problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
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Let \( A, B, C \) be positive integers such that the number \( 1212017ABC \) is divisible by 45. Find the difference between the largest and the smallest possible values of the two-digit number \( AB \). | 85 |
Let $a,$ $b,$ $c$ be distinct, nonzero real numbers such that
\[a + \frac{1}{b} = b + \frac{1}{c} = c + \frac{1}{a}.\]Find $|abc|.$
Note: Intermediate Algebra writing problem, week 12. | 1 |
Given the equation about $x$, $2x^{2}-( \sqrt {3}+1)x+m=0$, its two roots are $\sin θ$ and $\cos θ$, where $θ∈(0,π)$. Find:
$(1)$ the value of $m$;
$(2)$ the value of $\frac {\tan θ\sin θ}{\tan θ-1}+ \frac {\cos θ}{1-\tan θ}$;
$(3)$ the two roots of the equation and the value of $θ$ at this time. | \frac {1}{2} |
An acronym XYZ is drawn within a 2x4 rectangular grid with grid lines spaced 1 unit apart. The letter X is formed by two diagonals crossing in a $1 \times 1$ square. Y consists of a vertical line segment and two slanted segments each forming 45° with the vertical line, making up a symmetric letter. Z is formed by a horizontal segment at the top and bottom of a $1 \times 2$ rectangle, with a diagonal connecting these segments. In units, what is the total length of the line segments forming the acronym XYZ?
A) $5 + 4\sqrt{2} + \sqrt{5}$
B) $5 + 2\sqrt{2} + 3\sqrt{5}$
C) $6 + 3\sqrt{2} + 2\sqrt{5}$
D) $7 + 4\sqrt{2} + \sqrt{3}$
E) $4 + 5\sqrt{2} + \sqrt{5}$ | 5 + 4\sqrt{2} + \sqrt{5} |
Given \(\triangle DEF\), where \(DE=28\), \(EF=30\), and \(FD=16\), calculate the area of \(\triangle DEF\). | 221.25 |
There are 7 line segments with integer lengths in centimeters: $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, $a_6$, $a_7$, satisfying $a_1 < a_2 < a_3 < a_4 < a_5 < a_6 < a_7$, and any three of these line segments cannot form a triangle. If $a_1 = 1$ cm and $a_7 = 21$ cm, then $a_6 = \ $. | 13 |
Along a road, there are 25 poles in a single row. Sometimes a siskin lands on one of the poles, and immediately a siskin flies off from one of the neighboring poles (if there was at least one siskin sitting on the neighboring poles at that moment). Also, each pole can hold no more than one siskin.
Initially, there are no birds on the poles. What is the maximum number of siskins that can be simultaneously on the poles? | 24 |
Our basketball team has 12 members, each of whom can play any position. In how many ways can we choose a starting lineup consisting of a center, a power forward, a shooting forward, a point guard, and a shooting guard? | 95,\!040 |
Let $p, q, r, s$ be distinct primes such that $p q-r s$ is divisible by 30. Find the minimum possible value of $p+q+r+s$. | 54 |
How many lattice points lie on the hyperbola \( x^2 - y^2 = 1800^2 \)? | 150 |
1. When a die (with faces numbered 1 through 6) is thrown twice in succession, find the probability that the sum of the numbers facing up is at least 10.
2. On a line segment MN of length 16cm, a point P is chosen at random. A rectangle is formed with MP and NP as adjacent sides. Find the probability that the area of this rectangle is greater than 60cm². | \frac{1}{4} |
Let \( p(x) = x^{4} + a x^{3} + b x^{2} + c x + d \), where \( a, b, c, d \) are constants, and \( p(1) = 1993 \), \( p(2) = 3986 \), \( p(3) = 5979 \). Calculate \( \frac{1}{4}[p(11) + p(-7)] \). | 5233 |
Given that $\theta=\arctan \frac{5}{12}$, find the principal value of the argument of the complex number $z=\frac{\cos 2 \theta+i \sin 2 \theta}{239+i}$. | \frac{\pi}{4} |
If $9:y^3 = y:81$, what is the value of $y$? | 3\sqrt{3} |
Given the parabola $C$: $x^2 = 2py (p > 0)$ and the line $2x-y+2=0$, they intersect at points $A$ and $B$. A vertical line is drawn from the midpoint of the line segment $AB$ to the $x$-axis, intersecting the parabola $C$ at point $Q$. If $\overrightarrow{QA} \cdot \overrightarrow{QB}=0$, calculate the value of $p$. | \frac{1}{4} |
In an enterprise, no two employees have jobs of the same difficulty and no two of them take the same salary. Every employee gave the following two claims:
(i) Less than $12$ employees have a more difficult work;
(ii) At least $30$ employees take a higher salary.
