problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Find the $y$-intercept point of the line $3x+5y=20$. Provide your answer as an ordered pair. | (0,4) |
In this diagram, both polygons are regular. What is the value, in degrees, of the sum of the measures of angles $ABC$ and $ABD$?
[asy]
draw(10dir(0)--10dir(60)--10dir(120)--10dir(180)--10dir(240)--10dir(300)--10dir(360)--cycle,linewidth(2));
draw(10dir(240)--10dir(300)--10dir(300)+(0,-10)--10dir(240)+(0,-10)--10dir(240)--cycle,linewidth(2));
draw(10dir(300)+(-1,0)..9dir(300)..10dir(300)+dir(60),linewidth(2));
draw(10dir(300)+(-1.5,0)..10dir(300)+1.5dir(-135)..10dir(300)+(0,-1.5),linewidth(2));
label("A",10dir(240),W);
label("B",10dir(300),E);
label("C",10dir(0),E);
label("D",10dir(300)+(0,-10),E);
draw(10dir(300)+2dir(-135)--10dir(300)+dir(-135),linewidth(2));
[/asy] | 210 |
Jon runs a website where he gets paid for every person who visits. He gets paid $0.10 for every person who visits. Each hour he gets 50 visits. His website operates 24 hours a day. How many dollars does he make in a 30 day month? | He makes 50*$.10=$<<50*.10=5>>5 per hour
So he makes $5*24=$<<5*24=120>>120 per day
So he makes $120*30=$<<120*30=3600>>3600 a month
#### 3600 |
Compute $\begin{pmatrix} 2 & 0 \\ 5 & -3 \end{pmatrix} \begin{pmatrix} 8 & -2 \\ 1 & 1 \end{pmatrix}.$ | \begin{pmatrix} 16 & -4 \\ 37 & -13 \end{pmatrix} |
In a school garden, there are flowers in different colors. In total there are 96 flowers of four colors: green, red, blue, and yellow. There are 9 green flowers and three times more red flowers. Blue flowers make up to 50% of the total flower count. How many yellow flowers are there? | There are 3 * 9 = <<3*9=27>>27 red flowers.
50/100 * 96 = <<50/100*96=48>>48 flowers are blue.
Knowing that, there are 96 - 48 - 27 - 9 = <<96-48-27-9=12>>12 yellow flowers in a school's garden.
#### 12 |
The students' written work has a binary grading system, i.e., a work will either be accepted if it is done well or not accepted if done poorly. Initially, the works are checked by a neural network which makes an error in 10% of the cases. All works identified as not accepted by the neural network are then rechecked manually by experts who do not make mistakes. The neural network can misclassify good work as not accepted and bad work as accepted. It is known that among all the submitted works, 20% are actually bad. What is the minimum percentage of bad works among those rechecked by the experts after the selection by the neural network? Indicate the integer part of the number in your answer. | 66 |
In right triangle $GHI$, we have $\angle G = 40^\circ$, $\angle H = 90^\circ$, and $HI = 12$. Find the length of $GH$ and $GI$. | 18.7 |
If $27^8=9^q$, what is $q$? | 12 |
Suppose $A, B, C$, and $D$ are four circles of radius $r>0$ centered about the points $(0, r),(r, 0)$, $(0,-r)$, and $(-r, 0)$ in the plane. Let $O$ be a circle centered at $(0,0)$ with radius $2 r$. In terms of $r$, what is the area of the union of circles $A, B, C$, and $D$ subtracted by the area of circle $O$ that is not contained in the union of $A, B, C$, and $D$? | 8 r^{2} |
Simplify first, then evaluate: $(1-\frac{m}{{m+3}})÷\frac{{{m^2}-9}}{{{m^2}+6m+9}}$, where $m=\sqrt{3}+3$. | \sqrt{3} |
In an isosceles triangle, one of the angles opposite an equal side is $40^{\circ}$. How many degrees are in the measure of the triangle's largest angle? [asy] draw((0,0)--(6,0)--(3,2)--(0,0)); label("$\backslash$",(1.5,1)); label("{/}",(4.5,1));
label("$40^{\circ}$",(.5,0),dir(45));
[/asy] | 100 |
What is the greatest integer not exceeding the number $\left( 1 + \frac{\sqrt 2 + \sqrt 3 + \sqrt 4}{\sqrt 2 + \sqrt 3 + \sqrt 6 + \sqrt 8 + 4}\right)^{10}$ ? | 32 |
A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is planning to offer a discount on the selling price, but the profit margin must not be less than $10\%$. Find the maximum discount rate that can be offered. | 12\% |
