problem
stringlengths 10
5.15k
| answer
stringlengths 0
1.23k
|
|---|---|
The numbers from 1 to 150, inclusive, are placed in a bag and a number is randomly selected from the bag. What is the probability it is not a perfect power (integers that can be expressed as $x^{y}$ where $x$ is an integer and $y$ is an integer greater than 1. For example, $2^{4}=16$ is a perfect power, while $2\times3=6$ is not a perfect power)? Express your answer as a common fraction.
|
\frac{133}{150}
|
Billy buys a 12 pack of soda from the store. If he has twice as many brothers as sisters, and he has 2 sisters, how many sodas can he give to each of his siblings if he wants to give out the entire 12 pack while giving each the same number?
|
First, we need to determine how many siblings Billy has. To do this, we perform 2*2= <<2*2=4>>4 brothers, as he has 2 sisters and twice as many brothers as sisters.
Then we divide the total number of sodas in the 12 pack, which is 12, by the number of siblings, performing 12/6=<<12/6=2>>2 cans per person.
#### 2
|
There are 2017 jars in a row on a table, initially empty. Each day, a nice man picks ten consecutive jars and deposits one coin in each of the ten jars. Later, Kelvin the Frog comes back to see that $N$ of the jars all contain the same positive integer number of coins (i.e. there is an integer $d>0$ such that $N$ of the jars have exactly $d$ coins). What is the maximum possible value of $N$?
|
2014
|
Given a function $f(x) = \frac {1}{2}\sin x - \frac {\sqrt {3}}{2}\cos x$ defined on the interval $[a, b]$, the range of $f(x)$ is $[-\frac {1}{2}, 1]$. Find the maximum value of $b-a$.
|
\frac{4\pi}{3}
|
The operation $*$ is defined by
\[a * b = \frac{a - b}{1 - ab}.\]Compute
\[1 * (2 * (3 * (\dotsb (999 * 1000) \dotsb))).\]
|
1
|
A container holds $47\frac{2}{3}$ cups of sugar. If one recipe requires $1\frac{1}{2}$ cups of sugar, how many full recipes can be made with the sugar in the container? Express your answer as a mixed number.
|
31\frac{7}{9}
|
A semicircle is inscribed in an isosceles triangle with base 16 and height 15 so that the diameter of the semicircle is contained in the base of the triangle. What is the radius of the semicircle?
|
\frac{120}{17}
|
The midpoints of the sides of a regular octagon \( ABCDEFGH \) are joined to form a smaller octagon. What fraction of the area of \( ABCDEFGH \) is enclosed by the smaller octagon?
|
\frac{1}{2}
|
Circles with centers $P, Q$ and $R$, having radii $1, 2$ and $3$, respectively, lie on the same side of line $l$ and are tangent to $l$ at $P', Q'$ and $R'$, respectively, with $Q'$ between $P'$ and $R'$. The circle with center $Q$ is externally tangent to each of the other two circles. What is the area of triangle $PQR$?
|
0
|
In Flower Town, there are $99^{2}$ residents, some of whom are knights (who always tell the truth) and others are liars (who always lie). The houses in the town are arranged in the cells of a $99 \times 99$ square grid (totaling $99^{2}$ houses, arranged on 99 vertical and 99 horizontal streets). Each house is inhabited by exactly one resident. The house number is denoted by a pair of numbers $(x ; y)$, where $1 \leq x \leq 99$ is the number of the vertical street (numbers increase from left to right), and $1 \leq y \leq 99$ is the number of the horizontal street (numbers increase from bottom to top). The flower distance between two houses numbered $\left(x_{1} ; y_{1}\right)$ and $\left(x_{2} ; y_{2}\right)$ is defined as the number $\rho=\left|x_{1}-x_{2}\right|+\left|y_{1}-y_{2}\right|$. It is known that on every vertical or horizontal street, at least $k$ residents are knights. Additionally, all residents know which house Knight Znayka lives in, but you do not know what Znayka looks like. You want to find Znayka's house and you can approach any house and ask the resident: "What is the flower distance from your house to Znayka’s house?". What is the smallest value of $k$ that allows you to guarantee finding Znayka’s house?
|
75
|
Three positive integers differ from each other by at most 6. The product of these three integers is 2808. What is the smallest integer among them?
|
12
|
The angle that has the same terminal side as $- \frac{\pi}{3}$ is $\frac{\pi}{3}$.