Assuming that an employee either always lies or always tells the truth, find how many employees are there in the enterprise. | 42 |
Given the set $S=\{A, A_1, A_2, A_3, A_4\}$, define the operation $\oplus$ on $S$ as: $A_i \oplus A_j = A_k$, where $k=|i-j|$, and $i, j = 0, 1, 2, 3, 4$. Calculate the total number of ordered pairs $(i, j)$ that satisfy the condition $(A_i \oplus A_j) \oplus A_2 = A_1$ (where $A_i, A_j \in S$). | 12 |
How many ordered pairs of real numbers $(x, y)$ are there such that $x^2+y^2 = 200$ and
\[\sqrt{(x-5)^2+(y-5)^2}+\sqrt{(x+5)^2+(y+5)^2}\]
is an integer? | 12 |
Evaluate the expression: \\( \frac {\cos 40 ^{\circ} +\sin 50 ^{\circ} (1+ \sqrt {3}\tan 10 ^{\circ} )}{\sin 70 ^{\circ} \sqrt {1+\cos 40 ^{\circ} }}\\) | \sqrt {2} |
Compute $\cos \left( \arcsin \frac{5}{13} \right).$ | \frac{12}{13} |
Given a family of sets \(\{A_{1}, A_{2}, \ldots, A_{n}\}\) that satisfies the following conditions:
(1) Each set \(A_{i}\) contains exactly 30 elements;
(2) For any \(1 \leq i < j \leq n\), the intersection \(A_{i} \cap A_{j}\) contains exactly 1 element;
(3) The intersection \(A_{1} \cap A_{2} \cap \ldots \cap A_{n} = \varnothing\).
Find the maximum number \(n\) of such sets. | 871 |
Out of the 500 marbles that Cindy had, she gave her four friends 80 marbles each. What's four times the number of marbles she has remaining? | Cindy gave her four friends 80 marbles each, a total of 4*80 = <<80*4=320>>320 marbles
She was left with 500-320 = <<500-320=180>>180 marbles
Four times the number of marbles she has remaining is 180*4 = <<180*4=720>>720
#### 720 |
The function $f(n)$ is defined on the positive integers and takes non-negative integer values. $f(2)=0,f(3)>0,f(9999)=3333$ and for all $m,n:$ \[ f(m+n)-f(m)-f(n)=0 \text{ or } 1. \] Determine $f(1982)$. | 660 |
There are numbers $1, 2, \cdots, 36$ to be filled into a $6 \times 6$ grid, with each cell containing one number. Each row must be in increasing order from left to right. What is the minimum sum of the six numbers in the third column? | 63 |
The medians of a right triangle which are drawn from the vertices of the acute angles are $5$ and $\sqrt{40}$. The value of the hypotenuse is: | 2\sqrt{13} |
Given the numbers 1, 2, 3, 4, find the probability that $\frac{a}{b}$ is not an integer, where $a$ and $b$ are randomly selected numbers from the set $\{1, 2, 3, 4\}$. | \frac{2}{3} |
Find the positive root of
\[x^3 - 3x^2 - x - \sqrt{2} = 0.\] | 2 + \sqrt{2} |
Let $A$ be a point on the circle $x^2 + y^2 - 12x + 31 = 0,$ and let $B$ be a point on the parabola $y^2 = 4x.$ Find the smallest possible distance $AB.$ | \sqrt{5} |
Steve has an isosceles triangle with base 8 inches and height 10 inches. He wants to cut it into eight pieces that have equal areas, as shown below. To the nearest hundredth of an inch what is the number of inches in the greatest perimeter among the eight pieces? [asy]
size(150);
defaultpen(linewidth(0.7));
draw((0,0)--(8,0));
for(int i = 0; i < 9; ++i){
draw((4,10)--(i,0));
}
draw((0,-0.5)--(8,-0.5),Bars(5));
label("$8''$",(0,-0.5)--(8,-0.5),S);
[/asy] | 22.21 |
Twenty-seven players are randomly split into three teams of nine. Given that Zack is on a different team from Mihir and Mihir is on a different team from Andrew, what is the probability that Zack and Andrew are on the same team? | \frac{8}{17} |
A circle passes through the vertices $K$ and $P$ of triangle $KPM$ and intersects its sides $KM$ and $PM$ at points $F$ and $B$, respectively. Given that $K F : F M = 3 : 1$ and $P B : B M = 6 : 5$, find $K P$ given that $B F = \sqrt{15}$. | 2 \sqrt{33} |
Chelsea has 24 kilos of sugar. She divides them into 4 bags equally. Then one of the bags gets torn and half of the sugar falls to the ground. How many kilos of sugar remain? | Each bag has 24/4=<<24/4=6>>6 kilos of sugar.