Let $ABC$ be a triangle with $\angle BAC=40^\circ $ , $O$ be the center of its circumscribed circle and $G$ is its centroid. Point $D$ of line $BC$ is such that $CD=AC$ and $C$ is between $B$ and $D$ . If $AD\parallel OG$ , find $\angle ACB$ . | 70 |
For positive integers $n$ and $k$, let $\mho(n, k)$ be the number of distinct prime divisors of $n$ that are at least $k$. Find the closest integer to $$\sum_{n=1}^{\infty} \sum_{k=1}^{\infty} \frac{\mho(n, k)}{3^{n+k-7}}$$ | 167 |
In rectangle \(ABCD\), \(BE = 5\), \(EC = 4\), \(CF = 4\), and \(FD = 1\), as shown in the diagram. What is the area of triangle \(\triangle AEF\)? | 42.5 |
$\triangle ABC$ is similar to $\triangle DEF$ . What is the number of centimeters in the length of $\overline{EF}$ ? Express your answer as a decimal to the nearest tenth.
[asy]
draw((0,0)--(8,-2)--(5,4)--cycle);
label("8cm",(2.5,2),NW);
label("5cm",(6.1,1),NE);
draw((12,0)--(18,-1.5)--(15.7,2.5)--cycle);
label("$A$",(8,-2),SE);
label("3cm",(16.9,0.5),NE);
label("$B$",(5,4),N);
label("$C$",(0,0),SW);
label("$D$",(18,-1.5),SE);
label("$E$",(15.7,2.5),N);
label("$F$",(12,0),N);
[/asy] | 4.8 |
Solve \[\frac{2x+4}{x^2+4x-5}=\frac{2-x}{x-1}\]for $x$. | -6 |
In a small pond there are eleven lily pads in a row labeled 0 through 10. A frog is sitting on pad 1. When the frog is on pad $N$, $0<N<10$, it will jump to pad $N-1$ with probability $\frac{N}{10}$ and to pad $N+1$ with probability $1-\frac{N}{10}$. Each jump is independent of the previous jumps. If the frog reaches pad 0 it will be eaten by a patiently waiting snake. If the frog reaches pad 10 it will exit the pond, never to return. What is the probability that the frog will escape without being eaten by the snake? | \frac{63}{146} |
Given the quadratic function \( y = ax^2 + bx + c \) where \( a \neq 0 \), its vertex is \( C \), and it intersects the x-axis at points \( A \) and \( B \). If triangle \( \triangle ABC \) is an acute triangle and \(\sin \angle BCA = \frac{4}{5}\), find the discriminant \(\Delta = b^2 - 4ac\). | 16 |
About 40% of students in a certain school are nearsighted, and about 30% of the students in the school use their phones for more than 2 hours per day, with a nearsighted rate of about 50% among these students. If a student who uses their phone for no more than 2 hours per day is randomly selected from the school, calculate the probability that the student is nearsighted. | \frac{5}{14} |
A bag contains 4 red, 3 blue, and 6 yellow marbles. One marble is drawn and removed from the bag but is only considered in the new count if it is yellow. What is the probability, expressed as a fraction, of then drawing one marble which is either red or blue from the updated contents of the bag? | \frac{91}{169} |
What is the sum of the positive factors of 24? | 60 |
Given the function $f(x)=\sqrt{3}\sin \omega x+\cos \omega x (\omega > 0)$, where the x-coordinates of the points where the graph of $f(x)$ intersects the x-axis form an arithmetic sequence with a common difference of $\frac{\pi}{2}$, determine the probability of the event "$g(x) \geqslant \sqrt{3}$" occurring, where $g(x)$ is the graph of $f(x)$ shifted to the left along the x-axis by $\frac{\pi}{6}$ units, when a number $x$ is randomly selected from the interval $[0,\pi]$. | \frac{1}{6} |
Consider a unit square $ABCD$ whose bottom left vertex is at the origin. A circle $\omega$ with radius $\frac{1}{3}$ is inscribed such that it touches the square's bottom side at point $M$. If $\overline{AM}$ intersects $\omega$ at a point $P$ different from $M$, where $A$ is at the top left corner of the square, find the length of $AP$. | \frac{1}{3} |
Donna has $n$ boxes of doughnuts. Each box contains $13$ doughnuts.