|
\frac{5\pi}{3}
|
To get free delivery, Alice needs to spend a minimum of $35.00 online at her favorite grocery store. In her cart she has 1.5 pounds of chicken at $6.00 per pound, 1 pack of lettuce for $3.00, cherry tomatoes for $2.50, 4 sweet potatoes at $0.75 each, 2 heads of broccoli for $2.00 each and a pound of Brussel sprouts for $2.50. How much more does she need to spend in order to get free delivery?
|
She bought 1.5 pounds of chicken for $6.00/pound so that's 1.5*$6 = $<<1.5*6=9.00>>9.00
She bought 4 sweet potatoes at $0.75 each so that's 4*$0.75 = $<<4*0.75=3.00>>3.00
She bought 2 heads of broccoli for $2.00 each so that's 2*$2 = $<<2*2=4.00>>4.00
The chicken is $9.00, lettuce is $3.00, tomatoes are $2.50, the sweet potatoes are $3.00, broccoli is $4.00 and Brussel sprouts are $2.50 for a total of $9 + $3 + $2.50 + $3 + $4 + $2.5 = $<<9+3+2.5+3+4+2.5=24.00>>24.00
To get free delivery she needs to spend $35.00, and she already has $24.00 in her cart, so she needs to spend an extra $35-$24 = $<<35-24=11.00>>11.00
#### 11
|
On the side AB of triangle ABC with a $100^{\circ}$ angle at vertex C, points P and Q are taken such that $AP = BC$ and $BQ = AC$. Let M, N, and K be the midpoints of segments AB, CP, and CQ respectively. Find the angle $NMK$.
|
40
|
A school cafeteria uses ground mince to cook lasagnas and cottage pies. They make 100 lasagnas, which use 2 pounds of ground mince each, and cottage pies, which use 3 pounds of ground mince each. If the cafeteria has used 500 pounds of ground mince in total, how many cottage pies did they make?
|
The cafeteria used 100 lasagnas * 2 pounds of ground mince = <<100*2=200>>200 pounds of ground mince for the lasagnas.
This means they used 500 pounds of ground mince in total – 200 pounds of ground mince for the lasagnas = 300 pounds of ground mince for the cottage pies.
So they must have made 300 pounds of ground mince for the cottage pies / 3 pounds of ground mince per cottage pie = <<300/3=100>>100 cottage pies.
#### 100
|
How many ways are there to partition $7$ students into the groups of $2$ or $3$ ?
|
105
|
$100$ children stand in a line each having $100$ candies. In one move, one of them may take some of their candies and distribute them to a non-empty set of the remaining children. After what least number of moves can it happen that no two children have the same number of candies?
|
30
|
What number is a multiple of every integer?
|
0
|
Hannah fills her kids' stockings with 4 candy canes, 2 beanie babies and 1 book. If she has 3 kids, how many stocking stuffers does she buy total?
|
First find the total number of stocking stuffers one kid gets: 4 candy canes + 2 beanie babies + 1 book = <<4+2+1=7>>7 stocking stuffers
Then multiply the number of stocking stuffers per kid by the number of kids to find the total number: 7 stocking stuffers/kid * 3 kids = <<7*3=21>>21 stocking stuffers
#### 21
|
What is the smallest positive integer $n$ such that all the roots of $z^5 - z^3 + z = 0$ are $n^{\text{th}}$ roots of unity?
|
12
|
Michael has $42. His brother has $17. Michael gives away half the money to his brother. His brother then buys 3 dollars worth of candy. How much money, in dollars, did his brother have in the end?
|
Michael gives away 42/2=<<42/2=21>>21 dollars.
His brother then has 21+17=<<21+17=38>>38 dollars.
In the end, his brother has 38-3=<<38-3=35>>35 dollars.
#### 35
|
A circle of radius $2$ is cut into four congruent arcs. The four arcs are joined to form the star figure shown. What is the ratio of the area of the star figure to the area of the original circle?