6/2=<<6/2=3>>3 kilos of sugar falls to the ground.
24-3=<<24-3=21>>21 kilos of sugar remains.
#### 21 |
The cubic polynomial $q(x)$ satisfies $q(1) = 5,$ $q(6) = 20,$ $q(14) = 12,$ and $q(19) = 30.$ Find
\[q(0) + q(1) + q(2) + \dots + q(20).\] | 357 |
At a tennis tournament there were $2n$ boys and $n$ girls participating. Every player played every other player. The boys won $\frac 75$ times as many matches as the girls. It is knowns that there were no draws. Find $n$ . | \[ n \in \{ \mathbf{N} \equiv 0, 3 \pmod{8} \} \] |
The line \(y = -\frac{1}{2}x + 8\) crosses the \(x\)-axis at \(P\) and the \(y\)-axis at \(Q\). Point \(T(r, s)\) is on line segment \(PQ\). If the area of \(\triangle POQ\) is twice the area of \(\triangle TOP\), then what is the value of \(r+s\)? | 12 |
Marvin had a birthday on Tuesday, May 27 in the leap year $2008$. In what year will his birthday next fall on a Saturday? | 2017 |
Assume that $x_1,x_2,\ldots,x_7$ are real numbers such that \begin{align*} x_1 + 4x_2 + 9x_3 + 16x_4 + 25x_5 + 36x_6 + 49x_7 &= 1, \\ 4x_1 + 9x_2 + 16x_3 + 25x_4 + 36x_5 + 49x_6 + 64x_7 &= 12, \\ 9x_1 + 16x_2 + 25x_3 + 36x_4 + 49x_5 + 64x_6 + 81x_7 &= 123. \end{align*} Find the value of $16x_1+25x_2+36x_3+49x_4+64x_5+81x_6+100x_7$. | 334 |
How many positive two-digit integers have an odd number of positive factors? | 6 |
Let $a, b, c, d$ be the four roots of $X^{4}-X^{3}-X^{2}-1$. Calculate $P(a)+P(b)+P(c)+P(d)$, where $P(X) = X^{6}-X^{5}-X^{4}-X^{3}-X$. | -2 |
A positive integer $n$ is known as an [i]interesting[/i] number if $n$ satisfies
\[{\ \{\frac{n}{10^k}} \} > \frac{n}{10^{10}} \]
for all $k=1,2,\ldots 9$.
Find the number of interesting numbers. | 999989991 |
Let $G$ be the centroid of triangle $ABC.$ If $GA^2 + GB^2 + GC^2 = 58,$ then find $AB^2 + AC^2 + BC^2.$ | 174 |
Given that the terminal side of angle $\alpha$ passes through point $P\left(\sin \frac{7\pi }{6},\cos \frac{11\pi }{6}\right)$, find the value of $\frac{1}{3\sin ^{2}\alpha -\cos ^{2}\alpha }=\_\_\_\_\_\_\_\_\_\_.$ | \frac{1}{2} |
Given the parabola $C:y^{2}=8x$ with focus $F$ and directrix $l$. $P$ is a point on $l$ and the line $(PF)$ intersects the parabola $C$ at points $M$ and $N$. If $\overrightarrow{{PF}}=3\overrightarrow{{MF}}$, find the length of the segment $MN$. | \frac{32}{3} |
We will call a two-digit number power-less if neither of its digits can be written as an integer to a power greater than 1. For example, 53 is power-less, but 54 is not power-less since \(4 = 2^{2}\). Which of the following is a common divisor of the smallest and the largest power-less numbers?