After eating one doughnut, Donna is able to rearrange the remaining doughnuts into bags so that each bag contains $9$ doughnuts, and none are left over. What is the smallest possible value of $n$? | 7 |
Triangle $A B C$ is given with $A B=13, B C=14, C A=15$. Let $E$ and $F$ be the feet of the altitudes from $B$ and $C$, respectively. Let $G$ be the foot of the altitude from $A$ in triangle $A F E$. Find $A G$. | \frac{396}{65} |
Tessa has a figure created by adding a semicircle of radius 1 on each side of an equilateral triangle with side length 2, with semicircles oriented outwards. She then marks two points on the boundary of the figure. What is the greatest possible distance between the two points? | 3 |
For the ellipse $25x^2 - 100x + 4y^2 + 8y + 16 = 0,$ find the distance between the foci. | \frac{2\sqrt{462}}{5} |
Gina is considered a bad tipper because she tipped 5%. If good tippers tip at least 20%, how many more cents than normal would Gina have to tip on a bill of $26 to be considered a good tipper? | First convert the bill amount to cents: $26 * 100 cents/dollar = <<26*100=2600>>2600 cents
Then multiply that amount by 5% to find the amount Gina normally tips: 2600 cents * 5% = <<2600*5*.01=130>>130 cents
Then multiply the bill amount by 20% to find the amount of a good tip: 2600 cents * 20% = <<2600*20*.01=520>>520 cents
Then subtract the good tip amount from the bad tip amount to find how much more Gina needs to tip: 520 cents - 130 cents = <<520-130=390>>390 cents
#### 390 |
On her previous five attempts Sarah had achieved times, in seconds, of 86, 94, 97, 88 and 96, for swimming 50 meters. After her sixth try she brought her median time down to 92 seconds. What was her time, in seconds, for her sixth attempt? | 90 |
As shown in the diagram, in the tetrahedron \(A B C D\), the face \(A B C\) intersects the face \(B C D\) at a dihedral angle of \(60^{\circ}\). The projection of vertex \(A\) onto the plane \(B C D\) is \(H\), which is the orthocenter of \(\triangle B C D\). \(G\) is the centroid of \(\triangle A B C\). Given that \(A H = 4\) and \(A B = A C\), find \(G H\). | \frac{4\sqrt{21}}{9} |
Milly has twelve socks, three of each color: red, blue, green, and yellow. She randomly draws five socks. What is the probability that she has exactly two pairs of socks of different colors and one sock of a third color? | \frac{9}{22} |
Express the quotient $1121_5 \div 12_5$ in base $5$. | 43_5. |
A boss schedules a meeting at a cafe with two of his staff, planning to arrive randomly between 1:00 PM and 4:00 PM. Each staff member also arrives randomly within the same timeframe. If the boss arrives and any staff member isn't there, he leaves immediately. Each staff member will wait for up to 90 minutes for the other to arrive before leaving. What is the probability that the meeting successfully takes place? | \frac{1}{4} |
\( S \) is a set of 5 coplanar points, no 3 of which are collinear. \( M(S) \) is the largest area of a triangle with vertices in \( S \). Similarly, \( m(S) \) is the smallest area of such a triangle. What is the smallest possible value of \( \frac{M(S)}{m(S)} \) as \( S \) varies? | \frac{1 + \sqrt{5}}{2} |
A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of lengths 3, 4, and 5. What is the area of the triangle?