[asy] size(0,50); draw((-1,1)..(-2,2)..(-3,1)..(-2,0)..cycle); dot((-1,1)); dot((-2,2)); dot((-3,1)); dot((-2,0)); draw((1,0){up}..{left}(0,1)); dot((1,0)); dot((0,1)); draw((0,1){right}..{up}(1,2)); dot((1,2)); draw((1,2){down}..{right}(2,1)); dot((2,1)); draw((2,1){left}..{down}(1,0));[/asy]
|
\frac{4-\pi}{\pi}
|
Susie Q has 1000 dollars to invest. She invests some of the money at the Pretty Penny Bank, which compounds annually at 3 percent. She invests the rest of the money at the Five and Dime Bank, which compounds annually at 5 percent. After two years, Susie has a total of $\$1090.02$. How much did Susie Q originally invest at the Pretty Penny Bank, in dollars?
|
300
|
If $\log_{25}(x-4)=\frac{1}{2}$, find $\frac{1}{\log_{x}3}$.
|
2
|
In a particular game, each of $4$ players rolls a standard $8$-sided die. The winner is the player who rolls the highest number. If there is a tie for the highest roll, those involved in the tie will roll again continuing until one player wins. Hugo is one of the players. What is the probability that Hugo's first roll was a $7$, given that he won the game?
A) $\frac{25}{256}$
B) $\frac{27}{128}$
C) $\frac{13}{64}$
D) $\frac{17}{96}$
|
\frac{27}{128}
|
Kris has been suspended for bullying many times. For every instance of bullying, she was suspended for 3 days. If she has been suspended for three times as many days as a typical person has fingers and toes, how many instances of bullying is she responsible for?
|
Three times as many days as a typical person has fingers and toes is 3*20=<<3*20=60>>60 days
Three days per instance of bullying is 60/3=<<60/3=20>>20 instances of bullying.
#### 20
|
Some boys and girls are having a car wash to raise money for a class trip to China. Initially $40\%$ of the group are girls. Shortly thereafter two girls leave and two boys arrive, and then $30\%$ of the group are girls. How many girls were initially in the group?
|
8
|
Compute the number of intersection points of the graphs of
\[(x - \lfloor x \rfloor)^2 + y^2 = x - \lfloor x \rfloor\]and $y = \frac{1}{5} x.$
|
11
|
Laura typically performs her routine using two 10-pound dumbbells for 30 repetitions. If she decides to use two 8-pound dumbbells instead, how many repetitions must she complete to lift the same total weight as her usual routine?
|
37.5
|
Four cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is an Ace, the second card is a 2, the third card is a 3, and the fourth card is a 4? Assume that the dealing is done without replacement.
|
\frac{16}{405525}
|
Multiply $(x^4 +18 x^2 + 324) (x^2-18)$.
|
x^6-5832
|
If $7^{4x}=343$, what is the value of $7^{4x-3}$?
|
1
|
The two squares shown share the same center $O$ and have sides of length 1. The length of $\overline{AB}$ is $43/99$ and the area of octagon $ABCDEFGH$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
[asy] //code taken from thread for problem real alpha = 25; pair W=dir(225), X=dir(315), Y=dir(45), Z=dir(135), O=origin; pair w=dir(alpha)*W, x=dir(alpha)*X, y=dir(alpha)*Y, z=dir(alpha)*Z; draw(W--X--Y--Z--cycle^^w--x--y--z--cycle); pair A=intersectionpoint(Y--Z, y--z), C=intersectionpoint(Y--X, y--x), E=intersectionpoint(W--X, w--x), G=intersectionpoint(W--Z, w--z), B=intersectionpoint(Y--Z, y--x), D=intersectionpoint(Y--X, w--x), F=intersectionpoint(W--X, w--z), H=intersectionpoint(W--Z, y--z); dot(O); label("$O$", O, SE); label("$A$", A, dir(O--A)); label("$B$", B, dir(O--B)); label("$C$", C, dir(O--C)); label("$D$", D, dir(O--D)); label("$E$", E, dir(O--E)); label("$F$", F, dir(O--F)); label("$G$", G, dir(O--G)); label("$H$", H, dir(O--H));[/asy]
|
185
|
If $AAA_4$ can be expressed as $33_b$, where $A$ is a digit in base 4 and $b$ is a base greater than 5, what is the smallest possible sum $A+b$?
|
7
|
Given that $f(x) = x^{2}-2x+5$ and $g(x) =x+3$, what is the value of $f(g(5)) -g(f(5))$?
|
30
|
The faces of a 12-sided die are numbered 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12 such that the sum of the numbers on opposite faces is 13. The die is meticulously carved so that it is biased: the probability of obtaining a particular face \( F \) is greater than \( \frac{1}{12} \), the probability of obtaining the face opposite \( F \) is less than \( \frac{1}{12} \) while the probability of obtaining any one of the other ten faces is \( \frac{1}{12} \).
When two such dice are rolled, the probability of obtaining a sum of 13 is \( \frac{29}{384} \).