A 3
B 5
C 7
D 11
E 13 | 11 |
There is a hemispherical raw material. If this material is processed into a cube through cutting, the maximum value of the ratio of the volume of the obtained workpiece to the volume of the raw material is ______. | \frac { \sqrt {6}}{3\pi } |
What is the constant term of the expansion of $\left(5x + \dfrac{1}{3x}\right)^8$? | \frac{43750}{81} |
A right pyramid with a square base has total surface area 432 square units. The area of each triangular face is half the area of the square face. What is the volume of the pyramid in cubic units? | 288\sqrt{3} |
Segment $AB$ is both a diameter of a circle of radius $1$ and a side of an equilateral triangle $ABC$. The circle also intersects $AC$ and $BC$ at points $D$ and $E$, respectively. The length of $AE$ is | \sqrt{3} |
A regular hexagon \( A B C D E K \) is inscribed in a circle of radius \( 3 + 2\sqrt{3} \). Find the radius of the circle inscribed in the triangle \( B C D \). | \frac{3}{2} |
Find the largest \( n \) so that the number of integers less than or equal to \( n \) and divisible by 3 equals the number divisible by 5 or 7 (or both). | 65 |
In the diagram, $AB$ is a line segment. What is the value of $x$?
[asy]
draw((0,0)--(10,0),black+linewidth(1));
draw((4,0)--(4,8),black+linewidth(1));
draw((4,0)--(3.5,0)--(3.5,0.5)--(4,0.5)--cycle,black+linewidth(1));
draw((4,0)--(9,7),black+linewidth(1));
label("$A$",(0,0),W);
label("$B$",(10,0),E);
label("$x^\circ$",(4.75,2.25));
label("$52^\circ$",(5.5,0.75));
[/asy] | 38 |
The volume of the solid generated by rotating the circle $x^2 + (y + 1)^2 = 3$ around the line $y = kx - 1$ for one complete revolution is what? | 4\sqrt{3}\pi |
The taxi fare in Gotham City is $2.40 for the first $\frac{1}{2}$ mile and additional mileage charged at the rate $0.20 for each additional 0.1 mile. You plan to give the driver a $2 tip. How many miles can you ride for $10? | 3.3 |
Dolly wants to ride the Ferris wheel twice, the roller coaster three times, and the log ride seven times. The Ferris wheel costs 2 tickets, the roller coaster costs 5 tickets and the log ride costs 1 ticket. Dolly has 20 tickets. How many more tickets should Dolly buy? | It costs 2 rides x 2 tickets/ride = <<2*2=4>>4 tickets to ride the Ferris wheel twice.
It costs 3 rides x 5 tickets/ride = <<3*5=15>>15 tickets to ride the roller coaster three times.
It costs 7 rides x 1 ticket/ride = <<7*1=7>>7 tickets to ride the roller coaster seven times.
In total Dolly needs 4 tickets + 15 tickets + 7 tickets = <<4+15+7=26>>26 tickets.
Dolly needs to buy 26 tickets - 20 tickets = <<26-20=6>>6 more tickets.
#### 6 |
Points \(P, Q, R,\) and \(S\) lie in the plane of the square \(EFGH\) such that \(EPF\), \(FQG\), \(GRH\), and \(HSE\) are equilateral triangles. If \(EFGH\) has an area of 25, find the area of quadrilateral \(PQRS\). Express your answer in simplest radical form. | 100 + 50\sqrt{3} |
How many 5-digit numbers $\overline{a b c d e}$ exist such that digits $b$ and $d$ are each the sum of the digits to their immediate left and right? (That is, $b=a+c$ and $d=c+e$.) | 330 |
In the quadrilateral $MARE$ inscribed in a unit circle $\omega,$ $AM$ is a diameter of $\omega,$ and $E$ lies on the angle bisector of $\angle RAM.$ Given that triangles $RAM$ and $REM$ have the same area, find the area of quadrilateral $MARE.$ | \frac{8\sqrt{2}}{9} |
Caleb has 3 dozen jellybeans. Sophie has half as many jellybeans as Caleb. How many jellybeans do they have in total? | Caleb has 3 x 12 = <<3*12=36>>36 jellybeans.
Sophie has 36 / 2 = <<36/2=18>>18 jellybeans.