$\mathrm{(A) \ 6 } \qquad \mathrm{(B) \frac{18}{\pi^2} } \qquad \mathrm{(C) \frac{9}{\pi^2}(\sqrt{3}-1) } \qquad \mathrm{(D) \frac{9}{\pi^2}(\sqrt{3}-1) } \qquad \mathrm{(E) \frac{9}{\pi^2}(\sqrt{3}+3) }$
| \frac{9}{\pi^2}(\sqrt{3}+3) |
The points $A = (3,-4,2),$ $B = (5,-8,5),$ $C = (4,-3,0),$ and $D = (6,-7,3)$ in space form a flat quadrilateral. Find the area of this quadrilateral. | \sqrt{110} |
Four equal circles with diameter $6$ are arranged such that three circles are tangent to one side of a rectangle and the fourth circle is tangent to the opposite side. All circles are tangent to at least one other circle with their centers forming a straight line that is parallel to the sides of the rectangle they touch. The length of the rectangle is twice its width. Calculate the area of the rectangle. | 648 |
What is the remainder when $5^{207}$ is divided by 7? | 6 |
A slant asymptote of the rational expression $y = \frac{2x^2 + 3x - 7}{x-3}$ is the line approached by the equation as $x$ approaches $\infty$ or $-\infty$. If this line is of the form $y = mx + b$, find $m+b$. | 11 |
Let $V$ be a 10-dimensional real vector space and $U_1,U_2$ two linear subspaces such that $U_1 \subseteq U_2, \dim U_1 =3, \dim U_2=6$ . Let $\varepsilon$ be the set of all linear maps $T: V\rightarrow V$ which have $T(U_1)\subseteq U_1, T(U_2)\subseteq U_2$ . Calculate the dimension of $\varepsilon$ . (again, all as real vector spaces) | 67 |
Given a parallelepiped $A B C D A_1 B_1 C_1 D_1$, points $M, N, K$ are midpoints of edges $A B$, $B C$, and $D D_1$ respectively. Construct the cross-sectional plane of the parallelepiped with the plane $MNK$. In what ratio does this plane divide the edge $C C_1$ and the diagonal $D B_1$? | 3:7 |
Convert $6532_8$ to base 5. | 102313_5 |
There are three two-digit numbers $A$, $B$, and $C$.
- $A$ is a perfect square, and each of its digits is also a perfect square.
- $B$ is a prime number, and each of its digits is also a prime number, and their sum is also a prime number.
- $C$ is a composite number, and each of its digits is also a composite number, the difference between its two digits is also a composite number. Furthermore, $C$ is between $A$ and $B$.
What is the sum of these three numbers $A$, $B$, and $C$?
| 120 |
Let the set \(T = \{0,1,2,3,4,5,6\}\) and \(M=\left\{\frac{a_{1}}{7}+\frac{a_{2}}{7^{2}}+\frac{a_{3}}{7^{3}}+\frac{a_{4}}{7^{4}}\right\}\), where \(a_{i} \in \mathbf{T}, i=\{1,2,3,4\}\). Arrange the numbers in \(M\) in descending order. Determine the 2005th number. | \frac{1}{7} + \frac{1}{7^2} + \frac{0}{7^3} + \frac{4}{7^4} |
Let $x,$ $y,$ $z$ be real numbers such that
\begin{align*}
x + y + z &= 4, \\
x^2 + y^2 + z^2 &= 6.
