problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
In triangle $ABC,$ the midpoint of $\overline{BC}$ is $(1,5,-1),$ the midpoint of $\overline{AC}$ is $(0,4,-2),$ and the midpoint of $\overline{AB}$ is $(2,3,4).$ Find the coordinates of vertex $A.$ | (1, 2, 3) |
What is the radius of the circle inscribed in triangle $ABC$ if $AB = AC=7$ and $BC=6$? Express your answer in simplest radical form. | \frac{3\sqrt{10}}{5} |
Consider parallelogram $A B C D$ with $A B>B C$. Point $E$ on $\overline{A B}$ and point $F$ on $\overline{C D}$ are marked such that there exists a circle $\omega_{1}$ passing through $A, D, E, F$ and a circle $\omega_{2}$ passing through $B, C, E, F$. If $\omega_{1}, \omega_{2}$ partition $\overline{B D}$ into segmen... | 51 |
Hilton had a box of 26 marbles that he was playing with. He found 6 marbles while he was playing, but afterward realized that he had lost 10 marbles. Lori felt bad and gave Hilton twice as many marbles as he lost. How many marbles did Hilton have in the end? | Hilton had 26 marbles + 6 marbles - 10 marbles = <<26+6-10=22>>22 marbles.
Lori gave Hilton 2 * the 10 marbles that he lost = <<2*10=20>>20 marbles.
In total Hilton had 22 marbles + 20 marbles = <<22+20=42>>42 marbles.
#### 42 |
Four consecutive positive integers have a product of 840. What is the largest of the four integers? | 7 |
The quadratic $ax^2 + bx + c$ can be expressed in the form $2(x - 4)^2 + 8$. When the quadratic $3ax^2 + 3bx + 3c$ is expressed in the form $n(x - h)^2 + k$, what is $h$? | 4 |
A point \((x, y)\) is a distance of 14 units from the \(x\)-axis. It is a distance of 8 units from the point \((1, 8)\). Given that \(x > 1\), what is the distance \(n\) from this point to the origin? | 15 |
A whole number, $N$, is chosen so that $\frac{N}{3}$ is strictly between 7.5 and 8. What is the value of $N$ ? | 23 |
Given that the graph of a power function passes through the points $(2,16)$ and $(\frac{1}{2},m)$, find the value of $m$. | \frac{1}{16} |
When Cheenu was a boy, he could run $15$ miles in $3$ hours and $30$ minutes. As an old man, he can now walk $10$ miles in $4$ hours. How many minutes longer does it take for him to walk a mile now compared to when he was a boy? | 10 |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(f(x) + y) = f(x^2 - y) + 4f(x) y\]for all real numbers $x$ and $y.$
Let $n$ be the number of possible values of $f(3),$ and let $s$ be the sum of all possible values of $f(3).$ Find $n \times s.$ | 18 |
Place 6 balls, labeled from 1 to 6, into 3 different boxes with each box containing 2 balls. If the balls labeled 1 and 2 cannot be placed in the same box, the total number of different ways to do this is \_\_\_\_\_\_. | 72 |
The squares of a chessboard are labelled with numbers, as shown below.