Together, they have 36 + 18 = <<36+18=54>>54 jellybeans.
#### 54 |
Find the numbers $\mathbf{1 5 3 , 3 7 0 , 3 7 1 , 4 0 7}$. | 153, 370, 371, 407 |
What is the mean of the set $\{m, m + 6, m + 8, m + 11, m + 18, m + 20\}$ if the median of the set is 19? | 20 |
Find all values of $x$ with $0 \le x < 2 \pi$ that satisfy $\sin x + \cos x = \sqrt{2}.$ Enter all the solutions, separated by commas. | \frac{\pi}{4} |
In a circle, $15$ equally spaced points are drawn and arbitrary triangles are formed connecting $3$ of these points. How many non-congruent triangles can be drawn? | 19 |
Find the maximum number of white dominoes that can be cut from the board shown on the left. A domino is a $1 \times 2$ rectangle. | 16 |
Given is a isosceles triangle ABC so that AB=BC. Point K is in ABC, so that CK=AB=BC and <KAC=30°.Find <AKB=? | 150 |
Suppose \( S = \{1,2, \cdots, 2005\} \). Find the minimum value of \( n \) such that every subset of \( S \) consisting of \( n \) pairwise coprime numbers contains at least one prime number. | 16 |
A colony of bees can contain up to 80000 individuals. In winter they are more exposed to death, and if the winter is really cold the bees can begin to die slowly. If the colony starts to lose 1200 bees per day, how many days will pass until the number of bees in the colony reaches a fourth of its initial number? | A fourth of the initial number of bees is 80000 bees / 4 = <<80000/4=20000>>20000 bees
The number of bees that has to die is 80000 bees – 20000 bees = <<80000-20000=60000>>60000 bees
For the colony to reach a fourth of its population, 60000 bees ÷ 1200 bees/day = <<60000/1200=50>>50 days have to pass.
#### 50 |
John buys 5 toys that each cost $3. He gets a 20% discount. How much did he pay for everything? | The toys cost 5*3=$<<5*3=15>>15
His discount is worth 15*.2=$<<15*.2=3>>3
So he pays 15-3=$<<15-3=12>>12
#### 12 |
Simplify: $(\sqrt{5})^4$. | 25 |
Given $x^3y = k$ for a positive constant $k$, find the percentage decrease in $y$ when $x$ increases by $20\%$. | 42.13\% |
Xiao Ming's home is 30 minutes away from school by subway and 50 minutes by bus. One day, Xiao Ming took the subway first and then transferred to the bus, taking a total of 40 minutes to reach school, with the transfer process taking 6 minutes. How many minutes did Xiao Ming take the bus that day? | 10 |
Jim is baking loaves of bread. He has 200g of flour in the cupboard, 100g of flour on the kitchen counter, and 100g in the pantry. If one loaf of bread requires 200g of flour, how many loaves can Jim bake? | Altogether, Jim has 200g + 100g + 100g = <<200+100+100=400>>400g of flour.
He can therefore bake 400g / 200g = <<400/200=2>>2 loaves of bread.
#### 2 |
If Layla scored 104 goals in four hockey games and Kristin scored 24 fewer goals in the same four games, calculate the average number of goals the two scored. | If Layla scored 104 goals in four hockey games and Kristin scored 24 fewer goals in the same four games, Kristin scored 104-24 = <<104-24=80>>80 goals.
The combined number of goals they scored is 80+104 = <<80+104=184>>184
The average number of goals they scored is 184/2 = <<184/2=92>>92
#### 92 |
Leila eats cake almost every week. Last week, she ate 6 cakes on Monday, 9 cakes on Friday, and on Saturday, she ate triple the number of cakes she ate on Monday. How many cakes does Leila eat? | Combining Monday and Friday, Leila ate 6 + 9 = <<6+9=15>>15 cakes.
On Saturday, she ate 6 * 3 = <<6*3=18>>18 cakes.
Altogether, Leila ate 15 + 18 = <<15+18=33>>33 cakes.