\end{align*}Let $m$ and $M$ be the smallest and largest possible values of $x,$ respectively. Find $m + M.$ | \frac{8}{3} |
What is the probability, expressed as a decimal, of drawing one marble which is either red or blue from a bag containing 3 red, 2 blue, and 5 yellow marbles? | 0.5 |
Let $x$ and $y$ be real numbers such that $\frac{\sin x}{\sin y} = 3$ and $\frac{\cos x}{\cos y} = \frac12$. Find the value of
\[\frac{\sin 2x}{\sin 2y} + \frac{\cos 2x}{\cos 2y}.\] | \frac{49}{58} |
Two non-collinear vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are given, and $\overrightarrow{a}+2\overrightarrow{b}$ is perpendicular to $2\overrightarrow{a}-\overrightarrow{b}$, $\overrightarrow{a}-\overrightarrow{b}$ is perpendicular to $\overrightarrow{a}$. The cosine of the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is __________. | \frac{\sqrt{10}}{5} |
If the first digit of a four-digit number, which is a perfect square, is decreased by 3, and the last digit is increased by 3, it also results in a perfect square. Find this number. | 4761 |
Call a three-digit number $\overline{ABC}$ $\textit{spicy}$ if it satisfies $\overline{ABC}=A^3+B^3+C^3$ . Compute the unique $n$ for which both $n$ and $n+1$ are $\textit{spicy}$ . | 370 |
Let $x$ and $y$ be positive real numbers such that $x^{2}+y^{2}=1$ and \left(3 x-4 x^{3}\right)\left(3 y-4 y^{3}\right)=-\frac{1}{2}$. Compute $x+y$. | \frac{\sqrt{6}}{2} |
Select three digits from 1, 3, 5, 7, 9, and two digits from 0, 2, 4, 6, 8 to form a five-digit number without any repeating digits. How many such numbers can be formed? | 11040 |
Compute $\begin{pmatrix} 1 & -1 \\ 1 & 0 \end{pmatrix}^3.$ | \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}. |
Given that $\frac{x}{2} = y^2$ and $\frac{x}{5} = 3y$, solve for $x$. | 112.5 |
It is given polygon with $2013$ sides $A_{1}A_{2}...A_{2013}$ . His vertices are marked with numbers such that sum of numbers marked by any $9$ consecutive vertices is constant and its value is $300$ . If we know that $A_{13}$ is marked with $13$ and $A_{20}$ is marked with $20$ , determine with which number is marked $A_{2013}$ | 67 |
A rival football team has won twice as many matches as the home team they will be playing did. If records show that the home team has won three matches, and each team drew four matches and lost none, how many matches have both teams played in total? | The rival team won twice as many as the home team who won 3 matches so they won 2*3 = <<2*3=6>>6 matches
They both drew 4 matches each for a total of 2*4 = 8 matches
The total number of matches played is 3+6+8 = <<3+6+8=17>>17 matches
#### 17 |
Given vectors $\overrightarrow{a}=(\cos α,\sin α)$ and $\overrightarrow{b}=(-2,2)$.
(1) If $\overrightarrow{a}\cdot \overrightarrow{b}= \frac {14}{5}$, find the value of $(\sin α+\cos α)^{2}$;
(2) If $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$, find the value of $\sin (π-α)\cdot\sin ( \frac {π}{2}+α)$. | -\frac{1}{2} |
The lengths of the edges of a rectangular parallelepiped extending from one vertex are 8, 8, and 27. Divide the parallelepiped into four parts that can be assembled into a cube. | 12 |
There are some identical square pieces of paper. If a part of them is paired up to form rectangles with a length twice their width, the total perimeter of all the newly formed rectangles is equal to the total perimeter of the remaining squares. Additionally, the total perimeter of all shapes after pairing is 40 centimeters less than the initial total perimeter. What is the initial total perimeter of all square pieces of paper in centimeters? | 280 |
Given $\overrightarrow{a}=(\sin \pi x,1)$, $\overrightarrow{b}=( \sqrt {3},\cos \pi x)$, and $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}$:
(I) If $x\in[0,2]$, find the interval(s) where $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}$ is monotonically increasing.