[asy]
unitsize(0.8 cm);
int i, j;
for (i = 0; i <= 8; ++i) {
draw((i,0)--(i,8));
draw((0,i)--(8,i));
}
for (i = 0; i <= 7; ++i) {
for (j = 0; j <= 7; ++j) {
label("$\frac{1}{" + string(i + 8 - j) + "}$", (i + 0.5, j + 0.5));
}}
[/asy]
Eig... | 1 |
Find the cosine of the angle between the non-intersecting diagonals of two adjacent lateral faces of a regular triangular prism, where the lateral edge is equal to the side of the base. | \frac{1}{4} |
How many ways are there to color every integer either red or blue such that \(n\) and \(n+7\) are the same color for all integers \(n\), and there does not exist an integer \(k\) such that \(k, k+1\), and \(2k\) are all the same color? | 6 |
A solid triangular prism is made up of 27 identical smaller solid triangular prisms. The length of every edge of each of the smaller prisms is 1. If the entire outer surface of the larger prism is painted, what fraction of the total surface area of all the smaller prisms is painted? | 1/3 |
A school organizes a social practice activity during the summer vacation and needs to allocate 8 grade 10 students to company A and company B evenly. Among these students, two students with excellent English scores cannot be allocated to the same company, and neither can the three students with strong computer skills. ... | 36 |
Let $\ell$ and $m$ be two non-coplanar lines in space, and let $P_{1}$ be a point on $\ell$. Let $P_{2}$ be the point on $m$ closest to $P_{1}, P_{3}$ be the point on $\ell$ closest to $P_{2}, P_{4}$ be the point on $m$ closest to $P_{3}$, and $P_{5}$ be the point on $\ell$ closest to $P_{4}$. Given that $P_{1} P_{2}=5... | \frac{\sqrt{39}}{4} |
Let $r$ be the speed in miles per hour at which a wheel, $13$ feet in circumference, travels. If the time for a complete rotation of the wheel is shortened by $\frac{1}{3}$ of a second, the speed $r$ is increased by $6$ miles per hour. Find $r$.
A) 10
B) 11
C) 12
D) 13
E) 14 | 12 |
If $a=2 \int_{-3}^{3} (x+|x|) \, dx$, determine the total number of terms in the expansion of $(\sqrt{x} - \frac{1}{\sqrt[3]{x}})^a$ where the power of $x$ is not an integer. | 14 |
Let $x$ be a real number between $0$ and $\tfrac{\pi}2$ such that \[\dfrac{\sin^4(x)}{42}+\dfrac{\cos^4(x)}{75} = \dfrac{1}{117}.\] Find $\tan(x)$ . | \frac{\sqrt{14}}{5} |
Simplify
\[\tan x + 2 \tan 2x + 4 \tan 4x + 8 \cot 8x.\]The answer will be a trigonometric function of some simple function of $x,$ like "$\cos 2x$" or "$\sin (x^3)$". | \cot x |
Given that $θ$ is an angle in the third quadrant and $\sin (θ- \frac {π}{4})= \frac {3}{5}$, find $\tan (θ+ \frac {π}{4})=$____. | \frac {4}{3} |
Find the area of the ellipse given by $x^2 + 6x + 4y^2 - 8y + 9 = 0.$ | 2 \pi |
Compute $\frac{x^6-16x^3+64}{x^3-8}$ when $x=6$. | 208 |
Thomas wants to throw a party for his best friend Casey. He needs to order enough chairs for all the party guests to sit in, one for each guest. First he orders 3 dozen chairs for 3 dozen guests he invites. Then he finds out that 1/3 of the guests want to bring a guest of their own, so Thomas needs to order more chairs... | Thomas originally orders chairs for 3 dozen guests, 12 x 3 = 36 guests
Thomas finds out that 1/3 of the guests want to bring a guest of their own, 1/3 of 36 = <<1/3*36=12>>12
The 12 guests bring a guest of their own each, 36 guests + 12 guests = <<36+12=48>>48
Thomas finds out that 20 of the guests can't make it, 48 gu... |
Davonte is trying to figure out how much space his art collection takes up. He measures his paintings and finds he has three square 6-foot by 6-foot paintings, four small 2-foot by 3-foot paintings, and one large 10-foot by 15-foot painting. How many square feet does his collection take up? | His square paintings take up 36 square feet each because 6 x 6 = <<6*6=36>>36
His small paintings take 6 square feet each because 2 x 3 = <<2*3=6>>6
His large painting takes up 150 square feet because 10 x 15 = <<10*15=150>>150.