#### 33 |
Calculate the integrals:
1) \(\int_{0}^{\frac{\pi}{2}} \sin ^{3} x \, dx\);
2) \(\int_{0}^{\ln 2} \sqrt{e^{x}-1} \, dx\);
3) \(\int_{-a}^{a} x^{2} \sqrt{a^{2}-x^{2}} \, dx\);
4) \(\int_{1}^{2} \frac{\sqrt{x^{2}-1}}{x} \, dx\). | \sqrt{3} - \frac{\pi}{3} |
Amerigo Vespucci has a map of America drawn on the complex plane. The map does not distort distances. Los Angeles corresponds to $0$ on this complex plane, and Boston corresponds to $2600i$. Meanwhile, Knoxville corresponds to the point $780+1040i$. With these city-point correspondences, how far is it from Knoxville to Los Angeles on this complex plane? | 1300 |
A cube has eight vertices (corners) and twelve edges. A segment, such as $x$, which joins two vertices not joined by an edge is called a diagonal. Segment $y$ is also a diagonal. How many diagonals does a cube have? [asy]
/* AMC8 1998 #17 Problem */
pair A=(0,48), B=(0,0), C=(48,0), D=(48,48);
pair E=(24,72), F=(24,24), G=(72,24), H=(72,72);
pen d = linetype("8 8");
draw(A--D--C--B--cycle);
draw(D--H--G--C);
draw(A--E--H);
draw(B--F--E);
draw(F--G);
draw(H--A--G, d);
label("$x$", (40, 61), N);
label("$y$", (40, 35), N);
[/asy] | 16 |
The ratio of boys to girls in a classroom is 3:5. If there are 4 more girls than boys, how many students are in the classroom? | Since the ratio of boys to girls is 3:5, then the class is divided into 3 + 5 = <<3+5=8>>8 parts.
The girls have 5 - 3 = <<5-3=2>>2 parts more than the boys.
Since 2 parts are equal to 4, then 1 part is equal to 4/2 = 2 students.
Hence, there are 8 parts x 2 = <<8*2=16>>16 students.
#### 16 |
Given the function $f(x) = 4\sin^2 x + \sin\left(2x + \frac{\pi}{6}\right) - 2$,
$(1)$ Determine the interval over which $f(x)$ is strictly decreasing;
$(2)$ Find the maximum value of $f(x)$ on the interval $[0, \frac{\pi}{2}]$ and determine the value(s) of $x$ at which the maximum value occurs. | \frac{5\pi}{12} |
Calculate the volume of the tetrahedron with vertices at points \( A_{1}, A_{2}, A_{3}, A_{4} \) and its height, dropped from vertex \( A_{4} \) onto the face \( A_{1} A_{2} A_{3} \).
\( A_{1}(7, 2, 4) \)
\( A_{2}(7, -1, -2) \)
\( A_{3}(3, 3, 1) \)
\( A_{4}(-4, 2, 1) \) | 21.5 |
A bear is in the center of the left down corner of a $100*100$ square .we call a cycle in this grid a bear cycle if it visits each square exactly ones and gets back to the place it started.Removing a row or column with compose the bear cycle into number of pathes.Find the minimum $k$ so that in any bear cycle we can remove a row or column so that the maximum length of the remaining pathes is at most $k$ . | 5000 |
If $f^{-1}(g(x))=x^2-4$ and $g$ has an inverse, find $g^{-1}(f(10))$. | \sqrt{14} |
One angle of a triangle is twice another, and the sides opposite these angles have lengths 15 and 9. Compute the length of the third side of the triangle. | 16 |
A pen costs $9 more than a pencil. If a pencil costs $2, find the total cost of both items. | A pen costs $9 more than $2 which is $9+$2 = $<<9+2=11>>11
Altogether they cost $11+$2 = $<<11+2=13>>13
#### 13 |
What value of $x$ satisfies
\[x- \frac{3}{4} = \frac{5}{12} - \frac{1}{3}?\] | \frac{5}{6} |
The natural domain of the function \( y = f\left(\frac{2x}{3x^2 + 1}\right) \) is \(\left[\frac{1}{4}, a\right]\). Find the value of \( a \). | \frac{4}{3} |
The remainder when 111 is divided by 10 is 1. The remainder when 111 is divided by the positive integer $n$ is 6. How many possible values of $n$ are there? | 5 |
If $\frac{x-y}{x+y}=5$, what is the value of $\frac{2x+3y}{3x-2y}$? | 0 |
Find the smallest natural number $n$ with the following property: in any $n$-element subset of $\{1, 2, \cdots, 60\}$, there must be three numbers that are pairwise coprime. | 41 |
A company has 1000 employees. There will be three rounds of layoffs. For each round of layoff, 10% of the remaining employees will be laid off. How many employees will be laid off in total? | The first round will result in 1000 * 10% = <<1000*10*.01=100>>100 employee layoffs.
