCDEF$ has a height of $16,$ as shown. Also, its bases are equilateral triangles with side length $12.$ Points $X,$ $Y,$ and $Z$ are the midpoints of edges $AC,$ $BC,$ and $DC,$ respectively. A part of the prism above is sliced off with a straight cut through points $X,$ $Y,$ and $Z.$ Determine th... | 48+9\sqrt{3}+3\sqrt{91} |
Little Pang, Little Dingding, Little Ya, and Little Qiao's four families, totaling 8 parents and 4 children, went to the amusement park together. The ticket prices are as follows: adult tickets are 100 yuan per person; children's tickets are 50 yuan per person; if there are 10 or more people, they can buy group tickets... | 800 |
A positive integer with 3 digits $\overline{ABC}$ is $Lusophon$ if $\overline{ABC}+\overline{CBA}$ is a perfect square. Find all $Lusophon$ numbers. | 110,143,242,341,440,164,263,362,461,560,198,297,396,495,594,693,792,891,990 |
How many positive integers divide $5n^{11}-2n^5-3n$ for all positive integers $n$. | 12 |
Let $A=H_{1}, B=H_{6}+1$. A real number $x$ is chosen randomly and uniformly in the interval $[A, B]$. Find the probability that $x^{2}>x^{3}>x$. | \frac{1}{4} |
Given that the terminal side of angle $θ$ is symmetric to the terminal side of a $480^\circ$ angle with respect to the $x$-axis, and point $P(x,y)$ is on the terminal side of angle $θ$ (not the origin), then the value of $\frac{xy}{{x}^2+{y}^2}$ is equal to __. | \frac{\sqrt{3}}{4} |
Hanna has twice as many erasers as Rachel. Rachel has three less than one-half as many erasers as Tanya has red erasers. If Tanya has 20 erasers, and half of them are red, how many erasers does Hanna have? | Half of Tanya's 20 erasers are red, so she has 20/2=<<20/2=10>>10 red erasers.
One-half as many erasers as Tanya has red erasers is 10/2=<<10/2=5>>5 red erasers.
Rachel's three less than one-half as many erasers as Tanya's red erasers, which is 5-3=2 erasers.
Hanna has twice as many erasers as Rachel, for a total of 2*... |
Determine all the pairs $ (p , n )$ of a prime number $ p$ and a positive integer $ n$ for which $ \frac{ n^p + 1 }{p^n + 1} $ is an integer. | $(p,n)=(p,p),(2,4)$ |
Express $0.000 000 04$ in scientific notation. | 4 \times 10^{-8} |
Olga purchases a rectangular mirror (the shaded region) that fits exactly inside a frame. The outer perimeter of the frame measures 60 cm by 80 cm. The width of each side of the frame is 10 cm. What is the area of the mirror?
[asy]
unitsize(0.15inch);
defaultpen(black);
draw(((0,0)--(8,0)--(8,6)--(0,6)--cycle));
draw(... | 2400 \mbox{ cm}^2 |
The base three number $12012_3$ is equal to which base ten number? | 140 |
Determine the value of
\[\frac{\frac{2016}{1} + \frac{2015}{2} + \frac{2014}{3} + \dots + \frac{1}{2016}}{\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots + \frac{1}{2017}}.\] | 2017 |
Let $A_{1} A_{2} A_{3}$ be a triangle. Construct the following points:
- $B_{1}, B_{2}$, and $B_{3}$ are the midpoints of $A_{1} A_{2}, A_{2} A_{3}$, and $A_{3} A_{1}$, respectively.
- $C_{1}, C_{2}$, and $C_{3}$ are the midpoints of $A_{1} B_{1}, A_{2} B_{2}$, and $A_{3} B_{3}$, respectively.
- $D_{1}$ is the interse... | 25/49 |
Let \[f(x) = \left\{
\begin{array}{cl}
-x + 3 & \text{if } x \le 0, \\
2x - 5 & \text{if } x > 0.
\end{array}
\right.\]How many solutions does the equation $f(f(x)) = 4$ have? | 3 |
Given triangle $ABC$ with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$, respectively. It is known that $a=5$ and $\sin A= \frac{\sqrt{5}}{5}$.
(1) If the area of triangle $ABC$ is $\sqrt{5}$, find the minimum value of the perimeter $l$.