(II) Let $P$ be the coordinates of the first highest point and $Q$ be the coordinates of the first lowest point on the graph of $y=f(x)$ to the right of the $y$-axis. Calculate the cosine value of $\angle POQ$. | -\frac{16\sqrt{481}}{481} |
Simplify $\dfrac{18}{17}\cdot\dfrac{13}{24}\cdot\dfrac{68}{39}$. | 1 |
Express $326_{13} + 4C9_{14}$ as a base 10 integer, where $C = 12$ in base 14. | 1500 |
Let $ S $ be the set of all sides and diagonals of a regular hexagon. A pair of elements of $ S $ are selected at random without replacement. What is the probability that the two chosen segments have the same length? | \frac{17}{35} |
Compute $$100^{2}+99^{2}-98^{2}-97^{2}+96^{2}+95^{2}-94^{2}-93^{2}+\ldots+4^{2}+3^{2}-2^{2}-1^{2}$$ | 10100 |
Arrange the 9 numbers 12, 13, ..., 20 in a row such that the sum of every three consecutive numbers is a multiple of 3. How many such arrangements are there? | 216 |
Let \( f(n) \) be the sum of the squares of the digits of positive integer \( n \) (in decimal). For example, \( f(123) = 1^{2} + 2^{2} + 3^{2} = 14 \). Define \( f_{1}(n) = f(n) \), and \( f_{k+1}(n) = f\left(f_{k}(n)\right) \) for \( k = 1, 2, 3, \ldots \). What is the value of \( f_{2005}(2006) \)? | 145 |
Compute
$$
\int_{1}^{2} \frac{9x+4}{x^{5}+3x^{2}+x} \, dx. | \ln \frac{80}{23} |
Let $0 \le a,$ $b,$ $c \le 1.$ Find the maximum value of
\[\sqrt{abc} + \sqrt{(1 - a)(1 - b)(1 - c)}.\] | 1 |
The sum of Mario and Maria's ages now is 7. Mario is 1 year older than Maria. How old is Mario? | If x is Maria's age, then Mario's age is x + 1.
The equation that represents the sum of their ages is x + x + 1 = 7.
By combining like terms, the equation becomes 2x = 6.
Hence, the value of x which represents the age of Maria is 6/2=<<6/2=3>>3.
So Mariu is 3 + 1 = <<3+1=4>>4 years old.
#### 4 |
A force of $60 \mathrm{H}$ stretches a spring by 2 cm. The initial length of the spring is $14 \mathrm{~cm}$. How much work is required to stretch it to 20 cm? | 5.4 |
In the diagram, what is the perimeter of polygon $PQRST$? [asy]
import olympiad;
size(6cm); // ADJUST
pair p = (0, 6);
pair q = (3, 6);
pair r = (3, 3);
pair t = (0, 0);
pair s = (7, 0);
draw(p--q--r--s--t--cycle);
label("$P$", p, NW);
label("$Q$", q, NE);
label("$R$", r, E + NE);
label("$S$", s, SE);
label("$T$", t, SW);
label("$6$", p / 2, W);
label("$3$", p + (q - p) / 2, 2 * N);
label("$7$", s / 2, S);
draw(rightanglemark(p, t, s));
draw(rightanglemark(t, p, q));
draw(rightanglemark(p, q, r));
add(pathticks(p--q, s=6));
add(pathticks(q--r, s=6));
[/asy] | 24 |
Evaluate $|(4\sqrt{2}-4i)(\sqrt{3}+3i)|$ | 24 |
Mary needs school supplies. She has 6 classes and needs 1 folder for each class. She also needs 3 pencils for each class. She decides that for every 6 pencils she should have 1 eraser. She also needs a set of paints for an art class. Folders cost $6, pencils cost $2, and erasers cost $1. If she spends $80, how much did the set of paints cost in dollars? | Mary needs 6*1= <<6*1=6>>6 folders.