There are 1000 - 100 = <<1000-100=900>>900 employees remaining.
The second round will result in 900 * 10% = <<900*10*.01=90>>90 employee layoffs.
There are 900 - 90 = <<900-90=810>>810 employees remaining.
The third round will result in 810 * 10% = <<810*10*.01=81>>81 employee layoffs.
The total number of of employees laid off is 100 + 90 + 81 = <<100+90+81=271>>271.
#### 271 |
In a convex 13-gon, all diagonals are drawn. They divide it into polygons. Consider the polygon with the largest number of sides among them. What is the greatest number of sides that it can have? | 13 |
A company is hosting a seminar. So far, 30 attendees from company A have been registered; company B has twice the number of attendees of company A; company C has 10 more attendees than company A; company D has 5 fewer attendees than company C. If a total of 185 attendees have registered, how many attendees who registered are not from either company A, B, C or D? | There are 30 x 2 = <<30*2=60>>60 attendees from company B.
Company C has 30 + 10 = <<30+10=40>>40 attendees.
Company D has 40 - 5 = <<40-5=35>>35 registered attendees.
So, the total attendees from company A, B, C, and D is 30 + 60 + 40 + 35 = <<30+60+40+35=165>>165.
Therefore there are 185 - 165 = <<185-165=20>>20 attendees that are not from companies A, B, C, or D.
#### 20 |
If $a$ and $b$ are elements of the set ${ 1,2,3,4,5,6 }$ and $|a-b| \leqslant 1$, calculate the probability that any two people playing this game form a "friendly pair". | \dfrac{4}{9} |
In a regular tetrahedron the centers of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to that of the larger is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
| 28 |
What is the product of all real numbers that are doubled when added to their reciprocals? | -1 |
Lark has forgotten her locker combination. It is a sequence of three numbers, each in the range from 1 to 30, inclusive. She knows that the first number is odd, the second number is even, and the third number is a multiple of 3. How many combinations could possibly be Lark's? | 2250 |
Given the quadratic function \( y = ax^2 + bx + c \) with its graph intersecting the \( x \)-axis at points \( A \) and \( B \), and its vertex at point \( C \):
(1) If \( \triangle ABC \) is a right-angled triangle, find the value of \( b^2 - 4ac \).
(2) Consider the quadratic function
\[ y = x^2 - (2m + 2)x + m^2 + 5m + 3 \]
with its graph intersecting the \( x \)-axis at points \( E \) and \( F \), and it intersects the linear function \( y = 3x - 1 \) at two points, with the point having the smaller \( y \)-coordinate denoted as point \( G \).
(i) Express the coordinates of point \( G \) in terms of \( m \).
(ii) If \( \triangle EFG \) is a right-angled triangle, find the value of \( m \). | -1 |
Mr. Sean has a veterinary clinic where he charges clients $60 to treat a dog and $40 to care for a cat. If Khalil took 20 dogs and 60 cats to the clinic for treatment, how much money did he pay Mr. Sean? | To treat his 20 dogs, Mr. Sean charged Khalil 20*$60 = $<<20*60=1200>>1200
Khalil also paid $40*60 = $<<40*60=2400>>2400 to Mr. Sean to treat his cats.
Altogether, Khalil paid Mr. Sean $2400+$1200 = $<<2400+1200=3600>>3600 to treat his pets.
#### 3600 |
In a trapezoid with bases 3 and 4, find the length of the segment parallel to the bases that divides the area of the trapezoid in the ratio $5:2$, counting from the shorter base. | \sqrt{14} |
Altitudes $\overline{AX}$ and $\overline{BY}$ of acute triangle $ABC$ intersect at $H$. If $\angle BAC = 61^\circ$ and $\angle ABC = 73^\circ$, then what is $\angle CHX$? | 73^\circ |
Sadie has 140 math homework problems for the week. 40 percent are Algebra problems, and half of the Algebra problems are solving linear equations. How many solving linear equations problems does Sadie have to solve? | Algebra:140(.40)=56 problems
Linear:56/2=<<56/2=28>>28 problems
#### 28 |
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