Combined, his square paintings take up 108 square feet because 3 x 36 = <<3*36=108>>108
Com... |
Sami remembers that the digits in her new three-digit area code contain a 9, 8, and 7, but she can't recall the order. How many possibilities are there for her to try? | 6 |
Let $S$ be a subset of $\{1,2,3, \ldots, 12\}$ such that it is impossible to partition $S$ into $k$ disjoint subsets, each of whose elements sum to the same value, for any integer $k \geq 2$. Find the maximum possible sum of the elements of $S$. | 77 |
Mary divides a circle into 12 sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle? | 8 |
Helena needs to save 40 files onto disks, each with 1.44 MB space. 5 of the files take up 1.2 MB, 15 of the files take up 0.6 MB, and the rest take up 0.3 MB. Determine the smallest number of disks needed to store all 40 files. | 16 |
Jake trips over his dog 40% percent of mornings. 25% of the time he trips, he drops his coffee. What percentage of mornings does he NOT drop his coffee? | First multiply the chance of each event happening to find the chance they both happen: 40% * 25% = 10%
Then subtract that percentage from 100% to find the percentage of the time Jake doesn't drop his coffee: 100% - 10% = 90%
#### 90 |
What is the measure, in degrees, of the acute angle formed by the hour hand and the minute hand of a 12-hour clock at 6:48? | 84 |
If 7:30 a.m. was 16 minutes ago, how many minutes will it be until 8:00 a.m.? | 14 |
In triangle $\triangle ABC$, $A=60^{\circ}$, $a=\sqrt{6}$, $b=2$.
$(1)$ Find $\angle B$;
$(2)$ Find the area of $\triangle ABC$. | \frac{3 + \sqrt{3}}{2} |
Tamika selects two different numbers at random from the set $\{8,9,10\}$ and adds them. Carlos takes two different numbers at random from the set $\{3,5,6\}$ and multiplies them. What is the probability that Tamika's result is greater than Carlos' result? Express your answer as a common fraction. | \frac{4}{9} |
Cara is sitting at a circular table with six friends. Assume there are three males and three females among her friends. How many different possible pairs of people could Cara sit between if each pair must include at least one female friend? | 12 |
Given two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ with an acute angle between them, and satisfying $|\overrightarrow{a}|= \frac{8}{\sqrt{15}}$, $|\overrightarrow{b}|= \frac{4}{\sqrt{15}}$. If for any $(x,y)\in\{(x,y)| |x \overrightarrow{a}+y \overrightarrow{b}|=1, xy > 0\}$, it holds that $|x+y|\leqslant ... | \frac{8}{15} |
The school nurse must conduct lice checks at the elementary school. She must check 26 Kindergarteners, 19 first graders, 20 second graders, and 25 third graders. If each check takes 2 minutes, how many hours will it take the nurse to complete all the checks? | The school nurse must check a total of 26 Kindergarteners + 19 first graders + 20 second graders + 25 third graders = <<26+19+20+25=90>>90 students in all.
If every check takes 2 minutes, it will take 90 students x 2 minutes = <<90*2=180>>180 minutes.
There are 60 minutes in one hour, so it will take 180 minutes / 60 =... |
The science club has 25 members: 10 boys and 15 girls. A 5-person committee is chosen at random. What is the probability that the committee has at least 1 boy and at least 1 girl? | \dfrac{475}{506} |
Alex gets paid $500 a week and 10% of his weekly income is deducted as tax. He also pays his weekly water bill for $55 and gives away another 10% of his weekly income as a tithe. How much money does Alex have left? | His income tax is 10% of 500 which is 500*10% = $<<500*10*.01=50>>50.
His tithe costs 10% of $500 which is 500*10% = $<<500*10*.01=50>>50.
The total expenses are 50 + 55 + 50 = $155
He is then left with $500 - $ 155= $<<500-155=345>>345.