Mary needs 6*3= <<6*3=18>>18 pencils.
Mary needs 18/6= <<18/6=3>>3 erasers.
Mary spends 6*6= $<<6*6=36>>36 on folders.
Mary spends 18*2= $<<18*2=36>>36 on pencils.
Mary spends 3*1= $<<3*1=3>>3 on erasers.
Mary spends 36+36+3= $<<36+36+3=75>>75 on all the supplies except the paints.
Mary spends 80-75= $<<80-75=5>>5 on the paint set.
#### 5 |
If $x = 3$, what is the value of $2x + 3$? | 9 |
Let $a$ and $b$ be real numbers randomly (and independently) chosen from the range $[0,1]$. Find the probability that $a, b$ and 1 form the side lengths of an obtuse triangle. | \frac{\pi-2}{4} |
A fair coin is flipped every second and the results are recorded with 1 meaning heads and 0 meaning tails. What is the probability that the sequence 10101 occurs before the first occurrence of the sequence 010101? | \frac{21}{32} |
Let $a_{10} = 10$, and for each positive integer $n >10$ let $a_n = 100a_{n - 1} + n$. Find the least positive $n > 10$ such that $a_n$ is a multiple of $99$.
| 45 |
Determine the smallest possible product when three different numbers from the set $\{-4, -3, -1, 5, 6\}$ are multiplied. | 15 |
A battery of three guns fired a volley, and two shells hit the target. Find the probability that the first gun hit the target, given that the probabilities of hitting the target by the first, second, and third guns are $p_{1}=0,4$, $p_{2}=0,3$, and $p_{3}=0,5$, respectively. | 20/29 |
What is the greatest product obtainable from two integers whose sum is 246? | 15129 |
Rory has 30 more jellybeans than her sister Gigi who has 15 jellybeans. Lorelai has already eaten three times the number of jellybeans that both girls have. How many jellybeans has Lorelai eaten? | Rory has 30 more jellybeans than Gigi who has 15 so Rory has 30+15 = <<30+15=45>>45 jellybeans
Rory has 45 jellybeans and Gigi has 15 so that's 45+15 = <<45+15=60>>60 jellybeans total
Lorelai has eaten three times the amount of jellybeans that both girls have so she has eaten 3*60 = <<3*60=180>>180 jellybeans
#### 180 |
Given eight distinguishable rings, let $n$ be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand. The order of rings on each finger is significant, but it is not required that each finger have a ring. Find the leftmost three nonzero digits of $n$.
| 376 |
The mean of the set of numbers $\{87,85,80,83,84,x\}$ is 83.5. What is the median of the set of six numbers? Express your answer as a decimal to the nearest tenth. | 83.5 |
Brachycephalus frogs have three toes on each foot and two fingers on each hand. The common frog has five toes on each foot and four fingers on each hand. Some Brachycephalus and common frogs are in a bucket. Each frog has all its fingers and toes. Between them they have 122 toes and 92 fingers. How many frogs are in the bucket?