(2) If $\cos B= \frac{3}{5}$, find the value of side $c$. | 11 |
How many even integers are there between $\frac{9}{2}$ and $\frac{24}{1}$? | 10 |
At the start of this month, Mathilde and Salah each had 100 coins. For Mathilde, this was $25 \%$ more coins than she had at the start of last month. For Salah, this was $20 \%$ fewer coins than he had at the start of last month. What was the total number of coins that they had at the start of last month? | 205 |
Compute $\binom{17}{9}$. You are told that $\binom{15}{6} = 5005$ and $\binom{15}{8} = 6435$. | 24310 |
The distance on the map is 3.6 cm, and the actual distance is 1.2 mm. What is the scale of this map? | 30:1 |
Determine the smallest positive integer $n$ such that $n^2$ is divisible by 50 and $n^3$ is divisible by 294. | 210 |
What is $\left(\dfrac{3}{4}\right)^5$? | \dfrac{243}{1024} |
Determine the number of ways to arrange the letters of the word "PERCEPTION". | 907,200 |
Form five-digit numbers without repeating digits using the numbers \\(0\\), \\(1\\), \\(2\\), \\(3\\), and \\(4\\).
\\((\\)I\\()\\) How many of these five-digit numbers are even?
\\((\\)II\\()\\) How many of these five-digit numbers are less than \\(32000\\)? | 54 |
Rachel is 12 years old, and her grandfather is 7 times her age. Her mother is half grandfather's age, and her father is 5 years older than her mother. How old will Rachel's father be when she is 25 years old? | Rachel's grandfather is 12 x 7 = <<12*7=84>>84 years old.
Her mother is 84/2 = <<84/2=42>>42 years old.
Her father is 42 + 5 = <<42+5=47>>47 years old.
Rachel will be 25 years old in 25 - 12 = <<25-12=13>>13 years.
Rachel's father will be 47 + 13 = <<47+13=60>>60 years old.
#### 60 |
The product of the first and third terms of the geometric sequence $\{a_n\}$, given that $a_1$ and $a_4$ are the roots of the equation $x^2-2x-3=0$. | -3 |
Two angles of a triangle measure 30 and 45 degrees. If the side of the triangle opposite the 30-degree angle measures $6\sqrt2$ units, what is the sum of the lengths of the two remaining sides? Express your answer as a decimal to the nearest tenth. | 28.4 |
The decimal expansion of $8/11$ is a repeating decimal. What is the least number of digits in a repeating block of 8/11? | 2 |
Egor wrote a number on the board and encoded it according to the rules of letter puzzles (different letters correspond to different digits, the same letters to the same digits). The result was the word "ГВАТЕМАЛА". How many different numbers could Egor have originally written if his number was divisible by 30? | 21600 |
A solid right prism $ABCDEF$ has a height of 16, as shown. Also, its bases are equilateral triangles with side length 12. Points $X$, $Y$, and $Z$ are the midpoints of edges $AC$, $BC$, and $DC$, respectively. Determine the perimeter of triangle $XYZ$. [asy]
pair A, B, C, D, E, F, X, Y, Z;
A=(0,0);
B=(12,0);
C=(6,-6);
... | 26 |
Given a geometric sequence $\{a_n\}$, the sum of the first $n$ terms is $S_n$. Given that $a_1 + a_2 + a_3 = 3$ and $a_4 + a_5 + a_6 = 6$, calculate the value of $S_{12}$. | 45 |
A seminar is offered to a school for their teachers. The regular seminar fee is $150 but they offer a 5% discount if they register 2 days before the scheduled seminar. The school registered 10 teachers for the seminar a week before the schedule and it also released a $10 food allowance for each of the teachers. How muc... | The school got a $150 x 5/100 = $<<150*5/100=7.50>>7.50 seminar fee discount for each teacher.
Each teacher's seminar fee costs $150 - $7.50 = $<<150-7.5=142.50>>142.50.
So, ten teachers' seminar fees cost $142.50 x 10 = $1425.
The food allowance for 10 teachers amounts to $10 x 10 = $<<10*10=100>>100.
Therefore, the s... |
Compute the number of ways to select 99 cells of a $19 \times 19$ square grid such that no two selected cells share an edge or vertex. | 1000 |
In triangle $ABC$, $AB = AC = 100$, and $BC = 56$. Circle $P$ has radius $16$ and is tangent to $\overline{AC}$ and $\overline{BC}$. Circle $Q$ is externally tangent to $P$ and is tangent to $\overline{AB}$ and $\overline{BC}$. No point of circle $Q$ lies outside of $\triangle ABC$. The radius of circle $Q$ can be expr... | 254 |
In triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $2a^{2}\sin B\sin C=\sqrt{3}(a^{2}+b^{2}-c^{2})\sin A$. Find:
$(1)$ Angle $C$;
$(2)$ If $a=1$, $b=2$, and the midpoint of side $AB$ is $D$, find the length of $CD$. | \frac{\sqrt{7}}{2} |
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