#### 345 |
What is the smallest positive integer with exactly 14 positive divisors? | 192 |
A circle passes through the vertices of a triangle with side-lengths $7\tfrac{1}{2},10,12\tfrac{1}{2}.$ The radius of the circle is: | \frac{25}{4} |
Stefan goes to a restaurant to eat dinner with his family. They order an appetizer that costs $10 and 4 entrees that are $20 each. If they tip 20% of the total for the waiter, what is the total amount of money that they spend at the restaurant? | The total cost of the entrees is 4 * $20 = $<<4*20=80>>80.
The total cost of the dinner is $80 + $10 = $<<80+10=90>>90.
The tip is $90 * 0.20 = $<<90*0.20=18>>18
The total cost with tip is $90 + $18 = $<<90+18=108>>108
#### 108 |
A necklace consists of 50 blue beads and some quantity of red beads. It is known that on any segment of the necklace containing 8 blue beads, there are at least 4 red beads. What is the minimum number of red beads that can be in this necklace? (The beads in the necklace are arranged cyclically, meaning the last bead is... | 29 |
How many times does the digit 9 appear in the list of all integers from 1 to 1000? | 300 |
There is a unique two-digit positive integer $u$ for which the last two digits of $15\cdot u$ are $45$, and $u$ leaves a remainder of $7$ when divided by $17$. | 43 |
A parabola has vertex $V = (0,0)$ and focus $F = (0,1).$ Let $P$ be a point in the first quadrant, lying on the parabola, so that $PF = 101.$ Find $P.$ | (20,100) |
Each vertex of a cube is to be labeled with an integer 1 through 8, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How ma... | 6 |
Eight chairs are arranged in a circle. Find the number of subsets of this set of chairs that contain at least four adjacent chairs. | 288 |
In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively, and $a=2$, $b=3$, $\cos C=\frac{1}{3}$. The radius of the circumcircle is ______. | \frac{9 \sqrt{2}}{8} |
The slant height of a cone forms an angle $\alpha$ with the plane of its base, where $\cos \alpha = \frac{1}{4}$. A sphere is inscribed in the cone, and a plane is drawn through the circle of tangency of the sphere and the lateral surface of the cone. The volume of the part of the cone enclosed between this plane and t... | 37 |
How many three-digit natural numbers are there such that the sum of their digits is equal to 24? | 10 |
John was 66 inches tall. He had a growth spurt and grew 2 inches per month for 3 months. How tall is he in feet? | He gained 2*3=<<2*3=6>>6 inches
So his height is 66+6=<<66+6=72>>72 inches
So he is 72/12=<<72/12=6>>6 feet tall
#### 6 |
Kira is making breakfast for herself. She fries 3 sausages then scrambles 6 eggs and cooks each item of food separately. If it takes 5 minutes to fry each sausage and 4 minutes to scramble each egg then how long, in minutes, did it take for Kira to make her breakfast? | The sausages take 5 minutes per sausage * 3 sausages = <<5*3=15>>15 minutes to cook.
The eggs take 4 minutes per egg * 6 eggs = <<4*6=24>>24 minutes to cook.
So in total, it takes 15 minutes for sausages + 24 minutes for eggs = <<15+24=39>>39 minutes to cook breakfast.
#### 39 |
A point has rectangular coordinates $(-5,-7,4)$ and spherical coordinates $(\rho, \theta, \phi).$ Find the rectangular coordinates of the point with spherical coordinates $(\rho, \theta, -\phi).$ | (5,7,4) |
The sequence is defined as \( a_{0}=134, a_{1}=150, a_{k+1}=a_{k-1}-\frac{k}{a_{k}} \) for \( k=1,2, \cdots, n-1 \). Determine the value of \( n \) for which \( a_{n}=0 \). | 201 |
Solve $x=\sqrt{x-\frac{1}{x}}+\sqrt{1-\frac{1}{x}}$ for $x$. | \frac{1+\sqrt{5}}{2} |
Jerry works as an independent contractor for a company that pays him $40 per task. If each task takes him two hours to complete and he works 10 hours a day for a whole week, calculate the total amount of money he would earn at the end of the week. | If a task takes two hours to complete and Briar works for ten hours a day, he manages to complete 10/2= five tasks a day.