What is the probability of obtaining face \( F \)?
|
\frac{7}{48}
|
How many times does 24 divide into 100! (factorial)?
|
32
|
Let \( S = \{1, 2, \ldots, 1963\} \). What is the maximum number of elements that can be chosen from \( S \) such that the sum of any two chosen numbers is not divisible by their difference?
|
655
|
The decimal representation of \(\frac{1}{20^{20}}\) consists of a string of zeros after the decimal point, followed by a 9 and then several more digits. How many zeros are in that initial string of zeros after the decimal point?
|
26
|
A student walks from intersection $A$ to intersection $B$ in a city layout as described: the paths allow him to move only east or south. Every morning, he walks from $A$ to $B$, but now he needs to pass through intersections $C$ and then $D$. The paths from $A$ to $C$ have 3 eastward and 2 southward moves, from $C$ to $D$ have 2 eastward and 1 southward move, and from $D$ to $B$ have 1 eastward and 2 southward moves. If at each intersection where the student has a choice, he chooses with probability $\frac{1}{2}$ whether to go east or south, find the probability that through any given morning, he goes through $C$ and then $D$.
A) $\frac{15}{77}$
B) $\frac{90}{462}$
C) $\frac{10}{21}$
D) $\frac{1}{2}$
E) $\frac{3}{4}$
|
\frac{15}{77}
|
A baseball league consists of two four-team divisions. Each team plays every other team in its division $N$ games. Each team plays every team in the other division $M$ games with $N>2M$ and $M>4$. Each team plays a $76$ game schedule. How many games does a team play within its own division?
|
48
|
Compute the definite integral:
$$
\int_{1}^{2} \frac{x+\sqrt{3 x-2}-10}{\sqrt{3 x-2}+7} d x
$$
|
-\frac{22}{27}
|
Let $m \ge 3$ be an integer and let $S = \{3,4,5,\ldots,m\}$. Find the smallest value of $m$ such that for every partition of $S$ into two subsets, at least one of the subsets contains integers $a$, $b$, and $c$ (not necessarily distinct) such that $ab = c$.
Note: a partition of $S$ is a pair of sets $A$, $B$ such that $A \cap B = \emptyset$, $A \cup B = S$.
|
243
|
A six-digit number can be tripled by reducing the first digit by three and appending a three at the end. What is this number?
|
428571
|
Find the least positive integer $n$ such that the prime factorizations of $n$ , $n + 1$ , and $n + 2$ each have exactly two factors (as $4$ and $6$ do, but $12$ does not).
|
33
|
Form a three-digit number using the digits 4, 5, and 6. The probability that the number formed is a common multiple of 3 and 5 is ____.
|
\frac{1}{3}
|
Let $(a_n)_{n \ge 1}$ be a sequence of positive real numbers such that the sequence $(a_{n+1}-a_n)_{n \ge 1}$ is convergent to a non-zero real number. Evaluate the limit $$ \lim_{n \to \infty} \left( \frac{a_{n+1}}{a_n} \right)^n. $$
|
e
|
There are 10 questions available, among which 6 are Type A questions and 4 are Type B questions. Student Xiao Ming randomly selects 3 questions to solve.
(1) Calculate the probability that Xiao Ming selects at least 1 Type B question.
(2) Given that among the 3 selected questions, there are 2 Type A questions and 1 Type B question, and the probability of Xiao Ming correctly answering each Type A question is $\dfrac{3}{5}$, and the probability of correctly answering each Type B question is $\dfrac{4}{5}$. Assuming the correctness of answers to different questions is independent, calculate the probability that Xiao Ming correctly answers at least 2 questions.
|
\dfrac{93}{125}
|
The union of sets \( A \) and \( B \) is \( A \cup B = \{a_1, a_2, a_3\} \). When \( A \neq B \), \((A, B)\) and \((B, A)\) are considered different pairs. How many such pairs \((A, B)\) exist?
|
27
|
In solving the system of equations $y = 7$ and $x^2+ y^2= 100,$ what is the sum of the solutions for $x?$
|
0
|
Max fills up water balloons for 30 minutes at a rate of 2 water balloons every minute. Max’s friend Zach fills up water balloons for 40 minutes at a rate of 3 water balloons every minute. In the process, 10 of the water balloons pop on the ground. How many filled water balloons do Max and Zach have in total?
|
Max fills 30 * 2 = <<30*2=60>>60 water balloons
Zach fills 40 * 3 = <<40*3=120>>120 water balloons
Max and Zach have a total of 60 + 120 - 10 = <<60+120-10=170>>170 water balloons
#### 170
|
Given 2009 points, where no three points are collinear, along with three vertices, making a total of 2012 points, and connecting these 2012 points to form non-overlapping small triangles, calculate the total number of small triangles that can be formed.
|
4019
|
On a circle, there are 2018 points. Each of these points is labeled with an integer. Each number is greater than the sum of the two numbers immediately preceding it in a clockwise direction.