A 15
B 17
C 19
D 21
E 23 | 15 |
Let $S$ be the set of all positive integers whose prime factorizations only contain powers of the primes 2 and 2017 (1, powers of 2, and powers of 2017 are thus contained in $S$). Compute $\sum_{s \in S} \frac{1}{s}$. | \frac{2017}{1008} |
Find the interval of all $x$ such that both $2x$ and $3x$ are in the interval $(1,2)$. | \left(\frac{1}{2},\frac{2}{3}\right) |
Let $A B C$ be an isosceles triangle with $A B=A C$. Let $D$ and $E$ be the midpoints of segments $A B$ and $A C$, respectively. Suppose that there exists a point $F$ on ray $\overrightarrow{D E}$ outside of $A B C$ such that triangle $B F A$ is similar to triangle $A B C$. Compute $\frac{A B}{B C}$. | \sqrt{2} |
Let $a,$ $b,$ $c,$ $d$ be real numbers, none of which are equal to $-1,$ and let $\omega$ be a complex number such that $\omega^3 = 1$ and $\omega \neq 1.$ If
\[\frac{1}{a + \omega} + \frac{1}{b + \omega} + \frac{1}{c + \omega} + \frac{1}{d + \omega} = \frac{2}{\omega},\]then find
\[\frac{1}{a + 1} + \frac{1}{b + 1} + \frac{1}{c +1} + \frac{1}{d + 1}.\] | 2 |
Julia is performing in her high school musical this weekend and her family wants to come to the show. Tickets are $12 for adults and $10 for children. If her mom, dad, grandma, and three little sisters come to the show, how much will the total be for their tickets? | The cost will for 3 adults will be $12/adult x 3 adults = $<<12*3=36>>36
The cost for 3 children will be $10/child x 3 children = $<<10*3=30>>30
The total cost will be $36 + $30 = $<<36+30=66>>66
#### 66 |
What integer is closest to the value of $\sqrt[3]{6^3+8^3}$? | 9 |
For how many integers $n$ is $\frac n{20-n}$ the square of an integer? | 4 |
Tomorrow, Pete must finish paying off the last $90 he owes on a bike. He goes through his wallet and finds two $20 bills. Checking his pockets, he finds four $10 bills. Unhappy that he doesn't have the entire amount, he suddenly remembers that he has plastic bottles that can be returned to his local store for cash. If the store pays 50 cents for each bottle, how many bottles will Pete have to return to the store? | Pete’s wallet contains 2 * 20 = <<2*20=40>>40 dollars
The money in Pete’s pockets is 4 * 10 = <<4*10=40>>40 dollars
In total Pete has 40 + 40 = <<40+40=80>>80 dollars
He owes 90 dollars on the bike and therefore needs 90 - 80 = <<90-80=10>>10 more dollars
Since 50 cents is ½ dollar, then the number of bottles required to get 10 dollars is 10 / (1/2) = 10*2 = <<10/(1/2)=20>>20 bottles.
#### 20 |
Find all positive integers $a,b,c$ and prime $p$ satisfying that
\[ 2^a p^b=(p+2)^c+1.\] | (1, 1, 1, 3) |
Jeff's five assignment scores are 89, 92, 88, 95 and 91. What is the arithmetic mean of these five scores? | 91 |
Karl, Ryan, and Ben are fond of collecting stickers. Karl has 25 stickers. Ryan has 20 more stickers than Karl. Ben has 10 fewer stickers than Ryan. They placed all their stickers in one sticker book. How many stickers did they place altogether? | Ryan has 25 + 20 = <<25+20=45>>45 stickers.
Ben has 45 - 10 = <<45-10=35>>35 stickers.
Therefore, there are a total of 25 + 45 + 35 = <<25+45+35=105>>105 stickers placed all together in a sticker book.
#### 105 |
Let $a$, $b$, and $c$ be the roots of $x^3 - 20x^2 + 18x - 7 = 0$. Compute \[(a+b)^2 + (b+c)^2 + (c+a)^2.\] | 764 |
The function \( g(x) \) satisfies
\[ g(x) - 2 g \left( \frac{1}{x} \right) = 3^x + x \]
for all \( x \neq 0 \). Find \( g(2) \). | -4 - \frac{2\sqrt{3}}{3} |
Convert the base 2 number \(1011111010_2\) to its base 4 representation. | 23322_4 |
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