Since each task pays him $40, for the five tasks, he makes $40*5 = $<<40*5=200>>200
If he works for a whole week, he makes $200*7= $<<200*7=1400>>1400
#### 1400 |
Laura wants to bake a cake for her mother. She checks the recipe and the pantry and sees that she needs to buy 2 cups of flour, 2 cups of sugar, a cup of butter, and two eggs. The flour costs $4. The sugar costs $2. The eggs cost $.5, and the butter costs $2.5. When she is done baking it, she cuts the cake into 6 slice... | The cake cost $9 in total because 4 + 2 + .5 + 2.5 = <<4+2+.5+2.5=9>>9
Each slice costs $1.5 because 9 / 6 = <<9/6=1.5>>1.5
The dog ate 4 slices because 6 - 2 = <<6-2=4>>4
The amount the dog ate cost $6 because 4 x 1.5 = <<4*1.5=6>>6
#### 6 |
Given two real numbers \( p > 1 \) and \( q > 1 \) such that \( \frac{1}{p} + \frac{1}{q} = 1 \) and \( pq = 9 \), what is \( q \)? | \frac{9 + 3\sqrt{5}}{2} |
Given a parabola $y=x^{2}-7$, find the length of the line segment $|AB|$ where $A$ and $B$ are two distinct points on it that are symmetric about the line $x+y=0$. | 5 \sqrt{2} |
Given that in the expansion of \\((1+2x)^{n}\\), only the coefficient of the fourth term is the maximum, then the constant term in the expansion of \\((1+ \dfrac {1}{x^{2}})(1+2x)^{n}\\) is \_\_\_\_\_\_. | 61 |
A positive integer is called a perfect power if it can be written in the form \(a^b\), where \(a\) and \(b\) are positive integers with \(b \geq 2\). The increasing sequence \(2, 3, 5, 6, 7, 10, \ldots\) consists of all positive integers which are not perfect powers. Calculate the sum of the squares of the digits of th... | 21 |
Consider the sequence
$$
a_{n}=\cos (\underbrace{100 \ldots 0^{\circ}}_{n-1})
$$
For example, $a_{1}=\cos 1^{\circ}, a_{6}=\cos 100000^{\circ}$.
How many of the numbers $a_{1}, a_{2}, \ldots, a_{100}$ are positive? | 99 |
There are two natural ways to inscribe a square in a given isosceles right triangle. If it is done as in Figure 1 below, then one finds that the area of the square is $441 \text{cm}^2$. What is the area (in $\text{cm}^2$) of the square inscribed in the same $\triangle ABC$ as shown in Figure 2 below?
[asy] draw((0,0)--... | 392 |
Let $A B C$ be a triangle with $A B=A C=\frac{25}{14} B C$. Let $M$ denote the midpoint of $\overline{B C}$ and let $X$ and $Y$ denote the projections of $M$ onto $\overline{A B}$ and $\overline{A C}$, respectively. If the areas of triangle $A B C$ and quadrilateral $A X M Y$ are both positive integers, find the minimu... | 1201 |
For how many integer values of $b$ does the equation $$x^2 + bx + 12b = 0$$ have integer solutions for $x$? | 16 |
In a math test, Mark scored twice as much as the least score. If the highest score is 98 and the range of the scores is 75, what was Mark's score? | The range of scores is the difference between the highest and the least score so 75 = 98 - least score
Then the least score = 98-75 = <<98-75=23>>23
Mark scored twice as much as the least score which is 2*23 = <<2*23=46>>46