Determine the maximum possible number of positive numbers among these 2018 numbers.
(Walther Janous)
|
1009
|
What is the largest value of $n$ less than 50,000 for which the expression $3(n-3)^2 - 4n + 28$ is a multiple of 7?
|
49999
|
Jasmine had 2 paperclips on Monday, then she had 6 on Tuesday, and her number of paperclips proceeded to triple on each subsequent day. On what day of the week did she first have more than 100 paperclips?
|
\text{Friday}
|
The fourth power of $\sqrt{1+\sqrt{1+\sqrt{1}}}$ is
|
3+2\sqrt{2}
|
In an acute triangle $ABC$ , the points $H$ , $G$ , and $M$ are located on $BC$ in such a way that $AH$ , $AG$ , and $AM$ are the height, angle bisector, and median of the triangle, respectively. It is known that $HG=GM$ , $AB=10$ , and $AC=14$ . Find the area of triangle $ABC$ .
|
12\sqrt{34}
|
In triangle $DEF$, $DE=130$, $DF=110$, and $EF=140$. The angle bisector of angle $D$ intersects $\overline{EF}$ at point $T$, and the angle bisector of angle $E$ intersects $\overline{DF}$ at point $S$. Let $R$ and $U$ be the feet of the perpendiculars from $F$ to $\overline{ES}$ and $\overline{DT}$, respectively. Find $RU$.
|
60
|
A three-digit number has digits a, b, and c in the hundreds, tens, and units place respectively. If a < b and b > c, then the number is called a "convex number". If you randomly select three digits from 1, 2, 3, and 4 to form a three-digit number, what is the probability that it is a "convex number"?
|
\frac{1}{3}
|
Given that non-negative real numbers \( x \) and \( y \) satisfy
\[
x^{2}+4y^{2}+4xy+4x^{2}y^{2}=32,
\]
find the maximum value of \( \sqrt{7}(x+2y)+2xy \).
|
16
|
Find the largest possible value of \( n \) such that there exist \( n \) consecutive positive integers whose sum is equal to 2010.
|
60
|
Let $x$ and $y$ be real numbers such that $\frac{\sin x}{\sin y} = 3$ and $\frac{\cos x}{\cos y} = \frac12$. The value of $\frac{\sin 2x}{\sin 2y} + \frac{\cos 2x}{\cos 2y}$ can be expressed in the form $\frac pq$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
|
107
|
An environmental agency decides to expand its monitoring teams due to new regulations requiring more extensive testing. They estimate needing 120 new employees to monitor water pollution and 105 new employees to monitor air pollution. Additionally, they need 65 new employees capable of monitoring air and water pollution. On top of this, there should be another team where 40 of the new employees will also monitor soil pollution (including taking roles in air and water tasks). Determine the minimum number of new employees the agency must hire.
|
160
|
Find the least real number $k$ with the following property: if the real numbers $x$ , $y$ , and $z$ are not all positive, then \[k(x^{2}-x+1)(y^{2}-y+1)(z^{2}-z+1)\geq (xyz)^{2}-xyz+1.\]
|
\frac{16}{9}
|
A line $x=k$ intersects the graph of $y=\log_5 x$ and the graph of $y=\log_5 (x + 4)$. The distance between the points of intersection is $0.5$. Given that $k = a + \sqrt{b}$, where $a$ and $b$ are integers, what is $a+b$?