#### 46 |
James decides to go to prom with Susan. He pays for everything. The tickets cost $100 each. Dinner is $120. He leaves a 30% tip. He also charters a limo for 6 hours that cost $80 per hour. How much did it all cost? | He paid 100*2=$<<100*2=200>>200 for tickets
The tip for dinner was $120*.3=$<<120*.3=36>>36
So he paid $120+$36=$<<120+36=156>>156 for dinner
The limo cost $80*6=$<<80*6=480>>480
In total he paid $200+$156+$480=$<<200+156+480=836>>836
#### 836 |
Compute the number of ways there are to assemble 2 red unit cubes and 25 white unit cubes into a $3 \times 3 \times 3$ cube such that red is visible on exactly 4 faces of the larger cube. (Rotations and reflections are considered distinct.) | 114 |
Given that $x+y=12$, $xy=9$, and $x < y$, find the value of $\frac {x^{ \frac {1}{2}}-y^{ \frac {1}{2}}}{x^{ \frac {1}{2}}+y^{ \frac {1}{2}}}=$ ___. | - \frac { \sqrt {3}}{3} |
A right triangle $ABC$ is inscribed in the circular base of a cone. If two of the side lengths of $ABC$ are $3$ and $4$ , and the distance from the vertex of the cone to any point on the circumference of the base is $3$ , then the minimum possible volume of the cone can be written as $\frac{m\pi\sqrt{n}}{p}$ ,... | 60 |
In how many ways can you select two letters from the word "УЧЕБНИК" such that one of the letters is a consonant and the other is a vowel? | 12 |
Let $p$, $q$, $r$, $s$, and $t$ be positive integers such that $p+q+r+s+t=2022$. Let $N$ be the largest of the sum $p+q$, $q+r$, $r+s$, and $s+t$. What is the smallest possible value of $N$? | 506 |
Jamie counted the number of edges of a cube, Jimmy counted the numbers of corners, and Judy counted the number of faces. They then added the three numbers. What was the resulting sum? | 26 |
Marla has a large white cube that has an edge of 10 feet. She also has enough green paint to cover 300 square feet. Marla uses all the paint to create a white square centered on each face, surrounded by a green border. What is the area of one of the white squares, in square feet? | 50 |
Vasya has:
a) 2 different volumes from the collected works of A.S. Pushkin, each volume is 30 cm high;
b) a set of works by E.V. Tarle in 4 volumes, each volume is 25 cm high;
c) a book of lyrical poems with a height of 40 cm, published by Vasya himself.
Vasya wants to arrange these books on a shelf so that his own wo... | 144 |
Determine the positive real value of $x$ for which $$\sqrt{2+A C+2 C x}+\sqrt{A C-2+2 A x}=\sqrt{2(A+C) x+2 A C}$$ | 4 |
Find all functions $f$ from the set $\mathbb{R}$ of real numbers into $\mathbb{R}$ which satisfy for all $x, y, z \in \mathbb{R}$ the identity \[f(f(x)+f(y)+f(z))=f(f(x)-f(y))+f(2xy+f(z))+2f(xz-yz).\] | f(x) = 0 \text{ and } f(x) = x^2 |
Roger rode his bike for 2 miles this morning, and then 5 times that amount in the evening. How many miles did Roger ride his bike for? | Roger rode for 2 miles in the morning, and then 5 times that in the evening for 5*2=10 miles ridden in the evening
In total, that means Roger rode 2+10=<<2+10=12>>12 miles over the course of the day.
#### 12 |
Rationalize the denominator of $\frac{\sqrt[3]{27} + \sqrt[3]{2}}{\sqrt[3]{3} + \sqrt[3]{2}}$ and express your answer in simplest form. | 7 - \sqrt[3]{54} + \sqrt[3]{6} |
The first 14 terms of the sequence $\left\{a_{n}\right\}$ are $4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, \ldots$. Following this pattern, what is $a_{18}$? | 51 |
In triangle $ABC$, $AB = 5$, $AC = 7$, $BC = 9$, and $D$ lies on $\overline{BC}$ such that $\overline{AD}$ bisects $\angle BAC$. Find $\cos \angle BAD$. | \sqrt{0.45} |
The midpoint of a line segment is located at $(1, -2)$. If one of the endpoints is $(4, 5)$, what is the other endpoint? Express your answer as an ordered pair. | (-2,-9) |
Compute
\[\sin^2 6^\circ + \sin^2 12^\circ + \sin^2 18^\circ + \dots + \sin^2 174^\circ.\] | \frac{31}{2} |
How many distinct arrangements of the letters in the word "balloon" are there? | 1260 |
Determine the smallest integer $k$ such that $k>1$ and $k$ has a remainder of $3$ when divided by any of $11,$ $4,$ and $3.$ | 135 |
The polynomial equation \[x^3 + bx + c = 0,\]where $b$ and $c$ are rational numbers, has $5-\sqrt{2}$ as a root. It also has an integer root. What is it? | -10 |
Equilateral triangle $ABC$ has a side length of $\sqrt{144}$. There are four distinct triangles $AD_1E_1$, $AD_1E_2$, $AD_2E_3$, and $AD_2E_4$, each congruent to triangle $ABC$, with $BD_1 = BD_2 = \sqrt{12}$. Additionally, $BD_1$ and $BD_2$ are placed such that $\angle ABD_1 = 30^\circ$ and $\angle ABD_2 = 150^\circ$.... | 576 |
Given the point $P(-\sqrt{3}, y)$ is on the terminal side of angle $\alpha$ and $\sin\alpha = \frac{\sqrt{13}}{13}$, find the value of $y$. | \frac{1}{2} |
Find the maximum positive integer $r$ that satisfies the following condition: For any five 500-element subsets of the set $\{1,2, \cdots, 1000\}$, there exist two subsets that have at least $r$ common elements. | 200 |
The school committee has organized a "Chinese Dream, My Dream" knowledge speech competition. There are 4 finalists, and each contestant can choose any one topic from the 4 backup topics to perform their speech. The number of scenarios where exactly one of the topics is not selected by any of the 4 contestants is ______... | 324 |
In three-dimensional space, find the number of lattice points that have a distance of 4 from the origin. | 42 |
Given points \(A=(8,15)\) and \(B=(16,9)\) are on a circle \(\omega\), and the tangent lines to \(\omega\) at \(A\) and \(B\) meet at a point \(P\) on the x-axis, calculate the area of the circle \(\omega\). | 250\pi |
Let $b_n$ be the integer obtained by writing the integers from $5$ to $n+4$ from left to right. For example, $b_2 = 567$, and $b_{10} = 567891011121314$. Compute the remainder when $b_{25}$ is divided by $55$ (which is the product of $5$ and $11$ for the application of the Chinese Remainder Theorem). | 39 |
A social club has $2k+1$ members, each of whom is fluent in the same $k$ languages. Any pair of members always talk to each other in only one language. Suppose that there were no three members such that they use only one language among them. Let $A$ be the number of three-member subsets such that the three distinct ... | \binom{2k+1}{3} - k(2k+1) |
Bailey is making a rectangle out of a 100cm length of rope he has. If the longer sides of the rectangle were 28cm long, what is the length of each of the shorter sides? | The longer sides are 2 in number so their total length of 2*28cm = <<2*28=56>>56cm
The rest of the length making up the short sides is 100-56 = <<100-56=44>>44 cm
There are two equal short sides so each one is 44/2 = <<44/2=22>>22cm
#### 22 |
In square $ABCD$, point $M$ is the midpoint of side $AB$ and point $N$ is the midpoint of side $BC$. What is the ratio of the area of triangle $AMN$ to the area of square $ABCD$? Express your answer as a common fraction. | \frac{1}{8} |
In \\( \triangle ABC \\), \\( a \\), \\( b \\), and \\( c \\) are the sides opposite to angles \\( A \\), \\( B \\), and \\( C \\) respectively. The vectors \\( \overrightarrow{m} = (a, b+c) \\) and \\( \overrightarrow{n} = (1, \cos C + \sqrt{3} \sin C) \\) are given, and \\( \overrightarrow{m} \parallel \overrightarro... | \sqrt{3} |
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