|
6
|
In the plane rectangular coordinate system $xOy$, the graph of the quadratic function $y=a\left(x-h\right)^{2}+k\left(a \lt 0\right)$ passes through the points $A\left(-3,m\right)$ and $B\left(-1,n\right)$.<br/>$(1)$ When $m=n$, find the length of the line segment $AB$ and the value of $h$;<br/>$(2)$ If the point $C\left(1,0\right)$ also lies on the graph of the quadratic function $y=a\left(x-h\right)^{2}+k\left(a \lt 0\right)$, and $m \lt 0 \lt n$,<br/>① find the abscissa of the other intersection point of the graph of the quadratic function $y=a\left(x-h\right)^{2}+k\left(a \lt 0\right)$ with the $x$-axis (expressed in terms of $h$) and the range of values for $h$;<br/>② if $a=-1$, find the area of $\triangle ABC$;<br/>③ a line passing through point $D(0$,$h^{2})$ perpendicular to the $y$-axis intersects the parabola at points $P(x_{1}$,$y_{1})$ and $(x_{2}$,$y_{2})$ (where $P$ and $Q$ are not coincident), and intersects the line $BC$ at point $(x_{3}$,$y_{3})$. Is there a value of $a$ such that $x_{1}+x_{2}-x_{3}$ is always a constant? If so, find the value of $a$; if not, explain why.
|
-\frac{1}{4}
|
The number $1000!$ has a long tail of zeroes. How many zeroes are there? (Reminder: The number $n!$ is the product of the integers from 1 to $n$. For example, $5!=5\cdot 4\cdot3\cdot2\cdot 1= 120$.)
|
249
|
If one-fourth of $2^{30}$ is equal to $2^x$, what is $x$?
|
28
|
Consider an alphabet of 2 letters. A word is any finite combination of letters. We will call a word unpronounceable if it contains more than two identical letters in a row. How many unpronounceable words of 7 letters exist? Points for the problem: 8.
|
86
|
The sum of an infinite geometric series is $64$ times the series that results if the first four terms of the original series are removed. What is the value of the series' common ratio?
|
\frac{1}{2}
|
Suppose $ABC$ is a scalene right triangle, and $P$ is the point on hypotenuse $\overline{AC}$ such that $\angle{ABP} =
45^{\circ}$. Given that $AP = 1$ and $CP = 2$, compute the area of $ABC$.
|
\frac{9}{5}
|
The sides of rectangle $ABCD$ have lengths $10$ and $11$. An equilateral triangle is drawn so that no point of the triangle lies outside $ABCD$. The maximum possible area of such a triangle can be written in the form $p\sqrt{q}-r$, where $p$, $q$, and $r$ are positive integers, and $q$ is not divisible by the square of any prime number. Find $p+q+r$.
|
554
|
How many distinct four-digit positive integers are there such that the product of their digits equals 18?
|
24
|
Given $\sin\theta + \cos\theta = \frac{1}{5}$, with $\theta \in (0,\pi)$,
1. Find the value of $\tan\theta$;
2. Find the value of $\frac{1+\sin 2\theta + \cos 2\theta}{1+\sin 2\theta - \cos 2\theta}$.
|
-\frac{3}{4}
|
In the Cartesian coordinate system, given that point $P(3,4)$ is a point on the terminal side of angle $\alpha$, if $\cos(\alpha+\beta)=\frac{1}{3}$, where $\beta \in (0,\pi)$, then $\cos \beta =\_\_\_\_\_\_.$
|
\frac{3 + 8\sqrt{2}}{15}
|
31 cars simultaneously started from one point on a circular track: the first car at a speed of 61 km/h, the second at 62 km/h, and so on (the 31st at 91 km/h). The track is narrow, and if one car overtakes another on a lap, they collide, both go off the track, and are out of the race. In the end, one car remains. At what speed is it traveling?
|
76
|
Two students, A and B, are playing table tennis. They have agreed on the following rules: ① Each point won earns 1 point; ② They use a three-point serve system, meaning they switch serving every three points. Assuming that when A serves, the probability of A winning a point is $\frac{3}{5}$, and when B serves, the probability of A winning a point is $\frac{1}{2}$, and the outcomes of each point are independent. According to the draw result, A serves first.
$(1)$ Let $X$ represent the score of A after three points. Find the distribution table and mean of $X$;
$(2)$ Find the probability that A has more points than B after six points.
|
\frac{441}{1000}
|
How many lines in a three dimensional rectangular coordinate system pass through four distinct points of the form $(i, j, k)$, where $i$, $j$, and $k$ are positive integers not exceeding four?
|
76
|
$A,B,C$ are points in the plane such that $\angle ABC=90^\circ$ . Circles with diameters $BA$ and $BC$ meet at $D$ . If $BA=20$ and $BC=21$ , then the length of segment $BD$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$ ?
*Ray Li*
|
449
|
Claire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of $2017$. She then discovers that she forgot to include one angle. What is the degree measure of the forgotten angle?
|
143
|
Mary can mow a lawn in four hours and Tom can mow the lawn in 5 hours. If Tom works for 2 hours alone, what fractional part of the lawn remains to be mowed?
|
\frac{3}{5}
|
Equilateral triangle $\triangle ABC$ is inscribed in circle $\omega$ with radius $18.$ Circle $\omega_A$ is tangent to sides $\overline{AB}$ and $\overline{AC}$ and is internally tangent to $\omega.$ Circles $\omega_B$ and $\omega_C$ are defined analogously. Circles $\omega_A,$ $\omega_B,$ and $\omega_C$ meet in six points---two points for each pair of circles. The three intersection points closest to the vertices of $\triangle ABC$ are the vertices of a large equilateral triangle in the interior of $\triangle ABC,$ and the other three intersection points are the vertices of a smaller equilateral triangle in the interior of $\triangle ABC.$ The side length of the smaller equilateral triangle can be written as $\sqrt{a} - \sqrt{b},$ where $a$ and $b$ are positive integers. Find $a+b.$
Diagram
[asy] /* Made by MRENTHUSIASM */ size(250); pair A, B, C, W, WA, WB, WC, X, Y, Z; A = 18*dir(90); B = 18*dir(210); C = 18*dir(330); W = (0,0); WA = 6*dir(270); WB = 6*dir(30); WC = 6*dir(150); X = (sqrt(117)-3)*dir(270); Y = (sqrt(117)-3)*dir(30); Z = (sqrt(117)-3)*dir(150); filldraw(X--Y--Z--cycle,green,dashed); draw(Circle(WA,12)^^Circle(WB,12)^^Circle(WC,12),blue); draw(Circle(W,18)^^A--B--C--cycle); dot("$A$",A,1.5*dir(A),linewidth(4)); dot("$B$",B,1.5*dir(B),linewidth(4)); dot("$C$",C,1.5*dir(C),linewidth(4)); dot("$\omega$",W,1.5*dir(270),linewidth(4)); dot("$\omega_A$",WA,1.5*dir(-WA),linewidth(4)); dot("$\omega_B$",WB,1.5*dir(-WB),linewidth(4)); dot("$\omega_C$",WC,1.5*dir(-WC),linewidth(4)); [/asy] ~MRENTHUSIASM ~ihatemath123
|
378
|
Juniper, the Irish Setter, has 4 bones. Her master gives her enough bones to double her number of bones. Unfortunately, the neighbor's dog steals away two of Juniper's bones. How many bones does Juniper have remaining?
|
Doubling 4 bones gives her 4*2=<<4*2=8>>8 bones.
The neighbor's dog steals two bones, leaving 8-2=<<8-2=6>>6 bones.
#### 6
|
A root of unity is a complex number that is a solution to $z^n = 1$ for some positive integer $n$. Determine the number of roots of unity that are also roots of $z^2 + az + b = 0$ for some integers $a$ and $b$.
|
8
|
Along the southern shore of a boundless sea stretches an archipelago of an infinite number of islands. The islands are connected by an endless chain of bridges, and each island is connected by a bridge to the shore. In the event of a strong earthquake, each bridge independently has a probability $p=0.5$ of being destroyed. What is the probability that after a strong earthquake, one can travel from the first island to the shore using the remaining intact bridges?
|
2/3
|
Kendall is counting her change. She has a total of $4 in quarters, dimes, and nickels. If she has 10 quarters and 12 dimes, how many nickels does she have?
|
The total money from the quarters is 10 * $0.25 = $<<10*0.25=2.50>>2.50
The total money from the dimes is 12 * $0.10 = $<<12*0.10=1.20>>1.20
The total money from the nickels is $4 - $2.50 - $1.20 = $<<4-2.5-1.2=0.30>>0.30
The number of nickels Kendall has is $0.30 / $0.05 = <<0.30/0.05=6>>6 nickels
#### 6
|
Suppose that $S$ is a series of real numbers between $2$ and $8$ inclusive, and that for any two elements $y > x$ in $S,$ $$ 98y - 102x - xy \ge 4. $$ What is the maximum possible size for the set $S?$ $$ \mathrm a. ~ 12\qquad \mathrm b.~14\qquad \mathrm c. ~16 \qquad \mathrm d. ~18 \qquad \mathrm e. 20 $$
|
16
|
Calculate the surface integrals of the first kind:
a) \(\iint_{\sigma}|x| dS\), where \(\sigma\) is defined by \(x^2 + y^2 + z^2 = 1\), \(z \geqslant 0\).
b) \(\iint_{\sigma} (x^2 + y^2) dS\), where \(\sigma\) is defined by \(x^2 + y^2 = 2z\), \(z = 1\).
c) \(\iint_{\sigma} (x^2 + y^2 + z^2) dS\), where \(\sigma\) is the part of the cone defined by \(z^2 - x^2 - y^2 = 0\), \(z \geqslant 0\), truncated by the cylinder \(x^2 + y^2 - 2x = 0\).
|
3\sqrt{2} \pi
|
What is the area of the smallest square that can contain a circle of radius 6?
|
144
|
A club has 20 members and needs to choose 2 members to be co-presidents and 1 member to be a treasurer. In how many ways can the club select its co-presidents and treasurer?
|
3420
|
The sum of two numbers is 25 and their product is 126. What is the absolute value of the difference of the two numbers?
|
11
|
Alexander draws 9 pictures for an exhibition at a gallery. 5 new galleries also want Alexander to draw for them, so he draws pictures for their exhibitions too. Each of these new galleries receives the same amount of pictures. For each picture, Alexander needs 4 pencils. For each exhibition, he needs another 2 pencils for signing his signature on the pictures. If Alexander uses 88 pencils on drawing and signing his pictures for all of the exhibitions, how many paintings are at each of the 5 new galleries?
|
Alexander has sent pictures to a total of 1 original gallery + 5 new galleries = <<1+5=6>>6 galleries.
For signatures, he has used a total of 6 galleries * 2 pencils per gallery = <<6*2=12>>12 pencils.
So of the pencils that he used, 88 total pencils – 12 signature pencils = <<88-12=76>>76 pencils for drawings.
At the first gallery, he used 9 pictures * 4 pencils per picture = <<9*4=36>>36 pencils.
So in the new galleries, he used 76 total pencils for drawings – 36 pencils at the first gallery = <<76-36=40>>40 pencils.
The new galleries received a total of 40 pencils / 4 pencils per picture = <<40/4=10>>10 pictures.
As the new galleries all receive equal amounts of pictures, each gallery must have 10 pictures / 5 galleries = <<10/5=2>>2 pictures per gallery.
#### 2
|
Given the hyperbola $\dfrac {x^{2}}{a^{2}}- \dfrac {y^{2}}{b^{2}}=1$ ($a > 0$, $b > 0$), its left and right vertices are $A$ and $B$, respectively. The right focus is $F$, and the line $l$ passing through point $F$ and perpendicular to the $x$-axis intersects the hyperbola at points $M$ and $N$. $P$ is a point on line $l$. When $\angle APB$ is maximized, point $P$ is exactly at $M$ (or $N$). Determine the eccentricity of the hyperbola.
|
\sqrt{2}
|
What is the sum of the largest and smallest prime factors of 990?
|
13
|
All the roots of the polynomial $z^6-10z^5+Az^4+Bz^3+Cz^2+Dz+16$ are positive integers, possibly repeated. What is the value of $B$?
|
-88
|
Find all values of $r$ such that $\lfloor r \rfloor + r = 12.2$.
|
r=6.2
|
How many subsets of two elements can be removed from the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}$ so that the mean (average) of the remaining numbers is 6?
|
5
|
Let $p(x)=x^4-4x^3+2x^2+ax+b$ . Suppose that for every root $\lambda$ of $p$ , $\frac{1}{\lambda}$ is also a root of $p$ . Then $a+b=$ [list=1]
[*] -3
[*] -6
[*] -4
[*] -8
[/list]
|
-3
|
A circle with diameter $AB$ is drawn, and the point $ P$ is chosen on segment $AB$ so that $\frac{AP}{AB} =\frac{1}{42}$ . Two new circles $a$ and $b$ are drawn with diameters $AP$ and $PB$ respectively. The perpendicular line to $AB$ passing through $ P$ intersects the circle twice at points $S$ and $T$ . Two more circles $s$ and $t$ are drawn with diameters $SP$ and $ST$ respectively. For any circle $\omega$ let $A(\omega)$ denote the area of the circle. What is $\frac{A(s)+A(t)}{A(a)+A(b)}$ ?
|
205/1681
|
If $\frac{1}{9}+\frac{1}{18}=\frac{1}{\square}$, what is the number that replaces the $\square$ to make the equation true?
|
6
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.